Package 'NeuDist'

Description

Tools for univariate continuous distributions with model diagnostics, based on the Lindley, Logistic, Half-Cauchy, Half-Logistic, and Poisson families, providing functions for probability density, distribution, quantile, and hazard evaluation, random variate generation, and generic diagnostic tools such as Q–Q and P–P plots, goodness-of-fit tests, and model selection criteria, with support for 58 distributions and 15 data sets.

Details

Distributions in the 'NeuDist' package:

ChenExp                 Chen-Exponential Distribution.
ExpoExpPower            Exponentiated Exponential Power Distribution.  
ExpoInvChen             Exponentiated Inverse Chen Distribution. 
GompertzExt             Gompertz Extension Distribution. 
HCChen                  Half-Cauchy Chen Distribution. 
HCGenExp                Half-Cauchy Generalized Exponential Distribution. 
HCGenRayleigh           Half-Cauchy Generalized Rayleigh Distribution. 
HCGompertz              Half-Cauchy Gompertz Distribution. 
HCInvGPZ                Half-Cauchy Inverse Gompertz Distribution. 
HCInvNHE                Half-Cauchy Inverse NHE Distribution. 
HCNHE                   Half-Cauchy exponential extension Distribution. 
HLIW                    Half Logistic Inverted Weibull Distribution. 
HLNHE                   Half Logistic NHE Distribution.
InvEEP                  Inverse Exponentiated Exponential Poisson Distribution.  
InvExpPower             Inverse Exponential Power Distribution. 
InvGenGPZ               Inverse Generalized Gompertz Distribution. 
InvPham                 Inverse Pham Distribution. 
InvPowerCauchy          Inverse Power Cauchy Distribution.
InvSGZ                  Inverted Shifted Gompertz Distribution. 
InvUBD                  Inverse Upside Down Bathtub-Shaped Hazard Distribution. 
LindleyChen             Lindley-Chen Distribution. 
LindleyExpPower         Lindley Exponential Power Distribution. 
LindleyGenInvExp        Lindley Generalized Inverted Exponential Distribution. 
LindleyGompertz         Lindley Gompertz Distribution.
LindleyHC               New Lindley Half Cauchy Distribution. 
LindleyInvExp           Lindley Inverse Exponential Distribution. 
LindleyInvWeibull       Lindley inverse Weibull Distribution. 
LindleyRayleigh         New Lindley-Rayleigh Distribution. 
LogisChen               Logistic Chen Distribution Distribution. 
LogisExpExt             Logistic Exponential Extension Distribution. 
LogisExpPower           Logistic-Exponential Power Distribution. 
LogisGompertz           Logistic Gompertz Distribution.  
LogisInvExp             Logistic Inverse Exponential Distribution. 
LogisInvLomax           Logistic Inverse Lomax Distribution. 
LogisInvWeibull         Logistic Inverse Weibull Distribution. 
LogisLomax              Logistic Lomax Distribution. 
LogisModExp             Logistic-Modified Exponential Distribution.
LogisNHE                Logistic-NHE Distribution. 
LogisRayleigh           Logistic-Rayleigh Distribution.
LogisWeib               Logistic-Weibull Distribution. 
ModAtanExp              Modified Arctan Exponential Distribution. 
ModGE                   Modified Generalized Exponential Distribution. 
ModInvGE                Modified Inverse Generalized Exponential Distribution. 
ModInvLomax             Modified Inverse Lomax Distribution.
ModInvNHE               Modified Inverse NHE Distribution. 
ModUbd                  Modified Upside Down Bathtub Shaped Hazard Function 
NewLindleyHC            New Lindley Half Cauchy Distribution. 
Perks                   Perks Distribution. 
PoisInvWeib             Poisson Inverse Weibull Distribution. 
PoissonChen             Poisson Chen Distribution. 
PoissonExpPower         Poisson Exponential Power Distribution. 
PoissonGenRayleigh      Poisson Generalized Rayleigh Distribution.
PoissonGPZ              Poisson Gompertz Distribution. 
PoissonInvLomax         Poisson Inverted Lomax Distribution.
PoissonInvNHE           Poisson Inverse NHE Distribution.  
PoissonInvSGZ           Poisson Inverse Shifted Gompertz Distribution.  
PoissonNHE              Poisson NHE Distribution. 
PoissonSGZ              Poisson Shifted Gompertz Distribution.

General functions:

gofic              Generic Goodness-of-Fit(GoF) and Model Diagnostics Function 
pp.plot            Generic Probability-Probability(P–P) Plot Function  
qq.plot            Generic Quantile-Quantile(Q-Q) Plot Function

Data:

bladder            Bladder Cancer Recurrence Times
conductors         Electromigration Failure Times of Microcircuit Conductors
fibers63           Strength of 63 Carbon Fibers at 10 mm Gauge Length
fibers65           Strength of 65 Carbon Fibers at 50 mm Gauge Length
fibers69           Tensile Strength of 69 Carbon Fibers at 20 mm Gauge Length
headneck44         Head and Neck Cancer Survival Times
rainfall           March Rainfall in Minneapolis/St. Paul
reactorpump        Failure Time Intervals of Secondary Reactor Pumps
relief             Relief Times of Patients Receiving an Analgesic
stress             Breaking Stress of Carbon Fibres
stress31           Fatigue Life of 6061-T6 Aluminum Coupons under 31,000 psi
stress66           Breaking Stress of 66 Carbon Fibers of Length 50 mm
survtimes          Survival Times of Guinea Pigs Infected with Tubercle Bacilli
waiting            Waiting Times of 100 Bank Customers
windshield         Service Times of Aircraft Windshields

Author(s)

Vijay Kumar <[email protected]>, Laxmi Prasad Sapkota <[email protected]>, Pankaj Kumar <[email protected]>, Lal Babu Sah <[email protected]>

Maintainer: Vijay Kumar <[email protected]>

Description

Recurrence times (in months) for bladder cancer patients, reported in Lee and Wang (2003). The dataset contains observed survival times without censoring information and is commonly used in survival analysis examples.

Usage

data(bladder)

Format

A numeric vector giving recurrence times (in months) for bladder cancer patients. A total of 128 observations are included.

Details

These recurrence times are widely used in demonstrations of survival analysis methods, including Kaplan–Meier estimation, hazard rate modelling, accelerated failure-time (AFT) models, and parametric distribution fitting. The dataset originally appears in Lee and Wang's Statistical Methods for Survival Data Analysis (3rd ed.), a standard reference text in biostatistics.

Note: The dataset provided here contains recurrence times only and does not include censoring indicators or covariates found in extended versions of the bladder cancer data.

Value

An object of class "numeric".

The vector consists of 128 observed recurrence times (in months), each corresponding to a single bladder cancer patient. Each value represents the time from treatment or diagnosis to documented cancer recurrence. The dataset is commonly used in survival analysis and biostatistics to illustrate time-to-event modeling, including Kaplan–Meier estimation, hazard rate analysis, accelerated failure-time (AFT) models, and parametric survival distributions.

References

Lee, E. T., & Wang, J. W. (2003). Statistical Methods for Survival Data Analysis (3rd ed.). Wiley, New York.

Examples

data(bladder)

# Basic summary
summary(bladder)

# Histogram of recurrence times
hist(
  bladder,
  main = "Bladder Cancer Recurrence Times",
  xlab = "Time (months)"
)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Chen-Exponential distribution.

Usage

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

Arguments

Details

The Chen-Exponential distribution is parameterized by the parameters $\alpha > 0$, $\beta > 0$, and $\lambda > 0$.

The Chen-Exponential distribution has CDF:

$F(x;\,\alpha,\beta,\lambda) = \, 1-\exp \left\{\lambda\left[1-\exp \left\{\left(e^{\beta x}-1\right)^\alpha\right\} \right] \right\}, \quad x > 0.$

where $\alpha$, $\beta$, and $\lambda$ are the parameters.

