Package 'reliaR'

Description

The function abic.burrX() gives the loglikelihood, AIC and BIC values assuming an BurrX distribution with parameters alpha and lambda.

Usage

abic.burrX(x, alpha.est, lambda.est)

Arguments

Value

The function abic.burrX() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.burrX for PP plot and qq.burrX for QQ plot

Examples

## Load data sets
data(bearings)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.1989515, lambda.est = 0.0130847

## Values of AIC, BIC and LogLik for the data(bearings)
abic.burrX(bearings, 1.1989515, 0.0130847)

Description

The function abic.chen() gives the loglikelihood, AIC and BIC values assuming Chen distribution with parameters beta and lambda. The function is based on the invariance property of the MLE.

Usage

abic.chen(x, beta.est, lambda.est)

Arguments

Value

The function abic.chen() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.chen for PP plot and qq.chen for QQ plot

Examples

## Load data sets

data(sys2)
## Maximum Likelihood(ML) Estimates of beta & lambda for the data(sys2)
## beta.est = 0.262282404, lambda.est = 0.007282371

## Values of AIC, BIC and LogLik for the data(sys2) 
abic.chen(sys2, 0.262282404, 0.007282371)

Description

The function abic.exp.ext() gives the loglikelihood, AIC and BIC values assuming an Exponential Extension(EE) distribution with parameters alpha and lambda.

Usage

abic.exp.ext(x, alpha.est, lambda.est)

Arguments

Value

The function abic.exp.ext() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.exp.ext for PP plot and qq.exp.ext for QQ plot

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.0126e+01, lambda.est = 1.5848e-04

## Values of AIC, BIC and LogLik for the data(sys2)
abic.exp.ext(sys2, 1.0126e+01, 1.5848e-04)

Description

The function abic.exp.power() gives the loglikelihood, AIC and BIC values assuming Chen distribution with parameters alpha and lambda. The function is based on the invariance property of the MLE.

Usage

abic.exp.power(x, alpha.est, lambda.est)

Arguments

Value

The function abic.exp.power() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.exp.power for PP plot and qq.exp.power for QQ plot

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## alpha.est = 0.905868898, lambda.est =  0.001531423

## Values of AIC, BIC and LogLik for the data(sys2) 

abic.exp.power(sys2, 0.905868898, 0.001531423)

Description

The function abic.expo.logistic() gives the loglikelihood, AIC and BIC values assuming an Exponentiated Logistic(EL) distribution with parameters alpha and beta.

Usage

abic.expo.logistic(x, alpha.est, beta.est)

Arguments

Value

The function abic.expo.logistic() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.expo.logistic for PP plot and qq.expo.logistic for QQ plot

Examples

## Load data sets
data(dataset2)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(dataset2)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 5.31302, beta.est = 139.04515

## Values of AIC, BIC and LogLik for the data(dataset2)
abic.expo.logistic(dataset2, 5.31302, 139.04515)

Description

The function abic.expo.weibull() gives the loglikelihood, AIC and BIC values assuming an Exponentiated Weibull(EW) distribution with parameters alpha and theta.

Usage

abic.expo.weibull(x, alpha.est, theta.est)

Arguments

Value

The function abic.expo.weibull() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.expo.weibull for PP plot and qq.expo.weibull for QQ plot

Examples

## Load data sets
data(stress)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(stress)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est =1.026465, theta.est = 7.824943

## Values of AIC, BIC and LogLik for the data(stress)
abic.expo.weibull(stress, 1.026465, 7.824943)

Description

The function abic.flex.weibull() gives the loglikelihood, AIC and BIC values assuming an flexible Weibull(FW) distribution with parameters alpha and beta.

Usage

abic.flex.weibull(x, alpha.est, beta.est)

Arguments

Value

The function abic.flex.weibull() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.flex.weibull for PP plot and qq.flex.weibull for QQ plot

Examples

## Load data sets
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(repairtimes)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 0.07077507, beta.est = 1.13181535

## Values of AIC, BIC and LogLik for the data(repairtimes)
abic.flex.weibull(repairtimes, 0.07077507, 1.13181535)

Description

The function abic.gen.exp() gives the loglikelihood, AIC and BIC values assuming an Generalized Exponential distribution with parameters alpha and lambda. The function is based on the invariance property of the MLE.

Usage

abic.gen.exp(x, alpha.est, lambda.est)

Arguments

Value

The function abic.gen.exp() gives the loglikelihood, AIC and BIC values.

References

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

See Also

pp.gen.exp for PP plot and qq.gen.exp for QQ plot

Examples

## Load data set
data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 5.28321139, lambda.est = 0.03229609
abic.gen.exp(bearings, 5.28321139, 0.03229609)

Description

The function abic.gompertz() gives the loglikelihood, AIC and BIC values assuming an Gompertz distribution with parameters alpha and theta.

Usage

abic.gompertz(x, alpha.est, theta.est)

Arguments

Value

The function abic.gompertz() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.gompertz for PP plot and qq.gompertz for QQ plot

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(sys2)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est = 0.00121307, theta.est = 0.00173329

## Values of AIC, BIC and LogLik for the data(sys2)
abic.gompertz(sys2, 0.00121307, 0.00173329)

Description

The function abic.gp.weibull() gives the loglikelihood, AIC and BIC values assuming an generalized power Weibull(GPW) distribution with parameters alpha and theta.

Usage

abic.gp.weibull(x, alpha.est, theta.est)

Arguments

Value

The function abic.gp.weibull() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.gp.weibull for PP plot and qq.gp.weibull for QQ plot

Examples

## Load data sets
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(repairtimes)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est = 1.566093, theta.est = 0.355321

## Values of AIC, BIC and LogLik for the data(repairtimes)
abic.gp.weibull(repairtimes, 1.566093, 0.355321)

Description

The function abic.gumbel() gives the loglikelihood, AIC and BIC values assuming an Gumbel distribution with parameters mu and sigma.

Usage

abic.gumbel(x, mu.est, sigma.est)

Arguments

Value

The function abic.gumbel() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.gumbel for PP plot and qq.gumbel for QQ plot

Examples

## Load data sets
data(dataset2)
## Maximum Likelihood(ML) Estimates of mu & sigma for the data(dataset2)
## Estimates of mu & sigma using 'maxLik' package
## mu.est = 212.157, sigma.est = 151.768

## Values of AIC, BIC and LogLik for the data(dataset2)
abic.gumbel(dataset2, 212.157, 151.768)

Description

The function abic.inv.genexp() gives the loglikelihood, AIC and BIC values assuming an Inverse Generalized Exponential(IGE) distribution with parameters alpha and lambda.

Usage

abic.inv.genexp(x, alpha.est, lambda.est)

Arguments

Value

The function abic.inv.genexp() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.inv.genexp for PP plot and qq.inv.genexp for QQ plot

Examples

## Load data sets
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(repairtimes)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.097807, lambda.est = 1.206889

## Values of AIC, BIC and LogLik for the data(repairtimes)
abic.inv.genexp(repairtimes, 1.097807, 1.206889)

Description

The function abic.lfr() gives the loglikelihood, AIC and BIC values assuming an linear failure rate(LFR) distribution with parameters alpha and beta.

Usage

abic.lfr(x, alpha.est, beta.est)

Arguments

Value

The function abic.lfr() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.lfr for PP plot and qq.lfr for QQ plot

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(sys2)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 1.77773e-03,  beta.est = 2.77764e-06

## Values of AIC, BIC and LogLik for the data(sys2)
abic.lfr(sys2, 1.777673e-03, 2.777640e-06)

Description

The function abic.log.gamma() gives the loglikelihood, AIC and BIC values assuming an log-gamma(LG) distribution with parameters alpha and lambda.

