Version: 0.1.0
Date: 2025-09-22
Title: Zero-Inflated Discrete Weibull Models
Depends: R (≥ 4.1.0)
Imports: DWreg, actuar, maxLik, COUNT, gtools, matrixcalc, DiscreteWeibull, dplyr, ggplot2, purrr, tibble
URL: https://github.com/dsy109/ZIDW
Description: Parameter estimation for zero-inflated discrete Weibull (ZIDW) regression models, the univariate setting, distribution functions, functions to generate randomized quantile residuals a pseudo R2, and plotting of rootograms. For more details, see Kalktawi (2017) https://bura.brunel.ac.uk/handle/2438/14476, Taconeli and Rodrigues de Lara (2022) <doi:10.1080/00949655.2021.2005597>, and Yeh and Young (2025) <doi:10.1080/03610918.2025.2464076>.
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation: no
Packaged: 2025-09-22 16:02:02 UTC; derekyoung
Author: Peng Yeh [aut], Derek S. Young ORCID iD [aut, cre]
Maintainer: Derek S. Young <derek.young@uky.edu>
Repository: CRAN
Date/Publication: 2025-09-30 07:20:22 UTC

Zero-Inflated Discrete Weibull Models

Description

Parameter estimation for zero-inflated discrete Weibull (ZIDW) regression models, the univariate setting, distribution functions, functions to generate randomized quantile residuals a pseudo R2, and plotting of rootograms.

Details

Package: ZIDW
Type: Package
Version: 0.1.0
Date: 2025-09-22
Imports: DWreg, actuar, maxLik, COUNT, gtools, matrixcalc, DiscreteWeibull, dplyr, ggplot2, purrr, tibble
License: GPL (>= 2)

Author(s)

Peng Yeh, Ph.D.

Derek S. Young, Ph.D.

Maintainer: Derek S. Young <derek.young@uky.edu>

References

Kalktawi, H. S. (2017), Discrete Weibull Regression Model for Count Data. Ph.D. Thesis, Brunel University London.

Taconeli, C. A. and Rodrigues de Lara, I. A. (2022), Discrete Weibull Distribution: Different Estimation Methods Under Ranked Set Sampling and Simple Random Sampling. Journal of Statistical Computation and Simulation, 92:8, 1740–1762.

Yeh, P. and Young, D. S. (2025), Some Estimation and Inference Considerations for the Zero-Inflated Discrete Weibull Distribution. Communications in Statistics - Simulation and Computation (in press), 1–22.

AIC and BIC for ZIDW Model Fits

Description

Generic function calculating AIC or BIC for ZIDW model fits.

Usage

## S3 method for class 'zidw'
AIC(object, ..., k = 2)
## S3 method for class 'zidw'
BIC(object, ...)

Arguments

object

A fitted ZIDW model object for which there exists a logLik method to extract the corresponding log-likelihood, or an object inheriting from class logLik.

...

Optionally more fitted model objects.

k

Numeric value for the penalty per parameter to be used; default = 2.

Value

A numeric value with the corresponding AIC or BIC.

References

Sakamoto, Y., Ishiguro, M., and Kitagawa G. (1986), Akaike Information Criterion Statistics. D. Reidel Publishing Company.

See Also

logLik AIC

Examples

 
## data
data("rwm1984", package = 'COUNT')

zidw_out <- zidw_reg(docvis ~ outwork + female + age, 
                     ziformula = ~ age + female,
                     betaformula = ~ 1, data = rwm1984[1:100, ], beta = .1)

AIC(zidw_out)
BIC(zidw_out)

Bootstrap Likelihood Ratio Test for Zero-Inflation

Description

Perform the bootstrap likelihood ratio test for comparing DW and ZIDW model fits.

Usage

bootstrap_lrt(data, B, tol = -1)

Arguments

data

Data to test.

B

Number of bootstrap samples to draw.

tol

Threshold of bootstrap likelihood ratio test statistics. See details.

Details

Theoretically, the likelihood ratio is always positive. In practice, however, if a negative (but close to zero) likelihood ratio occurrs, then the calculations are typically numerically unstable or the fitting procedure failed to converge to the MLEs. We, therefore, set the threshold to -1 by default. If the bootstrap likelihood ratio is less than tol, we drop this sample and resample. If it is between tol and 0, we truncate it at 0.

