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17 pages, 844 KiB

Open AccessArticle

Cumulative Histograms under Uncertainty: An Application to Dose–Volume Histograms in Radiotherapy Treatment Planning

by Flavia Gesualdi and Niklas Wahl

Stats 2024, 7(1), 284-300; https://doi.org/10.3390/stats7010017 - 6 Mar 2024

Viewed by 1875

Abstract

In radiotherapy treatment planning, the absorbed doses are subject to executional and preparational errors, which propagate to plan quality metrics. Accurately quantifying these uncertainties is imperative for improved treatment outcomes. One approach, analytical probabilistic modeling (APM), presents a highly computationally efficient method. This [...] Read more.

In radiotherapy treatment planning, the absorbed doses are subject to executional and preparational errors, which propagate to plan quality metrics. Accurately quantifying these uncertainties is imperative for improved treatment outcomes. One approach, analytical probabilistic modeling (APM), presents a highly computationally efficient method. This study evaluates the empirical distribution of dose–volume histogram points (a typical plan metric) derived from Monte Carlo sampling to quantify the accuracy of modeling uncertainties under different distribution assumptions, including Gaussian, log-normal, four-parameter beta, gamma, and Gumbel distributions. Since APM necessitates the bivariate cumulative distribution functions, this investigation also delves into approximations using a Gaussian or an Ali–Mikhail–Haq Copula. The evaluations are performed in a one-dimensional simulated geometry and on patient data for a lung case. Our findings suggest that employing a beta distribution offers improved modeling accuracy compared to a normal distribution. Moreover, the multivariate Gaussian model outperforms the Copula models in patient data. This investigation highlights the significance of appropriate statistical distribution selection in advancing the accuracy of uncertainty modeling in radiotherapy treatment planning, extending an understanding of the analytical probabilistic modeling capacities in this crucial medical domain. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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13 pages, 533 KiB

Open AccessArticle

Importance and Uncertainty of λ-Estimation for Box–Cox Transformations to Compute and Verify Reference Intervals in Laboratory Medicine

by Frank Klawonn, Neele Riekeberg and Georg Hoffmann

Stats 2024, 7(1), 172-184; https://doi.org/10.3390/stats7010011 - 9 Feb 2024

Cited by 3 | Viewed by 2097

Abstract

Reference intervals play an important role in medicine, for instance, for the interpretation of blood test results. They are defined as the central 95% values of a healthy population and are often stratified by sex and age. In recent years, so-called indirect methods [...] Read more.

Reference intervals play an important role in medicine, for instance, for the interpretation of blood test results. They are defined as the central 95% values of a healthy population and are often stratified by sex and age. In recent years, so-called indirect methods for the computation and validation of reference intervals have gained importance. Indirect methods use all values from a laboratory, including the pathological cases, and try to identify the healthy sub-population in the mixture of values. This is only possible under certain model assumptions, i.e., that the majority of the values represent non-pathological values and that the non-pathological values follow a normal distribution after a suitable transformation, commonly a Box–Cox transformation, rendering the parameter

λ

of the Box–Cox transformation as a nuisance parameter for the estimation of the reference interval. Although indirect methods put high effort on the estimation of

λ

, they come to very different estimates for

λ

, even though the estimated reference intervals are quite coherent. Our theoretical considerations and Monte-Carlo simulations show that overestimating

λ

can lead to intolerable deviations of the reference interval estimates, whereas

λ = 0

produces usually acceptable estimates. For

λ

close to 1, its estimate has limited influence on the estimate for the reference interval, and with reasonable sample sizes, the uncertainty for the

λ

-estimate remains quite high. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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27 pages, 450 KiB

Open AccessArticle

Process Monitoring Using Truncated Gamma Distribution

by Sajid Ali, Shayaan Rajput, Ismail Shah and Hassan Houmani

Stats 2023, 6(4), 1298-1322; https://doi.org/10.3390/stats6040080 - 1 Dec 2023

Cited by 3 | Viewed by 1879

Abstract

The time-between-events idea is commonly used for monitoring high-quality processes. This study aims to monitor the increase and/or decrease in the process mean rapidly using a one-sided exponentially weighted moving average (EWMA) chart for the detection of upward or downward mean shifts using [...] Read more.

