Research

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14 pages, 940 KiB

Open AccessArticle

Ratio Test for Mean Changes in Time Series with Heavy-Tailed AR(p) Noise Based on Multiple Sampling Methods

by Tianming Xu and Yuesong Wei

Mathematics 2023, 11(18), 3988; https://doi.org/10.3390/math11183988 - 20 Sep 2023

Viewed by 681

This paper discusses the problem of the mean changes in time series with heavy-tailed AR(p) noise. Firstly, it proposes a modified ratio-type test statistic, and the results show that under the null hypothesis of no mean change, the asymptotic distribution of [...] Read more.

This paper discusses the problem of the mean changes in time series with heavy-tailed AR(p) noise. Firstly, it proposes a modified ratio-type test statistic, and the results show that under the null hypothesis of no mean change, the asymptotic distribution of the modified statistic is a functional of Lévy processes and the consistency under the alternative hypothesis is obtained. However, a heavy-tailed index exists in the asymptotic distribution and is difficult to estimate. This paper uses bootstrap sampling, jackknife sampling, and subsampling to approximate the distribution under the null hypothesis, and obtain more accurate critical values and empirical power. In addition, some results from a small simulation study and a practical example give an idea of the finite sample behavior of the proposed statistic. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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16 pages, 414 KiB

Open AccessArticle

Discriminating among Generalized Exponential, Weighted Exponential and Weibull Distributions

by Ruizheng Niu, Weizhong Tian and Yunchu Zhang

Mathematics 2023, 11(18), 3847; https://doi.org/10.3390/math11183847 - 08 Sep 2023

Viewed by 695

Abstract

In this paper, we consider the problem of discriminating among three different positively skewed lifetime distributions, namely the generalized exponential distribution, the weighted exponential distribution, and the Weibull distribution. All of these distributions have been used quite effectively to analyze positively skewed lifetime [...] Read more.

In this paper, we consider the problem of discriminating among three different positively skewed lifetime distributions, namely the generalized exponential distribution, the weighted exponential distribution, and the Weibull distribution. All of these distributions have been used quite effectively to analyze positively skewed lifetime data. We use the methods of the ratio of maximized likelihood, the minimum Kolmogorov distance, and the sequential probability ratio test to discriminate among these three distributions. The probability of correct selection is considered for each hypothesis based on several scenarios with Monte Carlo simulation. Real data applications are studied to illustrate the effectiveness of these proposed methods. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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30 pages, 661 KiB

Open AccessArticle

Statistical Inference on the Entropy Measures of Gamma Distribution under Progressive Censoring: EM and MCMC Algorithms

by Essam A. Ahmed, Mahmoud El-Morshedy, Laila A. Al-Essa and Mohamed S. Eliwa

Mathematics 2023, 11(10), 2298; https://doi.org/10.3390/math11102298 - 15 May 2023

Cited by 1 | Viewed by 1003

Abstract

Studying the ages of mobile phones is considered one of the most important things in the recent period in the field of shopping and modern technology. In this paper, we will consider that the ages of these phones follow a gamma distribution under [...] Read more.

Studying the ages of mobile phones is considered one of the most important things in the recent period in the field of shopping and modern technology. In this paper, we will consider that the ages of these phones follow a gamma distribution under progressive first-failure (PFF) censoring. All of the unknown parameters, as well as Shannon and Rényi entropies, were estimated for this distribution. The maximum likelihood (ML) approach was utilized to generate point estimates for the target parameters based on the considered censoring strategy. The asymptotic confidence intervals of the ML estimators (MLEs) of the targeted parameters were produced using the normal approximation to ML and log-transformed ML. We employed the delta method to approximate the variances of the Shannon and Rényi functions to obtain their asymptotic confidence intervals. Additionally, all parameter estimates utilized in this study were determined using the successful expectation–maximization (EM) method. The Metropolis–Hastings (MH) algorithm was applied to construct the Bayes estimators and related highest posterior density (HPD) credible intervals under various loss functions. Further, the proposed methodologies were contrasted using Monte Carlo simulations. Finally, the radio transceiver dataset was analyzed to substantiate our results. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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69 pages, 751 KiB

