Research

7 pages, 231 KiB

Open AccessFeature PaperArticle

The Law of the Iterated Logarithm for L_p-Norms of Kernel Estimators of Cumulative Distribution Functions

by Fuxia Cheng

Mathematics 2024, 12(7), 1063; https://doi.org/10.3390/math12071063 - 01 Apr 2024

Viewed by 503

In this paper, we consider the strong convergence of

L_{p}

-norms (

p \geq 1

) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the

L_{p}

-norms [...] Read more.

In this paper, we consider the strong convergence of

L_{p}

-norms (

p \geq 1

) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the

L_{p}

-norms of empirical processes is extended to the kernel estimator of the CDF. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

28 pages, 409 KiB

Open AccessArticle

5th-Order Multivariate Edgeworth Expansions for Parametric Estimates

by C. S. Withers

Mathematics 2024, 12(6), 905; https://doi.org/10.3390/math12060905 - 19 Mar 2024

Viewed by 397

Abstract

The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need [...] Read more.

The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need the Edgeworth expansion for the distribution of the standardised estimate. This is an expansion in

n^{- 1 / 2}

about the normal distribution, where n is typically the sample size. The first few terms of this expansion were originally given for the special case of a sample mean. In earlier work we derived it for any standard estimate, hugely expanding its application. We define an estimate

\hat{w}

of an unknown vector w in

R^{p}

, as a standard estimate, if

E \hat{w} \to w

as

n \to \infty

, and for

r \geq 1

the rth-order cumulants of

\hat{w}

have magnitude

n^{1 - r}

and can be expanded in

n^{- 1} .

Here we present a significant extension. We give the expansion of the distribution of any smooth function of

\hat{w}

, say

t (\hat{w})

in

R^{q},

giving its distribution to

n^{- 5 / 2}

. We do this by showing that

t (\hat{w})

, is a standard estimate of

t (w)

. This provides far more accurate approximations for the distribution of

t (\hat{w})

than its asymptotic normality. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

13 pages, 302 KiB

Open AccessArticle

A Continuous-Time Urn Model for a System of Activated Particles

by Rafik Aguech and Hanene Mohamed

Mathematics 2023, 11(24), 4967; https://doi.org/10.3390/math11244967 - 15 Dec 2023

Viewed by 563

Abstract

We study a system of M particles with jump dynamics on a network of N sites. The particles can exist in two states, active or inactive. Only the former can jump. The state of each particle depends on its position. A given particle [...] Read more.

We study a system of M particles with jump dynamics on a network of N sites. The particles can exist in two states, active or inactive. Only the former can jump. The state of each particle depends on its position. A given particle is inactive when it is at a given site, and active when it moves to a change site. Indeed, each sleeping particle activates at a rate

λ > 0

, leaves its initial site, and moves for an exponential random time of parameter

μ > 0

before uniformly landing at a site and immediately returning to sleep. The behavior of each particle is independent of that of the others. These dynamics conserve the total number of particles; there is no limit on the number of particles at a given site. This system can be represented by a continuous-time Pólya urn with M balls where the colors are the sites, with an additional color to account for particles on the move at a given time t. First, using this Pólya interpretation for fixed M and N, we obtain the average number of particles at each site over time and, therefore, those on the move due to mass conservation. Secondly, we consider a large system in which the number of particles M and the number of sites N grow at the same rate, so that the

M / N

ratio tends to a scaling constant

α > 0

. Using the moment-generating function technique added to some probabilistic arguments, we obtain the long-term distribution of the number of particles at each site. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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18 pages, 1104 KiB

Open AccessArticle

On Rank Selection in Non-Negative Matrix Factorization Using Concordance

by Paul Fogel, Christophe Geissler, Nicolas Morizet and George Luta

Mathematics 2023, 11(22), 4611; https://doi.org/10.3390/math11224611 - 10 Nov 2023

Viewed by 813

Abstract

The choice of the factorization rank of a matrix is critical, e.g., in dimensionality reduction, filtering, clustering, deconvolution, etc., because selecting a rank that is too high amounts to adjusting the noise, while selecting a rank that is too low results in the [...] Read more.