The time-between-events idea is commonly used for monitoring high-quality processes. This study aims to monitor the increase and/or decrease in the process mean rapidly using a one-sided exponentially weighted moving average (EWMA) chart for the detection of upward or downward mean shifts using a truncated gamma distribution. The use of the truncation method helps to enhance and improve the sensitivity of the proposed chart. The performance of the proposed chart with known and estimated parameters is analyzed by using the run length properties, including the average run length (ARL) and standard deviation run length (SDRL), through extensive Monte Carlo simulation. The numerical results show that the proposed scheme is more sensitive than the existing ones. Finally, the chart is implemented in real-world situations to highlight the significance of the proposed chart. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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18 pages, 609 KiB

Open AccessArticle

Social Response and Measles Dynamics

by Atinuke O. Adebanji, Franz Aschl, Ednah Chepkemoi Chumo, Emmanuel Odame Owiredu, Johannes Müller and Tukae Mbegalo

Stats 2023, 6(4), 1280-1297; https://doi.org/10.3390/stats6040079 - 29 Nov 2023

Viewed by 2179

Abstract

Measles remains one of the leading causes of death among young children globally, even though a safe and cost-effective vaccine is available. Vaccine hesitancy and social response to vaccination continue to undermine efforts to eradicate measles. In this study, we consider data about [...] Read more.

Measles remains one of the leading causes of death among young children globally, even though a safe and cost-effective vaccine is available. Vaccine hesitancy and social response to vaccination continue to undermine efforts to eradicate measles. In this study, we consider data about measles vaccination and measles prevalence in Germany for the years 2008–2012 in 345 districts. In the first part of the paper, we show that the probability of a local outbreak does not significantly depend on the vaccination coverage, but—if an outbreak does take place—the scale of the outbreak depends significantly on the vaccination coverage. Additionally, we show that the willingness to be vaccinated is significantly increased by local outbreaks, with a delay of about one year. In the second part of the paper, we consider a deterministic delay model to investigate the consequences of the statistical findings on the dynamics of the infection. Here, we find that the delay might induce oscillations if the vaccination coverage is rather low and the social response to an outbreak is sufficiently strong. The relevance of our findings is discussed at the end of the paper. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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19 pages, 366 KiB

Open AccessArticle

Adjustment of Anticipatory Covariates in Retrospective Surveys: An Expected Likelihood Approach

by Gebrenegus Ghilagaber and Rolf Larsson

Stats 2023, 6(4), 1179-1197; https://doi.org/10.3390/stats6040074 - 1 Nov 2023

Viewed by 1683

Abstract

We address an inference issue where the value of a covariate is measured at the date of the survey but is used to explain behavior that has occurred long before the survey. This causes bias because the value of the covariate does not [...] Read more.

We address an inference issue where the value of a covariate is measured at the date of the survey but is used to explain behavior that has occurred long before the survey. This causes bias because the value of the covariate does not follow the temporal order of events. We propose an expected likelihood approach to adjust for such bias and illustrate it with data on the effects of educational level achieved by the time of marriage on risks of divorce. For individuals with anticipatory educational level (whose reported educational level was completed after marriage), conditional probabilities of having attained the reported level before marriage are computed. These are then used as weights in the expected likelihood to obtain adjusted estimates of relative risks. For our illustrative data set, the adjusted estimates of relative risks of divorce did not differ significantly from those obtained from anticipatory analysis that ignores the temporal order of events. Our results are slightly different from those in two other studies that analyzed the same data set in a Bayesian framework, though the studies are not fully comparable to each other. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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21 pages, 407 KiB

Open AccessArticle

The Semi-Hyperbolic Distribution and Its Applications

by Roman V. Ivanov

Stats 2023, 6(4), 1126-1146; https://doi.org/10.3390/stats6040071 - 21 Oct 2023

Cited by 2 | Viewed by 2013

Abstract

This paper studies a subclass of the class of generalized hyperbolic distribution called the semi-hyperbolic distribution. We obtain analytical expressions for the cumulative distribution function and, specifically, their first and second lower partial moments. Using the received formulas, we compute the value at [...] Read more.

