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An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications

  • Mustafa Ç. Korkmaz EMAIL logo and G. G. Hamedani
Published/Copyright: July 24, 2020
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 70 Issue 4

Abstract

This paper proposes a new extended Lindley distribution, which has a more flexible density and hazard rate shapes than the Lindley and Power Lindley distributions, based on the mixture distribution structure in order to model with new distribution characteristics real data phenomena. Its some distributional properties such as the shapes, moments, quantile function, Bonferonni and Lorenz curves, mean deviations and order statistics have been obtained. Characterizations based on two truncated moments, conditional expectation as well as in terms of the hazard function are presented. Different estimation procedures have been employed to estimate the unknown parameters and their performances are compared via Monte Carlo simulations. The flexibility and importance of the proposed model are illustrated by two real data sets.

Keywords: Lindley distribution; extended Weibull distribution; generalized Power Lindley distribution; non-monotone hazard rate; characterizations
MSC 2010: Primary 60E05; 62E10; Secondary 62F10; 62H12

  1. (Communicated by Gejza Wimmer)

Acknowledgement

The authors would like to thank the editor and an anonymous reviewer for carefully reading of the article and for their invaluable suggestions improving the content of the article.

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Appendix A

Theorem 1

Let (Ω, 𝓕, P) be a given probability space and letH = [a, b] be an interval for somed < b (a = −∞, b = ∞ might as well be allowed). LetX : Ω → Hbe a continuous random variable with the distribution functionFand letq1andq2be two real functions defined onHsuch that

Eq2X|X≥x=Eq1X|X≥xξx,x∈H,

is defined with some real functionη. Assume thatq1, q2 ∈ C1(H), ξ ∈ C2(H) andFis twice continuously differentiable and strictly monotone function on the setH. Finally, assume that the equationξq1 = q2has no real solution in the interior ofH. ThenFis uniquely determined by the functionsq1, q2andξ, particularly

Fx=∫axCξ′uξuq1u−q2uexp−sudu,

where the functionsis a solution of the differential equations′=ξ′q1ξq1−q2andCis the normalization constant, such that∫HdF=1.

We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see, [22]), in particular, let us assume that there is a sequence {Xn} of random variables with distribution functions {Fn} such that the functions q1n, q2n and ξn (n ∈ ℕ) satisfy the conditions of Theorem 1 and let q1n → q1, q2n → q2 for some continuously differentiable real functions q1 and q2. Let, finally, X be a random variable with distribution F. Under the condition that q1n(X) and q2n(X) are uniformly integrable and the family {Fn} is relatively compact, the sequence Xn converges to X in distribution if and only if ξn converges to ξ, where

ξx=Eq2X|X≥xEq1X|X≥x.

This stability theorem makes sure that the convergence of distribution functions is reflected by corresponding convergence of the functions q1, q2 and ξ, respectively. It guarantees, for instance, the ’convergence’ of characterization of the Wald distribution to that of the Lévy-Smirnov distribution if α → ∞.

A further consequence of the stability property of Theorem 1 is the application of this theorem to special tasks in statistical practice such as the estimation of the parameters of discrete distributions. For such purpose, the functions q1, q2 and, specially, ξ should be as simple as possible. Since the function triplet is not uniquely determined it is often possible to choose ξ as a linear function. Therefore, it is worth analyzing some special cases which helps to find new characterizations reflecting the relationship between individual continuous univariate distributions and appropriate in other areas of statistics.

In some cases, one can take q1(x) ≡ 1, which reduces the condition of Theorem 1 to

E[q2(X) ∣ X ≥ x] = ξ(x),   x ∈ H. We, however, believe that employing three functions q1, q2 and ξ will enhance the domain of applicability of Theorem 1.

Received: 2019-06-21
Accepted: 2019-11-12
Published Online: 2020-07-24
Published in Print: 2020-08-26

© 2020 Mathematical Institute Slovak Academy of Sciences

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  1. Regular papers
  2. Some relative normality properties in locales
  3. Upper bounds of some special zeros for the Rankin-Selberg L-function
  4. Factorization of polynomials over valued fields based on graded polynomials
  5. Varieties of ∗-regular rings
  6. On reverse Hölder and Minkowski inequalities
  7. Coefficient inequalities related with typically real functions
  8. Existence of wandering and periodic domain in given angular region
  9. The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*
  10. Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space
  11. Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces
  12. Bn-maximal operator and Bn-singular integral operators on variable exponent Lebesgue spaces
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  15. The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data
  16. Multi-opponent James functions
  17. An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications
  18. A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models
  19. On the bond pricing partial differential equation in a convergence model of interest rates with stochastic correlation
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https://doi.org/10.1515/ms-2017-0406
Keywords for this article
Lindley distribution; extended Weibull distribution; generalized Power Lindley distribution; non-monotone hazard rate; characterizations

Articles in the same Issue

  1. Regular papers
  2. Some relative normality properties in locales
  3. Upper bounds of some special zeros for the Rankin-Selberg L-function
  4. Factorization of polynomials over valued fields based on graded polynomials
  5. Varieties of ∗-regular rings
  6. On reverse Hölder and Minkowski inequalities
  7. Coefficient inequalities related with typically real functions
  8. Existence of wandering and periodic domain in given angular region
  9. The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*
  10. Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space
  11. Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces
  12. Bn-maximal operator and Bn-singular integral operators on variable exponent Lebesgue spaces
  13. 𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms
  14. More on closed non-vanishing ideals in CB(X)
  15. The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data
  16. Multi-opponent James functions
  17. An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications
  18. A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models
  19. On the bond pricing partial differential equation in a convergence model of interest rates with stochastic correlation
  20. Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations
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