Open AccessArticle

Non-Parametric Conditional U-Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design

by Salim Bouzebda and Inass Soukarieh

Mathematics 2023, 11(1), 16; https://doi.org/10.3390/math11010016 - 20 Dec 2022

Cited by 15 | Viewed by 1344

Abstract

Stute presented the so-called conditional U-statistics generalizing the Nadaraya–Watson estimates of the regression function. Stute demonstrated their pointwise consistency and the asymptotic normality. In this paper, we extend the results to a more abstract setting. We develop an asymptotic theory of conditional [...] Read more.

Stute presented the so-called conditional U-statistics generalizing the Nadaraya–Watson estimates of the regression function. Stute demonstrated their pointwise consistency and the asymptotic normality. In this paper, we extend the results to a more abstract setting. We develop an asymptotic theory of conditional U-statistics for locally stationary random fields

{X_{s, A_{n}} : s in R_{n}}

observed at irregularly spaced locations in

R_{n} = {[0, A_{n}]}^{d}

as a subset of

R^{d}

. We employ a stochastic sampling scheme that may create irregularly spaced sampling sites in a flexible manner and includes both pure and mixed increasing domain frameworks. We specifically examine the rate of the strong uniform convergence and the weak convergence of conditional U-processes when the explicative variable is functional. We examine the weak convergence where the class of functions is either bounded or unbounded and satisfies specific moment conditions. These results are achieved under somewhat general structural conditions pertaining to the classes of functions and the underlying models. The theoretical results developed in this paper are (or will be) essential building blocks for several future breakthroughs in functional data analysis. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

21 pages, 738 KiB

Open AccessFeature PaperArticle

A Semiparametric Bayesian Joint Modelling of Skewed Longitudinal and Competing Risks Failure Time Data: With Application to Chronic Kidney Disease

by Melkamu Molla Ferede, Samuel Mwalili, Getachew Dagne, Simon Karanja, Workagegnehu Hailu, Mahmoud El-Morshedy and Afrah Al-Bossly

Mathematics 2022, 10(24), 4816; https://doi.org/10.3390/math10244816 - 18 Dec 2022

Cited by 1 | Viewed by 1387

Abstract

In clinical and epidemiological studies, when the time-to-event(s) and the longitudinal outcomes are associated, modelling them separately may give biased estimates. A joint modelling approach is required to obtain unbiased results and to evaluate their association. In the joint model, a subject may [...] Read more.

In clinical and epidemiological studies, when the time-to-event(s) and the longitudinal outcomes are associated, modelling them separately may give biased estimates. A joint modelling approach is required to obtain unbiased results and to evaluate their association. In the joint model, a subject may be exposed to more than one type of failure event (competing risks). Considering the competing event as an independent censoring of the time-to-event process may underestimate the true survival probability and give biased results. Within the joint model, longitudinal outcomes may have nonlinear (irregular) trajectories over time and exhibit skewness with heavy tails. Accordingly, fully parametric mixed-effect models may not be flexible enough to model this type of complex longitudinal data. In addition, assuming a Gaussian distribution for model errors may be too restrictive to adequately represent within-individual variations and may lack robustness against deviation from distributional assumptions. To simultaneously overcome these issues, in this paper, we presented semiparametric joint models for competing risks failure time and skewed-longitudinal data by using a smoothing spline approach and a multivariate skew-t distribution. We also considered different parameterization approaches in the formulation of joint models and used a Bayesian approach to make the statistical inference. We illustrated the proposed methods by analyzing real data on a chronic kidney disease. To evaluate the performance of the methods, we also carried out simulation studies. The results of both the application and simulation studies revealed that the joint modelling approach proposed in this study performed well when the semiparametric, random-effects parameterization, and skew-t distribution specifications were taken into account. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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12 pages, 289 KiB

