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A bivariate Kumaraswamy-exponential distribution with application

  • Hassan S. Bakouch EMAIL logo , Fernando A. Moala , Abdus Saboor and Haniya Samad
Published/Copyright: October 5, 2019
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 69 Issue 5

Abstract

In this paper, we introduce a new bivariate Kumaraswamy exponential distribution, whose marginals are univariate Kumaraswamy exponential. Some probabilistic properties of this bivariate distribution are derived, such as joint density function, marginal density functions, conditional density functions, moments and stress-strength reliability. Also, we provide the expected information matrix with its elements in a closed form. Estimation of the parameters is investigated by the maximum likelihood, Bayesian and least squares estimation methods. A simulation study is carried out to compare the performance of the estimators by estimation methods. Further, one data set have been analyzed to show how the proposed distribution works in practice.

MSC 2010: Primary 60E05; Secondary: 62E15; 62E20
Keywords: bivariate Kumaraswamy-exponential distribution; marginal and conditional density functions; moments; stress-strength; maximum likelihood; Fisher information matrix; Bayesian estimation

  1. (Communicated by Gejza Wimmer)

References

[1] Adepoju, K. A.—Chukwu, O. I.: Maximum likelihood estimation of the Kumaraswamy exponential distribution with applications, Journal of Modern Applied Statistical Methods 14 (2015), #18.10.22237/jmasm/1430453820Search in Google Scholar

[2] Amin, E. A.: Estimation of stress-strength reliability for Kumaraswamy exponential distribution based on upper record values, International Journal of Contemporary Mathematical Sciences 12 (2017), 59–71.10.12988/ijcms.2017.7210Search in Google Scholar

[3] Arnold. B.: A note on multivariate distributions with specified marginals, J. Amer. Statist. Assoc. 62 (1967), 1460–1461.10.1080/01621459.1967.10500946Search in Google Scholar

[4] Block, H. W.—Basu, A. P.: A continuous bivariate exponential extension, J. Amer. Statist. Assoc. 69 (1974), 1031–1037.Search in Google Scholar

[5] Chacko, M.—Mohan, R.: Estimation of parameters of Kumaraswamy-Exponential distribution under progressive type-II censoring, J. Stat. Comput. Simul. 87 (2017), 1951–1963.10.1080/00949655.2017.1300662Search in Google Scholar

[6] Chib, S.—Greenberg, E.: Understanding the metropolis-hasting algorithm, Amer. Statist. 49 (1995), 327–335.Search in Google Scholar

[7] Gelfand, A. E.—Smith, F. M.: Sampling-based approaches to calculating marginal densities, J. Amer. Statist. Assoc. 85 (1990), 398–409.10.21236/ADA208388Search in Google Scholar

[8] Johnson, N. L.—Kotz, S.: A vector multivariate hazard rate, J. Multivariate Anal. 5 (1975), 53–66.10.1016/0047-259X(75)90055-XSearch in Google Scholar

[9] Karlin, S. Total Positivity, Stanford University Press, Stanford, California, 1968.Search in Google Scholar

[10] Kundu, D.—Gupta, R.: Modified Sarhan–Balakrishnan singular bivariate distribution, J. Statist. Plann. Inference 140 (2010), 526–538.10.1016/j.jspi.2009.07.026Search in Google Scholar

[11] Lu, J. C.: Weibull extensions of the Freund and Marshall–Olkin bivariate exponential, IEEE Transactions on Reliability 38 (1989), 615–619.10.1109/24.46492Search in Google Scholar

[12] Marshall, A. W.—Olkin, I.: A multivariate exponential distribution, J. Amer. Statist. Assoc. 62 (1967), 30–41.10.21236/AD0634335Search in Google Scholar

[13] Marshall, A. W.: Some comments on hazard gradient, Stochastic Process. Appl. 3 (1975), 293–300.10.1016/0304-4149(75)90028-9Search in Google Scholar

12 Appendix

This Appendix provides the calculations needed to derive the elements of the expected information matrix given by

IF=IββIβα1Iβα2Iβα3Iα1α1Iα1α2Iα1α3Iα2α2Iα2α3Iα3α3,(12.1)

Suppose (x1, y1), …, (xn, yn) is a random sample of lifetimes from BVKE and n1, n2, n3, I1, I2 and I3 are the same as defined in Section 5. For brevity we further denote α = α1 + α2 + α3.