Value

Return a list containing four vectors:

pvalue

P-value of the test.

Observe likelihood ratio test statistics

Likelihood ratio test statistics from the data.

Bootstrap likelihood ratio test statistics

Bootstrap likelihood ratio test statistics.

count

Number of times that bootstrap LRT is less than tol.

See Also

glm, lm

Examples

 

## data
data("rwm1984", package = 'COUNT')

set.seed(1)
test <- suppressWarnings(bootstrap_lrt(rwm1984, B = 20))
test

Print Coefficients from a ZIDW Model Fit

Description

Print the coefficients of a ZIDW object.

Usage

## S3 method for class 'zidw'
coef(object, ...)

Arguments

object

A ZIDW object to extract the model coefficients.

...

Further arguments.

Value

Coefficients extracted from the ZIDW object object. This will be a named numeric vector.

Examples

 
## data
data("rwm1984", package = 'COUNT')

zidw_out <- zidw_reg(docvis ~ outwork + female + age, 
                     ziformula = ~ age + female,
                     betaformula = ~ 1, data = rwm1984[1:100, ], beta = .1)

coef.zidw(zidw_out)

COVID-19 cases at the 2020 Summer Olympics and 2020 Summer Paralympics

Description

This dataset contains the number of cases from 2020 Summer Olympics and 2020 Summer Paralympics in Tokyo.

Usage

data(covid)

Format

This data frame consists of 5 variables on 70 dates:

Source

Wikipedia contributors. (2022, December 5). COVID-19 cases at the 2020 Summer Olympics and 2020 Summer Paralympics. In Wikipedia, The Free Encyclopedia. Retrieved 13:01, December 20, 2022, https://en.wikipedia.org/wiki/COVID-19_cases_at_the_2020_Summer_Olympics_and_2020_Summer_Paralympics#cite_note-Tokyo2020CovidList-1

Hurdle Discrete Weibull Distribution

Description

Density, distribution function, quantile function and random generation for the hurdle Discrete Weibull distribution with parameter q_{par}, \beta and hurdle crossing probability \lambda (i.e., 1 - \lambda is the probability for observed zeros).

Usage

dhdw(x, q_par, beta, lam, log = FALSE)
phdw(q, q_par, beta, lam, lower.tail = TRUE, log.p = FALSE)
qhdw(p, q_par, beta, lam, lower.tail = TRUE, log.p = FALSE)
rhdw(n, q_par, beta, lam)

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observation.

q_par

Shape parameter.

beta

Shape parameter.

lam

Zero-infation parameter.

log, log.p

Logical; if TRUE, probabilities are returned on log-scale.

lower.tail

Logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x].

Details

The hurdle discrete Weibull has the mass function

f(y) = \lambda + (1 - \lambda)\frac{q^{y^\beta} - q^{(y + 1)^\beta}}{q},

for y=0, 1, 2, ... where \lambda \in (0, 1) is the zero-inflation parameter, q \in (0, 1), and \beta > 0.

Value

dhdw

Gives the density.

phdw

Gives the cumulative probability.

qhdw

Gives the quantile value.

rhdw

Generates random numbers.

References

Kalktawi, H. S. (2017), Discrete Weibull Regression Model for Count Data. Ph.D. Thesis, Brunel University London.

Yeh, P. and Young, D. S. (2025), Some Estimation and Inference Considerations for the Zero-Inflated Discrete Weibull Distribution. Communications in Statistics - Simulation and Computation (in press), 1–22.

Examples

 
dhdw(.5, .6, 1, .4)
phdw(.5, .6, 1, .4)
qhdw(.8, .6, 1, .4)
rhdw(100, .6, 1, .4)

Extract Log-Likelihood for ZIDW Model Fits

Description

Extract the log-likelihood of a ZIDW model fit.

Usage

## S3 method for class 'zidw'
logLik(object, ...)

Arguments

object

A fitted ZIDW model object for which there exists a logLik method to extract the corresponding log-likelihood, or an object inheriting from class logLik.

...

Some methods for this generic function require additional arguments.

Value

A numeric value with the corresponding log-likelihood.

