The sine extended odd Fréchet-G family of distribution with applications to complete and censored data

Farrukh Jamal; Christophe Chesneau; Khaoula Aidi

doi:10.1515/ms-2021-0033

Article

The sine extended odd Fréchet-G family of distribution with applications to complete and censored data

Farrukh Jamal , Christophe Chesneau and Khaoula Aidi

Published/Copyright: August 4, 2021

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From the journal Mathematica Slovaca Volume 71 Issue 4

Abstract

In this paper, we introduce the sine extended odd Fréchet-G family of distributions, obtained from two well-established families of distributions of completely different nature: the sine-G and the extended odd Fréchet-G families. A particular focus is put on a very flexible member of this family defin ed with the Nadarajah-Haghighi distribution as a baseline, called the sine extended odd Fréchet Nadarajah-Haghighi distribution. For the theoretical part, the interesting mathematical properties of the family are investigated, including asymptotes, quantile function, linear representations and moments, with application to the introduced special member. Then, the inferential aspects of the sine extended odd Fréchet Nadarajah-Haghighi model are examined. In particular, the parameters are estimated by the maximum likelihood method. Two complementary cases are distinguished: the complete data case and the right censored data case, with the development of appropriate statistical tests. A simulation study is carried out to illustrate the convergence of the obtained estimates. Applications are given for three practicaldata sets, including one having the right censored property, illustrating the applicability of the proposed model.

Keywords: sine-G family of distributions; extended odd Fréchet-G family of distributions; Nadarajah-Haghighi distribution; maximum likelihood estimation; data analysis; censored data

MSC 2010: Primary 60E05; Secondary 62E15; 62F10

(Communicated by Gejza Wimmer)

Acknowledgement

We thank the reviewers for the constructive comments and suggestions which made the paper more substantial and interesting.

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Appendix A

After long calculations, we obtain the expressions of the first derivatives of the log-likelihood function with respect to the parameters for complete data. They are given below.

∂Ln(γ)∂α=nα−(θ−1)∑i=1nuiαln⁡(ui)1−uiα−θ∑i=1nln⁡(ui)+θ∑i=1nviθln⁡(ui)1−ui−πθ2∑i=1nviθtanπ2e−viθln⁡(ui)e−viθ1−uiα,

∂Ln(γ)∂λ=nλ+β−1∑i=1nXizi−β∑i=1nziβ−1Xi+αβ(θ−1)∑i=1nXi(1−ui)ziβ−1(1−uiα)−(αθ+1)∑i=1nXi(1−ui)ziβ−1ui+αβθ∑i=1nXiviθ(1−ui)ziβ−1ui(1−uiα)−π2αβθ∑i=1nXi(1−ui)ziβ−1viθe−viθui1−uiαtanπ2e−viθ,

∂Ln(γ)∂β=nβ+∑i=1nln⁡(zi)−∑i=1nziβln⁡(zi)−(αθ−1)∑i=1nziβln⁡(zi)(1−ui)ui−α(θ−1)∑i=1nziβln⁡(zi)(1−ui)uiα(1−uiα)+αθ∑i=1nviθziβln⁡(zi)(1−ui)ui(1−uiα)−π2αθ∑i=1nziβviθln⁡(zi)(1−ui)e−viθui1−uiαtanπ2e−viθ

and

∂Ln(γ)∂θ=nθ+∑i=1nlnvi−∑i=1nviθln⁡(vi)+π2∑i=1nviθlnvie−viθtanπ2e−viθ.

Appendix B

After long calculations, we obtain the expressions of the first derivatives of the log-likelihood function with respect to the parameters for censored data. They are given below.

∂Ln(γ)∂α=∑i=1nδi[1α+θviθ−πθ2e−viθviθln⁡(ui)tan⁡(π2e−viθ)1−uiαuiα−θln⁡(ui)1−uiα+πθ2viθlnuie−viθcos⁡(π2e−viθ)1−uiα1−sinπ2e−vθ]−πθ2∑i=1nviθlnuie−viθcos⁡(π2e−viθ)1−uiα1−sin⁡(π2e−viθ),

∂Ln(γ)∂λ=∑i=1nδi[1λ−βziβ−1xi+βxi(1−ui)ziβ−1(αuiα−αθ+uiα−1)ui(1−uiα)+(β−1)xizi−παβθxi(1−ui)ziβ−1e−viθviθtan⁡(π2e−viθ)2ui(1−uiα)+αβθxiviθziβ−1ui(1−uiα)+παβθxi(1−ui)ziβ−1e−viθviθcos⁡(π2e−viθ)2ui(1−uiα)[1−sin⁡(π2e−viθ)]]−∑i=1nπαβθxi(1−ui)ziβ−1e−viθviθcos⁡(π2e−viθ)2ui(1−uiα)[1−sin⁡(π2e−viθ)],

