Mathematics

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15 pages, 575 KiB

Open AccessArticle

Accumulative Pension Schemes with Various Decrement Factors

by Mohammed S. Al-Nator and Sofya V. Al-Nator

Mathematics 2020, 8(11), 2081; https://doi.org/10.3390/math8112081 - 22 Nov 2020

Cited by 5 | Viewed by 4852

Abstract

We consider accumulative defined contribution pension schemes with a lump sum payment on retirement. These schemes differ in relation to inheritance and provide various decrement factors. For each scheme, we construct the balance equation and obtain an expression for calculation of gross premium. [...] Read more.

We consider accumulative defined contribution pension schemes with a lump sum payment on retirement. These schemes differ in relation to inheritance and provide various decrement factors. For each scheme, we construct the balance equation and obtain an expression for calculation of gross premium. Payments are made at the end of the insurance event period (survival to retirement age or death or retirement for disability within the accumulation interval). A simulation model was developed to analyze the constructed schemes. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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8 pages, 256 KiB

Open AccessArticle

Local Limit Theorem for the Multiple Power Series Distributions

by Arsen L. Yakymiv

Mathematics 2020, 8(11), 2067; https://doi.org/10.3390/math8112067 - 19 Nov 2020

Cited by 1 | Viewed by 1303

Abstract

We study the behavior of multiple power series distributions at the boundary points of their existence. In previous papers, the necessary and sufficient conditions for the integral limit theorem were obtained. Here, the necessary and sufficient conditions for the corresponding local limit theorem [...] Read more.

We study the behavior of multiple power series distributions at the boundary points of their existence. In previous papers, the necessary and sufficient conditions for the integral limit theorem were obtained. Here, the necessary and sufficient conditions for the corresponding local limit theorem are established. This article is dedicated to the memory of my teacher, professor V.M. Zolotarev. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

18 pages, 1211 KiB

Open AccessArticle

Approximations in Performance Analysis of a Controllable Queueing System with Heterogeneous Servers

by Dmitry Efrosinin, Natalia Stepanova, Janos Sztrik and Andreas Plank

Mathematics 2020, 8(10), 1803; https://doi.org/10.3390/math8101803 - 16 Oct 2020

Cited by 7 | Viewed by 2440

Abstract

The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of [...] Read more.

The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of customers in the system reaches its minimal value. The Markov decision model and the policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, is of threshold type defined as a sequence of threshold levels which specifies the queue lengths for the usage of any slower server. To avoid time-consuming calculations for systems with a large number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this solution with the real values. We develop also the simple lower and upper bound methods based on approximation by an equivalent heterogeneous queueing system with a preemption to measure the mean number of customers in the system operating under the optimal policy. Finally, the simulation technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of inter-arrival and service time distributions. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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20 pages, 350 KiB

Open AccessArticle

A Priority Queue with Many Customer Types, Correlated Arrivals and Changing Priorities

by Seokjun Lee, Sergei Dudin, Olga Dudina, Chesoong Kim and Valentina Klimenok

Mathematics 2020, 8(8), 1292; https://doi.org/10.3390/math8081292 - 5 Aug 2020

Cited by 11 | Viewed by 3058

Abstract

A single-server queueing system with a finite buffer, several types of impatient customers, and non-preemptive priorities is analyzed. The initial priority of a customer can increase during its waiting time in the queue. The behavior of the system is described by a multi-dimensional [...] Read more.

A single-server queueing system with a finite buffer, several types of impatient customers, and non-preemptive priorities is analyzed. The initial priority of a customer can increase during its waiting time in the queue. The behavior of the system is described by a multi-dimensional Markov chain. The generator of this chain, having essential dependencies between the components, is derived and formulas for computation of the most important performance indicators of the system are presented. The dependence of some of these indicators on the capacity of the buffer space is illustrated. The profound effect of the phenomenon of correlation of successive inter-arrival times and variance of the service time is numerically demonstrated. Results can be used for the optimization of dispatching various types of customers in information transmission systems, emergency departments and first aid stations, perishable foods supply chains, etc. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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16 pages, 341 KiB

Open AccessArticle

Sensitivity Analysis and Simulation of a Multiserver Queueing System with Mixed Service Time Distribution

by Evsey Morozov, Michele Pagano, Irina Peshkova and Alexander Rumyantsev

Mathematics 2020, 8(8), 1277; https://doi.org/10.3390/math8081277 - 3 Aug 2020

Cited by 7 | Viewed by 2226

Abstract

The motivation of mixing distributions in communication/queueing systems modeling is that some input data (e.g., service time in queueing models) may follow several distinct distributions in a single input flow. In this paper, we study the sensitivity of performance measures on proximity of [...] Read more.

