Mathematics

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13 pages, 287 KiB

Open AccessArticle

The Wasserstein Metric between a Discrete Probability Measure and a Continuous One

by Weihua Yang, Xu Zhang and Xia Wang

Mathematics 2024, 12(15), 2320; https://doi.org/10.3390/math12152320 - 24 Jul 2024

Viewed by 532

This paper examines the Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform measure in the d-dimensional ball, providing an asymptotic estimation of their expectations as n approaches infinity. Furthermore, we investigate this problem within a [...] Read more.

This paper examines the Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform measure in the d-dimensional ball, providing an asymptotic estimation of their expectations as n approaches infinity. Furthermore, we investigate this problem within a mixed process framework, where n discrete random variables are generated by the Poisson process. Full article

(This article belongs to the Special Issue Probability, Statistics and Random Processes)

17 pages, 270 KiB

Open AccessArticle

Birth–Death Processes with Two-Type Catastrophes

by Junping Li

Mathematics 2024, 12(10), 1468; https://doi.org/10.3390/math12101468 - 9 May 2024

Viewed by 985

Abstract

This paper concentrates on the general birth–death processes with two different types of catastrophes. The Laplace transform of transition probability function for birth–death processes with two-type catastrophes is successfully expressed with the Laplace transform of transition probability function of the birth–death processes without [...] Read more.

This paper concentrates on the general birth–death processes with two different types of catastrophes. The Laplace transform of transition probability function for birth–death processes with two-type catastrophes is successfully expressed with the Laplace transform of transition probability function of the birth–death processes without catastrophe. The first effective catastrophe occurrence time is considered. The Laplace transform of its probability density function, expectation and variance are obtained. Full article

(This article belongs to the Special Issue Probability, Statistics and Random Processes)

22 pages, 2318 KiB

Open AccessArticle

Asymptotic Form of the Covariance Matrix of Likelihood-Based Estimator in Multidimensional Linear System Model for the Case of Infinity Number of Nuisance Parameters

by Alexander Varypaev

Mathematics 2024, 12(3), 473; https://doi.org/10.3390/math12030473 - 1 Feb 2024

Viewed by 802

Abstract

This article is devoted to the synthesis and analysis of the quality of the statistical estimate of parameters of a multidimensional linear system (MLS) with one input and m outputs. A nontrivial case is investigated when the one-dimensional input signal of MLS is [...] Read more.

This article is devoted to the synthesis and analysis of the quality of the statistical estimate of parameters of a multidimensional linear system (MLS) with one input and m outputs. A nontrivial case is investigated when the one-dimensional input signal of MLS is a deterministic process, the values of which are unknown nuisance parameters. The estimate is based only on observations of MLS output signals distorted by random Gaussian stationary

m

-dimensional noise with a known spectrum. It is assumed that the likelihood function of observations of the output signals of MLS satisfies the conditions of local asymptotic normality. The

\sqrt{n}

-consistency of the estimate is established. Under the assumption of asymptotic normality of an objective function, the limiting covariance matrix of the estimate is calculated for case where the number of observations tends to infinity. Full article

(This article belongs to the Special Issue Probability, Statistics and Random Processes)

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25 pages, 400 KiB

Open AccessArticle

Modified Block Bootstrap Testing for Persistence Change in Infinite Variance Observations

by Si Zhang, Hao Jin and Menglin Su

Mathematics 2024, 12(2), 258; https://doi.org/10.3390/math12020258 - 12 Jan 2024

Cited by 2 | Viewed by 1013

Abstract

This paper investigates the properties of the change in persistence detection for observations with infinite variance. The innovations are assumed to be in the domain of attraction of a stable law with index

κ \in (0, 2]

. We provide [...] Read more.

