qaep function

Computing the quantile function of asymmetric exponential power (AEP) distribution.

Computing the quantile function of asymmetric exponential power (AEP) distribution.

Computes the quantile function of AEP distribution given by [REMOVE_ME]FX1(uΘ)=μσ(1ϵ)[γ(1ϵ2u1ϵ,1α)Γ(1α)]1α,  u1ϵ2,[REMOVEME2] F_{X}^{-1}(u|\Theta)=\mu-\sigma(1-\epsilon)\biggl[\frac{\gamma\bigl(\frac{1-\epsilon-2u}{1-\epsilon},\frac{1}{\alpha}\bigr)}{\Gamma\bigl(\frac{1}{\alpha}\bigr)}\biggr]^{\frac{1}{\alpha}},~{{}}~u\leq \frac{1-\epsilon}{2}, [REMOVE_ME_2]

[REMOVE_ME]FX1(uΘ)=μ+σ(1+ϵ)[γ(2u+ϵ11+ϵ,1α)Γ(1α)]1α,  u>1ϵ2.[REMOVEME2] F_{X}^{-1}(u|\Theta)=\mu+\sigma(1+\epsilon)\biggl[\frac{\gamma\bigl(\frac{2u+\epsilon-1}{1+\epsilon},\frac{1}{\alpha}\bigr)}{\Gamma\bigl(\frac{1}{\alpha}\bigr)}\biggr]^{\frac{1}{\alpha}},~{{}}~u> \frac{1-\epsilon}{2}.\\ [REMOVE_ME_2]

where <x<+-\infty<x<+\infty, Θ=(α,σ,μ,ϵ)T\Theta=(\alpha,\sigma,\mu,\epsilon)^T with 0<α2,σ>00<\alpha \leq 2, \sigma> 0, <μ<-\infty<\mu<\infty, 1<ϵ<1-1<\epsilon<1, and [REMOVE_ME]γ(u,ν)=0utν1exp{t}dt, ν>0.[REMOVEME2] \gamma(u,\nu) =\int_{0}^{u}t^{\nu-1}\exp\bigl\{-t\bigr\}dt, ~\nu>0. [REMOVE_ME_2]

Description

Computes the quantile function of AEP distribution given by

FX1(uΘ)=μσ(1ϵ)[γ(1ϵ2u1ϵ,1α)Γ(1α)]1α,  u1ϵ2, F_{X}^{-1}(u|\Theta)=\mu-\sigma(1-\epsilon)\biggl[\frac{\gamma\bigl(\frac{1-\epsilon-2u}{1-\epsilon},\frac{1}{\alpha}\bigr)}{\Gamma\bigl(\frac{1}{\alpha}\bigr)}\biggr]^{\frac{1}{\alpha}},~{{}}~u\leq \frac{1-\epsilon}{2}, FX1(uΘ)=μ+σ(1+ϵ)[γ(2u+ϵ11+ϵ,1α)Γ(1α)]1α,  u>1ϵ2. F_{X}^{-1}(u|\Theta)=\mu+\sigma(1+\epsilon)\biggl[\frac{\gamma\bigl(\frac{2u+\epsilon-1}{1+\epsilon},\frac{1}{\alpha}\bigr)}{\Gamma\bigl(\frac{1}{\alpha}\bigr)}\biggr]^{\frac{1}{\alpha}},~{{}}~u> \frac{1-\epsilon}{2}.\\

where <x<+-\infty<x<+\infty, Θ=(α,σ,μ,ϵ)T\Theta=(\alpha,\sigma,\mu,\epsilon)^T with 0<α2,σ>00<\alpha \leq 2, \sigma> 0, <μ<-\infty<\mu<\infty, 1<ϵ<1-1<\epsilon<1, and

γ(u,ν)=0utν1exp{t}dt, ν>0. \gamma(u,\nu) =\int_{0}^{u}t^{\nu-1}\exp\bigl\{-t\bigr\}dt, ~\nu>0.
qaep(u, alpha, sigma, mu, epsilon)

Arguments

  • u: Numeric vector with values in (0,1)(0,1) whose quantiles are desired.
  • alpha: Tail thickness parameter.
  • sigma: Scale parameter.
  • mu: Location parameter.
  • epsilon: Skewness parameter.

Returns

A vector of length n, consists of the random generated values from AEP distribution.

Author(s)

Mahdi Teimouri

Examples

qaep(runif(1), alpha = 1.5, sigma = 1, mu = 0, epsilon = 0.5)
  • Maintainer: Mahdi Teimouri
  • License: GPL (>= 2)
  • Last published: 2022-09-07

Useful links