Usage

abic.log.gamma(x, alpha.est, lambda.est)

Arguments

Value

The function abic.log.gamma() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.log.gamma for PP plot and qq.log.gamma for QQ plot

Examples

## Load data sets
data(conductors)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(conductors)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 0.0088741, lambda.est = 0.6059935

## Values of AIC, BIC and LogLik for the data(conductors)
abic.log.gamma(conductors, 0.0088741, 0.6059935)

Description

The function abic.logis.exp() gives the loglikelihood, AIC and BIC values assuming an Logistic-Exponential(LE) distribution with parameters alpha and lambda.

Usage

abic.logis.exp(x, alpha.est, lambda.est)

Arguments

Value

The function abic.logis.exp() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.logis.exp for PP plot and qq.logis.exp for QQ plot

Examples

## Load data sets
data(bearings)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 2.36754, lambda.est = 0.01059

## Values of AIC, BIC and LogLik for the data(bearings)
abic.logis.exp(bearings, 2.36754, 0.01059)

Description

The function abic.logis.rayleigh() gives the loglikelihood, AIC and BIC values assuming an Logistic-Rayleigh(LR) distribution with parameters alpha and lambda.

Usage

abic.logis.rayleigh(x, alpha.est, lambda.est)

Arguments

Value

The function abic.logis.rayleigh() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.logis.rayleigh for PP plot and qq.logis.rayleigh for QQ plot

Examples

## Load data sets
data(stress)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(stress)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.4779388, lambda.est = 0.2141343

## Values of AIC, BIC and LogLik for the data(stress)
abic.logis.rayleigh(stress, 1.4779388, 0.2141343)

Description

The function abic.loglog( ) gives the loglikelihood, AIC and BIC values assuming Loglog distribution with parameters alpha and lambda. The function is based on the invariance property of the MLE.

Usage

abic.loglog(x, alpha.est, lambda.est)

Arguments

Value

The function abic.loglog( ) gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

qq.loglog for QQ plot and ks.loglog function

Examples

## Load data set
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## alpha.est = 0.9058689 lambda.est = 1.0028228

## Values of AIC, BIC and LogLik for the data(sys2) 
abic.loglog(sys2, 0.9058689, 1.0028228)

Description

The function abic.moee() gives the loglikelihood, AIC and BIC values assuming an MOEE distribution with parameters alpha and lambda.

Usage

abic.moee(x, alpha.est, lambda.est)

Arguments

Value

The function abic.moee() gives the loglikelihood, AIC and BIC values.

References

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

See Also

pp.moee for PP plot and qq.moee for QQ plot

Examples

## Load data set
data(stress)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 75.67982, lambda.est = 1.67576
abic.moee(stress, 75.67982, 1.67576)

Description

The function abic.moew() gives the loglikelihood, AIC and BIC values assuming an MOEW distribution with parameters alpha and lambda.

Usage

abic.moew(x, alpha.est, lambda.est)

Arguments

Value

The function abic.moew() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.moew for PP plot and qq.moew for QQ plot

Examples

## Load data set
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## alpha.est = 0.3035937,  lambda.est = 279.2177754

## Values of AIC, BIC and LogLik for the data(sys2) 
abic.moew(sys2, 0.3035937, 279.2177754)

Description

The function abic.weibull.ext() gives the loglikelihood, AIC and BIC values assuming an Weibull Extension(WE) distribution with parameters alpha and beta.

Usage

abic.weibull.ext(x, alpha.est, beta.est)

Arguments

Value

The function abic.weibull.ext() gives the loglikelihood, AIC and BIC values.

References

Akaike, H. (1978). A new look at the Bayes procedure, Biometrika, 65, 53-59.

Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging, Cambridge University Press, London.

Konishi., S. and Kitagawa, G.(2008). Information Criteria and Statistical Modeling, Springer Science+Business Media, LLC.

Schwarz, S. (1978). Estimating the dimension of the model, Annals of Statistics, 6, 461-464.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of complexity and fit, Journal of the Royal Statistical Society Series B 64, 1-34.

See Also

pp.weibull.ext for PP plot and qq.weibull.ext for QQ plot

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(sys2)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 0.00019114, beta.est = 0.14696242

## Values of AIC, BIC and LogLik for the data(sys2)
abic.weibull.ext(sys2, 0.00019114, 0.14696242)

Description

Several data sets related to life test are available in the reliaR package, which have been taken from the literature.

Usage

data(bearings)

Format

A vector containing 23 observations.

Details

The data given here arose in tests on endurance of deep groove ball bearings. The data are the number of million revolutions before failure for each of the 23 ball bearings in the life test.

References

Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data, 2nd ed., John Wiley and Sons, New York.

Examples

## Load data sets
data(bearings)
## Histogram for bearings
hist(bearings)

Description

Density, distribution function, quantile function and random generation for the BurrX distribution with shape parameter alpha and scale parameter lambda.

Usage

dburrX(x, alpha, lambda, log = FALSE)
pburrX(q, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
qburrX(p, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
rburrX(n, alpha, lambda)

Arguments

Details

The BurrX distribution has density

$f(x; \alpha, \lambda) = 2 \alpha \lambda^2 x e^{-(\lambda x)^2} \left\{1-e^{-(\lambda x)^2} \right\}^{\alpha -1}; (\alpha, \lambda) > 0, x >0.$

where $\alpha$ and $\lambda$ are the shape and scale parameters, respectively.

Value

dburrX gives the density, pburrX gives the distribution function, qburrX gives the quantile function, and rburrX generates random deviates.

References

Kundu, D., and Raqab, M.Z. (2005). Generalized Rayleigh Distribution: Different Methods of Estimation, Computational Statistics and Data Analysis, 49, 187-200.

Surles, J.G., and Padgett, W.J. (2005). Some properties of a scaled Burr type X distribution, Journal of Statistical Planning and Inference, 128, 271-280.

Raqab, M.Z., and Kundu, D. (2006). Burr Type X distribution: revisited, Journal of Probability and Statistical Sciences, 4(2), 179-193.

See Also

.Random.seed about random number; sburrX for BurrX survival / hazard etc. functions

Examples

## Load data sets
data(bearings)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.1989515, lambda.est = 0.0130847

dburrX(bearings, 1.1989515, 0.0130847, log = FALSE)
pburrX(bearings, 1.1989515, 0.0130847, lower.tail = TRUE, log.p = FALSE)
qburrX(0.25, 1.1989515, 0.0130847, lower.tail=TRUE, log.p = FALSE)
rburrX(30, 1.1989515, 0.0130847)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the BurrX distribution with shape parameter alpha and scale parameter lambda.

Usage

crf.burrX(x, t = 0, alpha, lambda)
hburrX(x, alpha, lambda)
hra.burrX(x, alpha, lambda)
sburrX(x, alpha, lambda)

Arguments

Details

The hazard function is defined by

$h(x) = \frac{f(x)}{1 - F(x)},\, t > 0, 0 < F(x) < 1,$

where $f(\cdot)$ and $F(\cdot)$ are the pdf and cdf, respectively. The behavior of $h(x)$ allows one to characterize the aging of the units. For example, if the failure rate is increasing (IFR class), then the units age with time. If $h(x)$ is decreasing (DFR class), then the units improve in performance with time. Finally, if $h(x)$ is constant, then the lifetime distribution is necessarily exponential.

There are two more aging indicators which are the following:

The failure rate average (FRA) of X is given by

$FRA(x) = \frac{H(x)}{x} = \frac{\int^{x}_{0} h(x)\,dx}{x},\, x > 0,$

where $H(x)$ is the cumulative hazard function. An analysis for FRA($x$) on $x$ permits to obtain the IFRA and DFRA classes.

The survival/reliability function (s.f.) and the conditional survival of X are defined by

$R(x) = 1 - F(x) \quad {\rm and} \quad R(x|t) = \frac{R(x+t)}{R(x)},\, x > 0,\, t > 0,\, R(\cdot) > 0,$

respectively, where $F(\cdot)$ is the cdf of X. Similarly to $h(x)$ and $FRA(x)$, the distribution of X belongs to the new better than used (NBU), exponential, or new worse than used (NWU) classes, when $R(x|t) < R(x)$, $R(x|t) = R(x)$, or $R(x|t) > R(x)$, respectively.