The following functions are included:

• dchen.exp() — Density function

• pchen.exp() — Distribution function

• qchen.exp() — Quantile function

• rchen.exp() — Random generation

• hchen.exp() — Hazard function

Value

• dchen.exp: numeric vector of (log-)densities

• pchen.exp: numeric vector of probabilities

• qchen.exp: numeric vector of quantiles

• rchen.exp: numeric vector of random variates

• hchen.exp: numeric vector of hazard values

References

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

Sapkota, L.P., & Kumar, V. (2023). Chen Exponential Distribution with Applications to Engineering Data. International Journal of Statistics and Reliability Engineering, 10(1), 33–47.

Sapkota, L.P., Alsahangiti, A.M., Kumar, V. Gemeay, A.M., Bakr, M.E., Balogun, O.S., & Muse, A.H. (2023). Arc-Tangent Exponential Distribution With Applications to Weather and Chemical Data Under Classical and Bayesian Approach, IEEE Access, 11, 115462–115476. doi:10.1109/ACCESS.2023.3324293

Examples

x <- seq(0.1, 1, 0.1)
dchen.exp(x, 1.5, 0.8, 2)
pchen.exp(x, 1.5, 0.8, 2)
qchen.exp(0.5, 1.5, 0.8, 2)
rchen.exp(10, 1.5, 0.8, 2)
hchen.exp(x, 1.5, 0.8, 2)
#Data
x <- stress
#ML Estimates    
params = list(alpha=2.5462, beta=0.0537, lambda=87.6028)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pchen.exp, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qchen.exp, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
# Display plot; numerical summary stored in 'out'
out <- gofic(x, params = params, dfun = dchen.exp, 
             pfun = pchen.exp, plot=TRUE)
print.gofic(out)

Description

Failure-time data from an accelerated life test involving 59 microcircuit conductors. Electromigration refers to the movement of atoms in conductors under high current density, leading to eventual failure. The dataset contains observed failure times (in hours), with no censored observations.

Usage

data(conductors)

Format

A numeric vector of length 59 giving failure times in hours.

Details

Electromigration is a major wear-out mechanism in thin-film microelectronic circuits. Because electric current accelerates atomic migration, accelerated life tests are widely used to study the reliability of conductors. This dataset has been used extensively in the reliability literature, including analyses involving Weibull, lognormal, and power-lognormal lifetime models.

Value

An object of class "numeric".

The vector consists of 59 observed failure times (in hours), each corresponding to a single microcircuit conductor subjected to an accelerated life test. Each value represents the elapsed operating time until failure caused by electromigration. The dataset is commonly used in reliability engineering and lifetime data analysis to illustrate wear-out mechanisms and to fit and compare parametric lifetime models such as the Weibull, lognormal, and power-lognormal distributions.

References

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

Nelson, W., & Doganaksoy, N. (1995). Statistical analysis of life or strength data from specimens of various sizes using the power-(log)normal model. Recent Advances in Life-Testing and Reliability, 377–408.

Examples

data(conductors)

# Summary statistics
summary(conductors)

# Histogram of failure times
hist(conductors)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Exponentiated Exponential Power (EEP) distribution.

Usage

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

Arguments

Details

The EEP distribution is parameterized by the parameters $\alpha > 0$, $\lambda > 0$, and $\theta > 0$.

The Exponentiated Exponential Power (EEP) distribution has CDF:

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

where $\alpha$, $\lambda$, and $\theta$ are the parameters.

The implementation includes the following functions:

• dgen.exp.power() — Density function

• pgen.exp.power() — Distribution function

• qgen.exp.power() — Quantile function

• rgen.exp.power() — Random generation

• hgen.exp.power() — Hazard function

Value

• dgen.exp.power: numeric vector of (log-)densities

• pgen.exp.power: numeric vector of probabilities

• qgen.exp.power: numeric vector of quantiles

• rgen.exp.power: numeric vector of random variates

• hgen.exp.power: numeric vector of hazard values

References

Sapkota, L.P., & Kumar, V.(2024). Bayesian Analysis of Exponentiated Exponential Power Distribution under Hamiltonian Monte Carlo Method, Statistics and Applications. Statistics and Applications, 22(2), 231–258.

Srivastava, A.K., & Kumar, V.(2011). Analysis of Software Reliability Data using Exponential Power Model. International Journal of Advanced Computer Science and Applications, 2(2), 38–45, doi:10.14569/IJACSA.2011.020208

Chen, Z.(1999). Statistical inference about the shape parameter of the exponential power distribution, Statistical Papers, 40, 459–468.

Smith, R.M., & Bain, L.J. (1975). An exponential power life-test distribution. IEEE Communications in Statistics, 4, 469–481.

Examples

x <- seq(0.1, 1, 0.1)
dgen.exp.power(x, 1.5, 0.8, 2)
pgen.exp.power(x, 1.5, 0.8, 2)
qgen.exp.power(0.5, 1.5, 0.8, 2)
rgen.exp.power(10, 1.5, 0.8, 2)
hgen.exp.power(x, 1.5, 0.8, 2)
#Data
x <- waiting
#ML Estimates    
params = list(alpha=0.3407, lambda=0.6068, theta=7.6150)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pgen.exp.power, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qgen.exp.power, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
# Neither plot nor console output; results stored in 'out'
out <- gofic(x, params = params,
             dfun = dgen.exp.power, pfun = pgen.exp.power, plot=FALSE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Exponentiated Inverse Chen distribution.

Usage

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

Arguments

Details

The Exponentiated Inverse Chen distribution is parameterized by the parameters $\alpha > 0$, $\lambda > 0$, and $\theta > 0$.

The Exponentiated Inverse Chen distribution has CDF:

$F(x; \alpha, \lambda, \theta) = 1 - \left[ 1 - \exp\left( \lambda \left( 1 - \exp(x^{-\alpha}) \right) \right) \right]^{\theta}, \quad x > 0.$

where $\alpha$, $\lambda$, and $\theta$ are the parameters.

The functions available are listed below:

• dexpo.inv.chen() — Density function

• pexpo.inv.chen() — Distribution function

• qexpo.inv.chen() — Quantile function

• rexpo.inv.chen() — Random generation

• hexpo.inv.chen() — Hazard function

Value

• dexpo.inv.chen: numeric vector of (log-)densities

• pexpo.inv.chen: numeric vector of probabilities

• qexpo.inv.chen: numeric vector of quantiles

• rexpo.inv.chen: numeric vector of random variates

• hexpo.inv.chen: numeric vector of hazard values

References

Telee, L. B. S., & Kumar, V. (2023). Exponentiated Inverse Chen distribution: Properties and applications. Journal of Nepalese Management Academia, 1(1), 53–62. doi:10.3126/jnma.v1i1.62033

Srivastava, A.K., & Kumar, V.(2011). Markov Chain Monte Carlo Methods for Bayesian Inference of the Chen Model. International Journal of Computer Information Systems, 2(2), 7–14.

Examples

x <- seq(2, 5, 0.25)
dexpo.inv.chen(x, 0.5, 2.5, 1.5)
pexpo.inv.chen(x, 0.5, 2.5, 1.5)
qexpo.inv.chen(0.5, 0.5, 2.5, 1.5)
rexpo.inv.chen(10, 0.5, 2.5, 1.5)
hexpo.inv.chen(x, 0.5, 2.5, 1.5)

# Data
x <- headneck44
# ML estimates
params = list(alpha=0.3947, lambda=15.5330, theta=8.1726)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pexpo.inv.chen, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qexpo.inv.chen, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
# Display plot and print numerical summary
gofic(x, params = params,
      dfun = dexpo.inv.chen, pfun=pexpo.inv.chen, plot=TRUE, verbose = TRUE)

Description

Measurements of tensile strength (in gigapascals, GPa) for 63 single carbon fibers tested at a gauge length of 10 mm. These data were originally reported by Bader and Priest (1982) in their study of fibre and bundle strength in hybrid composites.

