bwd {new.dist}

Bimodal Weibull Distribution

Description

Density, distribution function, quantile function and random generation for a Bimodal Weibull distribution with parameters shape and scale.

Usage

dbwd(x, alpha, beta = 1, sigma, log = FALSE)

pbwd(q, alpha, beta = 1, sigma, lower.tail = TRUE, log.p = FALSE)

qbwd(p, alpha, beta = 1, sigma, lower.tail = TRUE)

rbwd(n, alpha, beta = 1, sigma)

Arguments

x, q

vector of quantiles.

alpha

a shape parameter.

beta

a scale parameter.

sigma

a control parameter that controls the uni- or bimodality of the distribution.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P\left[ X\leq x\right], otherwise, P\left[ X>x\right] .

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

A Bimodal Weibull distribution with shape parameter \alpha, scale parameter \beta,and the control parameter \sigma that determines the uni- or bimodality of the distribution, has density

f\left( x\right) =\frac{\alpha }{\beta Z_{\theta }} \left[ 1+\left( 1-\sigma~x\right) ^{2}\right] \left( \frac{x}{\beta } \right) ^{\alpha -1}\exp \left( -\left( \frac{x}{\beta }\right) ^{\alpha } \right),

where

Z_{\theta }=2+\sigma ^{2}\beta ^{2}\Gamma \left( 1+\left( 2/\alpha \right)\right) -2\sigma \beta \Gamma \left( 1+\left( 1/\alpha \right) \right)

and

x\geq 0,~\alpha ,\beta >0~ and ~\sigma \in\mathbb{R}.

Value

dbwd gives the density, pbwd gives the distribution function, qbwd gives the quantile function and rbwd generates random deviates.

References

Vila, R. ve Niyazi Çankaya, M., 2022, A bimodal Weibull distribution: properties and inference, Journal of Applied Statistics, 49 (12), 3044-3062.

Examples

library(new.dist)
dbwd(1,alpha=2,beta=3,sigma=4)
pbwd(1,alpha=2,beta=3,sigma=4)
qbwd(.7,alpha=2,beta=3,sigma=4)
rbwd(10,alpha=2,beta=3,sigma=4)