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.
Value
dbwd gives the density, pbwd gives the distribution
function, qbwd gives the quantile function and rbwd generates
random deviates.
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
$$x\geq 0,~\alpha ,\beta >0~ and ~\sigma \in\mathbb{R}.$$
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)
#> [1] 0.01594262
pbwd(1,alpha=2,beta=3,sigma=4)
#> [1] 0.003859685
qbwd(.7,alpha=2,beta=3,sigma=4)
#> [1] 4.759942
rbwd(10,alpha=2,beta=3,sigma=4)
#> [1] 5.549071 4.330503 4.975350 5.421088 4.053200 3.965505 5.833866 3.590104
#> [9] 3.555097 2.385201