Usage

fibers63

Format

A numeric vector of length 63 containing tensile strength measurements (in GPa).

Details

The dataset contains tensile strength values for individual carbon fibers cut to a gauge length of 10 mm. This dataset has been used extensively in materials science and reliability studies for modeling strength distributions and assessing variability in carbon fiber performance.

The data originate from the same experimental study that produced several related carbon-fiber datasets (e.g., fibers65, fibers69).

Value

An object of class "numeric".

The vector consists of 63 observed tensile strength measurements (in gigapascals), each corresponding to an individual carbon fiber tested at a gauge length of 10 mm. Each value represents the breaking strength of a single fiber specimen. The dataset is commonly used in materials science and reliability engineering for modeling strength distributions, assessing variability, and fitting parametric lifetime or strength models.

References

Bader, M. G., & Priest, A. M. (1982). Statistical aspects of fibre and bundle strength in hybrid composites. Progress in Science and Engineering of Composites, 1129–1136.

Examples

data(fibers63)

summary(fibers63)

hist(
  fibers63,
  main = "Tensile Strength of Carbon Fibers (10 mm Gauge Length)",
  xlab = "Strength (GPa)"
)

Description

Tensile strength measurements (in gigapascals, GPa) for 65 carbon fibers tested under tension at a gauge length of 50 mm. These data were originally reported by Bader and Priest (1982) in their foundational study on fibre and bundle strength in hybrid composites.

Usage

fibers65

Format

A numeric vector of length 65 containing tensile strength values (in GPa).

Details

The fibers were tested at a gauge length of 50 mm to study the variability of carbon fiber strength under controlled conditions. This dataset is frequently used in reliability analysis, composite material modeling, and strength distribution studies. It is one of several datasets originating from the Bader and Priest (1982) carbon-fiber experiments.

Value

An object of class "numeric".

The vector contains 65 observed tensile strength measurements (in gigapascals) of individual carbon fibers tested at a gauge length of 50 mm. Each element represents the breaking strength of a single fiber specimen. The dataset is typically used as input for statistical modeling, reliability analysis, and lifetime or strength distribution studies in composite materials research.

References

Bader, M. G., & Priest, A. M. (1982). Statistical aspects of fibre and bundle strength in hybrid composites. Progress in Science and Engineering of Composites, 1129–1136.

Examples

data(fibers65)

summary(fibers65)

plot(
  fibers65,
  ylab = "Strength (GPa)",
  main = "Carbon Fiber Strength (50 mm Gauge Length)"
)

hist(
  fibers65,
  main = "Histogram of Carbon Fiber Strength",
  xlab = "Strength (GPa)"
)

Description

Measurements of tensile strength (in gigapascals, GPa) for 69 carbon fibers tested under tension at a gauge length of 20 mm. These data were originally reported by Bader and Priest (1982) in their study of fibre and bundle strength in hybrid composites.

Usage

fibers69

Format

A numeric vector of length 69 containing tensile strength values (in GPa).

Details

This dataset has been widely used in composite-material and reliability studies, particularly for modeling strength distributions of carbon fibers. The original experiment measured the tensile strength of individual fibers at a gauge length of 20 mm, providing insight into the statistical behavior of fiber strength under tension.

Value

An object of class "numeric".

The vector consists of 69 tensile strength measurements (in gigapascals) corresponding to individual carbon fiber specimens tested at a gauge length of 20 mm. Each value represents the breaking strength of a single fiber. The dataset is commonly used for statistical analysis of strength distributions, reliability modeling, and comparative studies of gauge-length effects in composite materials.

References

Bader, M. G., & Priest, A. M. (1982). Statistical aspects of fibre and bundle strength in hybrid composites. Progress in Science and Engineering of Composites, 1129–1136.

Examples

data(fibers69)

summary(fibers69)

hist(
  fibers69,
  main = "Tensile Strength of Carbon Fibers (20 mm Gauge Length)",
  xlab = "Strength (GPa)"
)

Description

Computes log-likelihood, information criteria (AIC, BIC, AICC, HQIC) and classical goodness-of-fit statistics (Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling) for a given numeric data vector and user-supplied density and distribution functions.

Usage

gofic(x, params, dfun, pfun, plot = TRUE, verbose = FALSE)

Arguments

Details

Optionally plots the empirical cumulative distribution function (ECDF) against the theoretical cumulative distribution function.

The supplied dfun and pfun must accept arguments x and q respectively, followed by named model parameters. Density values must be finite and positive; non-positive densities trigger a warning but computation proceeds.

Value

An object of class "gofic" containing:

• logLik Numeric; log-likelihood value.

• AIC Akaike Information Criterion.

• BIC Bayesian Information Criterion.

• AICC Corrected Akaike Information Criterion.

• HQIC Hannan–Quinn Information Criterion.

• KS Object returned by stats::ks.test().

• CVM Object returned by goftest::cvm.test().

• AD Object returned by goftest::ad.test().

• n Sample size.

• params Model parameters supplied.

The object is returned invisibly.

See Also

print.gofic, ks.test, cvm.test, ad.test

Examples

# Example 1 with built-in Weibull distribution

set.seed(123)
x <- rweibull(100, shape = 2, scale = 1)
out <- gofic(x, params = list(shape = 2, scale = 1),
      dfun = dweibull, pfun = pweibull, plot=FALSE)
out


# Example 2: For a user defined distribution
# Goodness-of-Fit(GoF) and Model Diagnostics for Chen-Exponential distribution
#Data
x <- stress
#ML Estimates    
params = list(alpha=2.5462, beta=0.0537, lambda=87.6028)
# Display plot and print numerical summary
gofic(x, params = params,
        dfun = dchen.exp, pfun = pchen.exp, plot = TRUE, verbose = TRUE)

# Display plot only (no numerical summary)
gofic(x, params = params,
        dfun = dchen.exp, pfun = pchen.exp, plot = TRUE, verbose = FALSE)

# Print numerical summary only (no plot)
gofic(x, params = params,
        dfun = dchen.exp, pfun = pchen.exp, plot = FALSE, verbose = TRUE)

# Display plot; numerical summary stored in 'out'
out <- gofic(x, params = params,
              dfun = dchen.exp, pfun = pchen.exp, plot = TRUE, verbose = FALSE)
print.gofic(out)

# Neither plot nor console output; results stored in 'out'
out <- gofic(x, params = params,
               dfun = dchen.exp, pfun = pchen.exp, plot = FALSE, verbose = FALSE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Gompertz Extension distribution.

Usage

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

Arguments

Details

The Gompertz Extension distribution is parameterized by the parameters $\alpha > 0$, $\lambda > 0$, and $\theta > 0$.

The Gompertz Extension distribution has CDF:

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

where $\alpha$, $\lambda$, and $\theta$ are the parameters.

The functions available are listed below:

• dgompertz.ext() — Density function

• pgompertz.ext() — Distribution function

• qgompertz.ext() — Quantile function

• rgompertz.ext() — Random generation

• hgompertz.ext() — Hazard function

Value

• dgompertz.ext: numeric vector of (log-)densities

• pgompertz.ext: numeric vector of probabilities

• qgompertz.ext: numeric vector of quantiles

• rgompertz.ext: numeric vector of random variates

• hgompertz.ext: numeric vector of hazard values

References

Chaudhary, A.K., & Kumar, V. (2020). A Bayesian Estimation and Prediction of Gompertz Extension Distribution Using the MCMC Method. Nepal Journal of Science and Technology(NJST), 19(1), 142–160. doi:10.3126/njst.v19i1.29795

Examples

x <- seq(1.0, 10, 0.25)
dgompertz.ext(x, 0.1, 5.0, 2.5)
pgompertz.ext(x, 0.1, 5.0, 2.5)
qgompertz.ext(0.5, 0.1, 5.0, 2.5)
rgompertz.ext(10, 0.1, 5.0, 2.5)
hgompertz.ext(x, 0.1, 5.0, 2.5)

# Data
x <- stress
# ML estimates
params = list(alpha=0.0678, lambda=44.34760, theta=2.5225)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pgompertz.ext, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qgompertz.ext, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dgompertz.ext, pfun=pgompertz.ext, plot=TRUE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Half-Cauchy Chen distribution.