See Also

logLik AIC

Examples

 
## data
data("rwm1984", package = 'COUNT')

zidw_out <- zidw_reg(docvis ~ outwork + female + age, 
                     ziformula = ~ age + female,
                     betaformula = ~ 1, data = rwm1984[1:100, ], beta = .1)

logLik(zidw_out)

Prediction for ZIDW Model Fits

Description

Obtains predictions from the fitted ZIDW model object.

Usage

## S3 method for class 'zidw'
predict(object, newdata, 
                       type = c("response", "prob", "count", "zero"), 
                       at = NULL, ...)

Arguments

object

A fitted object of class inheriting from "zidw".

newdata

Optionally, a data frame in which to look for variables with which to predict. If omitted, the fitted predictors are used.

type

The type of prediction required. For details see below.

at

Optionally, if type = "prob", a numeric vector at which the probabilities are evaluated. By default 0:max(y) is used where y is the original observed response.

...

Currently not used.

Details

The default is type = "response", which is on the scale of the response variable. type = "prob" provides the predicted density (i.e., probabilities for the observed counts). type = "count" offers the predicted mean from the count component without zero-inflation and type = "zero" predicts the probability for the zero-component.

Value

If type = "response", a vector of estimated conditional mean values from the ZIDW model is returned. If type = "prob", a matrix of predicted probabilities is returned, where each row is a vector of predicted probabilities over the range of responses seen in the data (i.e., min(y):max(y)). If type = "count", a vector of the predicted means from the count component (without zero-inflation) is returned. If type = "zero", the predicted probability for the zero-component is returned.

Examples

 
## data
data("rwm1984", package = 'COUNT')

## model with covariates
zidw_out <- zidw_reg(docvis ~ outwork + female + age, 
                     ziformula = ~ age + female,
                     betaformula = ~ 1, data = rwm1984[1:100, ], beta = .1)
                     
predict.zidw(zidw_out)

Print Values from a ZIDW Object

Description

Print the output of a ZIDW object.

Usage

## S3 method for class 'zidw'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x

A ZIDW object used to select a method.

digits

Minimal number of significant digits.

...

Further arguments passed to or from other methods.

Value

Returns an object of class zidw based on the object x with mostly the same output when printing output from an object of class lm. Whereas an lm object has one model.matrix in its output, use of print.zidw returns three model matrices, one each for the model parameters q and beta as well as for the zero-inflation component.

See Also

print.lm

Examples

 
## data
data("rwm1984", package = 'COUNT')

zidw_out <- zidw_reg(docvis ~ outwork + female + age, 
                     ziformula = ~ age + female,
                     betaformula = ~ 1, data = rwm1984[1:100, ], beta = .1)

print(zidw_out)

Rootogram for a ZIDW fit

Description

A rootogram is a model diagnostic tool that assesses the goodness-of-fit of a statistical model. The rootogram is drawn using ggplot2::ggplot() graphics.

Usage

rootogram_zidw(object, type = c("hanging", "standing", "suspended"),
               sqrt = TRUE, ref_line = TRUE, warn_limits = TRUE, 
               fitted_colour = "steelblue", bar_colour = NA, 
               bar_fill = "grey", ref_line_colour = "black",
               warn_line_colour = "black", ylab = NULL, xlab = NULL, ...)

Arguments

object

A ZIDW model object.

type

Type of rootogram; default is type = "hanging".

sqrt

Logical; show the observed and fitted frequencies (default is sqrt = TRUE).

ref_line

Logical; draw a reference line at zero (default is ref_line = TRUE).

warn_limits

Logical; draw Tukey's warning limit lines at +/- 1 (default is warn_limits = TRUE).

fitted_colour, bar_colour, bar_fill, ref_line_colour, warn_line_colour

Colors used to draw the respective element of the rootogram.

xlab, ylab

Character; labels for the x and y axis of the rootogram. May be missing (NULL), in which case suitable labels will be used.

...

Arguments passed to other methods.

Value

A ggplot object.

References

Kleiber, C. and Zeileis, A. (2016). Visualizing Count Data Regressions Using Rootograms. The American Statistician, 70:3, 296–303.