Open AccessArticle

A Markov Chain Model for Approximating the Run Length Distributions of Poisson EWMA Charts under Linear Drifts

by Honghao Zhao, Huajun Tang, Chuan Pang and Huimin Jiang

Mathematics 2022, 10(24), 4786; https://doi.org/10.3390/math10244786 - 16 Dec 2022

Viewed by 1077

Abstract

In addition to monitoring the Poisson mean rate with step shifts, increasing attention has been given to monitoring Poisson processes subject to linear trends. The exponentially weighted moving average (EWMA) control chart has been widely implemented to monitor normal processes, but it lacks [...] Read more.

In addition to monitoring the Poisson mean rate with step shifts, increasing attention has been given to monitoring Poisson processes subject to linear trends. The exponentially weighted moving average (EWMA) control chart has been widely implemented to monitor normal processes, but it lacks investigation for detecting the Poisson mean change under a linear trend. In this paper, we analyze the performance of the EWMA chart by extending the Markov chain model from monitoring Poisson processes under a step shift to a Poisson process with linear drift. The results demonstrate that the proposed method is able to provide accurate average run length approximation, compared with the Monte Carlo simulation. Optimal design tables and sensitivity analysis are presented to facilitate the use of the EWMA chart in practice. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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22 pages, 1453 KiB

Open AccessArticle

Simultaneous Confidence Intervals for the Ratios of the Means of Zero-Inflated Gamma Distributions and Its Application

by Theerapong Kaewprasert, Sa-Aat Niwitpong and Suparat Niwitpong

Mathematics 2022, 10(24), 4724; https://doi.org/10.3390/math10244724 - 12 Dec 2022

Cited by 3 | Viewed by 925

Abstract

Heavy rain in September (the middle of the rainy season in Thailand) can cause unexpected events and natural disasters such as flooding in many areas of the country. Rainfall series that contain both zero and positive values belong to the zero-inflated gamma distribution, [...] Read more.

Heavy rain in September (the middle of the rainy season in Thailand) can cause unexpected events and natural disasters such as flooding in many areas of the country. Rainfall series that contain both zero and positive values belong to the zero-inflated gamma distribution, which combines the binomial and gamma distributions. Precipitation in various areas of a country can be estimated by using simultaneous confidence intervals (CIs) for the ratios of the means of multiple zero-inflated gamma populations. Herein, we propose six simultaneous CIs constructed using the fiducial generalized CI method, Bayesian and highest posterior density (HPD) interval methods based on the Jeffreys’rule or uniform prior, and method of variance estimates recovery (MOVER). The performances of the proposed simultaneous CI methods were evaluated using a Monte Carlo simulation in terms of the coverage probabilities and expected lengths. The results from a comparative simulation study show that the HPD interval based on the Jeffreys’rule prior performed the best in most cases, while in some situations, the fiducial generalized CI performed well. All of the methods were applied to estimate the simultaneous CIs for the ratios of the means of natural rainfall data from six regions in Thailand. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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22 pages, 334 KiB

Open AccessArticle

A Zero-and-One Inflated Cosine Geometric Distribution and Its Application

by Sunisa Junnumtuam, Sa-Aat Niwitpong and Suparat Niwitpong

Mathematics 2022, 10(21), 4012; https://doi.org/10.3390/math10214012 - 28 Oct 2022

Cited by 1 | Viewed by 1320

Abstract

Count data containing both excess zeros and ones occur in many fields, and the zero-and-one inflated distribution is suitable for analyzing them. Herein, we construct confidence intervals (CIs) for the parameters of the zero-and-one inflated cosine geometric (ZOICG) distribution constructed by using five [...] Read more.