Usage

dhc.chen(x, beta, lambda, theta, log = FALSE)
phc.chen(q, beta, lambda, theta, lower.tail = TRUE, log.p = FALSE)
qhc.chen(p, beta, lambda, theta, lower.tail = TRUE, log.p = FALSE)
rhc.chen(n, beta, lambda, theta)
hhc.chen(x, beta, lambda, theta)

Arguments

Details

The Half-Cauchy Chen distribution is parameterized by the parameters $\beta > 0$, $\lambda > 0$, and $\theta > 0$.

The Half-Cauchy Chen distribution has CDF:

$F(x; \beta, \lambda, \theta) = \quad \frac{2}{\pi }\arctan \left\{ { - \frac{\lambda }{\theta } (1 - {e^{{x^\beta }}})} \right\} \quad ;\;x > 0.$

where $\beta$, $\lambda$, and $\theta$ are the parameters.

Included functions are:

• dhc.chen() — Density function

• phc.chen() — Distribution function

• qhc.chen() — Quantile function

• rhc.chen() — Random generation

• hhc.chen() — Hazard function

Value

• dhc.chen: numeric vector of (log-)densities

• phc.chen: numeric vector of probabilities

• qhc.chen: numeric vector of quantiles

• rhc.chen: numeric vector of random variates

• hhc.chen: numeric vector of hazard values

References

Chaudhary, A.K., Yadav, R.S., & Kumar, V.(2023). Half-Cauchy Chen Distribution with Theories and Applications. Journal of Institute of Science and Technology, 28(1), 45–55. doi:10.3126/jist.v28i1.56494

Polson, N.G., & Scott, J.G. (2012). On the half-Cauchy prior for a global scale parameter. Bayesian Analysis, 7(4), 887–902.

Telee, L.B.S., & Kumar, V.(2024). Arctan-Chen Distribution with Properties and Application. International Journal of Statistics and Reliability Engineering, 11(1), 93–100.

Examples

x <- seq(1.0, 5, 0.25)
dhc.chen(x, 2.0, 0.5, 2.5)
phc.chen(x, 2.0, 0.5, 2.5)
qhc.chen(0.5, 2.0, 0.5, 2.5)
rhc.chen(10, 2.0, 0.5, 2.5)
hhc.chen(x, 2.0, 0.5, 2.5)

# Data
x <- conductors
# ML estimates
params = list(beta=0.9753, lambda=0.0398, theta=29.0272)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = phc.chen, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qhc.chen, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
res <- gofic(x, params = params,
             dfun = dhc.chen, pfun=phc.chen, plot=FALSE)
print.gofic(res)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Half-Cauchy Generalized Exponential(HCGE) distribution.

Usage

dhc.gen.exp(x, alpha, lambda, theta, log = FALSE)
phc.gen.exp(q, alpha, lambda, theta, lower.tail = TRUE, log.p = FALSE)
qhc.gen.exp(p, alpha, lambda, theta, lower.tail = TRUE, log.p = FALSE)
rhc.gen.exp(n, alpha, lambda, theta)
hhc.gen.exp(x, alpha, lambda, theta)

Arguments

Details

The HCGE distribution is parameterized by the parameters $\alpha > 0$, $\lambda > 0$, and $\theta > 0$.

The HCGE distribution has CDF:

$F(x; \alpha, \lambda, \theta) = \quad 1 - \frac{2}{\pi }\arctan \left[ { - \frac{\alpha }{\theta }\ln \left( {1 - {e^{ - \lambda x}}} \right)} \right] \quad ;\;x > 0.$

where $\alpha$, $\lambda$, and $\theta$ are the parameters.

The implementation includes the following functions:

• dhc.gen.exp() — Density function

• phc.gen.exp() — Distribution function

• qhc.gen.exp() — Quantile function

• rhc.gen.exp() — Random generation

• hhc.gen.exp() — Hazard function

Value

• dhc.gen.exp: numeric vector of (log-)densities

• phc.gen.exp: numeric vector of probabilities

• qhc.gen.exp: numeric vector of quantiles

• rhc.gen.exp: numeric vector of random variates

• hhc.gen.exp: numeric vector of hazard values

References

Chaudhary, A.K., Sapkota, L.P. & Kumar, V. (2022). Half-Cauchy Generalized Exponential Distribution:Theory and Application. Journal of Nepal Mathematical Society (JNMS), 5(2), 1–10. doi:10.3126/jnms.v5i2.50018

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

Examples

x <- seq(0.1, 10, 0.2)
dhc.gen.exp(x, 2.0, 0.5, 0.1)
phc.gen.exp(x, 2.0, 0.5, 0.1)
qhc.gen.exp(0.5, 2.0, 0.5, 0.1)
rhc.gen.exp(10, 2.0, 0.5, 0.1)
hhc.gen.exp(x, 2.0, 0.5, 0.1)

# Data
x <- conductors
# ML estimates
params = list(alpha=6.6141, lambda=0.9352, theta=0.0103)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = phc.gen.exp, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qhc.gen.exp, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
res <- gofic(x, params = params,
             dfun = dhc.gen.exp, pfun=phc.gen.exp, plot=FALSE)
print.gofic(res)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Half-Cauchy Generalized Rayleigh distribution.

Usage

dhc.gen.rayleigh(x, alpha, lambda, theta, log = FALSE)
phc.gen.rayleigh(q, alpha, lambda, theta, lower.tail = TRUE, log.p = FALSE)
qhc.gen.rayleigh(p, alpha, lambda, theta, lower.tail = TRUE, log.p = FALSE)
rhc.gen.rayleigh(n, alpha, lambda, theta)
hhc.gen.rayleigh(x, alpha, lambda, theta)

Arguments

Details

The Half-Cauchy Generalized Rayleigh distribution is parameterized by the parameters $\alpha > 0$, $\lambda > 0$, and $\theta > 0$.

The Half-Cauchy Generalized Rayleigh distribution has CDF:

$F(x; \alpha, \lambda, \theta) = \quad 1 - \frac{2}{\pi }\arctan \left\{ { - \frac{\alpha }{\theta } \log \left \{ {1 - {e^{ - {{\left( {\lambda x} \right)}^2}}}} \right\}} \right\} \quad ;\;x > 0.$

where $\alpha$, $\lambda$, and $\theta$ are the parameters.

The implementation includes the following functions:

• dhc.gen.rayleigh() — Density function

• phc.gen.rayleigh() — Distribution function

• qhc.gen.rayleigh() — Quantile function

• rhc.gen.rayleigh() — Random generation

• hhc.gen.rayleigh() — Hazard function

Value

• dhc.gen.rayleigh: numeric vector of (log-)densities

• phc.gen.rayleigh: numeric vector of probabilities

• qhc.gen.rayleigh: numeric vector of quantiles

• rhc.gen.rayleigh: numeric vector of random variates

• hhc.gen.rayleigh: numeric vector of hazard values

References

Sapkota, L.P., & Kumar, V. (2023). Half-Cauchy Generalized Rayleigh : Theory and Applications.South East Asian J. Math. & Math. Sc., 19(1), 335–360. doi:10.56827/SEAJMMS.2023.1901.27

Shrestha, S.K., & Kumar, V. (2014). Bayesian Analysis for the Generalized Rayleigh Distribution. International Journal of Statistika and Mathematika, 9(3), 118–131.