See Also

rootogram

Examples

 
## data
data("rwm1984", package = 'COUNT')

zidw_out <- zidw_reg(docvis ~ outwork + female + age, 
                     ziformula = ~ age + female,
                     betaformula = ~ 1, data = rwm1984[1:100, ], beta = .1)
rootogram_zidw(zidw_out)

Randomized Quantile Residuals for a ZIDW Regression Fit

Description

Calculate randomized quantile residuals from a ZIDW regression fit.

Usage

rqres_zidw_reg(test, plot = FALSE)

Arguments

test

Zero-inflated discrete Weibull regresssion output.

plot

If plot = TRUE, produce the randomized quantile residuals plot.

Value

Return a vector of randomized quantile resuduals or a randomized quantile residuals plot:

rqr

Randomized quantile resuduals.

References

Dunn, P. K. and Smyth, G. K. (1996), Randomzied Quantile Residuals. Journal of Computational and Graphical Statistics, 5:3, 236–244.

Yeh, P. and Young, D. S. (2025), Some Estimation and Inference Considerations for the Zero-Inflated Discrete Weibull Distribution. Communications in Statistics - Simulation and Computation (in press), 1–22.

See Also

glm, lm

Examples

 
## data
data("rwm1984", package = 'COUNT')

zidw_out <- zidw_reg(docvis ~ outwork + female + age, 
                     ziformula = ~ age + female,
                     betaformula = ~ 1, data = rwm1984[1:100, ], beta = .1)
rqr <- rqres_zidw_reg(zidw_out)

Object Summaries

Description

summary method to produce results for objects of class "zidw".

Usage

## S3 method for class 'zidw'
summary(object, ...)
## S3 method for class 'summary.zidw'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

object

An object of class "zidw" for which a summary is desired.

...

Additional arguments affecting the summary produced.

x

A summary.zidw object.

digits

The number of digits in the output.

Details

Additional information about the ZIDW fit represented by object is extracted and included as components of object. The returned object has a print method.

Value

Currently, limited results from an object of class "zidw" is returned. The output is of class summary.zidw. Future versions of this code will attempt to mimic summary.glm.

See Also

summary.glm

Examples

 
## data
data("rwm1984", package = 'COUNT')

## model with covariates
zidw_out <- zidw_reg(docvis ~ outwork + female + age, 
                     ziformula = ~ age + female, betaformula = ~ 1, 
                     data = rwm1984[1:100, ], beta = .1)
                     
summary(zidw_out)

Zero-Inflated Discrete Weibull Distribution

Description

Density, distribution function, quantile function and random generation for the zero-inflated Discrete Weibull distribution with parameters q_{par}, \beta, \lambda.

Usage

dzidw(x, q_par, beta, lam, log = FALSE)
pzidw(q, q_par, beta, lam, lower.tail = TRUE, log.p = FALSE)
qzidw(p, q_par, beta, lam, lower.tail = TRUE, log.p = FALSE)
rzidw(n, q_par, beta, lam)

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observation.

q_par

Shape parameter.

beta

Shape parameter.

lam

Zero-infation parameter.

log, log.p

Logical; if TRUE, probabilities are returned on log-scale.

lower.tail

Logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x].

Details

The zero-inflated discrete Weibull has the mass function

f(y) = \lambda + (1 - \lambda)(q^{y^\beta} - q^{(y + 1)^\beta}),

for y = 0, 1, 2, ... where \lambda \in (0, 1) is zero-inflation parameter, q \in (0, 1), and \beta > 0.

Value

dzidw

Gives the density.

pzidw

Gives the cumulative probability.

qzidw

Gives the quantile value.

rzidw

Generates random numbers.

References

Kalktawi, H. S. (2017), Discrete Weibull Regression Model for Count Data. Ph.D. Thesis, Brunel University London.

Yeh, P. and Young, D. S. (2025), Some Estimation and Inference Considerations for the Zero-Inflated Discrete Weibull Distribution. Communications in Statistics - Simulation and Computation (in press), 1–22.

Examples

 
dzidw(.5, .6, 1, .4)
pzidw(.5, .6, 1, .4)
qzidw(.8, .6, 1, .4)
rzidw(100, .6, 1, .4)

Internal 'ZIDW' Functions

Description

Internal functions for the package ZIDW.