Count data containing both excess zeros and ones occur in many fields, and the zero-and-one inflated distribution is suitable for analyzing them. Herein, we construct confidence intervals (CIs) for the parameters of the zero-and-one inflated cosine geometric (ZOICG) distribution constructed by using five methods: a Wald CI based on the maximum likelihood estimate, equal-tailed Bayesian CIs based on the uniform or Jeffreys prior, and the highest posterior density intervals based on the uniform or Jeffreys prior. Their efficiencies were compared in terms of their coverage probabilities and average lengths via a simulation study. The results show that the highest posterior density intervals based on the uniform prior performed the best in most cases. The number of new daily COVID-19-related deaths in Luxembourg in 2020 involving data with a high proportion of zeros and ones were analyzed. It was found that the ZOICG model was appropriate for this scenario. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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42 pages, 542 KiB

Open AccessArticle

Exchangeably Weighted Bootstraps of General Markov U-Process

by Inass Soukarieh and Salim Bouzebda

Mathematics 2022, 10(20), 3745; https://doi.org/10.3390/math10203745 - 12 Oct 2022

Cited by 14 | Viewed by 1213

Abstract

We explore an exchangeably weighted bootstrap of the general function-indexed empirical U-processes in the Markov setting, which is a natural higher-order generalization of the weighted bootstrap empirical processes. As a result of our findings, a considerable variety of bootstrap resampling strategies arise. [...] Read more.

We explore an exchangeably weighted bootstrap of the general function-indexed empirical U-processes in the Markov setting, which is a natural higher-order generalization of the weighted bootstrap empirical processes. As a result of our findings, a considerable variety of bootstrap resampling strategies arise. This paper aims to provide theoretical justifications for the exchangeably weighted bootstrap consistency in the Markov setup. General structural conditions on the classes of functions (possibly unbounded) and the underlying distributions are required to establish our results. This paper provides the first general theoretical study of the bootstrap of the empirical U-processes in the Markov setting. Potential applications include the symmetry test, Kendall’s tau and the test of independence. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

17 pages, 972 KiB

Open AccessArticle

An Extended Weibull Regression for Censored Data: Application for COVID-19 in Campinas, Brazil

by Gabriela M. Rodrigues, Edwin M. M. Ortega, Gauss M. Cordeiro and Roberto Vila

Mathematics 2022, 10(19), 3644; https://doi.org/10.3390/math10193644 - 05 Oct 2022

Cited by 4 | Viewed by 1585

Abstract

This work aims to study the factors that increase the risk of death of hospitalized patients diagnosed with COVID-19 through the odd log-logistic regression model for censored data with two systematic components, as well as provide new mathematical properties of this distribution. To [...] Read more.

This work aims to study the factors that increase the risk of death of hospitalized patients diagnosed with COVID-19 through the odd log-logistic regression model for censored data with two systematic components, as well as provide new mathematical properties of this distribution. To achieve this, a dataset of individuals residing in the city of Campinas (Brazil) was used and simulations were performed to investigate the accuracy of the maximum likelihood estimators in the proposed regression model. The provided properties, such as stochastic representation, identifiability, and moments, among others, can help future research since they provide important information about the distribution structure. The simulation results revealed the consistency of the estimates for different censoring percentages and show that the empirical distribution of the modified deviance residuals converge to the standard normal distribution. The proposed model proved to be efficient in identifying the determinant variables for the survival of the individuals in this study, which can help to find more opportune treatments and medical interventions. Therefore, the new model can be considered an interesting alternative for future works that evaluate censored lifetimes. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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29 pages, 14256 KiB

Open AccessArticle

A Discrete Exponential Generalized-G Family of Distributions: Properties with Bayesian and Non-Bayesian Estimators to Model Medical, Engineering and Agriculture Data

by Mohamed S. Eliwa, Mahmoud El-Morshedy and Haitham M. Yousof

Mathematics 2022, 10(18), 3348; https://doi.org/10.3390/math10183348 - 15 Sep 2022

Cited by 5 | Viewed by 1226

Abstract

This paper introduces a new flexible probability tool for modeling extreme and zero-inflated count data under different shapes of hazard rates. Many relevant mathematical and statistical properties are derived and analyzed. The new tool can be used to discuss several kinds of data, [...] Read more.