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

Examples

x <- seq(1.0, 5, 0.25)
dhc.gen.rayleigh(x, 2.0, 0.5, 0.1)
phc.gen.rayleigh(x, 2.0, 0.5, 0.1)
qhc.gen.rayleigh(0.5, 2.0, 0.5, 0.1)
rhc.gen.rayleigh(10, 2.0, 0.5, 0.1)
hhc.gen.rayleigh(x, 2.0, 0.5, 0.1)

# Data
x <- stress66
# ML estimates
params = list(alpha=1.4585, lambda=0.5300, theta=0.1655)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = phc.gen.rayleigh, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qhc.gen.rayleigh, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dhc.gen.rayleigh, pfun=phc.gen.rayleigh, plot=FALSE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Half-Cauchy Gompertz distribution.

Usage

dhc.gpz(x, alpha, lambda, theta, log = FALSE)
phc.gpz(q, alpha, lambda, theta, lower.tail = TRUE, log.p = FALSE)
qhc.gpz(p, alpha, lambda, theta, lower.tail = TRUE, log.p = FALSE)
rhc.gpz(n, alpha, lambda, theta)
hhc.gpz(x, alpha, lambda, theta)

Arguments

Details

The Half-Cauchy Gompertz distribution is parameterized by the parameters $\alpha > 0$, $\lambda > 0$, and $\theta > 0$.

The Half-Cauchy Gompertz distribution has CDF:

$F(x; \alpha, \lambda, \theta) = \quad \frac{2}{\pi }\arctan \left\{ { - \frac{\lambda }{{\alpha \theta }} \left( {1 - {e^{\alpha x}}} \right)} \right\} \quad ;\;x > 0.$

where $\alpha$, $\lambda$, and $\theta$ are the parameters.

The implementation includes the following functions:

• dhc.gpz() — Density function

• phc.gpz() — Distribution function

• qhc.gpz() — Quantile function

• rhc.gpz() — Random generation

• hhc.gpz() — Hazard function

Value

• dhc.gpz: numeric vector of (log-)densities

• phc.gpz: numeric vector of probabilities

• qhc.gpz: numeric vector of quantiles

• rhc.gpz: numeric vector of random variates

• hhc.gpz: numeric vector of hazard values

References

Sah, L.B., & Kumar, V. (2019). Half-Cauchy Gompertz Distribution : Different Methods of Estimation, Journal of National Academy of Mathematics, 33, 51–65.

Examples

x <- seq(1.0, 5, 0.25)
dhc.gpz(x, 2.0, 0.5, 2.5)
phc.gpz(x, 2.0, 0.5, 2.5)
qhc.gpz(0.5, 2.0, 0.5, 2.5)
rhc.gpz(10, 2.0, 0.5, 2.5)
hhc.gpz(x, 2.0, 0.5, 2.5)

# Data
x <- stress66
# ML estimates
params = list(alpha=1.6660, lambda=0.0328, theta=2.0578)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = phc.gpz, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qhc.gpz, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params, dfun=dhc.gpz, pfun=phc.gpz, plot=TRUE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Half-Cauchy Inverse Gompertz distribution.

Usage

dhc.inv.gpz(x, alpha, lambda, theta, log = FALSE)
phc.inv.gpz(q, alpha, lambda, theta, lower.tail = TRUE, log.p = FALSE)
qhc.inv.gpz(p, alpha, lambda, theta, lower.tail = TRUE, log.p = FALSE)
rhc.inv.gpz(n, alpha, lambda, theta)
hhc.inv.gpz(x, alpha, lambda, theta)

Arguments

Details

The Half-Cauchy Inverse Gompertz distribution is parameterized by the parameters $\alpha > 0$, $\lambda > 0$, and $\theta > 0$.

The Half-Cauchy Inverse Gompertz distribution has CDF:

$F(x; \alpha, \lambda, \theta) = \quad 1 - \frac{2}{\pi }\arctan \left\{ { - \frac{\lambda }{{\alpha \theta }} \left( {1 - {e^{\alpha /x}}} \right)} \right\} \quad ;\;x > 0.$

where $\alpha$, $\lambda$, and $\theta$ are the parameters.

The implementation includes the following functions:

• dhc.inv.gpz() — Density function

• phc.inv.gpz() — Distribution function

• qhc.inv.gpz() — Quantile function

• rhc.inv.gpz() — Random generation

• hhc.inv.gpz() — Hazard function

Value

• dhc.inv.gpz: numeric vector of (log-)densities

• phc.inv.gpz: numeric vector of probabilities

• qhc.inv.gpz: numeric vector of quantiles

• rhc.inv.gpz: numeric vector of random variates

• hhc.inv.gpz: numeric vector of hazard values

References

Chaudhary, A. K., Yadav, R. S., & Kumar, V. (2022). Half-Cauchy Inverse Gompertz distribution: Theory and applications. International Journal of Statistics and Applied Mathematics, 7(5), 94–102. doi:10.22271/maths.2022.v7.i5b.885

Examples

x <- seq(1.0, 10, 0.25)
dhc.inv.gpz(x, 2.0, 0.5, 2.5)
phc.inv.gpz(x, 2.0, 0.5, 2.5)
qhc.inv.gpz(0.5, 2.0, 0.5, 2.5)
rhc.inv.gpz(10, 2.0, 0.5, 2.5)
hhc.inv.gpz(x, 2.0, 0.5, 2.5)

# Data
x <- relief
# ML estimates
params = list(alpha=9.0830, lambda=0.8369, theta=17.9925)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = phc.inv.gpz, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qhc.inv.gpz, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dhc.inv.gpz, pfun=phc.inv.gpz, plot=TRUE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Half-Cauchy Inverse NHE distribution.

Usage

dhc.inv.NHE(x, beta, lambda, theta, log = FALSE)
phc.inv.NHE(q, beta, lambda, theta, lower.tail = TRUE, log.p = FALSE)
qhc.inv.NHE(p, beta, lambda, theta, lower.tail = TRUE, log.p = FALSE)
rhc.inv.NHE(n, beta, lambda, theta)
hhc.inv.NHE(x, beta, lambda, theta)

Arguments

Details

The Half-Cauchy Inverse NHE distribution is parameterized by the parameters $\beta > 0$, $\lambda > 0$, and $\theta > 0$.

The Half-Cauchy Inverse NHE distribution has CDF:

$F(x; \beta, \lambda, \theta) = \quad 1 - \frac{2}{\pi }\arctan \left[ { - \frac{1}{\theta }\left\{ {1 - {{\left( {1 + \frac{\lambda }{x}} \right)}^\beta }} \right\}} \right] \quad ;\;x > 0.$

where $\beta$, $\lambda$, and $\theta$ are the parameters.

Included functions are:

• dhc.inv.NHE() — Density function

• phc.inv.NHE() — Distribution function

• qhc.inv.NHE() — Quantile function

• rhc.inv.NHE() — Random generation

• hhc.inv.NHE() — Hazard function

Value

• dhc.inv.NHE: numeric vector of (log-)densities

• phc.inv.NHE: numeric vector of probabilities

• qhc.inv.NHE: numeric vector of quantiles

• rhc.inv.NHE: numeric vector of random variates

• hhc.inv.NHE: numeric vector of hazard values

References

Chaudhary, A.K., Telee, L.B.S. & Kumar,V. (2022). Half-Cauchy Inverse NHE Distribution: Properties and Applications. Nepal Journal of Mathematical Sciences (NJMS), 3(2), 1–12. doi:10.3126/njmathsci.v3i2.49198

Chaudhary, A. K., Sapkota, L. P., & Kumar, V. (2022). Some properties and applications of half Cauchy extended exponential distribution. Int. J. Stat. Appl. Math., 7(4), 226–235. doi:10.22271/maths.2022.v7.i4c.866

Chaudhary, A.K., & Kumar, V. (2022). Half Cauchy-Modified Exponential Distribution: Properties and Applications. Nepal Journal of Mathematical Sciences (NJMS), 3(1), 47–58. doi:10.3126/njmathsci.v3i1.44125

Examples

x <- seq(1.0, 5, 0.25)
dhc.inv.NHE(x, 2.0, 0.5, 2.5)
phc.inv.NHE(x, 2.0, 0.5, 2.5)
qhc.inv.NHE(0.5, 2.0, 0.5, 2.5)
rhc.inv.NHE(10, 2.0, 0.5, 2.5)
hhc.inv.NHE(x, 2.0, 0.5, 2.5)

# Data
x <- relief
# ML estimates
params = list(beta=79.7799, lambda=0.1129, theta=154.1769)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = phc.inv.NHE, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qhc.inv.NHE, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
res <- gofic(x, params = params,
             dfun = dhc.inv.NHE, pfun=phc.inv.NHE, plot=FALSE)
print.gofic(res)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Half-Cauchy NHE distribution.

