Discrete Weibull

Description

Density, distribution function, quantile function and random generation for the discrete Weibull distribution with parameters q and beta.

Usage

ddw(x,q=exp(-1),beta=1)
pdw(x,q=exp(-1),beta=1)
qdw(p,q=exp(-1),beta=1)
rdw(n,q=exp(-1),beta=1)

Arguments

Details

The discrete Weibulll distribution has density

p(x,q,\beta) = q^{x^{\beta}}-q^{(x+1)^{\beta}}

for x = 0, 1, 2, \ldots. If q or beta are not specified they assume the default values of exp(-1) and 1, respectively. In this case, DW corresponds to a geometric distribution with p=1-q.

Value

ddw gives the density, pdw gives the distribution function, qdw gives the quantile function, and rdw generates random samples from a DW distribution with parameters q and beta.

Author(s)

Veronica Vinciotti

References

Nagakawa T, Osaki S. The discrete Weibull distribution. IEEE transactions on reliability 1975; R-24(5).

Examples

x<-rdw(1000,q=0.9,beta=1.5)
hist(x)
plot(x,unlist(lapply(x,ddw,q=0.9,beta=1.5)),ylab="density")
plot(x,unlist(lapply(x,pdw,q=0.9,beta=1.5)),ylab="cdf")

Mean and Variance of Discrete Weibull

Description

Mean and variance of a discrete Weibull distribution with parameters q and beta.

Usage

dw.meanvar(q,beta,M)

Arguments

Details

The mean and variance are computed using the following approximations:

E(X)=\sum_{k=1}^{M} q^{k^{\beta}}

E(X^2)=\sum_{k=1}^{M} (2k-1)q^{k^{\beta}} = 2\sum_{k=1}^{M} kq^{k^{\beta}}-E(X)

Value

The function returns the mean and variance of a DW distribution with parameters q and beta.

Author(s)

Veronica Vinciotti

References

Khan M, Khalique A, Abouammoth A. On estimating parameters in a discrete Weibull distribution. IEEE transactions on Reliability 1989; 38(3):348-350.

Examples

dw.meanvar(q=0.9,beta=1.5)
#compare with sample mean/variance from a random sample
x<-rdw(1000,q=0.9,beta=1.5)
mean(x)
var(x)

Parameter estimation for discrete Weibull

Description

Estimation of the parameters q and beta of a discrete Weibull distribution

Usage

dw.parest(data,method,method.opt)

Arguments

Details

If method="likelihood", the parameters q and beta are estimated by maximum likelihood.

If method="proportion", the method of Araujio Santos and Fraga Alves (2013) is used, based on count frequencies.

Value

The function returns the parameter estimates of q and beta.

Author(s)

Veronica Vinciotti

References

Araujo Santos P, Fraga Alves M. Improved shape parameter estimation in a discrete Weibull model. Recent Developments in Modeling and Applications in Statistics . Studies in Theoretical and Applied Statistics. Springer-Verlag, 2013; 71-80.

Examples

x<-rdw(1000,q=0.9,beta=1.5)
dw.parest(x) #maximum likelihood estimates
dw.parest(x,method="proportion") #proportion estimates

DW regression

Description

Parametric regression for discrete response data. The conditional distribution of the response given the predictors is assumed to be DW with parameters q and beta dependent on the predictors.

Usage

dw.reg(formula, data,tau=0.5,para.q1=NULL,para.q2=NULL,para.beta=NULL,...)

Arguments

Details

The conditional distribution of Y (response) given x (predictors) is assumed a DW(q(x),beta(x)).

If para.q1=TRUE,

log(q/(1-q))=\theta_0+\theta_1 X_1+\ldots+\theta_pX_p.

If para.q2=TRUE,

log(-log(q))=\theta_0+\theta_1 X_1+\ldots+\theta_pX_p.

This is equivalent to a continuous Weibull regression model with interval-censored data.

If para.q1=NULL and para.q2=NULL, then q(x) is constant.

If para.beta=TRUE,

log(\beta)= \gamma_0+\gamma_1 X_1+\ldots+\gamma_pX_p.

Otherwise beta(x) is constant.

Value

A list of class dw.reg containing the following components:

Author(s)

Veronica Vinciotti, Hadeel Kalktawi, Alina Peluso

References

Kalktawi, Vinciotti and Yu (2016) A simple and adaptive dispersion regression model for count data.

Examples

#simulated example (para.q1=TRUE, beta constant)
theta0 <- 2
theta1 <- 0.5
beta<-0.5
n<-500
x <- runif(n=n, min=0, max=1.5)
logq<-theta0 + theta1 * x - log(1+exp(theta0  + theta1 * x))		
y<-unlist(lapply(logq,function(x,beta) rdw(1,q=exp(x),beta),beta=beta)) 
data.sim<-data.frame(x,y) #simulated data
fit<-dw.reg(y~x,data=data.sim,para.q1=TRUE)
fit$tTable	

#simulated example (para.q2=TRUE, beta constant)
theta0 <- -2
theta1 <- -0.5
beta<-0.5
n<-500
x <- runif(n=n, min=0, max=1.5)
logq<--exp(theta0  + theta1 * x)		
y<-unlist(lapply(logq,function(x,beta) rdw(1,q=exp(x),beta),beta=beta)) 
data.sim<-data.frame(x,y) #simulated data
fit<-dw.reg(y~x,data=data.sim,para.q2=TRUE)
fit$tTable	
fit$survreg

#real example
library(Ecdat)
data(StrikeNb)
fit<-dw.reg(strikes~output,data=StrikeNb,para.q2=TRUE)
fit$tTable
fit$survreg

DW regression: Diagnostics

Description

Quantile-Quantile plot of the randomised quantile residuals of a DW regression fitted model with 95% simulated envelope.

Usage

res.dw(obj,k)

Arguments

Details

Diagnostic check for a DW regression model. The randomised quantile residuals should follow a standard normal distribution.

Value

A q-q plot of the residuals with 95% simulated envelope

Author(s)

Veronica Vinciotti, Hadeel Kalktawi

References

Kalktawi, Vinciotti and Yu (2016) A simple and adaptive dispersion regression model for count data.

Examples

#simulated example (para.q2=TRUE, beta constant)
theta0 <- -2
theta1 <- -0.5
beta<-0.5
n<-500
x <- runif(n=n, min=0, max=1.5)
logq<--exp(theta0  + theta1 * x)		
y<-unlist(lapply(logq,function(x,beta) rdw(1,q=exp(x),beta),beta=beta)) 
data.sim<-data.frame(x,y) #simulated data
fit<-dw.reg(y~x,data=data.sim,para.q2=TRUE)
res.dw(fit,k=5)
ks.test(fit$residuals,"pnorm")

#real example
library(Ecdat)
data(StrikeNb)
fit<-dw.reg(strikes~output,data=StrikeNb,para.q2=TRUE)
res.dw(fit,k=5)
ks.test(fit$residuals,"pnorm")