This paper investigates the properties of the change in persistence detection for observations with infinite variance. The innovations are assumed to be in the domain of attraction of a stable law with index

κ \in (0, 2]

. We provide a new test statistic and show that its asymptotic distribution under the null hypothesis of non-stationary

I (1)

series is a functional of a stable process. When the change point in persistence is not known, the consistency is always given under the alternative, either from stationary

I (0)

to non-stationary

I (1)

or vice versa. The proposed tests can be used to identify the direction of change and do not over-reject against constant

I (0)

series, even in relatively small samples. Furthermore, we also consider the change point estimator which is consistent and the asymptotic behavior of the test statistic in the case of near-integrated time series. A block bootstrap method is suggested to determine critical values because the null asymptotic distribution contains the unknown tail index, which results in critical values depending on it. The simulation study demonstrates that the block bootstrap-based test is robust against change in persistence for heavy-tailed series with infinite variance. Finally, we apply our methods to the two series of the US inflation rate and USD/CNY exchange rate, and find significant evidence for persistence changes, respectively, from

I (0)

to

I (1)

and from

I (1)

to

I (0)

. Full article

(This article belongs to the Special Issue Probability, Statistics and Random Processes)

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18 pages, 1046 KiB

Open AccessArticle

Validation of Stock Price Prediction Models in the Conditions of Financial Crisis

by Vesela Mihova, Ivan Georgiev, Elitsa Raeva, Slavi Georgiev and Velizar Pavlov

Mathematics 2024, 12(1), 33; https://doi.org/10.3390/math12010033 - 22 Dec 2023

Viewed by 1118

Abstract

The distribution laws of various natural and anthropogenic processes in the world around us are stochastic in nature. The development of mathematics and, in particular, of stochastic modeling allows us to study regularities in such processes. In practice, stochastic modeling finds a huge [...] Read more.

The distribution laws of various natural and anthropogenic processes in the world around us are stochastic in nature. The development of mathematics and, in particular, of stochastic modeling allows us to study regularities in such processes. In practice, stochastic modeling finds a huge number of applications in various fields, including finance and economics. In this work, some particular applications of stochastic processes in finance are examined in the conditions of financial crisis, aiming to provide a solid approach for stock price forecasting. More specifically, autoregressive integrated moving average (ARIMA) models and modified ordinary differential equation (ODE) models, previously developed by some of the authors to predict the asset prices of four Bulgarian companies, are validated against a time period during the crisis. Estimated rates of return are calculated from the models for one period ahead. The errors are estimated and the models are compared. The return values predicted with each of the two approaches are used to derive optimal risk portfolios based on the Markowitz model, which is the second major aim of this study. The third aim is to compare the resulting portfolios in terms of distribution (i.e., weights of the stocks), risk, and rate of return. Full article

(This article belongs to the Special Issue Probability, Statistics and Random Processes)

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10 pages, 302 KiB

Open AccessArticle

New Ways to Calculate the Probability in the Bertrand Problem

by Javier Rodrigo, Mariló López and Sagrario Lantarón

Mathematics 2024, 12(1), 3; https://doi.org/10.3390/math12010003 - 19 Dec 2023

Viewed by 784

Abstract

We give two new ways of calculating the probability of a chord of circumference randomly selected being larger than the side of an equilateral triangle inscribed in the circumference (this problem is known as the Bertrand paradox). The first one employs an immersion [...] Read more.

We give two new ways of calculating the probability of a chord of circumference randomly selected being larger than the side of an equilateral triangle inscribed in the circumference (this problem is known as the Bertrand paradox). The first one employs an immersion in R⁴, and the second one uses a direct probability measure over the set of chords. Full article

(This article belongs to the Special Issue Probability, Statistics and Random Processes)

14 pages, 472 KiB

Open AccessArticle

Estimation of Pianka Overlapping Coefficient for Two Exponential Distributions

by Suad Alhihi and Maalee Almheidat

Mathematics 2023, 11(19), 4152; https://doi.org/10.3390/math11194152 - 2 Oct 2023

Cited by 1 | Viewed by 1625

Abstract

Overlapping coefficients (OVL) are commonly used to estimate the similarity between populations in terms of their density functions. In this paper, we consider Pianka’s overlap coefficient for two exponential populations. The methods for statistical inference of Pianka’s coefficient are presented. The bias and [...] Read more.