Usage

dhc.NHE(x, beta, lambda, theta, log = FALSE)
phc.NHE(q, beta, lambda, theta, lower.tail = TRUE, log.p = FALSE)
qhc.NHE(p, beta, lambda, theta, lower.tail = TRUE, log.p = FALSE)
rhc.NHE(n, beta, lambda, theta)
hhc.NHE(x, beta, lambda, theta)

Arguments

Details

The Half-Cauchy NHE distribution is parameterized by the parameters $\beta > 0$, $\lambda > 0$, and $\theta > 0$.

The Half-Cauchy NHE distribution has CDF:

$F(x; \beta, \lambda, \theta) = \frac{2}{\pi} \arctan\left\{ -\frac{1}{\theta} \left( 1 - (1 + \lambda x)^{\beta} \right) \right\}, \quad x > 0.$

where $\beta$, $\lambda$, and $\theta$ are the parameters.

The implementation includes the following functions:

• dhc.NHE() — Density function

• phc.NHE() — Distribution function

• qhc.NHE() — Quantile function

• rhc.NHE() — Random generation

• hhc.NHE() — Hazard function

Value

• dhc.NHE: numeric vector of (log-)densities

• phc.NHE: numeric vector of probabilities

• qhc.NHE: numeric vector of quantiles

• rhc.NHE: numeric vector of random variates

• hhc.NHE: numeric vector of hazard values

References

Chaudhary, A. K., & Kumar, V.(2021). Arctan Exponential Extension Distribution with Properties and Applications. International Journal of Applied Research (IJAR), 7(1), 432–442. doi:10.22271/allresearch.2021.v7.i1f.8251

Telee, L. B. S., & Kumar, V. (2022). Some properties and applications of half-Cauchy exponential extension distribution. Int. J. Stat. Appl. Math., 7(6), 91–101. doi:10.22271/maths.2022.v7.i6b.902

Kumar, V. (2010). Bayesian analysis of exponential extension model. J. Nat. Acad. Math., 24, 109-128.

Examples

x <- seq(1.0, 5, 0.25)
dhc.NHE(x, 2.0, 0.5, 2.5)
phc.NHE(x, 2.0, 0.5, 2.5)
qhc.NHE(0.5, 2.0, 0.5, 2.5)
rhc.NHE(10, 2.0, 0.5, 2.5)
hhc.NHE(x, 2.0, 0.5, 2.5)

# Data
x <- stress66
# ML estimates
params = list(beta=95.2115, lambda=0.0184, theta=118.0656)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = phc.NHE, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qhc.NHE, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dhc.NHE, pfun=phc.NHE, plot=TRUE)
print.gofic(out)

Description

A dataset containing survival times (in days) of 44 patients with Head and Neck cancer who were treated using radiotherapy. The dataset was originally reported by Efron (1988) in his work on logistic regression, survival analysis, and Kaplan–Meier methods.

Usage

headneck44

Format

A numeric vector of length 44 containing survival times (in days).

Details

This dataset has been widely used in survival analysis literature, particularly for demonstrating Kaplan–Meier estimation and related nonparametric survival techniques. The patients in the study were treated with radiotherapy, and their survival times were recorded.

Value

An object of class "numeric".

The vector consists of 44 observed survival times (in days), each corresponding to a single patient diagnosed with Head and Neck cancer and treated with radiotherapy. Each value represents the time from treatment initiation to death or last follow-up. The dataset is commonly used as input for illustrating and comparing nonparametric survival analysis methods, including Kaplan–Meier estimation.

References

Efron, B. (1988). Logistic regression, survival analysis and the Kaplan–Meier curve. Journal of the American Statistical Association, 83(402), 414–425.

Examples

summary(headneck44)

plot(
  headneck44,
  main = "Head and Neck Cancer Survival Times",
  ylab = "Days"
)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Half-Logistic Inverted Weibull distribution.

Usage

dHL.inv.weib(x, alpha, beta, lambda, log = FALSE)
pHL.inv.weib(q, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE)
qHL.inv.weib(p, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE)
rHL.inv.weib(n, alpha, beta, lambda)
hHL.inv.weib(x, alpha, beta, lambda)

Arguments

Details

The HLIW distribution is parameterized by shape parameters $\alpha > 0$, $\beta > 0$, and $\lambda > 0$.

The Half-Logistic Inverted Weibull (HLIW) distribution has CDF:

$F(x; \alpha, \beta, \lambda) = \frac{1-\left\{1-e^{-\alpha x^{-\beta}}\right\}^\lambda} {1+\left\{1-e^{-\alpha x^{-\beta}}\right\}^\lambda} \, ; \quad x > 0.$

where $\alpha$, $\beta$, and $\lambda$ are the parameters.

The implementation includes the following functions:

• dHL.inv.weib() — Density function

• pHL.inv.weib() — Distribution function

• qHL.inv.weib() — Quantile function

• rHL.inv.weib() — Random generation

• hHL.inv.weib() — Hazard function

Value

• dHL.inv.weib: numeric vector of (log-)densities

• pHL.inv.weib: numeric vector of probabilities

• qHL.inv.weib: numeric vector of quantiles

• rHL.inv.weib: numeric vector of random variates

• hHL.inv.weib: numeric vector of hazard values

References

Elgarhy, M., ul Haq, M.A. & Perveen, I. (2019). Type II Half Logistic Exponential Distribution with Applications. Ann. Data. Sci., 6, 245–257 doi:10.1007/s40745-018-0175-y

Chaudhary, A. K., & Kumar, V. (2020). Half Logistic Exponential Extension Distribution with Properties and Applications. International Journal of Recent Technology and Engineering (IJRTE), 8(3), 506–512. doi:10.35940/ijrte.C4625.099320

Dhungana, G.P. & Kumar, V.(2022). Half Logistic Inverted Weibull Distribution: Properties and Applications. J. Stat. Appl. Pro. Lett., 9(3), 161–178. doi:10.18576/jsapl/090306

Examples

x <- seq(0.1, 5, 0.1)
dHL.inv.weib(x, 1.5, 0.8, 2)
pHL.inv.weib(x, 1.5, 0.8, 2)
qHL.inv.weib(0.5, 1.5, 0.8, 2)
rHL.inv.weib(10, 1.5, 0.8, 2)
hHL.inv.weib(x, 1.5, 0.8, 2)

#Data
x <- survtimes
gofic(x,
      params = list(alpha=31.1650, beta=0.4213, lambda=45.5485),
      dfun = dHL.inv.weib, pfun = pHL.inv.weib, plot=TRUE, verbose = TRUE)

pp.plot(x,
        params = list(alpha=31.1650, beta=0.4213, lambda=45.5485),
        pfun = pHL.inv.weib, fit.line=TRUE)

qq.plot(x,
        params = list(alpha=31.1650, beta=0.4213, lambda=45.5485),
        qfun = qHL.inv.weib, fit.line=TRUE)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Half-Logistic NHE distribution.

Usage

dHL.nhe(x, alpha, beta, lambda, log = FALSE)
pHL.nhe(q, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE)
qHL.nhe(p, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE)
rHL.nhe(n, alpha, beta, lambda)
hHL.nhe(x, alpha, beta, lambda)

Arguments

Details

The Half-Logistic NHE distribution is parameterized by the parameters $\alpha > 0$, $\beta > 0$, and $\lambda > 0$.

