Mathematics

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23 pages, 2567 KiB

Open AccessArticle

A Generalized Log Gamma Approach: Theoretical Contributions and an Application to Companies’ Life Expectancy

by José H. Dias Gonçalves, João J. Ferreira Gomes, Lihki Rubio and Filipe R. Ramos

Mathematics 2023, 11(23), 4792; https://doi.org/10.3390/math11234792 - 27 Nov 2023

Cited by 1 | Viewed by 1051

The survival of a company has been a topic of growing interest in the scientific community. Measuring the life expectancy of Portuguese telecommunications companies using generalized log-gamma (GLG) distribution is a new research endeavor. Regarding the new theoretical contributions, original expressions for the [...] Read more.

The survival of a company has been a topic of growing interest in the scientific community. Measuring the life expectancy of Portuguese telecommunications companies using generalized log-gamma (GLG) distribution is a new research endeavor. Regarding the new theoretical contributions, original expressions for the moments and mode of the GLG distribution are presented. In this empirical study, data on the entrepreneurial fabric in the Information and Communication sector from 2004 to 2018, when some companies were born or died, were used. In addition to the GLG, three other statistical distributions with two parameters are analyzed: gamma, Weibull, and log-normal. Maximum likelihood parameters and confidence intervals for survival probabilities are estimated and compared. The Akaike information criterion is used to compare the performance of the four estimated models. The results show that GLG distribution is a promising solution to assess the resilience and longevity of a firm. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

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14 pages, 298 KiB

Open AccessArticle

Percolation Problems on N-Ary Trees

by Tianxiang Ren and Jinwen Wu

Mathematics 2023, 11(11), 2571; https://doi.org/10.3390/math11112571 - 4 Jun 2023

Viewed by 771

Abstract

Percolation theory is a subject that has been flourishing in recent decades. Because of its simple expression and rich connotation, it is widely used in chemistry, ecology, physics, materials science, infectious diseases, and complex networks. Consider an infinite-rooted N-ary tree where each [...] Read more.

Percolation theory is a subject that has been flourishing in recent decades. Because of its simple expression and rich connotation, it is widely used in chemistry, ecology, physics, materials science, infectious diseases, and complex networks. Consider an infinite-rooted N-ary tree where each vertex is assigned an i.i.d. random variable. When the random variable follows a Bernoulli distribution, a path is called head run if all the random variables that are assigned on the path are 1. We obtain the weak law of large numbers for the length of the longest head run. In addition, when the random variable follows a continuous distribution, a path is called an increasing path if the sequence of random variables on the path is increasing. By Stein’s method and other probabilistic methods, we prove that the length of the longest increasing path with a probability of one focuses on three points. We also consider limiting behaviours for the longest increasing path in a special tree. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

19 pages, 2470 KiB

Open AccessEditor’s ChoiceArticle

Quasar Identification Using Multivariate Probability Density Estimated from Nonparametric Conditional Probabilities

by Jenny Farmer, Eve Allen and Donald J. Jacobs

Mathematics 2023, 11(1), 155; https://doi.org/10.3390/math11010155 - 28 Dec 2022

Cited by 2 | Viewed by 1498

Abstract

Nonparametric estimation for a probability density function that describes multivariate data has typically been addressed by kernel density estimation (KDE). A novel density estimator recently developed by Farmer and Jacobs offers an alternative high-throughput automated approach to univariate nonparametric density estimation based on [...] Read more.

Nonparametric estimation for a probability density function that describes multivariate data has typically been addressed by kernel density estimation (KDE). A novel density estimator recently developed by Farmer and Jacobs offers an alternative high-throughput automated approach to univariate nonparametric density estimation based on maximum entropy and order statistics, improving accuracy over univariate KDE. This article presents an extension of the single variable case to multiple variables. The univariate estimator is used to recursively calculate a product array of one-dimensional conditional probabilities. In combination with interpolation methods, a complete joint probability density estimate is generated for multiple variables. Good accuracy and speed performance in synthetic data are demonstrated by a numerical study using known distributions over a range of sample sizes from 100 to

10^{6}

for two to six variables. Performance in terms of speed and accuracy is compared to KDE. The multivariate density estimate developed here tends to perform better as the number of samples and/or variables increases. As an example application, measurements are analyzed over five filters of photometric data from the Sloan Digital Sky Survey Data Release 17. The multivariate estimation is used to form the basis for a binary classifier that distinguishes quasars from galaxies and stars with up to 94% accuracy. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

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26 pages, 417 KiB

Open AccessArticle

Bootstrapping Not Independent and Not Identically Distributed Data

by Martin Hrba, Matúš Maciak, Barbora Peštová and Michal Pešta

Mathematics 2022, 10(24), 4671; https://doi.org/10.3390/math10244671 - 9 Dec 2022

Viewed by 1508

Abstract

Classical normal asymptotics could bring serious pitfalls in statistical inference, because some parameters appearing in the limit distributions are unknown and, moreover, complicated to estimated (from a theoretical as well as computational point of view). Due to this, plenty of stochastic approaches for [...] Read more.