The following results are needed:

E(n1)=nP{X>Y}=nα2α,E(n2)=nP{X<Y}=nα1α,E(n3)=nP{X=Y}=nα3α.

Consider now the second derivatives of log L(θ |x, y):

∂2∂α12logL(θ|x,y)=−n2α12−n1(α1+α3)2,∂2∂α1∂α2logL(θ|x,y)=0,∂2∂α1∂α3logL(θ|x,y)=−n1(α1+α3)2,∂2∂α22logL(θ|x,y)=−n1α22−n2(α2+α3)2,∂2∂α2∂α3logL(θ|x,y)=−n2(α2+α3)2,∂∂α32logL(θ|x,y)=−n3α32−n1(α1+α3)2−n2(α2+α3)2,∂2∂α1∂βlogL(θ|x,y)=∑i∈I(1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β,∂2∂α1∂βlogL(θ|x,y)=∑i∈I(1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β,∂2∂α3∂βlogL(θ|x,y)=∑i∈I1∪I3(1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β+∑i∈I2(1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β,∂2∂β2logL(θ|x,y)=−2n1+2n2+n3β2+(α1+α3−1)∑i∈I1(1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β+(α2−1)∑i∈I1(1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β+(α1−1)∑i∈I2(1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β+(α2+α3−1)∑i∈I2(1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β+(α1+α2+α3−1)∑i∈I3(1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β.

Proposition 1

Assuming that data are from theBvKE(β, α1, α2, α3), we get:

E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I1)=α1+α3β(η(α1+α3)+ηα¯−η(α1+α3−1)−γ+ψα¯α),E((1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β|i∈I1)=α2β(η(α)−ηα¯),E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I2)=α1β(η(α)−ηα¯),E((1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β|i∈I2)=α2+α3β(η(α2+α3)+η(α−1)−η(α2+α3−1)−γ+ψα¯α),E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I3)=α1β(η(α)−ηα¯).

Proof

Let

E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I1)=∫0∞∫0x((1−e−x))βlog⁡((1−e−x))1−((1−e−x))βf1(x,y)dxdy=(α1+α3)∫0∞FY(x)e−x((1−e−x))2β−1(1−((1−e−x))β)α1+α3−2log⁡((1−e−x))βdx,

where FY(x) = 1 –(1 – ((1 – e–x))β)α2.

Then,

E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I1)=(α1+α3)[∫0∞e−x((1−e−x))2β−1(1−((1−e−x))β)α1+α3−2log⁡((1−e−x))βdx−∫0∞e−x((1−e−x))2β−1(1−((1−e−x))β)α1+α2+α3−2log⁡((1−e−x))βdx].

After using a substitution of variables as u = (1 – e–x)β and ∫01ualog⁡(1−u)du=−γ+ψ(α+2)α+1 we have

E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I1)=α1+α3β(γ+ψ(α1+α3+1)α1+α3+γ+ψ(α)α−1−γ+ψα¯α−γ+ψ(α1+α3)α1+α3−1),

and considering η(α)=γ+ψ(α+1)α the proof is completed.□

The entries Iθiθj=E(∂2∂θiθjlogL(θ|x,y)) of the information matrix IF are derived, after some algebra, as follows:

Iα1α1=−E(∂2∂α22log⁡L(θ|x,y))=E(n2)α12+E(n1)(α1+α3)2=nα(1α1+α2(α1+α3)2),Iα1α2=−E(∂2∂α1∂α2log⁡L(θ|,x,y))=0,Iα1α3=−E(∂2∂α1∂α3log⁡L(θ|x,y))=E(n1)(α1+α3)2=nα2α(α1+α3)2,Iα2α2=−E(∂2∂α22log⁡L(θ|x,y))=E(n1)α22+E(n2)(α2+α3)2=nα(1α2+α1(α2+α3)2),
Iα2α3=−E(∂2∂α2∂α3log⁡L(θ|x,y))=E(n2)(α2+α3)2=nα1α(α2+α3)2,Iα3α3=−E(∂∂α32log⁡L(θ|x,y))=E(n3)α32+E(n1)(α1+α3)2+E(n2)(α2+α3)2=nα(1α3+α2(α1+α3)2+α1(α2+α3)2),