Value

crf.burrX gives the conditional reliability function (crf), hburrX gives the hazard function, hra.burrX gives the hazard rate average (HRA) function, and sburrX gives the survival function for the BurrX distribution.

References

Kundu, D., and Raqab, M.Z. (2005). Generalized Rayleigh Distribution: Different Methods of Estimation, Computational Statistics and Data Analysis, 49, 187-200.

Lawless, J.F.(2003). Statistical Models and Methods for Lifetime Data, John Wiley and Sons, New York.

Marshall, A. W., Olkin, I.(2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families, Springer, New York.

See Also

dburrX for other BurrX distribution related functions;

Examples

## load data set
data(bearings)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.1989515, lambda.est = 0.0130847

## Reliability indicators for data(bearings):

## Reliability function
sburrX(bearings, 1.1989515, 0.0130847)

## Hazard function
hburrX(bearings, 1.1989515, 0.0130847)

## hazard rate average(hra)
hra.burrX(bearings, 1.1989515, 0.0130847)

## Conditional reliability function (age component=0)
crf.burrX(bearings, 0.00, 1.1989515, 0.0130847)

## Conditional reliability function (age component=3.0)
crf.burrX(bearings, 3.0, 1.1989515, 0.0130847)

Description

Density, distribution function, quantile function and random generation for the Chen distribution with shape parameter beta and scale parameter lambda.

Usage

dchen(x, beta, lambda, log = FALSE)
pchen(q, beta, lambda, lower.tail = TRUE, log.p = FALSE)
qchen(p, beta, lambda, lower.tail = TRUE, log.p = FALSE)
rchen(n, beta, lambda)

Arguments

Details

The Chen distribution has density

$f(x; \lambda, \beta) = \lambda \beta x^{\beta -1} \exp \left(x^{\beta} \right) \exp \left[\lambda \left\{1-\exp \left(x^{\beta} \right)\right\}\right];\; (\lambda ,\; \beta )>0,\; x > 0,$

where $\beta$ and $\lambda$ are the shape and scale parameters, respectively.

Value

dchen gives the density, pchen gives the distribution function, qchen gives the quantile function, and rchen generates random deviates.

References

Chen, Z. (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics & Probability Letters, 49, 155-161.

Murthy, D.N.P., Xie, M. and Jiang, R. (2004). Weibull Models, Wiley, New York.

Pham, H. (2006). System Software Reliability, Springer-Verlag.

Pham, H. and Lai, C.D. (2007). On recent generalizations of the Weibull distribution, IEEE Trans. on Reliability, Vol. 56(3), 454-458.

See Also

.Random.seed about random number; schen for Chen survival / hazard etc. functions

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of beta & lambda for the data(sys2)
## beta.est = 0.262282404, lambda.est = 0.007282371

dchen(sys2, 0.262282404, 0.007282371, log = FALSE)
pchen(sys2, 0.262282404, 0.007282371, lower.tail = TRUE, 
    log.p = FALSE)
qchen(0.25, 0.262282404, 0.007282371, lower.tail = TRUE, log.p = FALSE)
rchen(10, 0.262282404, 0.007282371)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the Chen distribution with shape parameter beta and scale parameter lambda.

Usage

crf.chen(x, t = 0, beta, lambda)
hchen(x, beta, lambda)
hra.chen(x, beta, lambda)
schen(x, beta, lambda)

Arguments

Value

crf.chen gives the conditional reliability function (crf), hchen gives the hazard function, hra.chen gives the hazard rate average (HRA) function, and schen gives the survival function for the Chen distribution.

References

Chen, Z.(2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics and Probability Letters, 49, 155-161.

Pham, H. (2003). Handbook of Reliability Engineering, Springer-Verlag.

See Also

dchen for other Chen distribution related functions

Examples

## Maximum Likelihood(ML) Estimates of beta & lambda 
## beta.est = 0.262282404, lambda.est = 0.007282371
## Load data sets
data(sys2)

## Reliability indicators:

## Reliability function
schen(sys2, 0.262282404, 0.007282371)

## Hazard function 
hchen(sys2, 0.262282404, 0.007282371)

## hazard rate average(hra)
hra.chen(sys2, 0.262282404, 0.007282371)

## Conditional reliability function (age component=0)
crf.chen(sys2, 0.00, 0.262282404, 0.007282371)

## Conditional reliability function (age component=3.0)
crf.chen(sys2, 3.0, 0.262282404, 0.007282371)

Description

Several data sets related to life test are available in the reliaR package, which have been taken from the literature.

Usage

data(conductors)

Format

A vector containing 59 observations.

Details

The data is obtained from Lawless(2003, pp. 267) and it represents the faiure times of 59 conductors from an accelerated life test. Failure times are in hours, and there are no censored observations.

References

Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data,2nd ed., John Wiley and Sons, New York.

Examples

## Load data sets
data(conductors)
## Histogram for conductors
hist(conductors)

Description

Several data sets related to life test are available in the reliaR package, which have been taken from the literature.

Usage

data(dataset2)

Format

A vector containing 111 observations.

Details

The data is obtained from Lyu(1996) and is given in chapter 11 as DATASET2. The data set contains 36 months of defect-discovery times for a release of Controller Software consisting of about 500,000 lines of code installed on over 100,000 controllers.

References

Lyu, M. R. (1996). Handbook of Software Reliability Engineering, IEEE Computer Society Press, http://www.cse.cuhk.edu.hk/~lyu/book/reliability/

Examples

## Load data sets
data(dataset2)
## Histogram for dataset2
hist(dataset2)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the Exponential Power distribution with shape parameter alpha and scale parameter lambda.

Usage

crf.exp.power(x, t = 0, alpha, lambda)
hexp.power(x, alpha, lambda)
hra.exp.power(x, alpha, lambda)
sexp.power(x, alpha, lambda)

Arguments

Value

crf.exp.power gives the conditional reliability function (crf), hexp.power gives the hazard function, hra.exp.power gives the hazard rate average (HRA) function, and sexp.power gives the survival function for the Exponential Power distribution.

References

Chen, Z.(1999). Statistical inference about the shape parameter of the exponential power distribution, Journal :Statistical Papers, Vol. 40(4), 459-468.

Pham, H. and Lai, C.D.(2007). On recent generalizations of the Weibull distribution, IEEE Trans. on Reliability, Vol. 56(3), 454-458.

Smith, R.M. and Bain, L.J.(1975). An exponential power life-test distribution, Communications in Statistics - Simulation and Computation, Vol.4(5), 469 - 481

See Also

dexp.power for other Exponential Power distribution related functions

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## alpha.est = 0.905868898, lambda.est =  0.001531423

## Reliability indicators:

## Reliability function
sexp.power(sys2, 0.905868898, 0.001531423)

## Hazard function 
hexp.power(sys2, 0.905868898, 0.001531423)

## hazard rate average(hra)
hra.exp.power(sys2, 0.905868898, 0.001531423)

## Conditional reliability function (age component=0)
crf.exp.power(sys2, 0.00, 0.905868898, 0.001531423)

## Conditional reliability function (age component=3.0)
crf.exp.power(sys2, 3.0, 0.905868898, 0.001531423)

Description

Density, distribution function, quantile function and random generation for the Exponential Extension(EE) distribution with shape parameter alpha and scale parameter lambda.

Usage

dexp.ext(x, alpha, lambda, log = FALSE)
pexp.ext(q, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
qexp.ext(p, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
rexp.ext(n, alpha, lambda)

Arguments

Details

The Exponential Extension(EE) distribution has density

$f(x) = \alpha \lambda \left(1+\lambda x\right)^{\alpha -1} \exp \left\{1-\left(1+\lambda x\right)^{\alpha } \right\} ;\, x\ge 0, \alpha >0, \lambda >0.$

where $\alpha$ and $\lambda$ are the shape and scale parameters, respectively.

Value

dexp.ext gives the density, pexp.ext gives the distribution function, qexp.ext gives the quantile function, and rexp.ext generates random deviates.