The Half-Logistic NHE distribution has CDF:

$F(x;\,\alpha,\beta,\lambda) = \frac{{1 - \exp \left[ {\lambda \left\{ {1 - {{(1 + \alpha x)}^\beta }} \right\}} \right]}}{{1 + \exp \left[ {\lambda \left\{ {1 - {{(1 + \alpha x)} ^\beta }} \right\}} \right]}} \, ;\quad x > 0.$

where $\alpha$, $\beta$, and $\lambda$ are the parameters.

The functions available are listed below:

• dHL.nhe() — Density function

• pHL.nhe() — Distribution function

• qHL.nhe() — Quantile function

• rHL.nhe() — Random generation

• hHL.nhe() — Hazard function

Value

• dHL.nhe: numeric vector of (log-)densities

• pHL.nhe: numeric vector of probabilities

• qHL.nhe: numeric vector of quantiles

• rHL.nhe: numeric vector of random variates

• hHL.nhe: numeric vector of hazard values

References

Almarashi, A. M., Elgarhy, M., Elsehetry, M. M., Kibria, B. G., & Algarni, A. (2019). A new extension of exponential distribution with statistical properties and applications. Journal of Nonlinear Sciences and Applications, 12, 135–145.

Chaudhary, A.K., & Kumar, V.(2020). Half Logistic Modified Exponential Distribution:Properties and Applications. EPRA International Journal of Multidisciplinary Research (IJMR), 6(12),276–286. doi:10.36713/epra3291

Joshi, R. K., & Kumar, V. (2020). Half Logistic NHE: Properties and Application. International Journal for Research in Applied Science & Engineering Technology (IJRASET), 8(9), 742–753. doi:10.22214/ijraset.2020.31557

Nadarajah, S., & Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45(6), 543–558.

Examples

x <- seq(0.1, 1, 0.1)
dHL.nhe(x, 1.5, 0.8, 2)
pHL.nhe(x, 1.5, 0.8, 2)
qHL.nhe(0.5, 1.5, 0.8, 2)
rHL.nhe(10, 1.5, 0.8, 2)
hHL.nhe(x, 1.5, 0.8, 2)

#Data
x <- windshield
#ML Estimates    
params = list(alpha =0.1649, beta=3.7152, lambda=0.5881)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pHL.nhe, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qHL.nhe, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dHL.nhe, pfun = pHL.nhe, plot=FALSE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Inverse Exponentiated Exponential Poisson distribution.

Usage

dinv.expo.exp.pois(x, alpha, beta, lambda, log = FALSE)
pinv.expo.exp.pois(q, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE)
qinv.expo.exp.pois(p, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE)
rinv.expo.exp.pois(n, alpha, beta, lambda)
hinv.expo.exp.pois(x, alpha, beta, lambda)

Arguments

Details

The Inverse Exponentiated Exponential Poisson distribution is parameterized by the parameters $\alpha > 0$, $\beta > 0$, and $\lambda > 0$.

The Inverse Exponentiated Exponential Poisson distribution has CDF:

$F(x;\,\alpha,\beta,\lambda) = 1 - \frac{1}{{\left( {1 - {e^{ - \lambda }}} \right)}} \left[ {1 - \exp \left\{ { - \lambda {{\left( {1 - {e^{ - \beta /x}}} \right)}^ \alpha }} \right\}} \right]\,; \quad x > 0.$

where $\alpha$, $\beta$, and $\lambda$ are the parameters.

The implementation includes the following functions:

• dinv.expo.exp.pois() — Density function

• pinv.expo.exp.pois() — Distribution function

• qinv.expo.exp.pois() — Quantile function

• rinv.expo.exp.pois() — Random generation

• hinv.expo.exp.pois() — Hazard function

Value

• dinv.expo.exp.pois: numeric vector of (log-)densities

• pinv.expo.exp.pois: numeric vector of probabilities

• qinv.expo.exp.pois: numeric vector of quantiles

• rinv.expo.exp.pois: numeric vector of random variates

• hinv.expo.exp.pois: numeric vector of hazard values

References

Ristic, M.M., & Nadarajah, S.(2014). A New Lifetime Distribution. Journal of Statistical Computation and Simulation, 84(1), 135–150. doi:10.1080/00949655.2012.697163

Telee, L. B. S., & Kumar, V. (2023). Inverse Exponentiated Exponential Poisson Distribution with Theory and Applications. International Journal of Engineering Science Technologies, 7(5), 17–36. doi:10.29121/IJOEST.v7.i5.2023.535

Examples

x <- seq(0.1, 1, 0.1)
dinv.expo.exp.pois(x, 1.5, 0.8, 2)
pinv.expo.exp.pois(x, 1.5, 0.8, 2)
qinv.expo.exp.pois(0.5, 1.5, 0.8, 2)
rinv.expo.exp.pois(10, 1.5, 0.8, 2)
hinv.expo.exp.pois(x, 1.5, 0.8, 2)

#Data
x <- conductors
#ML Estimates    
params = list(alpha =40.5895, beta=22.7519, lambda=2.9979)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pinv.expo.exp.pois, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qinv.expo.exp.pois, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
res <- gofic(x, params = params, dfun = dinv.expo.exp.pois, 
             pfun = pinv.expo.exp.pois, plot=FALSE)
print.gofic(res)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Inverse Exponential Power distribution.

Usage

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

Arguments

Details

The Inverse Exponential Power distribution is parameterized by the parameters $\alpha > 0$ and $\lambda > 0$.

The Inverse Exponential Power distribution has CDF:

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

where $\alpha$ and $\lambda$ are the parameters.

The implementation includes the following functions:

• dinv.exp.power() — Density function

• pinv.exp.power() — Distribution function

• qinv.exp.power() — Quantile function

• rinv.exp.power() — Random generation

• hinv.exp.power() — Hazard function

Value

• dinv.exp.power: numeric vector of (log-)densities

• pinv.exp.power: numeric vector of probabilities

• qinv.exp.power: numeric vector of quantiles

• rinv.exp.power: numeric vector of random variates

• hinv.exp.power: numeric vector of hazard values

References

Chaudhary, A.K., Sapkota,L.P. & Kumar, V.(2023). Inverse Exponential Power distribution: Theory and Applications. International Journal of Mathematics, Statistics and Operations Research, 3(1), 175–185.

Examples

x <- seq(1.0, 5.0, 0.2)
dinv.exp.power(x, 2.5, 0.5)
pinv.exp.power(x, 2.5, 0.5)
qinv.exp.power(0.5, 2.5, 0.5)
rinv.exp.power(10, 2.5, 0.5)
hinv.exp.power(x, 2.5, 0.5)

# Data
x <- relief
# ML estimates
params = list(alpha=2.8286, lambda=1.3346)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pinv.exp.power, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qinv.exp.power, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dinv.exp.power, pfun=pinv.exp.power, plot=FALSE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Inverse Generalized Gompertz distribution.

Usage

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

Arguments

Details

The Inverse Generalized Gompertz distribution is parameterized by the parameters $\alpha > 0$, $\lambda > 0$, and $\theta > 0$.

The Inverse Generalized Gompertz distribution has CDF:

$F(x; \alpha, \lambda, \theta) = 1 - \left[ 1 - \exp\left( \frac{\lambda}{\alpha} \left( 1 - \exp(\alpha / x) \right) \right) \right]^{\theta}, \quad x > 0.$

where $\alpha$, $\lambda$, and $\theta$ are the parameters.