∂Ln(γ)∂β=∑i=1nδi[ziβln⁡(zi)(1−ui)(αuiα−αθ+uiα−1)ui(1−uiα)+αθziβln⁡(zi)(1−ui)viθui(1−uiα)+1β+ln⁡(zi)−παθziβln⁡(zi)(1−ui)e−viθviθtan⁡(π2e−viθ)2ui(1−uiα)−ziβln⁡zi+π2αθziβln⁡(zi)(1−ui)e−viθviθcos⁡(π2e−viθ)ui(1−uiα)[−sin⁡(π2e−viθ)]]−∑i=1nπαθziβln⁡(zi)(1−ui)e−viθviθcos⁡(π2e−viθ)2ui(1−uiα)[1−sin⁡(π2e−viθ)]

and

∂Ln(γ)∂θ=∑i=1nδi[1θ+ln⁡(vi)+π2viθln⁡(vi)e−viθtan⁡(π2e−viθ)−viθln⁡(vi)−π2viθln⁡(vi)e−viθcos⁡(π2e−viθ)1−sin⁡(π2e−viθ)]+π2∑i=1nviθln⁡(vi)e−viθcos⁡(π2e−viθ)1−sin⁡(π2e−viθ).

Appendix C

C^1j=1n∑i:Ti∈Ijnδi[1α^+θ^v^iθ^−πθ^2e−v^iθ^v^iθ^ln⁡(u^i)tan⁡(π2e−v^iθ^)1−u^iα^(u^iα^−θ^)ln⁡u^i1−u^iα^+πθ^2v^iθ^ln⁡(u^i)e−v^iθ^cos⁡(π2e−v^iθ^)(1−u^iα^)[1−sin⁡(π2e−v^iθ^)]],

C^2j=1n∑i:Ti∈Ijnδi[1λ−β^z^iβ^−1xi+β^xi(1−u^i)z^iβ^−1(α^u^iα^−α^θ+u^iα^−1)u^i(1−u^iα^)+(β^−1)xiz^i−πα^β^θxi(1−u^i)z^iβ^−1e−v^iθ^v^iθ^tan⁡(π2e−v^iθ^)2u^i(1−u^iα^)+α^β^θ^xiv^iθ^z^iβ^−1u^i(1−u^iα^)+πα^β^θxi(1−u^i)z^iβ^−1e−v^iθ^v^iθ^cos⁡(π2e−v^iθ^)2u^i(1−u^iα^)[1−sin⁡(π2e−v^iθ^)]],

C^3j=1n∑i:Ti∈Ijnδi[z^iβ^ln⁡(z^i)(1−u^i)(α^u^iα^−α^θ^+u^iα^−1)u^i(1−u^iα^)+α^θ^z^iβ^ln⁡(z^i)(1−u^i)v^iθ^u^i(1−u^iα^)+1β^+ln⁡(z^i)−πα^θ^z^iβ^ln⁡(z^i)(1−u^i)e−v^iθ^v^iθ^tan⁡(π2e−v^iθ^)2u^i(1−u^iα^)−z^iβ^ln⁡z^i+π2α^θ^z^iβ^ln⁡(z^i)(1−u^i)e−v^iθ^v^iθ^cos⁡(π2e−v^iθ^)u^i(1−u^iα^)[1−sin⁡(π2e−v^iθ^)]],

C^4j=1n∑i:Ti∈Ijnδi[1θ^+ln⁡(v^i)+π2v^iθ^ln⁡(v^i)e−v^iθ^tan⁡(π2e−v^iθ^)−v^iθ^ln⁡(v^i)−π2v^iθln⁡(v^i)e−v^iθ^cos⁡(π2e−v^iθ^)1−sin⁡(π2e−v^iθ)],

Where

u^i=1−e1−(1+λ^Xi)β^,e1−(1+λ^Xi)β^=1−ui,vi=(1−uiα^)uiα^,zi=1+λ^Xi,

and

W^l=∑j=1rC^ljAj−1Zj,l=1,…,s.

Received: 2020-06-01

Accepted: 2020-09-03

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-2021-0033

Keywords for this article

sine-G family of distributions; extended odd Fréchet-G family of distributions; Nadarajah-Haghighi distribution; maximum likelihood estimation; data analysis; censored data