References

Nikulin, M. and Haghighi, F. (2006). A Chi-squared test for the generalized power Weibull family for the head-and-neck cancer censored data, Journal of Mathematical Sciences, Vol. 133(3), 1333-1341.

See Also

.Random.seed about random number; sexp.ext for ExpExt survival / hazard etc. functions

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.0126e+01, lambda.est = 1.5848e-04
dexp.ext(sys2, 1.012556e+01, 1.5848e-04, log = FALSE)
pexp.ext(sys2, 1.012556e+01, 1.5848e-04, lower.tail = TRUE, log.p = FALSE)
qexp.ext(0.25, 1.012556e+01, 1.5848e-04, lower.tail=TRUE, log.p = FALSE)
rexp.ext(30, 1.012556e+01, 1.5848e-04)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the Exponential Extension(EE) distribution with shape parameter alpha and scale parameter lambda.

Usage

crf.exp.ext(x, t = 0, alpha, lambda)
hexp.ext(x, alpha, lambda)
hra.exp.ext(x, alpha, lambda)
sexp.ext(x, alpha, lambda)

Arguments

Value

crf.exp.ext gives the conditional reliability function (crf), hexp.ext gives the hazard function, hra.exp.ext gives the hazard rate average (HRA) function, and sexp.ext gives the survival function for the Exponential Extension(EE) distribution.

References

Nikulin, M. and Haghighi, F.(2006). A Chi-squared test for the generalized power Weibull family for the head-and-neck cancer censored data, Journal of Mathematical Sciences, Vol. 133(3), 1333-1341.

See Also

dexp.ext for other Exponential Extension(EE) distribution related functions;

Examples

## load data set
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.0126e+01, lambda.est = 1.5848e-04

## Reliability indicators for data(sys2):

## Reliability function
sexp.ext(sys2, 1.0126e+01, 1.5848e-04)

## Hazard function
hexp.ext(sys2, 1.0126e+01, 1.5848e-04)

## hazard rate average(hra)
hra.exp.ext(sys2, 1.0126e+01, 1.5848e-04)

## Conditional reliability function (age component=0)
crf.exp.ext(sys2, 0.00, 1.0126e+01, 1.5848e-04)

## Conditional reliability function (age component=3.0)
crf.exp.ext(sys2, 3.0, 1.0126e+01, 1.5848e-04)

Description

Density, distribution function, quantile function and random generation for the Exponentiated Logistic(EL) distribution with shape parameter alpha and scale parameter beta.

Usage

dexpo.logistic(x, alpha, beta, log = FALSE)
pexpo.logistic(q, alpha, beta, lower.tail = TRUE, log.p = FALSE)
qexpo.logistic(p, alpha, beta, lower.tail = TRUE, log.p = FALSE)
rexpo.logistic(n, alpha, beta)

Arguments

Details

The Exponentiated Logistic(EL) distribution has density

$f(x; \alpha, \beta) = \frac{\alpha}{\beta} \exp\left(-\frac{x}{\beta}\right)\left\{1+\exp\left(-\frac{x}{\beta}\right)\right\}^{-(\alpha + 1)};\; (\alpha, \beta) > 0, x > 0$

where $\alpha$ and $\beta$ are the shape and scale parameters, respectively.

Value

dexpo.logistic gives the density, pexpo.logistic gives the distribution function, qexpo.logistic gives the quantile function, and rexpo.logistic generates random deviates.

References

Ali, M.M., Pal, M. and Woo, J. (2007). Some Exponentiated Distributions, The Korean Communications in Statistics, 14(1), 93-109.

Shirke, D.T., Kumbhar, R.R. and Kundu, D. (2005). Tolerance intervals for exponentiated scale family of distributions, Journal of Applied Statistics, 32, 1067-1074

See Also

.Random.seed about random number; sexpo.logistic for Exponentiated Logistic(EL) survival / hazard etc. functions

Examples

## Load data sets
data(dataset2)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(dataset2)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 5.31302, beta.est = 139.04515

dexpo.logistic(dataset2, 5.31302, 139.04515, log = FALSE)
pexpo.logistic(dataset2, 5.31302, 139.04515, lower.tail = TRUE, log.p = FALSE)
qexpo.logistic(0.25, 5.31302, 139.04515, lower.tail=TRUE, log.p = FALSE)
rexpo.logistic(30, 5.31302, 139.04515)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the Exponentiated Logistic(EL) distribution with shape parameter alpha and scale parameter beta.

Usage

crf.expo.logistic(x, t = 0, alpha, beta)
hexpo.logistic(x, alpha, beta)
hra.expo.logistic(x, alpha, beta)
sexpo.logistic(x, alpha, beta)

Arguments

Value

crf.expo.logistic gives the conditional reliability function (crf), hexpo.logistic gives the hazard function, hra.expo.logistic gives the hazard rate average (HRA) function, and sexpo.logistic gives the survival function for the Exponentiated Logistic(EL) distribution.

References

Ali, M.M., Pal, M. and Woo, J. (2007). Some Exponentiated Distributions, The Korean Communications in Statistics, 14(1), 93-109.

Shirke, D.T., Kumbhar, R.R. and Kundu, D.(2005). Tolerance intervals for exponentiated scale family of distributions, Journal of Applied Statistics, 32, 1067-1074

See Also

dexpo.logistic for other Exponentiated Logistic(EL) distribution related functions;

Examples

## load data set
data(dataset2)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(dataset2)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 5.31302, beta.est = 139.04515

## Reliability indicators for data(dataset2):

## Reliability function
sexpo.logistic(dataset2, 5.31302, 139.04515)

## Hazard function
hexpo.logistic(dataset2, 5.31302, 139.04515)

## hazard rate average(hra)
hra.expo.logistic(dataset2, 5.31302, 139.04515)

## Conditional reliability function (age component=0)
crf.expo.logistic(dataset2, 0.00, 5.31302, 139.04515)

## Conditional reliability function (age component=3.0)
crf.expo.logistic(dataset2, 3.0, 5.31302, 139.04515)

Description

Density, distribution function, quantile function and random generation for the Exponentiated Weibull(EW) distribution with shape parameters alpha and theta.

Usage

dexpo.weibull(x, alpha, theta, log = FALSE)
pexpo.weibull(q, alpha, theta, lower.tail = TRUE, log.p = FALSE)
qexpo.weibull(p, alpha, theta, lower.tail = TRUE, log.p = FALSE)
rexpo.weibull(n, alpha, theta)

Arguments

Details

The Exponentiated Weibull(EW) distribution has density

$f(x; \alpha, \theta) = \alpha \; \theta \; x^{\alpha - 1} \; e^{-x^{\alpha}} \left\{1-\exp \left(-x^{\alpha}\right)\right\}^{\theta -1};\; (\alpha, \theta) > 0, x > 0$

where $\alpha$ and $\theta$ are the shape and scale parameters, respectively.

Value

dexpo.weibull gives the density, pexpo.weibull gives the distribution function, qexpo.weibull gives the quantile function, and rexpo.weibull generates random deviates.

References

Mudholkar, G.S. and Srivastava, D.K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, 42(2), 299-302.

Murthy, D.N.P., Xie, M. and Jiang, R. (2003). Weibull Models, Wiley, New York.

Nassar, M.M., and Eissa, F. H. (2003). On the Exponentiated Weibull Distribution, Communications in Statistics - Theory and Methods, 32(7), 1317-1336.

See Also

.Random.seed about random number; sexpo.weibull for Exponentiated Weibull(EW) survival / hazard etc. functions

Examples

## Load data sets
data(stress)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(stress)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est =1.026465, theta.est = 7.824943

dexpo.weibull(stress, 1.026465, 7.824943, log = FALSE)
pexpo.weibull(stress, 1.026465, 7.824943, lower.tail = TRUE, log.p = FALSE)
qexpo.weibull(0.25, 1.026465, 7.824943, lower.tail=TRUE, log.p = FALSE)
rexpo.weibull(30, 1.026465, 7.824943)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the Exponentiated Weibull(EW) distribution with shape parameters alpha and theta.