The implementation includes the following functions:

• dinv.gen.gpz() — Density function

• pinv.gen.gpz() — Distribution function

• qinv.gen.gpz() — Quantile function

• rinv.gen.gpz() — Random generation

• hinv.gen.gpz() — Hazard function

Value

• dinv.gen.gpz: numeric vector of (log-)densities

• pinv.gen.gpz: numeric vector of probabilities

• qinv.gen.gpz: numeric vector of quantiles

• rinv.gen.gpz: numeric vector of random variates

• hinv.gen.gpz: numeric vector of hazard values

References

Chaudhary, A.K., & Kumar, V. (2017). Inverse Generalized Gompertz Distribution with Properties and Applications. Journal of National Academy of Mathematics, 31, 1–15.

Examples

x <- seq(2, 5, 0.25)
dinv.gen.gpz(x, 1.5, 2.5, 5.0)
pinv.gen.gpz(x, 1.5, 2.5, 5.0)
qinv.gen.gpz(0.5, 1.5, 2.5, 5.0)
rinv.gen.gpz(10, 1.5, 2.5, 5.0)
hinv.gen.gpz(x, 1.5, 2.5, 5.0)

# Data
x <- fibers63
# ML estimates
params = list(alpha=3.4106, lambda=5.4685, theta=20.9199)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pinv.gen.gpz, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qinv.gen.gpz, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dinv.gen.gpz, pfun=pinv.gen.gpz, plot=TRUE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Inverse Pham distribution.

Usage

dinv.pham(x, beta, delta, log = FALSE)
pinv.pham(q, beta, delta, lower.tail = TRUE, log.p = FALSE)
qinv.pham(p, beta, delta, lower.tail = TRUE, log.p = FALSE)
rinv.pham(n, beta, delta)
hinv.pham(x, beta, delta)

Arguments

Details

The Inverse Pham distribution is parameterized by the parameters $\beta > 0$, and $\delta > 0$.

The Inverse Pham distribution has CDF:

$F(x; \beta, \delta) = \exp \left( {1 - {\delta ^{{x^{ - \beta }}}}} \right) \quad ;\;x > 0.$

where$\beta$ and $\delta$ are the parameters.

The following functions are included:

• dinv.pham() — Density function

• pinv.pham() — Distribution function

• qinv.pham() — Quantile function

• rinv.pham() — Random generation

• hinv.pham() — Hazard function

Value

• dinv.pham: numeric vector of (log-)densities

• pinv.pham: numeric vector of probabilities

• qinv.pham: numeric vector of quantiles

• rinv.pham: numeric vector of random variates

• hinv.pham: numeric vector of hazard values

References

Elbatal, M., Araibi, M.I.A., Ocloo, S.K., Almetwally, E.M., Sapkota, L.P., & Gemeay, A.M. (2025). Classical and Bayesian Methodology for a New Inverse Statistical Model. Engineering Reports, 7(8), 1–33. doi:10.1002/eng2.70323

Srivastava, A.K., & Kumar, V. (2011). Analysis of Pham (Loglog) Reliability Model Using Bayesian Approach. Computer Science Journal, 1(2), 79–100.

Pham, H. (2002). A Vtub-Shaped Hazard Rate Function With Applications to System Safety. International Journal of Reliability and Applications, 3(1), 1–16.

Examples

x <- seq(1, 10, 0.5)
dinv.pham(x, 0.5, 1.5)
pinv.pham(x, 0.5, 1.5)
qinv.pham(0.5, 0.5, 1.5)
rinv.pham(10, 0.5, 1.5)
hinv.pham(x, 0.5, 1.5)

# Data
x <- relief
# ML estimates
params = list(beta=2.8287, delta=9.6044)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pinv.pham, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qinv.pham, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dinv.pham, pfun=pinv.pham, plot=FALSE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Inverse Power Cauchy distribution.

Usage

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

Arguments

Details

The Inverse Power Cauchy distribution is parameterized by the parameters $\alpha > 0$ and $\lambda > 0$.

The Inverse Power Cauchy distribution has CDF:

$F(x; \alpha, \lambda) = \quad 1-2 \pi^{-1} \tan ^{-1}\left[\left(\frac{\lambda}{x}\right) ^\alpha\right] \, ; \quad x > 0.$

where $\alpha$ and $\lambda$ are the parameters.

The following functions are included:

• dinv.pow.cauchy() — Density function

• pinv.pow.cauchy() — Distribution function

• qinv.pow.cauchy() — Quantile function

• rinv.pow.cauchy() — Random generation

• hinv.pow.cauchy() — Hazard function

Value

• dinv.pow.cauchy: numeric vector of (log-)densities

• pinv.pow.cauchy: numeric vector of probabilities

• qinv.pow.cauchy: numeric vector of quantiles

• rinv.pow.cauchy: numeric vector of random variates

• hinv.pow.cauchy: numeric vector of hazard values

References

Sapkota L. P., & Kumar V. (2023). Applications and Some Characteristics of Inverse Power Cauchy Distribution. Reliability: Theory & Applications. 18, 1(72), 301–315. doi:10.24412/1932-2321-2023-172-301-315

Chaudhary, A.K., Sapkota, L.P., & Kumar, V. (2020). Truncated Cauchy Power–Inverse Exponential distribution: Theory and Applications. IOSR Journal of Mathematics (IOSR-JM), 16(4), Ser.IV, 12–23.

Examples

x <- seq(0.1, 10, 0.2)
dinv.pow.cauchy(x, 2.0, 5.0)
pinv.pow.cauchy(x, 2.0, 5.0)
qinv.pow.cauchy(0.5, 2.0, 5.0)
rinv.pow.cauchy(10, 2.0, 5.0)
hinv.pow.cauchy(x, 2.0, 5.0)

# Data
x <- headneck44
# ML estimates
params = list(alpha=1.4271, lambda=123.5294)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pinv.pow.cauchy, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qinv.pow.cauchy, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
res <- gofic(x, params = params,
             dfun = dinv.pow.cauchy, pfun=pinv.pow.cauchy, plot=FALSE)
print.gofic(res)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Inverted Shifted Gompertz distribution.

Usage

dinv.sgz(x, alpha, theta, log = FALSE)
pinv.sgz(q, alpha, theta, lower.tail = TRUE, log.p = FALSE)
qinv.sgz(p, alpha, theta, lower.tail = TRUE, log.p = FALSE)
rinv.sgz(n, alpha, theta)
hinv.sgz(x, alpha, theta)

Arguments

Details

The Inverted Shifted Gompertz distribution is parameterized by the parameters $\alpha > 0$, and $\theta > 0$.

The Inverted Shifted Gompertz distribution has CDF:

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

where$\alpha$ and $\theta$ are the parameters.

The following functions are included:

• dinv.sgz() — Density function

• pinv.sgz() — Distribution function

• qinv.sgz() — Quantile function

• rinv.sgz() — Random generation

• hinv.sgz() — Hazard function

Value

• dinv.sgz: numeric vector of (log-)densities

• pinv.sgz: numeric vector of probabilities

• qinv.sgz: numeric vector of quantiles

• rinv.sgz: numeric vector of random variates

• hinv.sgz: numeric vector of hazard values

References

Chaudhary, A.K., Sapkota, L.P., & Kumar, V. (2020). Inverted Shifted Gompertz Distribution with Theory and Applications. Pravaha, 26(1), 1–10. doi:10.3126/pravaha.v26i1.41645

Jimenez T.F. (2014). Estimation of the Parameters of the Shifted Gompertz Distribution, Using Least Squares, Maximum Likelihood and Moments Methods. Journal of Computational and Applied Mathematics, 255(1) 867–877.

Examples

x <- seq(1.0, 5, 0.25)
dinv.sgz(x, 25, 10)
pinv.sgz(x, 25, 10)
qinv.sgz(0.5, 25, 10)
rinv.sgz(10, 25, 10)
hinv.sgz(x, 25, 10)

# Data
x <- fibers65
# ML estimates
params = list(alpha=215.8181, theta=12.7678)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pinv.sgz, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qinv.sgz, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dinv.sgz, pfun=pinv.sgz, plot=FALSE)
print.gofic(out)

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Inverse Upside Down Bathtub-shaped Hazard Function distribution.

Usage

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

Arguments

Details