The choice of the factorization rank of a matrix is critical, e.g., in dimensionality reduction, filtering, clustering, deconvolution, etc., because selecting a rank that is too high amounts to adjusting the noise, while selecting a rank that is too low results in the oversimplification of the signal. Numerous methods for selecting the factorization rank of a non-negative matrix have been proposed. One of them is the cophenetic correlation coefficient (

c c c

), widely used in data science to evaluate the number of clusters in a hierarchical clustering. In previous work, it was shown that

c c c

performs better than other methods for rank selection in non-negative matrix factorization (NMF) when the underlying structure of the matrix consists of orthogonal clusters. In this article, we show that using the ratio of

c c c

to the approximation error significantly improves the accuracy of the rank selection. We also propose a new criterion,

c o n c o r d a n c e

, which, like

c c c

, benefits from the stochastic nature of NMF; its accuracy is also improved by using its ratio-to-error form. Using real and simulated data, we show that

c o n c o r d a n c e

, with a CUSUM-based automatic detection algorithm for its original or ratio-to-error forms, significantly outperforms

c c c

. It is important to note that the new criterion works for a broader class of matrices, where the underlying clusters are not assumed to be orthogonal. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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21 pages, 421 KiB

Open AccessArticle

Statistical Analysis and Theoretical Framework for a Partially Accelerated Life Test Model with Progressive First Failure Censoring Utilizing a Power Hazard Distribution

by Amel Abd-El-Monem, Mohamed S. Eliwa, Mahmoud El-Morshedy, Afrah Al-Bossly and Rashad M. EL-Sagheer

Mathematics 2023, 11(20), 4323; https://doi.org/10.3390/math11204323 - 17 Oct 2023

Viewed by 580

Abstract

Monitoring life-testing trials for a product or substance often demands significant time and effort. To expedite this process, sometimes units are subjected to more severe conditions in what is known as accelerated life tests. This paper is dedicated to addressing the challenge of [...] Read more.

Monitoring life-testing trials for a product or substance often demands significant time and effort. To expedite this process, sometimes units are subjected to more severe conditions in what is known as accelerated life tests. This paper is dedicated to addressing the challenge of estimating the power hazard distribution, both in terms of point and interval estimations, during constant- stress partially accelerated life tests using progressive first failure censored samples. Three techniques are employed for this purpose: maximum likelihood, two parametric bootstraps, and Bayesian methods. These techniques yield point estimates for unknown parameters and the acceleration factor. Additionally, we construct approximate confidence intervals and highest posterior density credible intervals for both the parameters and acceleration factor. The former relies on the asymptotic distribution of maximum likelihood estimators, while the latter employs the Markov chain Monte Carlo technique and focuses on the squared error loss function. To assess the effectiveness of these estimation methods and compare the performance of their respective confidence intervals, a simulation study is conducted. Finally, we validate these inference techniques using real-life engineering data. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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15 pages, 367 KiB

Open AccessArticle

A Study of Assessment of Casinos’ Risk of Ruin in Casino Games with Poisson Distribution

by Ka-Meng Siu, Ka-Hou Chan and Sio-Kei Im

Mathematics 2023, 11(7), 1736; https://doi.org/10.3390/math11071736 - 05 Apr 2023

Viewed by 2617

Abstract

Gambling, as an uncertain business involving risks confronting casinos, is commonly analysed using the risk of ruin (ROR) formula. However, due to its brevity, the ROR does not provide any implication of nuances in terms of the distribution of wins/losses, thus causing the [...] Read more.

Gambling, as an uncertain business involving risks confronting casinos, is commonly analysed using the risk of ruin (ROR) formula. However, due to its brevity, the ROR does not provide any implication of nuances in terms of the distribution of wins/losses, thus causing the potential failure of unravelling exceptional and extreme cases. This paper discusses the mathematical model of ROR using Poisson distribution theory with the consideration of house advantage (a) and the law of large numbers in order to compensate for the insufficiency mentioned above. In this discussion, we explore the relationship between cash flow and max bet limits in the model and examine how these factors interact in influencing the risk of casino bankruptcy. In their business nature, casinos operate gambling businesses and capitalize on the house advantage favouring them. The house advantage of the games signifies casinos’ profitability, and in addition, the uncertainty inevitably poses a certain risk of bankruptcy to them even though the house advantage favours them. In this paper, the house advantage is incorporated into our model for a few popular casino games. Furthermore, a set of full-range scales is defined to facilitate effective judgment on the levels of risk confronted by casinos in certain settings. Some wagers of popular casino games are also exemplified with our proposed model. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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29 pages, 1863 KiB

Open AccessArticle

Exploring the Dynamics of COVID-19 with a Novel Family of Models

by Abdulaziz S. Alghamdi and M. M. Abd El-Raouf

Mathematics 2023, 11(7), 1641; https://doi.org/10.3390/math11071641 - 28 Mar 2023

Cited by 5 | Viewed by 889

Abstract

Much effort has recently been expended in developing efficient models that can depict the true picture for COVID-19 mortality data and help scientists choose the best-fit models. As a result, this research intends to provide a new G family for both theoretical and [...] Read more.