Usage

dw.parest2(data, method = "likelihood", method.opt = "NR")
ddw2(x, q = exp(-1), beta = 1)
pdw2(x, q = exp(-1), beta = 1)

Arguments

data

Data that is a vector.

method

Either "likelihood" or "proportion".

method.opt

Optimization criterion used in maxLik (default is "NR").

x

Vector of quantiles.

q, beta

Parameters of the distribution.

Details

These are usually not to be called by the user.

Value

These functions yield different outputs, which are called for their desired effects within other functions. Thus, they are usually not to be called by the user.

Pseudo-R^2 for ZIDW Regression

Description

Calculates the pseudo-R^2 for ZIDW regression fits. Modeling q and \lambda.

Usage

zidw_r_squared(object, adj = TRUE)

Arguments

object

An object from a ZIDW regression fit of class "zidw".

adj

An adjustment for calculating the pseudo-R^2. Default is adj = TRUE.

Value

A numerical value with the corresponding pseudo-R^2.

References

Martin, J. and Hall, D. B. (2016), R^2 Measures for Zero-Inflated Regression Models for Count Data with Excess Zeros. Journal of Statistical Computation and Simulation, 84:18, 3777–3790.

Examples

 
## data
data("rwm1984", package = 'COUNT')

## model with covariates
zidw_out <- zidw_reg(docvis ~ outwork + female + age, ziformula = ~ age + female,
                     betaformula = ~ 1, data = rwm1984[1:100, ], beta = .1)
                     
zidw_r_squared(zidw_out)

Fitting of Zero-Inflated Discrete Weibull Regression Models

Description

Fitting of the zero-inflated discrete Weibull regression model is done via constrOptim. Fitting of the univariate model can be accomplished via maximum likelihood or minimum distance estimation.

Usage

zidw_reg(qformula, betaformula = ~ 1, ziformula = ~ 1, data, lam = NULL, 
         beta = NULL, q = NULL, k = 1000, uni_method =  c('MLE', 'mde'), 
         max_method = NULL, constraint = TRUE, B = NULL)

Arguments

qformula

A symbolic description of the model to be fitted for the parameter q, including the response variable.

betaformula

A symbolic description of the model to be fitted for the parameter \beta.

ziformula

a symbolic description of the model to be fitted for the parameter \lambda.

data

A data frame containg model variables.

lam

The starting value for \lambda.

beta

The starting value for parameter \beta.

q

The starting value for parameter q.

k

A cutofff value used to calculate residuals.

uni_method

The univariate fitting method. Methods include uni_method = 'MLE' (default) for maximum likelihood estimation and uni_method = 'MDE' for minimum distance estimation.

max_method

Maximum likelihood estimation method for the univariate setting. See maxLik for the different methods.

constraint

Set constraint = FALSE for unconstrained maximization. constraint = TRUE is the default.

B

Number of bootstrap replciates for standard error estimation when modeling univariate data using uni_method = 'MDE'.

Value

zidw_reg An object of class "zidw", i.e., a list with components including the following:

call

The original function call.

coefficients

A list with elements "zero", "beta", and "q" containing the coefficients from the respective models.

loglik

The log-likehood of the fitted model.

SE

Estimated standard errors.

convergence

The convergence code of optim.

nall

The number of observations.

res

A vector of raw residuals (observed - fitted).

fitted_values

The fitted values.

model_matrix_q

The model matrix of parameter q.

model_matrix_beta

The model matrix of parameter \beta.

model_matrix_zi

The model matrix of parameter \lambda.

response

Vector of response values.

model

The dataset.

formula

A list containing three formulas for q, \beta, and \lambda.

References

Taconeli, C. A. and Rodrigues de Lara, I. A. (2022), Discrete Weibull Distribution: Different Estimation Methods Under Ranked Set Sampling and Simple Random Sampling. Journal of Statistical Computation and Simulation, 92:8, 1740–1762.

Yeh, P. and Young, D. S. (2025), Some Estimation and Inference Considerations for the Zero-Inflated Discrete Weibull Distribution. Communications in Statistics - Simulation and Computation (in press), 1–22.

