Density, distribution function, quantile function and random generation for the Kumaraswamy Complementary Weibull Geometric (Kw-CWG) probability distribution.
dkwcwg(x, alpha, beta, gamma, a, b, log = FALSE)pkwcwg(q, alpha, beta, gamma, a, b, lower.tail = TRUE, log.p = FALSE)
qkwcwg(p, alpha, beta, gamma, a, b, lower.tail = TRUE, log.p = FALSE)
rkwcwg(n, alpha, beta, gamma, a, b)
vector of quantiles.
Parameters of the distribution. 0 < alpha < 1, and the other parameters mustb e positive.
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are \(P[X \le x]\) otherwise, \(P[X > x]\).
vector of probabilities.
number of observations. If length(n) > 1,
the length is taken to be the number required.
Probability density function $$ f(x) = \alpha^a \beta \gamma a b (\gamma x)^{\beta - 1} \exp[-(\gamma x)^\beta] \cdot \frac{\{1 - \exp[-(\gamma x)^\beta]\}^{a-1}}{\{ \alpha + (1 - \alpha) \exp[-(\gamma x)^\beta] \}^{a+1}} \cdot $$ $$ \cdot \bigg\{ 1 - \frac{\alpha^a[1 - \exp[-(\gamma x)^\beta]]^a}{\{ \alpha + (1 - \alpha) \exp[-(\gamma x)^\beta] \}^a} \bigg\} $$
Cumulative density function $$ F(x) = 1 - \bigg\{ 1 - \bigg[ \frac{\alpha (1 - \exp[-(\gamma x)^\beta]) }{ \alpha + (1 - \alpha) \exp[-(\gamma x)^\beta] } \bigg]^a \bigg\}^b $$
Quantile function $$ Q(u) = \gamma^{-1} \bigg\{ \log\bigg[\frac{ \alpha + (1 - \alpha) \sqrt[a]{1 - \sqrt[b]{1 - u} } }{ \alpha (1 - \sqrt[a]{1 - \sqrt[b]{1 - u} } ) }\bigg] \bigg\}^{1/\beta}, 0 < u < 1 $$
Afify, A.Z., Cordeiro, G.M., Butt, N.S., Ortega, E.M. and Suzuki, A.K. (2017). A new lifetime model with variable shapes for the hazard rate. Brazilian Journal of Probability and Statistics