Usage

crf.expo.weibull(x, t = 0, alpha, theta)
hexpo.weibull(x, alpha, theta)
hra.expo.weibull(x, alpha, theta)
sexpo.weibull(x, alpha, theta)

Arguments

Value

crf.expo.weibull gives the conditional reliability function (crf), hexpo.weibull gives the hazard function, hra.expo.weibull gives the hazard rate average (HRA) function, and sexpo.weibull gives the survival function for the Exponentiated Weibull(EW) distribution.

References

Mudholkar, G.S. and Srivastava, D.K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, 42(2), 299-302.

Murthy, D.N.P., Xie, M. and Jiang, R. (2003). Weibull Models, Wiley, New York.

Nassar, M.M., and Eissa, F. H. (2003). On the Exponentiated Weibull Distribution, Communications in Statistics - Theory and Methods, 32(7), 1317-1336.

See Also

dexpo.weibull for other Exponentiated Weibull(EW) distribution related functions;

Examples

## load data set
data(stress)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(stress)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est =1.026465, theta.est = 7.824943

## Reliability indicators for data(stress):

## Reliability function
sexpo.weibull(stress, 1.026465, 7.824943)

## Hazard function
hexpo.weibull(stress, 1.026465, 7.824943)

## hazard rate average(hra)
hra.expo.weibull(stress, 1.026465, 7.824943)

## Conditional reliability function (age component=0)
crf.expo.weibull(stress, 0.00, 1.026465, 7.824943)

## Conditional reliability function (age component=3.0)
crf.expo.weibull(stress, 3.0, 1.026465, 7.824943)

Description

Density, distribution function, quantile function and random generation for the Exponential Power distribution with shape parameter alpha and scale parameter lambda.

Usage

dexp.power(x, alpha, lambda, log = FALSE)
pexp.power(q, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
qexp.power(p, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
rexp.power(n, alpha, lambda)

Arguments

Details

The probability density function of exponential power distribution is

$f(x; \alpha, \lambda) = \alpha \lambda^\alpha x^{\alpha - 1} e^{\left({\lambda x}\right)^\alpha} \exp\left\{{1 - e^{\left({\lambda x}\right)^\alpha}}\right\};\;(\alpha, \lambda) > 0, x > 0.$

where $\alpha$ and $\lambda$ are the shape and scale parameters, respectively.

Value

dexp.power gives the density, pexp.power gives the distribution function, qexp.power gives the quantile function, and rexp.power generates random deviates.

References

Chen, Z.(1999). Statistical inference about the shape parameter of the exponential power distribution, Journal :Statistical Papers, Vol. 40(4), 459-468.

Pham, H. and Lai, C.D.(2007). On Recent Generalizations of theWeibull Distribution, IEEE Trans. on Reliability, Vol. 56(3), 454-458.

Smith, R.M. and Bain, L.J.(1975). An exponential power life-test distribution, Communications in Statistics - Simulation and Computation, Vol.4(5), 469 - 481

See Also

.Random.seed about random number; sexp.power for Exponential Power distribution survival / hazard etc. functions;

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## alpha.est = 0.905868898, lambda.est =  0.001531423

dexp.power(sys2, 0.905868898, 0.001531423, log = FALSE)
pexp.power(sys2, 0.905868898, 0.001531423, lower.tail = TRUE, log.p = FALSE)
qexp.power(0.25, 0.905868898, 0.001531423, lower.tail=TRUE, log.p = FALSE)
rexp.power(30, 0.905868898, 0.001531423)

Description

Density, distribution function, quantile function and random generation for the flexible Weibull(FW) distribution with parameters alpha and beta.

Usage

dflex.weibull(x, alpha, beta, log = FALSE)
pflex.weibull(q, alpha, beta, lower.tail = TRUE, log.p = FALSE)
qflex.weibull(p, alpha, beta, lower.tail = TRUE, log.p = FALSE)
rflex.weibull(n, alpha, beta)

Arguments

Details

The flexible Weibull(FW) distribution has density

$f(x) = \left(\alpha + \frac{\beta}{x^2}\right) \exp\left(\alpha \, x - \frac{\beta}{x}\right)\, \exp\left\{-\exp\left(\alpha x - \frac{\beta}{x}\right)\right\};\, x \ge 0, \alpha > 0, \beta > 0.$

where $\alpha$ and $\beta$ are the shape and scale parameters, respectively.

Value

dflex.weibull gives the density, pflex.weibull gives the distribution function, qflex.weibull gives the quantile function, and rflex.weibull generates random deviates.

References

Bebbington, M., Lai, C.D. and Zitikis, R. (2007). A flexible Weibull extension, Reliability Engineering and System Safety, 92, 719-726.

See Also

.Random.seed about random number; sflex.weibull for flexible Weibull(FW) survival / hazard etc. functions

Examples

## Load data sets
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(repairtimes)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 0.07077507, beta.est = 1.13181535

dflex.weibull(repairtimes, 0.07077507, 1.13181535, log = FALSE)
pflex.weibull(repairtimes, 0.07077507, 1.13181535, lower.tail = TRUE, log.p = FALSE)
qflex.weibull(0.25, 0.07077507, 1.13181535, lower.tail=TRUE, log.p = FALSE)
rflex.weibull(30, 0.07077507, 1.13181535)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the flexible Weibull(FW) distribution with parameters alpha and beta.

Usage

crf.flex.weibull(x, t = 0, alpha, beta)
hflex.weibull(x, alpha, beta)
hra.flex.weibull(x, alpha, beta)
sflex.weibull(x, alpha, beta)

Arguments

Value

crf.flex.weibull gives the conditional reliability function (crf), hflex.weibull gives the hazard function, hra.flex.weibull gives the hazard rate average (HRA) function, and sflex.weibull gives the survival function for the flexible Weibull(FW) distribution.

References

Bebbington, M., Lai, C.D. and Zitikis, R. (2007). A flexible Weibull extension, Reliability Engineering and System Safety, 92, 719-726.

See Also

dflex.weibull for other flexible Weibull(FW) distribution related functions;

Examples

## load data set
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(repairtimes)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 0.07077507, beta.est = 1.13181535

## Reliability indicators for data(repairtimes):

## Reliability function
sflex.weibull(repairtimes, 0.07077507, 1.13181535)

## Hazard function
hflex.weibull(repairtimes, 0.07077507, 1.13181535)

## hazard rate average(hra)
hra.flex.weibull(repairtimes, 0.07077507, 1.13181535)

## Conditional reliability function (age component=0)
crf.flex.weibull(repairtimes, 0.00, 0.07077507, 1.13181535)

## Conditional reliability function (age component=3.0)
crf.flex.weibull(repairtimes, 3.0, 0.07077507, 1.13181535)

Description

Density, distribution function, quantile function and random generation for the Generalized Exponential (GE) distribution with shape parameter alpha and scale parameter lambda.

Usage

dgen.exp(x, alpha, lambda, log = FALSE)
pgen.exp(q, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
qgen.exp(p, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
rgen.exp(n, alpha, lambda)

Arguments

Details

The generalized exponential distribution has density

$f(x; \alpha, \lambda) = \alpha \lambda x\; e^{-\lambda x} \; \left\{1-e^{-\lambda x} \right\}^{\alpha -1};\; (\alpha, \lambda) > 0, x > 0.$

where $\alpha$ and $\lambda$ are the shape and scale parameters, respectively.

Value

dgen.exp gives the density, pgen.exp gives the distribution function, qgen.exp gives the quantile function, and rgen.exp generates random deviates.

References

Gupta, R. D. and Kundu, D. (2001). Exponentiated exponential family; an alternative to gamma and Weibull distributions. Biometrical Journal, 43(1), 117 - 130.

Gupta, R. D. and Kundu, D. (1999). Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173 - 188.