The motivation of mixing distributions in communication/queueing systems modeling is that some input data (e.g., service time in queueing models) may follow several distinct distributions in a single input flow. In this paper, we study the sensitivity of performance measures on proximity of the service time distributions of a multiserver system model with two-component Pareto mixture distribution of service times. The theoretical results are illustrated by numerical simulation of the

M / G / c

systems while using the perfect sampling approach. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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28 pages, 592 KiB

Open AccessArticle

Second Order Expansions for High-Dimension Low-Sample-Size Data Statistics in Random Setting

by Gerd Christoph and Vladimir V. Ulyanov

Mathematics 2020, 8(7), 1151; https://doi.org/10.3390/math8071151 - 14 Jul 2020

Cited by 4 | Viewed by 2120

Abstract

We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error [...] Read more.

We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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14 pages, 392 KiB

Open AccessArticle

On the Fractional Wave Equation

by Francesco Iafrate and Enzo Orsingher

Mathematics 2020, 8(6), 874; https://doi.org/10.3390/math8060874 - 31 May 2020

Cited by 2 | Viewed by 2540

Abstract

In this paper we study the time-fractional wave equation of order

1 < ν < 2

and give a probabilistic interpretation of its solution. In the case

0 < ν < 1

,

d = 1

, the solution can be interpreted as [...] Read more.

In this paper we study the time-fractional wave equation of order

1 < ν < 2

and give a probabilistic interpretation of its solution. In the case

0 < ν < 1

,

d = 1

, the solution can be interpreted as a time-changed Brownian motion, while for

1 < ν < 2

it coincides with the density of a symmetric stable process of order

2 / ν

. We give here an interpretation of the fractional wave equation for

d > 1

in terms of laws of stable d−dimensional processes. We give a hint at the case of a fractional wave equation for

ν > 2

and also at space-time fractional wave equations. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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38 pages, 912 KiB

Open AccessArticle

The Calculation of the Density and Distribution Functions of Strictly Stable Laws

by Viacheslav Saenko

Mathematics 2020, 8(5), 775; https://doi.org/10.3390/math8050775 - 12 May 2020

Cited by 1 | Viewed by 2515

Abstract

Integral representations for the probability density and distribution function of a strictly stable law with the characteristic function in the Zolotarev’s “C” parametrization were obtained in the paper. The obtained integral representations express the probability density and distribution function of standard strictly stable [...] Read more.

Integral representations for the probability density and distribution function of a strictly stable law with the characteristic function in the Zolotarev’s “C” parametrization were obtained in the paper. The obtained integral representations express the probability density and distribution function of standard strictly stable laws through a definite integral. Using the methods of numerical integration, the obtained integral representations allow us to calculate the probability density and distribution function of a strictly stable law for a wide range of admissible values of parameters

(α, θ)

. A number of cases were given when numerical algorithms had difficulty in calculating the density. Formulas were given to calculate the density and distribution function with an arbitrary value of the scale parameter

λ

. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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29 pages, 459 KiB

Open AccessArticle

Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

by Yury Khokhlov, Victor Korolev and Alexander Zeifman

Mathematics 2020, 8(5), 749; https://doi.org/10.3390/math8050749 - 8 May 2020

Cited by 6 | Viewed by 2330

Abstract

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding [...] Read more.

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag–Leffler distributions are considered in detail. Their relations with multivariate ‘ordinary’ Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag–Leffler and generalized Mittag–Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

14 pages, 263 KiB

Open AccessArticle

Statistical Indicators of the Scientific Publications Importance: A Stochastic Model and Critical Look

by Lev B. Klebanov, Yulia V. Kuvaeva and Zeev E. Volkovich

Mathematics 2020, 8(5), 713; https://doi.org/10.3390/math8050713 (registering DOI) - 3 May 2020

Viewed by 2008

Abstract

A model of scientific citation distribution is given. We apply it to understand the role of the Hirsch index as an indicator of scientific publication importance in Mathematics and some related fields. The proposed model is based on a generalization of such well-known [...] Read more.