Much effort has recently been expended in developing efficient models that can depict the true picture for COVID-19 mortality data and help scientists choose the best-fit models. As a result, this research intends to provide a new G family for both theoretical and practical scientists that solves the concerns typically encountered in both normal and non-normal random events. The new-G distribution family is able to generate efficient continuous univariate and skewed models that may outperform the baseline model. The analytic properties of the new-G family and its sub-model are investigated and described, as well as a theoretical framework. The parameters were estimated using a classical approach along with an extensive simulation study to assess the behaviour of the parameters. The efficiency of the new-G family is discussed using one of its sub-models on COVID-19 mortality data sets. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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20 pages, 684 KiB

Open AccessArticle

Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions

by Shaul K. Bar-Lev, Apostolos Batsidis, Jochen Einbeck, Xu Liu and Panpan Ren

Mathematics 2023, 11(7), 1603; https://doi.org/10.3390/math11071603 - 26 Mar 2023

Cited by 1 | Viewed by 893

Abstract

The class of natural exponential families (NEFs) of distributions having power variance functions (NEF-PVFs) is huge (uncountable), with enormous applications in various fields. Based on a characterization property that holds for the cumulants of the members of this class, we developed a novel [...] Read more.

The class of natural exponential families (NEFs) of distributions having power variance functions (NEF-PVFs) is huge (uncountable), with enormous applications in various fields. Based on a characterization property that holds for the cumulants of the members of this class, we developed a novel goodness-of-fit (gof) test for testing whether a given random sample fits fixed members of this class. We derived the asymptotic null distribution of the test statistic and developed an appropriate bootstrap scheme. As the content of the paper is mainly theoretical, we exemplify its applicability to only a few elements of the NEF-PVF class, specifically, the gamma and modified Bessel-type NEFs. A Monte Carlo study was executed for examining the performance of both—the asymptotic test and the bootstrap counterpart—in controlling the type I error rate and evaluating their power performance in the special case of gamma, while real data examples demonstrate the applicability of the gof test to the modified Bessel distribution. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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22 pages, 480 KiB

Open AccessArticle

Undirected Structural Markov Property for Bayesian Model Determination

by Xiong Kang, Yingying Hu and Yi Sun

Mathematics 2023, 11(7), 1590; https://doi.org/10.3390/math11071590 - 25 Mar 2023

Viewed by 774

Abstract

This paper generalizes the structural Markov properties for undirected decomposable graphs to arbitrary ones. This helps us to exploit the conditional independence properties of joint prior laws to analyze and compare multiple graphical structures, while being able to take advantage of the common [...] Read more.

This paper generalizes the structural Markov properties for undirected decomposable graphs to arbitrary ones. This helps us to exploit the conditional independence properties of joint prior laws to analyze and compare multiple graphical structures, while being able to take advantage of the common conditional independence constraints. This work provides a theoretical support for full Bayesian posterior updating about the structure of a graph using data from a certain distribution. We further investigate the ratio of graph law so as to simplify the acceptance probability of the Metropolis–Hastings sampling algorithms. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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28 pages, 3011 KiB

Open AccessArticle

A Novel Discrete Generator with Modeling Engineering, Agricultural and Medical Count and Zero-Inflated Real Data with Bayesian, and Non-Bayesian Inference

by Walid Emam, Yusra Tashkandy, G.G. Hamedani, Mohamed Abdelhamed Shehab, Mohamed Ibrahim and Haitham M. Yousof

Mathematics 2023, 11(5), 1125; https://doi.org/10.3390/math11051125 - 23 Feb 2023

Cited by 4 | Viewed by 1140

Abstract

This study introduces a unique flexible family of discrete probability distributions for modeling extreme count and zero-inflated count data with different failure rates. Certain significant mathematical properties, such as the cumulant generating function, moment generating function, dispersion index, L-moments, ordinary moments, and central [...] Read more.