See Also

.Random.seed about random number; sgen.exp for GE survival / hazard etc. functions

Examples

## Load data set
data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 5.28321139, lambda.est = 0.03229609

dgen.exp(bearings, 5.28321139, 0.03229609, log = FALSE)
pgen.exp(bearings, 5.28321139, 0.03229609, lower.tail = TRUE, 
    log.p = FALSE)
qgen.exp(0.25, 5.28321139, 0.03229609, lower.tail = TRUE, log.p = FALSE)
rgen.exp(10, 5.28321139, 0.03229609)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the Generalized Exponential (GE) distribution with shape parameter alpha and scale parameter lambda.

Usage

crf.gen.exp(x, t = 0, alpha, lambda)
hgen.exp(x, alpha, lambda)
hra.gen.exp(x, alpha, lambda)
sgen.exp(x, alpha, lambda)

Arguments

Value

crf.gen.exp gives the conditional reliability function (crf), hgen.exp gives the hazard function, hra.gen.exp gives the hazard rate average (HRA) function, and sgen.exp gives the survival function for the GE distribution.

References

Gupta, R. D. and Kundu, D. (2001). Exponentiated exponential family; an alternative to gamma and Weibull distributions. Biometrical Journal, 43(1), 117 - 130.

Gupta, R. D. and Kundu, D. (1999). Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173 - 188.

See Also

dgen.exp for other GE distribution related functions;

Examples

## load data set
data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 5.28321139, lambda.est = 0.03229609
sgen.exp(bearings, 5.28321139, 0.03229609)
hgen.exp(bearings, 5.28321139, 0.03229609)
hra.gen.exp(bearings, 5.28321139, 0.03229609)
crf.gen.exp(bearings, 20.0, 5.28321139, 0.03229609)

Description

Density, distribution function, quantile function and random generation for the Gompertz distribution with shape parameter alpha and scale parameter theta.

Usage

dgompertz(x, alpha, theta, log = FALSE)
pgompertz(q, alpha, theta, lower.tail = TRUE, log.p = FALSE)
qgompertz(p, alpha, theta, lower.tail = TRUE, log.p = FALSE)
rgompertz(n, alpha, theta)

Arguments

Details

The Gompertz distribution has density

$f(x) = \theta e^{\alpha x} \exp\left\{\frac{\theta}{\alpha}\left(1 - e^{\alpha x}\right)\right\};\, x \ge 0, \theta > 0, -\infty < \alpha < \infty.$

where $\alpha$ and $\theta$ are the shape and scale parameters, respectively.

Value

dgompertz gives the density, pgompertz gives the distribution function, qgompertz gives the quantile function, and rgompertz generates random deviates.

References

Marshall, A. W., Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families, Springer, New York.

See Also

.Random.seed about random number; sgompertz for Gompertz survival / hazard etc. functions

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(sys2)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est = 0.00121307, theta.est = 0.00173329

dgompertz(sys2, 0.00121307, 0.00173329, log = FALSE)
pgompertz(sys2, 0.00121307, 0.00173329, lower.tail = TRUE, log.p = FALSE)
qgompertz(0.25, 0.00121307, 0.00173329, lower.tail=TRUE, log.p = FALSE)
rgompertz(30, 0.00121307, 0.00173329)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the Gompertz distribution with shape parameter alpha and scale parameter theta.

Usage

crf.gompertz(x, t = 0, alpha, theta)
hgompertz(x, alpha, theta)
hra.gompertz(x, alpha, theta)
sgompertz(x, alpha, theta)

Arguments

Value

crf.gompertz gives the conditional reliability function (crf), hgompertz gives the hazard function, hra.gompertz gives the hazard rate average (HRA) function, and sgompertz gives the survival function for the Gompertz distribution.

References

Marshall, A. W., Olkin, I.(2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families, Springer, New York.

See Also

dgompertz for other Gompertz distribution related functions;

Examples

## load data set
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(sys2)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est = 0.00121307, theta.est = 0.00173329

## Reliability indicators for data(sys2):

## Reliability function
sgompertz(sys2, 0.00121307, 0.00173329)

## Hazard function
hgompertz(sys2, 0.00121307, 0.00173329)

## hazard rate average(hra)
hra.gompertz(sys2, 0.00121307, 0.00173329)

## Conditional reliability function (age component=0)
crf.gompertz(sys2, 0.00, 0.00121307, 0.00173329)

## Conditional reliability function (age component=3.0)
crf.gompertz(sys2, 3.0, 0.00121307, 0.00173329)

Description

Density, distribution function, quantile function and random generation for the generalized power Weibull(GPW) distribution with shape parameters alpha and theta.

Usage

dgp.weibull(x, alpha, theta, log = FALSE)
pgp.weibull(q, alpha, theta, lower.tail = TRUE, log.p = FALSE)
qgp.weibull(p, alpha, theta, lower.tail = TRUE, log.p = FALSE)
rgp.weibull(n, alpha, theta)

Arguments

Details

The generalized power Weibull(GPW) distribution has density

$f(x) = \alpha \theta x^{\alpha -1} \left(1 + x^{\alpha} \right)^{\theta - 1} \exp\left\{1-\left(1+x^{\alpha}\right)^{\theta}\right\};\, x \ge 0, \alpha > 0, \theta > 0.$

where $\alpha$ and $\theta$ are the shape and scale parameters, respectively.

Value

dgp.weibull gives the density, pgp.weibull gives the distribution function, qgp.weibull gives the quantile function, and rgp.weibull generates random deviates.

References

Nikulin, M. and Haghighi, F. (2006). A Chi-squared test for the generalized power Weibull family for the head-and-neck cancer censored data, Journal of Mathematical Sciences, Vol. 133(3), 1333-1341.

Pham, H. and Lai, C.D. (2007). On recent generalizations of the Weibull distribution, IEEE Trans. on Reliability, Vol. 56(3), 454-458.

See Also

.Random.seed about random number; sgp.weibull for generalized power Weibull(GPW) survival / hazard etc. functions

Examples

## Load data sets
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(repairtimes)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est = 1.566093, theta.est = 0.355321

dgp.weibull(repairtimes, 1.566093, 0.355321, log = FALSE)
pgp.weibull(repairtimes, 1.566093, 0.355321, lower.tail = TRUE, log.p = FALSE)
qgp.weibull(0.25, 1.566093, 0.355321, lower.tail=TRUE, log.p = FALSE)
rgp.weibull(30, 1.566093, 0.355321)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the generalized power Weibull(GPW) distribution with shape parameters alpha and theta.

Usage

crf.gp.weibull(x, t = 0, alpha, theta)
hgp.weibull(x, alpha, theta)
hra.gp.weibull(x, alpha, theta)
sgp.weibull(x, alpha, theta)

Arguments

Value

crf.gp.weibull gives the conditional reliability function (crf), hgp.weibull gives the hazard function, hra.gp.weibull gives the hazard rate average (HRA) function, and sgp.weibull gives the survival function for the generalized power Weibull(GPW) distribution.

References

Nikulin, M. and Haghighi, F.(2006). A Chi-squared test for the generalized power Weibull family for the head-and-neck cancer censored data, Journal of Mathematical Sciences, Vol. 133(3), 1333-1341.

Pham, H. and Lai, C.D.(2007). On recent generalizations of the Weibull distribution, IEEE Trans. on Reliability, Vol. 56(3), 454-458.

See Also

dgp.weibull for other generalized power Weibull(GPW) distribution related functions;

Examples

## load data set
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(repairtimes)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est = 1.566093, theta.est = 0.355321

## Reliability indicators for data(repairtimes):

## Reliability function
sgp.weibull(repairtimes, 1.566093, 0.355321)

## Hazard function
hgp.weibull(repairtimes, 1.566093, 0.355321)

## hazard rate average(hra)
hra.gp.weibull(repairtimes, 1.566093, 0.355321)

## Conditional reliability function (age component=0)
crf.gp.weibull(repairtimes, 0.00, 1.566093, 0.355321)

## Conditional reliability function (age component=3.0)
crf.gp.weibull(repairtimes, 3.0, 1.566093, 0.355321)

Description

Density, distribution function, quantile function and random generation for the Gumbel distribution with location parameter mu and scale parameter sigma.