A model of scientific citation distribution is given. We apply it to understand the role of the Hirsch index as an indicator of scientific publication importance in Mathematics and some related fields. The proposed model is based on a generalization of such well-known distributions as geometric and Sibuya laws. Real data analysis of the Hirsch index and corresponding citation numbers is given. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

17 pages, 256 KiB

Open AccessArticle

On Convergence Rates of Some Limits

by Edward Omey and Meitner Cadena

Mathematics 2020, 8(4), 634; https://doi.org/10.3390/math8040634 - 21 Apr 2020

Cited by 1 | Viewed by 1740

Abstract

In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each

y > 0

, [...] Read more.

In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each

y > 0

,

x (\frac{L (x + y)}{L (x)} - 1) \to 0 as x \to \infty

. Very recently, we have extended this result by considering a wider class of functions U related to the following more general condition. For each

y > 0

,

r (x) (\frac{U (x + y g (x))}{U (x)} - 1) \to 0 as x \to \infty

, for some functions r and g. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

30 pages, 5453 KiB

Open AccessArticle

Probability Models and Statistical Tests for Extreme Precipitation Based on Generalized Negative Binomial Distributions

by Victor Korolev and Andrey Gorshenin

Mathematics 2020, 8(4), 604; https://doi.org/10.3390/math8040604 - 16 Apr 2020

Cited by 15 | Viewed by 2750

Abstract

Mathematical models are proposed for statistical regularities of maximum daily precipitation within a wet period and total precipitation volume per wet period. The proposed models are based on the generalized negative binomial (GNB) distribution of the duration of a wet period. The GNB [...] Read more.

Mathematical models are proposed for statistical regularities of maximum daily precipitation within a wet period and total precipitation volume per wet period. The proposed models are based on the generalized negative binomial (GNB) distribution of the duration of a wet period. The GNB distribution is a mixed Poisson distribution, the mixing distribution being generalized gamma (GG). The GNB distribution demonstrates excellent fit with real data of durations of wet periods measured in days. By means of limit theorems for statistics constructed from samples with random sizes having the GNB distribution, asymptotic approximations are proposed for the distributions of maximum daily precipitation volume within a wet period and total precipitation volume for a wet period. It is shown that the exponent power parameter in the mixing GG distribution matches slow global climate trends. The bounds for the accuracy of the proposed approximations are presented. Several tests for daily precipitation, total precipitation volume and precipitation intensities to be abnormally extremal are proposed and compared to the traditional PoT-method. The results of the application of this test to real data are presented. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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21 pages, 375 KiB

Open AccessArticle

A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints

by Irina Shevtsova and Mikhail Tselishchev

Mathematics 2020, 8(4), 577; https://doi.org/10.3390/math8040577 - 13 Apr 2020

Cited by 8 | Viewed by 2415

Abstract

We introduce a generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with finite nonzero first moment and study its properties. In particular, we prove an optimal moment-type inequality for the Kantorovich distance between a distribution and its equilibrium transform. [...] Read more.

We introduce a generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with finite nonzero first moment and study its properties. In particular, we prove an optimal moment-type inequality for the Kantorovich distance between a distribution and its equilibrium transform. Using the introduced transform and Stein’s method, we investigate the rate of convergence in the Rényi theorem for the distributions of geometric sums of independent random variables with identical nonzero means and finite second moments without any constraints on their supports. We derive an upper bound for the Kantorovich distance between the normalized geometric random sum and the exponential distribution which has exact order of smallness as the expectation of the geometric number of summands tends to infinity. Moreover, we introduce the so-called asymptotically best constant and present its lower bound yielding the one for the Kantorovich distance under consideration. As a concluding remark, we provide an extension of the obtained estimates of the accuracy of the exponential approximation to non-geometric random sums of independent random variables with non-identical nonzero means. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

22 pages, 484 KiB

Open AccessArticle

Optimal Filtering of Markov Jump Processes Given Observations with State-Dependent Noises: Exact Solution and Stable Numerical Schemes

by Andrey Borisov and Igor Sokolov

Mathematics 2020, 8(4), 506; https://doi.org/10.3390/math8040506 - 2 Apr 2020

Cited by 10 | Viewed by 2278

Abstract

The paper is devoted to the optimal state filtering of the finite-state Markov jump processes, given indirect continuous-time observations corrupted by Wiener noise. The crucial feature is that the observation noise intensity is a function of the estimated state, which breaks forthright filtering [...] Read more.