and from the Proposition 1, we have

Iβα1=−E(∂2∂β∂α1log⁡L(θ|x,y))=E(n1)E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I1)+E(n2)E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I2)+E(n3)E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I3)=nα2(α1+α3)αβ(η(α1+α3)+ηα¯−η(α1+α3−1)−γ+ψα¯α)+nα12αβ(η(α)−ηα¯)+nα32α2β(η(α)−ηα¯),Iβα2=−E(∂2∂β∂α2log⁡L(θ|x,y))=E(n1)E((1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β|i∈I1)+E(n2)E((1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β|i∈I2)+E(n3)E((1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β|i∈I3)=nα22αβ(η(α)−ηα¯)+nα1(α2+α3)αβ(η(α2+α3)+ηα¯−η(α2+α3−1)−γ+ψα¯α)+nα32α2β(η(α)−ηα¯),Iβα3=−E(∂2∂β∂α3log⁡L(θ|x,y))=E(n1)E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I1)+E(n2)E((1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β|i∈I2)+E(n3)E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I3)=nα2(α1+α3)αβ(η(α1+α3)+ηα¯−η(α1+α3−1)−γ+ψα¯α)+nα1(α2+α3)αβ(η(α2+α3)+ηα¯−η(α2+α3−1)−γ+ψα¯α)+nα32α2β(η(α)−ηα¯),
Iββ=−E(∂2∂β2logL(θ|x,y))=2E(n1)+2E(n2)+E(n3)β2+(α1+α3−1)E(n1)E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I1)+(α2−1)E(n1)E((1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β|i∈I1)+(α1−1)E(n2)E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I2)+(α2+α3−1)E(n2)E((1−e−yi)βlog⁡(1−e−yi)1−(1−e−yi)β|i∈I2)+(α1+α2+α3−1)E(n3)E((1−e−xi)βlog⁡(1−e−xi)1−(1−e−xi)β|i∈I3)=nαβ2(α1+α2+α)+nα2(α1+α3)(α1+α3−1)αβ(η(α1+α3)+ηα¯−η(α1+α3−1)−γ+ψα¯α)+nα22(α2−1)αβ(η(α)−ηα¯)+nα12(α1−1)αβ(η(α)−ηα¯)+nα1(α2+α3)(α2+α3−1)αβ(η(α2+α3)+ηα¯−η(α2+α3−1)−γ+ψα¯α)+nα32α¯α2β(η(α)−ηα¯).
Received: 2018-07-19
Accepted: 2019-02-15
Published Online: 2019-10-05
Published in Print: 2019-10-25

© 2019 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0300
Keywords for this article
bivariate Kumaraswamy-exponential distribution; marginal and conditional density functions; moments; stress-strength; maximum likelihood; Fisher information matrix; Bayesian estimation

Articles in the same Issue

  1. Regular papers
  2. Professor Anatolij Dvurečenskij
  3. A generating theorem of punctured surface triangulations with inner degree at least 4
  4. Natural extension of EF-spaces and EZ-spaces to the pointfree context
  5. On weak isometries in directed groups
  6. On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
  7. On the extensions of discrete valuations in number fields
  8. On certain classes of generalized derivations
  9. Nilpotent elements in medial semigroups
  10. Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
  11. Sharp coefficient bounds for starlike functions associated with the Bell numbers
  12. Sufficient conditions for Carathéodory functions and applications to univalent functions
  13. Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
  14. Linear directional differential equations in the unit ball: solutions of bounded L-index
  15. On oscilatory fourth order nonlinear neutral differential equations – IV
  16. Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
  17. Lerch’s theorem on nabla time scales
  18. IK-convergence of sequences of functions
  19. Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
  20. Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
  21. On the amenability of a class of Banach algebras with application to measure algebra
  22. A bivariate Kumaraswamy-exponential distribution with application
  23. Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
  24. Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
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