Usage

dgumbel(x, mu, sigma, log = FALSE)
pgumbel(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qgumbel(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rgumbel(n, mu, sigma)

Arguments

Details

The Gumbel distribution has density

$f(x) = \frac{1}{\sigma} \; \exp\left\{-\left(\frac{x-\mu}{\sigma}\right)\right\} \; \exp\left[-\exp\left\{-\left(\frac{x-\mu}{\sigma}\right)\right\}\right];\, -\infty < x < \infty, \sigma > 0.$

where $\mu$ and $\sigma$ are the shape and scale parameters, respectively.

Value

dgumbel gives the density, pgumbel gives the distribution function, qgumbel gives the quantile function, and rgumbel generates random deviates.

References

Marshall, A. W., Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families, Springer, New York.

See Also

.Random.seed about random number; sgumbel for Gumbel survival / hazard etc. functions

Examples

## Load data sets
data(dataset2)
## Maximum Likelihood(ML) Estimates of mu & sigma for the data(dataset2)
## Estimates of mu & sigma using 'maxLik' package
## mu.est = 212.157, sigma.est = 151.768

dgumbel(dataset2, 212.157, 151.768, log = FALSE)
pgumbel(dataset2, 212.157, 151.768, lower.tail = TRUE, log.p = FALSE)
qgumbel(0.25, 212.157, 151.768, lower.tail=TRUE, log.p = FALSE)
rgumbel(30, 212.157, 151.768)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the Gumbel distribution with location parameter mu and scale parameter sigma.

Usage

crf.gumbel(x, t = 0, mu, sigma)
hgumbel(x, mu, sigma)
hra.gumbel(x, mu, sigma)
sgumbel(x, mu, sigma)

Arguments

Value

crf.gumbel gives the conditional reliability function (crf), hgumbel gives the hazard function, hra.gumbel gives the hazard rate average (HRA) function, and sgumbel gives the survival function for the Gumbel distribution.

References

Marshall, A. W., Olkin, I.(2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families, Springer, New York.

See Also

dgumbel for other Gumbel distribution related functions;

Examples

## load data set
data(dataset2)
## Maximum Likelihood(ML) Estimates of mu & sigma for the data(dataset2)
## Estimates of mu & sigma using 'maxLik' package
## mu.est = 212.157, sigma.est = 151.768

## Reliability indicators for data(dataset2):

## Reliability function
sgumbel(dataset2, 212.157, 151.768)

## Hazard function
hgumbel(dataset2, 212.157, 151.768)

## hazard rate average(hra)
hra.gumbel(dataset2, 212.157, 151.768)

## Conditional reliability function (age component=0)
crf.gumbel(dataset2, 0.00, 212.157, 151.768)

## Conditional reliability function (age component=3.0)
crf.gumbel(dataset2, 3.0, 212.157, 151.768)

Description

Density, distribution function, quantile function and random generation for the Inverse Generalized Exponential(IGE) distribution with shape parameter alpha and scale parameter lambda.

Usage

dinv.genexp(x, alpha, lambda, log = FALSE)
pinv.genexp(q, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
qinv.genexp(p, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
rinv.genexp(n, alpha, lambda)

Arguments

Details

The Inverse Generalized Exponential(IGE) distribution has density

$f(x; \alpha, \lambda) = \frac{\alpha \; \lambda}{x^2}\; e^{-\lambda /x} \; \left\{1-e^{-\lambda /x}\right\}^{\alpha - 1};\; (\alpha, \lambda) > 0, x > 0$

where $\alpha$ and $\lambda$ are the shape and scale parameters, respectively.

Value

dinv.genexp gives the density, pinv.genexp gives the distribution function, qinv.genexp gives the quantile function, and rinv.genexp generates random deviates.

References

Gupta, R. D. and Kundu, D. (2001). Exponentiated exponential family; an alternative to gamma and Weibull distributions, Biometrical Journal, 43(1), 117-130.

Gupta, R.D. and Kundu, D. (2007). Generalized exponential distribution: Existing results and some recent development, Journal of Statistical Planning and Inference. 137, 3537-3547.

See Also

.Random.seed about random number; sinv.genexp for Inverse Generalized Exponential(IGE) survival / hazard etc. functions

Examples

## Load data sets
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(repairtimes)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.097807, lambda.est = 1.206889
dinv.genexp(repairtimes, 1.097807, 1.206889, log = FALSE)
pinv.genexp(repairtimes, 1.097807, 1.206889, lower.tail = TRUE, log.p = FALSE)
qinv.genexp(0.25, 1.097807, 1.206889, lower.tail=TRUE, log.p = FALSE)
rinv.genexp(30, 1.097807, 1.206889)

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the Inverse Generalized Exponential(IGE) distribution with shape parameter alpha and scale parameter lambda.

Usage

crf.inv.genexp(x, t = 0, alpha, lambda)
hinv.genexp(x, alpha, lambda)
hra.inv.genexp(x, alpha, lambda)
sinv.genexp(x, alpha, lambda)

Arguments

Value

crf.inv.genexp gives the conditional reliability function (crf), hinv.genexp gives the hazard function, hra.inv.genexp gives the hazard rate average (HRA) function, and sinv.genexp gives the survival function for the Inverse Generalized Exponential(IGE) distribution.

References

Gupta, R. D. and Kundu, D. (2001). Exponentiated exponential family; an alternative to gamma and Weibull distributions, Biometrical Journal, 43(1), 117-130.

Gupta, R.D. and Kundu, D., (2007). Generalized exponential distribution: Existing results and some recent development, Journal of Statistical Planning and Inference. 137, 3537-3547.

See Also

dinv.genexp for other Inverse Generalized Exponential(IGE) distribution related functions;

Examples

## load data set
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(repairtimes)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.097807, lambda.est = 1.206889

## Reliability indicators for data(repairtimes):

## Reliability function
sinv.genexp(repairtimes, 1.097807, 1.206889)

## Hazard function
hinv.genexp(repairtimes, 1.097807, 1.206889)

## hazard rate average(hra)
hra.inv.genexp(repairtimes, 1.097807, 1.206889)

## Conditional reliability function (age component=0)
crf.inv.genexp(repairtimes, 0.00, 1.097807, 1.206889)

## Conditional reliability function (age component=3.0)
crf.inv.genexp(repairtimes, 3.0, 1.097807, 1.206889)

Description

The function ks.burrX() gives the values for the KS test assuming a BurrX with shape parameter alpha and scale parameter lambda. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.burrX(x, alpha.est, lambda.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.burrX() carries out the KS test for the BurrX

References

Kundu, D., and Raqab, M.Z. (2005). Generalized Rayleigh Distribution: Different Methods of Estimation, Computational Statistics and Data Analysis, 49, 187-200.

Surles, J.G., and Padgett, W.J. (2005). Some properties of a scaled Burr type X distribution, Journal of Statistical Planning and Inference, 128, 271-280.

Raqab, M.Z., and Kundu, D. (2006). Burr Type X distribution: revisited, Journal of Probability and Statistical Sciences, 4(2), 179-193.

See Also

pp.burrX for PP plot and qq.burrX for QQ plot

Examples

## Load data sets
data(bearings)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.1989515, lambda.est = 0.0130847

ks.burrX(bearings, 1.1989515, 0.0130847, alternative = "two.sided", plot = TRUE)

Description

The function ks.chen() gives the values for the KS test assuming the Chen distribution with shape parameter beta and scale parameter lambda. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.chen(x, beta.est, lambda.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.chen() carries out the KS test for the Chen.

References

Castillo, E., Hadi, A.S., Balakrishnan, N. and Sarabia, J.M.(2004). Extreme Value and Related Models with Applications in Engineering and Science, John Wiley and Sons, New York.

