A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations

Majid Hashempour

doi:10.1515/ms-2021-0034

Article

A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations

Majid Hashempour

Published/Copyright: August 4, 2021

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From the journal Mathematica Slovaca Volume 71 Issue 4

Abstract

In this paper, we introduce a new two-parameter lifetime distribution which is called extended Half-Logistic (EHL) distribution. Theoretical properties of this model including the hazard function, quantile function, asymptotic, extreme value, moments, conditional moments, mean residual life, mean past lifetime, residual entropy, cumulative residual entropy and order statistics are derived and studied in details. The maximum likelihood estimates of parameters are compared with various methods of estimations by conducting a simulation study. Finally, two real data sets are illustration the purposes.

Keywords: Entropy; half-logistic distribution; maximum likelihood estimation; moments

MSC 2010: Primary 60E05; 62G05; Secondary 62N05; 62P30

(Communicated by Gejza Wimmer)

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Appendix A pdf of competitive models in application section

fESHL(x;α)=2αe−x(1−e−x)α−1(1+e−x)α+1x>0,α>0,fNOLL−SHL(x;α,β)=2e−x(1+e−x)21−e−x1+e−xα−11−1−e−x1+e−xβ−1α+(β−α)1−e−x1+e−x1−e−x1+e−xα+1−1−e−x1+e−xβ2x>0,α>0,β>0,

fKwSHL(x;α,β)=2αβe−x(1−e−x)α−1(1+e−x)α+11−(1−e−x1+e−x)αβ−1x>0,α>0,β>0,fBSHL(x;α,β)=2βe−βx(1−e−x)α−1B(α,β)(1+e−x)α+βx>0,α>0,β>0,fMcSHL(x;α,β,c)=2ce−x(1−e−x)αc−1B(α,β)(1+e−x)αc+11−(1−e−x1+e−x)cβ−1x>0,α>0,β>0,c>0,fLi(x;α)=α21+α(1+x)e−αxx>0,α>0,fPL(x;α,β)=α2β1+αxβ−1(1+xβ)e−αxβx>0,α>0,β>0,fGE(x;α,β)=αβe−αx(1−e−αx)β−1x>0,α>0,β>0,fNH(x;α,β)=αβ(1+αx)β−1e1−(1+αx)βx>0,α>0,β>0,fLN(x;α,β)=1xβ2πe−(log⁡(x)−α)22β2x>0,α∈R,β>0,fGa(x;α,β)=βαΓ(α)xα−1e−βxx>0,α>0,β>0,fW(x;α,β)=αβ(xβ−1e−αxβx>0,α>0,β>0.

Appendix B R codes for obtaining start value and finding ML estimators

x <– scan()

1.32 12.37 6.56 5.05 11.58

10.56 21.82 3.60 1.33 12.62

5.36 7.71 3.53 19.61 36.63

0.39 21.35 7.22 12.42 8.92

hist(x, prob=T)

#————————-

pdf_EHL <– function(x, alpha, beta){

A=(1–exp(–alpha*x))/(1+exp(–beta*x))

B=(alpha*exp(–alpha*x)+beta*exp(–beta*x)+

(alpha–beta)*exp(–(alpha+beta)*x))/(1+exp(–beta*x))^2

return(B)

}

library(”GenSA”)

fit.sa4 <– function(data,density) {

minusllike <– function(x) –sum(log(density(data, x[1], x[2])))

lower <– c(0.001,0.001) #may need some changes here

upper <– c(1000,1000)

out <– GenSA(lower = lower, upper = upper,

fn = minusllike, control=list(verbose=TRUE,max.time=2))

return(out[c(”value”,”par”,”counts”)])

}

fit.sa4(x, pdf_EHL)

#———————-

library(”AdequacyModel”)

cdf_EHL <– function(par, x){

alpha=par[1]

beta=par[2]

A=(1–exp(–alpha*x))/(1+exp(–beta*x))

B=(alpha*exp(–alpha*x)+beta*exp(–beta*x)+

(alpha–beta)*exp(–(alpha+beta)*x))/(1+exp(–beta*x))^2

return(A)

}

pdf_EHL <– function(par, x){

alpha=par[1]

beta=par[2]

A=(1–exp(–alpha*x))/(1+exp(–beta*x))

B=(alpha*exp(–alpha*x)+beta*exp(–beta*x)+

(alpha–beta)*exp(–(alpha+beta)*x))/(1+exp(–beta*x))^2

return(B)

}

goodness.fit(pdf=pdf_EHL, cdf=cdf_EHL,

starts=c(0.1147411, 0.1751693), data = x,

method=”B”, domain=c(0, Inf), mle=NULL)

Received: 2020-06-13

Accepted: 2020-09-23

Published Online: 2021-08-04

Published in Print: 2021-08-26

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Articles in the same Issue

https://doi.org/10.1515/ms-2021-0034

Keywords for this article

Entropy; half-logistic distribution; maximum likelihood estimation; moments