The paper is devoted to the optimal state filtering of the finite-state Markov jump processes, given indirect continuous-time observations corrupted by Wiener noise. The crucial feature is that the observation noise intensity is a function of the estimated state, which breaks forthright filtering approaches based on the passage to the innovation process and Girsanov’s measure change. We propose an equivalent observation transform, which allows usage of the classical nonlinear filtering framework. We obtain the optimal estimate as a solution to the discrete–continuous stochastic differential system with both continuous and counting processes on the right-hand side. For effective computer realization, we present a new class of numerical algorithms based on the exact solution to the optimal filtering given the time-discretized observation. The proposed estimate approximations are stable, i.e., have non-negative components and satisfy the normalization condition. We prove the assertions characterizing the approximation accuracy depending on the observation system parameters, time discretization step, the maximal number of allowed state transitions, and the applied scheme of numerical integration. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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19 pages, 320 KiB

Open AccessArticle

Rates of Convergence in Laplace’s Integrals and Sums and Conditional Central Limit Theorems

by Vassili N. Kolokoltsov

Mathematics 2020, 8(4), 479; https://doi.org/10.3390/math8040479 - 1 Apr 2020

Cited by 1 | Viewed by 1789

Abstract

We obtained the exact estimates for the error terms in Laplace’s integrals and sums implying the corresponding estimates for the related laws of large number and central limit theorems including the large deviations approximation. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

8 pages, 263 KiB

Open AccessArticle

Wavelet Thresholding Risk Estimate for the Model with Random Samples and Correlated Noise

by Oleg Shestakov

Mathematics 2020, 8(3), 377; https://doi.org/10.3390/math8030377 - 8 Mar 2020

Cited by 3 | Viewed by 1920

Abstract

Signal de-noising methods based on threshold processing of wavelet decomposition coefficients have become popular due to their simplicity, speed, and ability to adapt to signal functions with spatially inhomogeneous smoothness. The analysis of the errors of these methods is an important practical task, [...] Read more.

Signal de-noising methods based on threshold processing of wavelet decomposition coefficients have become popular due to their simplicity, speed, and ability to adapt to signal functions with spatially inhomogeneous smoothness. The analysis of the errors of these methods is an important practical task, since it makes it possible to evaluate the quality of both methods and equipment used for processing. Sometimes the nature of the signal is such that its samples are recorded at random times. If the sample points form a variational series based on a sample from the uniform distribution on the data registration interval, then the use of the standard threshold processing procedure is adequate. The paper considers a model of a signal that is registered at random times and contains noise with long-term dependence. The asymptotic normality and strong consistency properties of the mean-square thresholding risk estimator are proved. The obtained results make it possible to construct asymptotic confidence intervals for threshold processing errors using only the observed data. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

Review

Jump to: Research

10 pages, 260 KiB

Open AccessReview

Highly Efficient Robust and Stable M-Estimates of Location

by Georgy Shevlyakov

Mathematics 2021, 9(1), 105; https://doi.org/10.3390/math9010105 - 5 Jan 2021

Cited by 3 | Viewed by 1719

Abstract

This article is partially a review and partially a contribution. The classical two approaches to robustness, Huber’s minimax and Hampel’s based on influence functions, are reviewed with the accent on distribution classes of a non-neighborhood nature. Mainly, attention is paid to the minimax [...] Read more.

This article is partially a review and partially a contribution. The classical two approaches to robustness, Huber’s minimax and Hampel’s based on influence functions, are reviewed with the accent on distribution classes of a non-neighborhood nature. Mainly, attention is paid to the minimax Huber’s M-estimates of location designed for the classes with bounded quantiles and Meshalkin-Shurygin’s stable M-estimates. The contribution is focused on the comparative performance evaluation study of these estimates, together with the classical robust M-estimates under the normal, double-exponential (Laplace), Cauchy, and contaminated normal (Tukey gross error) distributions. The obtained results are as follows: (i) under the normal, double-exponential, Cauchy, and heavily-contaminated normal distributions, the proposed robust minimax M-estimates outperform the classical Huber’s and Hampel’s M-estimates in asymptotic efficiency; (ii) in the case of heavy-tailed double-exponential and Cauchy distributions, the Meshalkin-Shurygin’s radical stable M-estimate also outperforms the classical robust M-estimates; (iii) for moderately contaminated normal, the classical robust estimates slightly outperform the proposed minimax M-estimates. Several directions of future works are enlisted. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

25 pages, 588 KiB

Open AccessFeature PaperReview

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

by Alexander Zeifman, Victor Korolev and Yacov Satin

Mathematics 2020, 8(2), 253; https://doi.org/10.3390/math8020253 - 14 Feb 2020

Cited by 18 | Viewed by 3297

Abstract

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the [...] Read more.

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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