Classical normal asymptotics could bring serious pitfalls in statistical inference, because some parameters appearing in the limit distributions are unknown and, moreover, complicated to estimated (from a theoretical as well as computational point of view). Due to this, plenty of stochastic approaches for constructing confidence intervals and testing hypotheses cannot be directly applied. Bootstrap seems to be a plausible alternative. A methodological framework for bootstrapping not independent and not identically distributed data is presented together with theoretical justification of the proposed procedures. Among others, bootstrap laws of large numbers and central limit theorems are provided. The developed methods are utilized in insurance and psychometry. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

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13 pages, 335 KiB

Open AccessArticle

A Look at the Primary Order Preserving Properties of Stochastic Orders: Theorems, Counterexamples and Applications in Cognitive Psychology

by Mohsen Soltanifar

Mathematics 2022, 10(22), 4362; https://doi.org/10.3390/math10224362 - 20 Nov 2022

Cited by 1 | Viewed by 1578

Abstract

In this paper, we prove that for a set of ten univariate stochastic orders including the usual order, a univariate stochastic order preserves either both, one or none of additivity and multiplication properties over the vector space of real-valued random variables. Then, classifying [...] Read more.

In this paper, we prove that for a set of ten univariate stochastic orders including the usual order, a univariate stochastic order preserves either both, one or none of additivity and multiplication properties over the vector space of real-valued random variables. Then, classifying participant’s quickness in a mental chronometry trial to “weakly faster” and “strongly faster”, we use the above results for the usual stochastic order to establish necessary and sufficient conditions for a participant to be strongly faster than the other in terms of the fitted Wald, Exponentially modified Wald(ExW), and Exponentially modified Gaussian(ExG) distributional parameters. This research field remains uncultivated for other univariate stochastic orders and in several directions. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

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38 pages, 1366 KiB

Open AccessArticle

Statistical Inference for Competing Risks Model with Adaptive Progressively Type-II Censored Gompertz Life Data Using Industrial and Medical Applications

by Muqrin A. Almuqrin, Mukhtar M. Salah and Essam A. Ahmed

Mathematics 2022, 10(22), 4274; https://doi.org/10.3390/math10224274 - 15 Nov 2022

Cited by 1 | Viewed by 1224

Abstract

This study uses the adaptive Type-II progressively censored competing risks model to estimate the unknown parameters and the survival function of the Gompertz distribution. Where the lifetime for each failure is considered independent, and each follows a unique Gompertz distribution with different shape [...] Read more.

This study uses the adaptive Type-II progressively censored competing risks model to estimate the unknown parameters and the survival function of the Gompertz distribution. Where the lifetime for each failure is considered independent, and each follows a unique Gompertz distribution with different shape parameters. First, the Newton-Raphson method is used to derive the maximum likelihood estimators (MLEs), and the existence and uniqueness of the estimators are also demonstrated. We used the stochastic expectation maximization (SEM) method to construct MLEs for unknown parameters, which simplified and facilitated computation. Based on the asymptotic normality of the MLEs and SEM methods, we create the corresponding confidence intervals for unknown parameters, and the delta approach is utilized to obtain the interval estimation of the reliability function. Additionally, using two bootstrap techniques, the approximative interval estimators for all unknowns are created. Furthermore, we computed the Bayes estimates of unknown parameters as well as the survival function using the Markov chain Monte Carlo (MCMC) method in the presence of square error and LINEX loss functions. Finally, we look into two real data sets and create a simulation study to evaluate the efficacy of the established approaches. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

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29 pages, 1828 KiB

Open AccessArticle

From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case

by Julia Adamska, Łukasz Bielak, Joanna Janczura and Agnieszka Wyłomańska

Mathematics 2022, 10(18), 3371; https://doi.org/10.3390/math10183371 - 16 Sep 2022

Cited by 5 | Viewed by 1664

Abstract

Multivariate modelling of economics data is crucial for risk and profit analyses in companies. However, for the final conclusions, a whole set of variables is usually transformed into a single variable describing a total profit/balance of company’s cash flows. One of the possible [...] Read more.

Multivariate modelling of economics data is crucial for risk and profit analyses in companies. However, for the final conclusions, a whole set of variables is usually transformed into a single variable describing a total profit/balance of company’s cash flows. One of the possible transformations is based on the product of market variables. Thus, in this paper, we study the distribution of products of Pareto or Student’s t random variables that are ubiquitous in various risk factors analysis. We review known formulas for the probability density functions and derive their explicit forms for the products of Pareto and Gaussian or log-normal random variables. We also study how the Pareto or Student’s t random variable influences the asymptotic tail behaviour of the distribution of their product with the Gaussian or log-normal random variables and discuss how the dependency between the marginal random variables of the same type influences the probabilistic properties of the final product. The theoretical results are then applied for an analysis of the distribution of transaction values, being a product of prices and volumes, from a continuous trade on the German intraday electricity market. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

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11 pages, 288 KiB

Open AccessArticle

Characterizations of the Exponential Distribution by Some Random Hazard Rate Sequences

by Mansour Shrahili and Mohamed Kayid

Mathematics 2022, 10(17), 3052; https://doi.org/10.3390/math10173052 - 24 Aug 2022

Cited by 1 | Viewed by 1078

Abstract

In this paper, several characterizations for exponential distribution are derived from a new relative hazard rate measure. This measure is closely related to the concept of remaining lifetime at a random time. It considers the random times specified by the order statistics of [...] Read more.

