Kw-CWG function

Kumaraswamy Complementary Weibull Geometric Probability Distribution

Kumaraswamy Complementary Weibull Geometric Probability Distribution

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)

Arguments

  • x, q: vector of quantiles.

  • alpha, beta, gamma, a, b: Parameters of the distribution. 0 < alpha < 1, and the other parameters mustb e positive.

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

  • lower.tail: logical; if TRUE (default), probabilities are P[X≤x]P[X \le x]

    otherwise, P[X>x]P[X > x].

  • p: vector of probabilities.

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

Details

Probability density function

f(x)=αaβγab(γx)β−1exp⁡[−(γx)β]⋅{1−exp⁡[−(γx)β]}a−1{α+(1−α)exp⁡[−(γx)β]}a+1⋅ 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 ⋅{1−αa[1−exp⁡[−(γx)β]]a{α+(1−α)exp⁡[−(γx)β]}a} \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−{1−[α(1−exp⁡[−(γx)β])α+(1−α)exp⁡[−(γx)β]]a}b 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)=γ−1{log⁡[α+(1−α)1−1−ubaα(1−1−1−uba)]}1/β,0<u<1 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

References

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

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  • Maintainer: Matheus H. J. Saldanha
  • License: MIT + file LICENSE
  • Last published: 2019-10-07

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