dbwd gives the density, pbwd gives the distribution
function, qbwd gives the quantile function and rbwd generates
random deviates.
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
$$x\geq 0,~\alpha ,\beta >0~ and ~\sigma \in\mathbb{R}.$$
References
Vila, R. ve Niyazi Çankaya, M., 2022,
A bimodal Weibull distribution: properties and inference,
Journal of Applied Statistics, 49 (12), 3044-3062.