See Also

glm, lm

Examples

 
## data
data("rwm1984", package = 'COUNT')

## model with covariates
zidw_out <- zidw_reg(docvis ~ outwork + female + age, ziformula = ~ age + female,
                     betaformula = ~ 1, data = rwm1984[1:100, ], beta = .1)
                     
## model without covariates
zidw_out2 <- zidw_reg(docvis ~ 1, data = rwm1984)

Parameter Estimation for the Zero-Inflated Discrete Weibull

Description

Parameter estimation for the univariate zero-inflated discrete Weibull distribution using maximum likelihood or minimum distance estimation.

Usage

zidw_uni(y, par = NULL, method = c('MLE', 'mde'), B = NULL, max_method = NULL, 
         constraint = TRUE)

Arguments

y

Vector of univariate counts.

par

Vector of starting values in the order of \lambda, \beta, and q.

method

Parameter estimation method. Methods include uni_method = 'MLE' (default) for maximum likelihood estimation and uni_method = 'MDE' for minimum distance estimation.

B

Number of bootstrap replciates for standard error estimation when modeling univariate data using uni_method = 'MDE'.

max_method

Maximum likelihood estimation method for the univariate setting. See maxLik for the different methods.

constraint

Logical value to constrain the parameter to be within the parameter space. Default is constraint = TRUE.

Value

Output for for maximum likelihood estimation (uni_method = 'MLE'):

MLE

Maximum likelihood estimates.

coefficients

Coefficients for q, \beta, and \lambda.

convergence

Convergence code of maxLik.

iteration

Number of iterations.

SE

Estimated standard errors.

Output for for minimum distance estimation (uni_method = 'MDE'):

lam

Parameter estimate for \lambda.

Beta

Parameter estimate for \beta.

q

Parameter estimate for q.

distance

The final distance value.

SE

Bootstrap standard error estimates for \lambda, \beta, and q.

References

Taconeli, C. A. and Rodrigues de Lara, I. A. (2022), Discrete Weibull Distribution: Different Estimation Methods Under Ranked Set Sampling and Simple Random Sampling. Journal of Statistical Computation and Simulation, 92:8, 1740–1762.

Yeh, P. and Young, D. S. (2025), Some Estimation and Inference Considerations for the Zero-Inflated Discrete Weibull Distribution. Communications in Statistics - Simulation and Computation (in press), 1–22.

See Also

glm, lm

Examples

 
## data
data("rwm1984", package = 'COUNT')

## MLE method
zidw_out <- zidw_uni(rwm1984$docvis)
                     
                     
## mde without standard error estimation
set.seed(1)
zidw_out2 <- zidw_uni(rwm1984$docvis, method = 'mde')

## mde with bootstrap standard error estimation
set.seed(1)
zidw_out2 <- zidw_uni(rwm1984$docvis, method = 'mde', B = 5)

Zero-Truncated Discrete Weibull Distribution

Description

Density, distribution function, quantile function and random generation for the zero-truncated Discrete Weibull distribution with parameter q_{par}, \beta.

Usage

dztdw(x, q_par, beta, log = FALSE)
pztdw(q, q_par, beta, lower.tail = TRUE, log.p = FALSE)
qztdw(p, q_par, beta, lower.tail = TRUE, log.p = FALSE)
rztdw(n, q_par, beta)

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observation.

q_par

Shape parameter.

beta

Shape parameter.

log, log.p

Logical; if TRUE, probabilities are returned on log-scale.

lower.tail

Logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x].

Details

The zero-truncated discrete Weibull has the mass function

f(y) = \frac{q^{y^\beta} - q^{(y + 1)^\beta}}{q},

for y = 1, 2, 3, ... where q \in (0, 1), and \beta > 0.

Value

dztdw

Gives the density.

pztdw

Gives the cumulative probability.

qztdw

Gives the quantile value.

rztdw

Generates random numbers.

References

Kalktawi, H. S. (2017), Discrete Weibull Regression Model for Count Data. Ph.D. Thesis, Brunel University London.

Yeh, P. and Young, D. S. (2025), Some Estimation and Inference Considerations for the Zero-Inflated Discrete Weibull Distribution. Communications in Statistics - Simulation and Computation (in press), 1–22.

Examples

 
dztdw(1, .6, 1)
pztdw(1, .6, 1)
qztdw(.8, .6, 1)
rztdw(100, .6, 1)