This study introduces a unique flexible family of discrete probability distributions for modeling extreme count and zero-inflated count data with different failure rates. Certain significant mathematical properties, such as the cumulant generating function, moment generating function, dispersion index, L-moments, ordinary moments, and central moment are derived. The new failure rate function offers a wide range of flexibility, including “upside down”, “monotonically decreasing”, “bathtub”, “monotonically increasing” and “decreasing-constant failure rate” and “constant”. Moreover, the new probability mass function accommodates many useful shapes including the “right skewed function with no peak”, “symmetric”, “right skewed with one peak” and “left skewed with one peak”. To obtain significant characterization findings, the hazard function and the conditional expectation of certain function of the random variable are both employed. Both Bayesian and non-Bayesian estimate methodologies are considered when estimating, assessing, and comparing inferential efficacy. The Bayesian estimation approach for the squared error loss function is suggested, and it is explained. Markov chain Monte Carlo simulation studies are performed using the Metropolis Hastings algorithm and the Gibbs sampler to compare non-Bayesian vs. Bayesian results. Four real-world applications of count data sets are used to evaluate the Bayesian versus non-Bayesian techniques. Four more real count data applications are used to illustrate the significance and versatility of the new discrete class. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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25 pages, 1128 KiB

Open AccessArticle

A New Alpha Power Cosine-Weibull Model with Applications to Hydrological and Engineering Data

by Abdulaziz S. Alghamdi and M. M. Abd El-Raouf

Mathematics 2023, 11(3), 673; https://doi.org/10.3390/math11030673 - 28 Jan 2023

Cited by 3 | Viewed by 1080

Abstract

Modifying the existing probability models in the literature and introducing new extensions of the existing probability models is a prominent and interesting research topic. However, in the most recent era, the extensions of the probability models via trigonometry methods have received great attention. [...] Read more.

Modifying the existing probability models in the literature and introducing new extensions of the existing probability models is a prominent and interesting research topic. However, in the most recent era, the extensions of the probability models via trigonometry methods have received great attention. This paper also offers a novel trigonometric version of the Weibull model called a new alpha power cosine-Weibull (for short, “NACos-Weibull”) distribution. The NACos-Weibull distribution is introduced by incorporating the cosine function. Certain distributional properties of the NACos-Weibull model are derived. The estimators of the NACos-Weibull model are derived by implementing the maximum likelihood approach. Three simulation studies are provided for different values of the parameters of the NACos-Weibull distribution. Finally, to demonstrate the effectiveness of the NACos-Weibull model, three applications from the hydrological and engineering sectors are considered. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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12 pages, 831 KiB

Open AccessArticle

Parameter Estimation and Hypothesis Testing of The Bivariate Polynomial Ordinal Logistic Regression Model

by Marisa Rifada, Vita Ratnasari and Purhadi Purhadi

Mathematics 2023, 11(3), 579; https://doi.org/10.3390/math11030579 - 21 Jan 2023

Viewed by 1321

Abstract

Logistic regression is one of statistical methods that used to analyze the correlation between categorical response variables and predictor variables which are categorical or continuous. Many studies on logistic regression have been carried out by assuming that the predictor variable and its logit [...] Read more.

Logistic regression is one of statistical methods that used to analyze the correlation between categorical response variables and predictor variables which are categorical or continuous. Many studies on logistic regression have been carried out by assuming that the predictor variable and its logit link function have a linear relationship. However, in several cases it was found that the relationship was not always linear, but could be quadratic, cubic, or in the form of other curves, so that the assumption of linearity was incorrect. Therefore, this study will develop a bivariate polynomial ordinal logistic regression (BPOLR) model which is an extension of ordinal logistic regression, with two correlated response variables in which the relationship between the continuous predictor variable and its logit is modeled as a polynomial form. There are commonly two correlated response variables in scientific research. In this study, each response variable used consisted of three categories. This study aims to obtain parameter estimators of the BPOLR model using the maximum likelihood estimation (MLE) method, obtain test statistics of parameters using the maximum likelihood ratio test (MLRT) method, and obtain algorithms of estimating and hypothesis testing for parameters of the BPOLR model. The results of the first partial derivatives are not closed-form, thus, a numerical optimization such as the Berndt–Hall–Hall–Hausman (BHHH) method is needed to obtain the maximum likelihood estimator. The distribution statistically test is followed the Chi-square limit distribution, asymptotically. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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17 pages, 675 KiB

Open AccessArticle

Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations

by John-Fritz Thony and Jean Vaillant

Mathematics 2022, 10(22), 4190; https://doi.org/10.3390/math10224190 - 09 Nov 2022

Viewed by 1228

Abstract

We consider a stochastic differential equation (SDE) governed by a fractional Brownian motion

(B_{t}^{H})

and a Poisson process

(N_{t})

associated with a stochastic process

(A_{t})

such that: [...] Read more.

We consider a stochastic differential equation (SDE) governed by a fractional Brownian motion

(B_{t}^{H})

and a Poisson process

(N_{t})

associated with a stochastic process

(A_{t})

such that:

d X_{t} = μ X_{t} d t + σ X_{t} d B_{t}^{H} + A_{t} X_{t^{-}} d N_{t}, X_{0} = x_{0} > 0 .

