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Mathematics
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Characterization of Probability Distributions
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Characterization of Probability Distributions

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (31 July 2021) | Viewed by 15679

Special Issue Editor

Prof. Dr. Lev Klebanov

E-Mail Website
Guest Editor
Department of Probability and Mathematical Statistics, Charles University, 12116 Prague, Czech Republic
Interests: probability and statistics; probability theory; probability; statistics; biostatistics

Special Issue Information

Dear Colleagues,

We invite you to contribute to a Special Issue of the MDPI journal Mathematics, devoted to the characterization of probability distributions. Problems of characterization probability distributions consist of describing all the laws for which suitable statistics have one or another desirable property. Although characterization problems have attracted the attention of specialists for a long time, they remain relevant now, especially to construct models associated with distributions other than normal, as well as for testing different hypotheses. Mathematically, characterization problems lead to interesting functional equations and non-standard approaches to their solutions. The practical value of these problems lies in model construction based on some properties of observed statistics. The articles in this Special Issue are devoted to problems of this kind. Characterization of the exponential distribution, distribution of the hyperbolic secant, two-point distribution, and some related laws will be presented. Results concerning the use of characterization theorems for constructing statistical hypotheses will also be included in the Special Issue.

Prof. Dr. Lev Klebanov
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Probability laws
  • Characterizations of distributions
  • Model construction
  • Exponential distribution
  • Hyperbolic secant distribution
  • Statistical testing

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Published Papers (6 papers)

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Research

20 pages, 347 KiB  
Article
On a Multivariate Analog of the Zolotarev Problem
by Yury Khokhlov and Victor Korolev
Mathematics 2021, 9(15), 1728; https://doi.org/10.3390/math9151728 - 22 Jul 2021
Viewed by 1545
Abstract
A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random [...] Read more.
A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric distribution are considered. Anisotropic-geometric stable distributions are introduced. It is demonstrated that these distributions are coordinate-wise scale mixtures of elliptically contoured stable distributions with the Marshall–Olkin mixing distributions. The corresponding “anisotropic” analogs of multivariate Laplace, Linnik and Mittag–Leffler distributions are introduced. Some relations between these distributions are presented. Full article
(This article belongs to the Special Issue Characterization of Probability Distributions)
4 pages, 198 KiB  
Article
On the Condition of Independence of Linear Forms with a Random Number of Summands
by Abram M. Kagan and Lev B. Klebanov
Mathematics 2021, 9(13), 1516; https://doi.org/10.3390/math9131516 - 29 Jun 2021
Viewed by 1310
Abstract
The property of independence of two random forms with a non-degenerate random number of summands contradicts the Gaussianity of the summands. Full article
(This article belongs to the Special Issue Characterization of Probability Distributions)
22 pages, 431 KiB  
Article
The Extended Log-Logistic Distribution: Inference and Actuarial Applications
by Nada M. Alfaer, Ahmed M. Gemeay, Hassan M. Aljohani and Ahmed Z. Afify
Mathematics 2021, 9(12), 1386; https://doi.org/10.3390/math9121386 - 15 Jun 2021
Cited by 33 | Viewed by 2921
Abstract
Actuaries are interested in modeling actuarial data using loss models that can be adopted to describe risk exposure. This paper introduces a new flexible extension of the log-logistic distribution, called the extended log-logistic (Ex-LL) distribution, to model heavy-tailed insurance losses data. The Ex-LL [...] Read more.
Actuaries are interested in modeling actuarial data using loss models that can be adopted to describe risk exposure. This paper introduces a new flexible extension of the log-logistic distribution, called the extended log-logistic (Ex-LL) distribution, to model heavy-tailed insurance losses data. The Ex-LL hazard function exhibits an upside-down bathtub shape, an increasing shape, a J shape, a decreasing shape, and a reversed-J shape. We derived five important risk measures based on the Ex-LL distribution. The Ex-LL parameters were estimated using different estimation methods, and their performances were assessed using simulation results. Finally, the performance of the Ex-LL distribution was explored using two types of real data from the engineering and insurance sciences. The analyzed data illustrated that the Ex-LL distribution provided an adequate fit compared to other competing distributions such as the log-logistic, alpha-power log-logistic, transmuted log-logistic, generalized log-logistic, Marshall–Olkin log-logistic, inverse log-logistic, and Weibull generalized log-logistic distributions. Full article
(This article belongs to the Special Issue Characterization of Probability Distributions)
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21 pages, 366 KiB  
Article
Characterization of Probability Distributions via Functional Equations of Power-Mixture Type
by Chin-Yuan Hu, Gwo Dong Lin and Jordan M. Stoyanov
Mathematics 2021, 9(3), 271; https://doi.org/10.3390/math9030271 - 29 Jan 2021
Viewed by 1955
Abstract
We study power-mixture type functional equations in terms of Laplace–Stieltjes transforms of probability distributions on the right half-line [0,∞). These equations arise when studying distributional equations of the type Z=dX+TZ, where [...] Read more.
We study power-mixture type functional equations in terms of Laplace–Stieltjes transforms of probability distributions on the right half-line [0,∞). These equations arise when studying distributional equations of the type Z=dX+TZ, where the random variable T≥0 has known distribution, while the distribution of the random variable Z≥0 is a transformation of that of X≥0, and we want to find the distribution of X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results that are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics. Full article
(This article belongs to the Special Issue Characterization of Probability Distributions)
10 pages, 273 KiB  
Article
Exponential and Hypoexponential Distributions: Some Characterizations
by George P. Yanev
Mathematics 2020, 8(12), 2207; https://doi.org/10.3390/math8122207 - 12 Dec 2020
Cited by 8 | Viewed by 4777
Abstract
The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some [...] Read more.
The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n≥2, X1,X2,…,Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables. Full article
(This article belongs to the Special Issue Characterization of Probability Distributions)
13 pages, 1247 KiB  
Article
A New Family of Extended Lindley Models: Properties, Estimation and Applications
by Abdulrahman Abouammoh and Mohamed Kayid
Mathematics 2020, 8(12), 2146; https://doi.org/10.3390/math8122146 - 2 Dec 2020
Cited by 3 | Viewed by 1930
Abstract
There are many proposed life models in the literature, based on Lindley distribution. In this paper, a unified approach is used to derive a general form for these life models. The present generalization greatly simplifies the derivation of new life distributions and significantly [...] Read more.
There are many proposed life models in the literature, based on Lindley distribution. In this paper, a unified approach is used to derive a general form for these life models. The present generalization greatly simplifies the derivation of new life distributions and significantly increases the number of lifetime models available for testing and fitting life data sets for biological, engineering, and other fields of life. Several distributions based on the disparity of the underlying weights of Lindley are shown to be special cases of these forms. Some basic statistical properties and reliability functions are derived for the general forms. In addition, comparisons among various forms are investigated. Moreover, the power distribution of this generalization has also been considered. Maximum likelihood estimator for complete and right-censored data has been discussed and in simulation studies, the efficiency and behavior of it have been investigated. Finally, the proposed models have been fit to some data sets. Full article
(This article belongs to the Special Issue Characterization of Probability Distributions)
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