Overlapping coefficients (OVL) are commonly used to estimate the similarity between populations in terms of their density functions. In this paper, we consider Pianka’s overlap coefficient for two exponential populations. The methods for statistical inference of Pianka’s coefficient are presented. The bias and mean square error (MSE) of the maximum likelihood estimator (MLE) and the Bayes estimator of Pianka’s overlap coefficient are investigated by simulation. Confidence intervals for Pianka’s overlap measure are constructed. Full article

(This article belongs to the Special Issue Probability, Statistics and Random Processes)

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21 pages, 543 KiB

Open AccessArticle

Bertrand’s Paradox Resolution and Its Implications for the Bing–Fisher Problem

by Richard A. Chechile

Mathematics 2023, 11(15), 3282; https://doi.org/10.3390/math11153282 - 26 Jul 2023

Cited by 1 | Viewed by 2867

Abstract

Bertrand’s paradox is a problem in geometric probability that has resisted resolution for more than one hundred years. Bertrand provided three seemingly reasonable solutions to his problem — hence the paradox. Bertrand’s paradox has also been influential in philosophical debates about frequentist versus [...] Read more.

Bertrand’s paradox is a problem in geometric probability that has resisted resolution for more than one hundred years. Bertrand provided three seemingly reasonable solutions to his problem — hence the paradox. Bertrand’s paradox has also been influential in philosophical debates about frequentist versus Bayesian approaches to statistical inference. In this paper, the paradox is resolved (1) by the clarification of the primary variate upon which the principle of maximum entropy is employed and (2) by imposing constraints, based on a mathematical analysis, on the random process for any subsequent nonlinear transformation to a secondary variable. These steps result in a unique solution to Bertrand’s problem, and this solution differs from the classic answers that Bertrand proposed. It is shown that the solutions proposed by Bertrand and others reflected sampling processes that are not purely random. It is also shown that the same two steps result in the resolution of the Bing–Fisher problem, which has to do with the selection of a consistent prior for Bayesian inference. The resolution of Bertrand’s paradox and the Bing–Fisher problem rebuts philosophical arguments against the Bayesian approach to statistical inference, which were based on those two ostensible problems. Full article

(This article belongs to the Special Issue Probability, Statistics and Random Processes)

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8 pages, 242 KiB

Open AccessArticle

Consistency of Decision in Finite and Numerable Multinomial Models

by Isaac Akoto and João T. Mexia

Mathematics 2023, 11(11), 2434; https://doi.org/10.3390/math11112434 - 24 May 2023

Viewed by 821

Abstract

The multinomial distribution is often used in modeling categorical data because it describes the probability of a random observation being assigned to one of several mutually exclusive categories. Given a finite or numerable multinomial model

M (| n, p)

whose decision is [...] Read more.

The multinomial distribution is often used in modeling categorical data because it describes the probability of a random observation being assigned to one of several mutually exclusive categories. Given a finite or numerable multinomial model

M (| n, p)

whose decision is indexed by a parameter

θ

and having a cost

c (θ, p)

depending on

θ

and on

p

, we show that, under general conditions, the probability of taking the least cost decision tends to 1 when n tends to ∞, i.e., we showed that the cost decision is consistent, representing a Statistical Decision Theory approach to the concept of consistency, which is not much considered in the literature. Thus, under these conditions, we have consistency in the decision making. The key result is that the estimator

{\tilde{p}}_{n}

with components

{\tilde{p}}_{n, i} = \frac{n_{i}}{n}, i = 1, \dots

, where

n_{i}

is the number of times we obtain the ith result when we have a sample of size n, is a consistent estimator of

p

. This result holds both for finite and numerable models. By this result, we were able to incorporate a more general form for consistency for the cost function of a multinomial model. Full article

(This article belongs to the Special Issue Probability, Statistics and Random Processes)

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