The solution of this SDE is analyzed and properties of its trajectories are presented. Estimators of the model parameters are proposed when the observations are carried out in discrete time. Some convergence properties of these estimators are provided according to conditions concerning the value of the Hurst index and the nonequidistance of the observation dates. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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19 pages, 1776 KiB

Open AccessArticle

Inferences for Nadarajah–Haghighi Parameters via Type-II Adaptive Progressive Hybrid Censoring with Applications

by Ahmed Elshahhat, Refah Alotaibi and Mazen Nassar

Mathematics 2022, 10(20), 3775; https://doi.org/10.3390/math10203775 - 13 Oct 2022

Cited by 5 | Viewed by 872

Abstract

This study aims to investigate the estimation problems when the parent distribution of the population under consideration is the Nadarajah–Haghighi distribution in the presence of an adaptive progressive Type-II hybrid censoring scheme. Two approaches are considered in this regard, namely, the maximum likelihood [...] Read more.

This study aims to investigate the estimation problems when the parent distribution of the population under consideration is the Nadarajah–Haghighi distribution in the presence of an adaptive progressive Type-II hybrid censoring scheme. Two approaches are considered in this regard, namely, the maximum likelihood and Bayesian estimation methods. From the classical point of view, the maximum likelihood estimates of the unknown parameters, reliability, and hazard rate functions are obtained as well as the associated approximate confidence intervals. On the other hand, the Bayes estimates are obtained based on symmetric and asymmetric loss functions. The Bayes point estimates and the highest posterior density Bayes credible intervals are computed using the Monte Carlo Markov Chain technique. A comprehensive simulation study is implemented by proposing different scenarios for sample sizes and progressive censoring schemes. Moreover, two applications are considered by analyzing two real data sets. The outcomes of the numerical investigations show that the Bayes estimates using the general entropy loss function are preferred over the other methods. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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15 pages, 1954 KiB

Open AccessArticle

Coherent Forecasting for a Mixed Integer-Valued Time Series Model

by Wooi Chen Khoo, Seng Huat Ong and Biswas Atanu

Mathematics 2022, 10(16), 2961; https://doi.org/10.3390/math10162961 - 16 Aug 2022

Cited by 3 | Viewed by 1282

Abstract

In commerce, economics, engineering and the sciences, quantitative methods based on statistical models for forecasting are very useful tools for prediction and decision. There is an abundance of papers on forecasting for continuous-time series but relatively fewer papers for time series of counts [...] Read more.

In commerce, economics, engineering and the sciences, quantitative methods based on statistical models for forecasting are very useful tools for prediction and decision. There is an abundance of papers on forecasting for continuous-time series but relatively fewer papers for time series of counts which require special consideration due to the integer nature of the data. A popular method for modelling is the method of mixtures which is known for its flexibility and thus improved prediction capability. This paper studies the coherent forecasting for a flexible stationary mixture of Pegram and thinning (MPT) process, and develops the likelihood-based asymptotic distribution. Score functions and the Fisher information matrix are presented. Numerical studies are used to assess the performance of the forecasting methods. Also, a comparison is made with existing discrete-valued time series models. Finally, the practical application is illustrated with two sets of real data. It is shown that the mixture model provides good forecasting performance. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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21 pages, 922 KiB

Open AccessArticle

Parametric Quantile Regression Models for Fitting Double Bounded Response with Application to COVID-19 Mortality Rate Data

by Diego I. Gallardo, Marcelo Bourguignon, Yolanda M. Gómez, Christian Caamaño-Carrillo and Osvaldo Venegas

Mathematics 2022, 10(13), 2249; https://doi.org/10.3390/math10132249 - 27 Jun 2022

Cited by 2 | Viewed by 1361

Abstract

In this paper, we develop two fully parametric quantile regression models, based on the power Johnson

S_{B}

distribution for modeling unit interval response in different quantiles. In particular, the conditional distribution is modeled by the power Johnson

S_{B}

distribution. The maximum [...] Read more.

In this paper, we develop two fully parametric quantile regression models, based on the power Johnson

S_{B}

distribution for modeling unit interval response in different quantiles. In particular, the conditional distribution is modeled by the power Johnson

S_{B}

distribution. The maximum likelihood (ML) estimation method is employed to estimate the model parameters. Simulation studies are conducted to evaluate the performance of the ML estimators in finite samples. Furthermore, we discuss influence diagnostic tools and residuals. The effectiveness of our proposals is illustrated with a data set of the mortality rate of COVID-19 in different countries. The results of our models with this data set show the potential of using the new methodology. Thus, we conclude that the results are favorable to the use of proposed quantile regression models for fitting double bounded data. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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