In this paper, several characterizations for exponential distribution are derived from a new relative hazard rate measure. This measure is closely related to the concept of remaining lifetime at a random time. It considers the random times specified by the order statistics of a sample, the convolution of random variables, and the record values of a sequence of random variables. The concept of completeness in functional analysis provides the technical background to obtain the main results. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

14 pages, 2091 KiB

Open AccessArticle

Some Statistical Aspects of the Truncated Multivariate Skew-t Distribution

by Raúl Alejandro Morán-Vásquez, Edwin Zarrazola and Daya K. Nagar

Mathematics 2022, 10(15), 2793; https://doi.org/10.3390/math10152793 - 6 Aug 2022

Cited by 2 | Viewed by 1543

Abstract

The multivariate skew-t distribution plays an important role in statistics since it combines skewness with heavy tails, a very common feature in real-world data. A generalization of this distribution is the truncated multivariate skew-t distribution which contains the truncated multivariate t [...] Read more.

The multivariate skew-t distribution plays an important role in statistics since it combines skewness with heavy tails, a very common feature in real-world data. A generalization of this distribution is the truncated multivariate skew-t distribution which contains the truncated multivariate t distribution and the truncated multivariate skew-normal distribution as special cases. In this article, we study several distributional properties of the truncated multivariate skew-t distribution involving affine transformations, marginalization, and conditioning. The generation of random samples from this distribution is described. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

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8 pages, 256 KiB

Open AccessArticle

On a Parametrization of Partial-Sums Discrete Probability Distributions

by Ján Mačutek, Gejza Wimmer and Michaela Koščová

Mathematics 2022, 10(14), 2476; https://doi.org/10.3390/math10142476 - 16 Jul 2022

Viewed by 1165

Abstract

For every discrete probability distribution, there is one and only one partial summation which leaves the distribution unchanged. This invariance property is reconsidered for distributions with one parameter. We show that if we change the parameter value in the function which defines the [...] Read more.

For every discrete probability distribution, there is one and only one partial summation which leaves the distribution unchanged. This invariance property is reconsidered for distributions with one parameter. We show that if we change the parameter value in the function which defines the summation, two families of distributions can be observed. The first of them consists of distributions which are resistant to the change. For these distributions, the change of the parameter is reversed by the normalization constant, and the distributions remain unchanged. The other contains distributions sensitive to the change. Partial summations with the changed parameter value applied to sensitive distributions result in new distributions with two parameters. A necessary and sufficient condition for a distribution to be resistant to the parameter change is presented. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

15 pages, 2212 KiB

Open AccessArticle

Modeling Electricity Price Dynamics Using Flexible Distributions

by Sherzod N. Tashpulatov

Mathematics 2022, 10(10), 1757; https://doi.org/10.3390/math10101757 - 21 May 2022

Cited by 1 | Viewed by 1703

Abstract

We consider the wholesale electricity market prices in England and Wales during its complete history, where price-cap regulation and divestment series were introduced at different points in time. We compare the impact of these regulatory reforms on the dynamics of electricity prices. For [...] Read more.

We consider the wholesale electricity market prices in England and Wales during its complete history, where price-cap regulation and divestment series were introduced at different points in time. We compare the impact of these regulatory reforms on the dynamics of electricity prices. For this purpose, we apply flexible distributions that account for asymmetry, heavy tails, and excess kurtosis usually observed in data or model residuals. The application of skew generalized error distribution is appropriate for our case study. We find that after the second series of divestments, price level and volatility are lower than during price-cap regulation and after the first series of divestments. This finding implies that a sufficient horizontal restructuring through divestment series may be superior to price-cap regulation. The conclusion could be interesting to other countries because the England and Wales electricity market served as the benchmark model for liberalizing energy markets worldwide. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

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17 pages, 828 KiB

Open AccessArticle

Survival with Random Effect

by Jonas Šiaulys and Rokas Puišys

Mathematics 2022, 10(7), 1097; https://doi.org/10.3390/math10071097 - 29 Mar 2022

Cited by 3 | Viewed by 1594

Abstract

The article focuses on mortality models with a random effect applied in order to evaluate human mortality more precisely. Such models are called frailty or Cox models. The main assertion of the paper shows that each positive random effect transforms the initial hazard [...] Read more.

The article focuses on mortality models with a random effect applied in order to evaluate human mortality more precisely. Such models are called frailty or Cox models. The main assertion of the paper shows that each positive random effect transforms the initial hazard rate (or density function) to a new absolutely continuous survival function. In particular, well-known Weibull and Gompertz hazard rates and corresponding survival functions are analyzed with different random effects. These specific models are presented with detailed calculations of hazard rates and corresponding survival functions. Six specific models with a random effect are applied to the same data set. The results indicate that the accuracy of the model depends on the data under consideration. Full article

(This article belongs to the Special Issue Probability Distributions and Their Applications)

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