Chen, Z.(2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics and Probability Letters, 49, 155-161.

Pham, H. (2003). Handbook of Reliability Engineering, Springer-Verlag.

See Also

pp.chen for PP plot and qq.chen for QQ plot

Examples

## Load data sets
data(sys2)
## Estimates of beta & lambda using 'maxLik' package
## beta.est = 0.262282404, lambda.est = 0.007282371

ks.chen(sys2, 0.262282404, 0.007282371, alternative = "two.sided", plot = TRUE)

Description

The function ks.exp.ext() gives the values for the KS test assuming a Exponential Extension(EE) with shape parameter alpha and scale parameter lambda. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.exp.ext(x, alpha.est, lambda.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.exp.ext() carries out the KS test for the Exponential Extension(EE)

References

Nikulin, M. and Haghighi, F. (2006). A Chi-squared test for the generalized power Weibull family for the head-and-neck cancer censored data, Journal of Mathematical Sciences, Vol. 133(3), 1333-1341.

See Also

pp.exp.ext for PP plot and qq.exp.ext for QQ plot

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.0126e+01, lambda.est = 1.5848e-04

ks.exp.ext(sys2, 1.0126e+01, 1.5848e-04, alternative = "two.sided", plot = TRUE)

Description

The function ks.exp.power() gives the values for the KS test assuming an Exponential Power distribution with shape parameter alpha and scale parameter lambda. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.exp.power(x, alpha.est, lambda.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.exp.power() carries out the KS test for the EP.

References

Smith, R.M. and Bain, L.J. (1975). An exponential power life-test distribution, Communications in Statistics - Simulation and Computation, Vol. 4(5), 469-481.

See Also

pp.exp.power for PP plot and qq.exp.power for QQ plot

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## alpha.est = 0.905868898, lambda.est =  0.001531423

ks.exp.power(sys2, 0.905868898, 0.001531423, alternative = "two.sided", plot = TRUE)

Description

The function ks.expo.logistic() gives the values for the KS test assuming a Exponentiated Logistic(EL) with shape parameter alpha and scale parameter beta. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.expo.logistic(x, alpha.est, beta.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.expo.logistic() carries out the KS test for the Exponentiated Logistic(EL)

References

Ali, M.M., Pal, M. and Woo, J. (2007). Some Exponentiated Distributions, The Korean Communications in Statistics, 14(1), 93-109.

Shirke, D.T., Kumbhar, R.R. and Kundu, D. (2005). Tolerance intervals for exponentiated scale family of distributions, Journal of Applied Statistics, 32, 1067-1074

See Also

pp.expo.logistic for PP plot and qq.expo.logistic for QQ plot

Examples

## Load data sets
data(dataset2)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(dataset2)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 5.31302, beta.est = 139.04515

ks.expo.logistic(dataset2, 5.31302, 139.04515, alternative = "two.sided", plot = TRUE)

Description

The function ks.expo.weibull() gives the values for the KS test assuming a Exponentiated Weibull(EW) with shape parameter alpha and scale parameter theta. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.expo.weibull(x, alpha.est, theta.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.expo.weibull() carries out the KS test for the Exponentiated Weibull(EW)

References

Mudholkar, G.S. and Srivastava, D.K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, 42(2), 299-302.

Murthy, D.N.P., Xie, M. and Jiang, R. (2003). Weibull Models, Wiley, New York.

Nassar, M.M., and Eissa, F. H. (2003). On the Exponentiated Weibull Distribution, Communications in Statistics - Theory and Methods, 32(7), 1317-1336.

See Also

pp.expo.weibull for PP plot and qq.expo.weibull for QQ plot

Examples

## Load data sets
data(stress)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(stress)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est =1.026465, theta.est = 7.824943

ks.expo.weibull(stress, 1.026465, 7.824943, alternative = "two.sided", plot = TRUE)

Description

The function ks.flex.weibull() gives the values for the KS test assuming a flexible Weibull(FW) with shape parameter alpha and scale parameter beta. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.flex.weibull(x, alpha.est, beta.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.flex.weibull() carries out the KS test for the flexible Weibull(FW)

References

Bebbington, M., Lai, C.D. and Zitikis, R. (2007). A flexible Weibull extension, Reliability Engineering and System Safety, 92, 719-726.

See Also

pp.flex.weibull for PP plot and qq.flex.weibull for QQ plot

Examples

## Load data sets
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & beta for the data(repairtimes)
## Estimates of alpha & beta using 'maxLik' package
## alpha.est = 0.07077507, beta.est = 1.13181535

ks.flex.weibull(repairtimes, 0.07077507, 1.13181535, 
    alternative = "two.sided", plot = TRUE)

Description

The function ks.gen.exp() gives the values for the KS test assuming an GE with shape parameter alpha and scale parameter lambda. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.gen.exp(x, alpha.est, lambda.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.gen.exp() carries out the KS test for the GE.

References

Gupta, R. D. and Kundu, D. (2001). Exponentiated exponential family; an alternative to gamma and Weibull distributions. Biometrical Journal, 43(1), 117 - 130.

Gupta, R. D. and Kundu, D. (1999). Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173 - 188.

See Also

pp.gen.exp for PP plot and qq.gen.exp for QQ plot

Examples

## Load data sets
data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 5.28321139, lambda.est = 0.03229609
ks.gen.exp(bearings, 5.28321139, 0.03229609, alternative = "two.sided", plot = TRUE)

Description

The function ks.gompertz() gives the values for the KS test assuming a Gompertz with shape parameter alpha and scale parameter theta. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.gompertz(x, alpha.est, theta.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.gompertz() carries out the KS test for the Gompertz

References

Marshall, A. W., Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families, Springer, New York.

See Also

pp.gompertz for PP plot and qq.gompertz for QQ plot

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(sys2)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est = 0.00121307, theta.est = 0.00173329

ks.gompertz(sys2, 0.00121307, 0.00173329, alternative = "two.sided", plot = TRUE)

Description

The function ks.gp.weibull() gives the values for the KS test assuming a generalized power Weibull(GPW) with shape parameter alpha and scale parameter theta. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.gp.weibull(x, alpha.est, theta.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.gp.weibull() carries out the KS test for the generalized power Weibull(GPW)

References

Nikulin, M. and Haghighi, F. (2006). A Chi-squared test for the generalized power Weibull family for the head-and-neck cancer censored data, Journal of Mathematical Sciences, Vol. 133(3), 1333-1341.

Pham, H. and Lai, C.D. (2007). On recent generalizations of the Weibull distribution, IEEE Trans. on Reliability, Vol. 56(3), 454-458.

See Also

pp.gp.weibull for PP plot and qq.gp.weibull for QQ plot

Examples

## Load data sets
data(repairtimes)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(repairtimes)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est = 1.566093, theta.est = 0.355321

ks.gp.weibull(repairtimes, 1.566093, 0.355321, alternative = "two.sided", plot = TRUE)

Description

The function ks.gumbel() gives the values for the KS test assuming a Gumbel with shape parameter mu and scale parameter sigma. In addition, optionally, this function allows one to show a comparative graph between the empirical and theoretical cdfs for a specified data set.

Usage

ks.gumbel(x, mu.est, sigma.est, 
    alternative = c("less", "two.sided", "greater"), plot = FALSE, ...)

Arguments

Details

The Kolmogorov-Smirnov test is a goodness-of-fit technique based on the maximum distance between the empirical and theoretical cdfs.

Value

The function ks.gumbel() carries out the KS test for the Gumbel

References

Marshall, A. W., Olkin, I.(2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families, Springer, New York.

See Also

pp.gumbel for PP plot and qq.gumbel for QQ plot

Examples

## Load data sets
data(dataset2)
## Maximum Likelihood(ML) Estimates of mu & sigma for the data(dataset2)
## Estimates of mu & sigma using 'maxLik' package
## mu.est = 212.157, sigma.est = 151.768

ks.gumbel(dataset2, 212.157, 151.768, alternative = "two.sided", plot = TRUE)