This paper introduces a new flexible probability tool for modeling extreme and zero-inflated count data under different shapes of hazard rates. Many relevant mathematical and statistical properties are derived and analyzed. The new tool can be used to discuss several kinds of data, such as “asymmetric and left skewed”, “asymmetric and right skewed”, “symmetric”, “symmetric and bimodal”, “uniformed”, and “right skewed with a heavy tail”, among other useful shapes. The failure rate of the new class can vary and can take the forms of “increasing-constant”, “constant”, “monotonically dropping”, “bathtub”, “monotonically increasing”, or “J-shaped”. Eight classical estimation techniques—including Cramér–von Mises, ordinary least squares, L-moments, maximum likelihood, Kolmogorov, bootstrapping, and weighted least squares—are considered, described, and applied. Additionally, Bayesian estimation under the squared error loss function is also derived and discussed. Comprehensive comparison between approaches is performed for both simulated and real-life data. Finally, four real datasets are analyzed to prove the flexibility, applicability, and notability of the new class. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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24 pages, 558 KiB

Open AccessArticle

New Class of Unit-Power-Skew-Normal Distribution and Its Associated Regression Model for Bounded Responses

by Guillermo Martínez-Flórez, Rafael B. Azevedo-Farias and Roger Tovar-Falón

Mathematics 2022, 10(17), 3035; https://doi.org/10.3390/math10173035 - 23 Aug 2022

Cited by 2 | Viewed by 999

Abstract

Several papers on distributions to model rates and proportions have been recently published; their fitting in numerous instances is better than the alternative beta distribution, which has been the distribution to follow when it is necessary to quantify the average of a response [...] Read more.

Several papers on distributions to model rates and proportions have been recently published; their fitting in numerous instances is better than the alternative beta distribution, which has been the distribution to follow when it is necessary to quantify the average of a response variable based on a set of covariates. Despite the great usefulness of this distribution to fit the responses on the

(0, 1)

unit interval, its relevance loses objectivity when the interest is quantifying the influence of these covariates on the quantiles of the variable response in

(0, 1)

; being the most critical situation when the distribution presents high asymmetry and/or kurtosis. The main objective of this work is to introduce a distribution for modeling rates and proportions. The introduced distribution is obtained from the alpha-power extension of the skew–normal distribution, which is known in the literature as the power–skew–normal distribution. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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Review

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23 pages, 1996 KiB

Open AccessReview

Parametric Distributions for Survival and Reliability Analyses, a Review and Historical Sketch

by Nanami Taketomi, Kazuki Yamamoto, Christophe Chesneau and Takeshi Emura

Mathematics 2022, 10(20), 3907; https://doi.org/10.3390/math10203907 - 21 Oct 2022

Cited by 17 | Viewed by 3693

Abstract

During its 330 years of history, parametric distributions have been useful for survival and reliability analyses. In this paper, we comprehensively review the historical backgrounds and statistical properties of a number of parametric distributions used in survival and reliability analyses. We provide encyclopedic [...] Read more.

During its 330 years of history, parametric distributions have been useful for survival and reliability analyses. In this paper, we comprehensively review the historical backgrounds and statistical properties of a number of parametric distributions used in survival and reliability analyses. We provide encyclopedic coverage of the important parametric distributions, which is more extensive than the existing textbooks on survival and reliability analyses. We also explain how these distributions have been adopted in survival and reliability analyses with original and state-of-the-art references. We cover the exponential, Weibull, Rayleigh, lognormal, log-logistic, gamma, generalized gamma, Pareto (types I, II, and IV), Hjorth, Burr (types III and XII), Dagum, exponential power, Gompertz, Birnbaum-Saunders, exponential-logarithmic, piecewise exponential, generalized exponential, exponentiated Weibull, generalized modified Weibull, and spline distributions. We analyze a real dataset for illustration. Full article

(This article belongs to the Special Issue Current Developments in Theoretical and Applied Statistics)

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