This paper studies a subclass of the class of generalized hyperbolic distribution called the semi-hyperbolic distribution. We obtain analytical expressions for the cumulative distribution function and, specifically, their first and second lower partial moments. Using the received formulas, we compute the value at risk, the expected shortfall, and the semivariance in the semi-hyperbolic model of the financial market. The formulas depend on the values of generalized hypergeometric functions and modified Bessel functions of the second kind. The research illustrates the possibility of analysis of generalized hyperbolic models using the same methodology as is employed for the well-established variance-gamma model. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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14 pages, 599 KiB

Open AccessArticle

A Family of Finite Mixture Distributions for Modelling Dispersion in Count Data

by Seng Huat Ong, Shin Zhu Sim, Shuangzhe Liu and Hari M. Srivastava

Stats 2023, 6(3), 942-955; https://doi.org/10.3390/stats6030059 - 18 Sep 2023

Cited by 1 | Viewed by 1742

Abstract

This paper considers the construction of a family of discrete distributions with the flexibility to cater for under-, equi- and over-dispersion in count data using a finite mixture model based on standard distributions. We are motivated to introduce this family because its simple [...] Read more.

This paper considers the construction of a family of discrete distributions with the flexibility to cater for under-, equi- and over-dispersion in count data using a finite mixture model based on standard distributions. We are motivated to introduce this family because its simple finite mixture structure adds flexibility and facilitates application and use in analysis. The family of distributions is exemplified using a mixture of negative binomial and shifted negative binomial distributions. Some basic and probabilistic properties are derived. We perform hypothesis testing for equi-dispersion and simulation studies of their power and consider parameter estimation via maximum likelihood and probability-generating-function-based methods. The utility of the distributions is illustrated via their application to real biological data sets exhibiting under-, equi- and over-dispersion. It is shown that the distribution fits better than the well-known generalized Poisson and COM–Poisson distributions for handling under-, equi- and over-dispersion in count data. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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7 pages, 243 KiB

Open AccessCommunication

Some More Results on Characterization of the Exponential and Related Distributions

by Lev B. Klebanov

Stats 2023, 6(3), 740-746; https://doi.org/10.3390/stats6030047 - 29 Jun 2023

Viewed by 1132

Abstract

There are given characterizations of the exponential distribution based on the properties of independence of linear forms with random coefficients. Results based on the constancy of regression of one statistic in a linear form are obtained. Related characterizations based on the property of [...] Read more.

There are given characterizations of the exponential distribution based on the properties of independence of linear forms with random coefficients. Results based on the constancy of regression of one statistic in a linear form are obtained. Related characterizations based on the property of the identical distribution of statistics are also provided. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

17 pages, 475 KiB

Open AccessArticle

Modeling Model Misspecification in Structural Equation Models

by Alexander Robitzsch

Stats 2023, 6(2), 689-705; https://doi.org/10.3390/stats6020044 - 14 Jun 2023

Cited by 3 | Viewed by 2414

Abstract

Structural equation models constrain mean vectors and covariance matrices and are frequently applied in the social sciences. Frequently, the structural equation model is misspecified to some extent. In many cases, researchers nevertheless intend to work with a misspecified target model of interest. In [...] Read more.

Structural equation models constrain mean vectors and covariance matrices and are frequently applied in the social sciences. Frequently, the structural equation model is misspecified to some extent. In many cases, researchers nevertheless intend to work with a misspecified target model of interest. In this article, a simultaneous statistical inference for sampling errors and model misspecification errors is discussed. A modified formula for the variance matrix of the parameter estimate is obtained by imposing a stochastic model for model errors and applying M-estimation theory. The presence of model errors is quantified in increased standard errors in parameter estimates. The proposed inference is illustrated with several analytical examples and an empirical application. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

9 pages, 272 KiB

Open AccessArticle

Combining Probability and Nonprobability Samples by Using Multivariate Mass Imputation Approaches with Application to Biomedical Research

by Sixia Chen, Alexandra May Woodruff, Janis Campbell, Sara Vesely, Zheng Xu and Cuyler Snider

Stats 2023, 6(2), 617-625; https://doi.org/10.3390/stats6020039 - 8 May 2023

Cited by 2 | Viewed by 3218

Abstract

Nonprobability samples have been used frequently in practice including public health study, economics, education, and political polls. Naïve estimates based on nonprobability samples without any further adjustments may suffer from serious selection bias. Mass imputation has been shown to be effective in practice [...] Read more.

Nonprobability samples have been used frequently in practice including public health study, economics, education, and political polls. Naïve estimates based on nonprobability samples without any further adjustments may suffer from serious selection bias. Mass imputation has been shown to be effective in practice to improve the representativeness of nonprobability samples. It builds an imputation model based on nonprobability samples and generates imputed values for all units in the probability samples. In this paper, we compare two mass imputation approaches including latent joint multivariate normal model mass imputation (e.g., Generalized Efficient Regression-Based Imputation with Latent Processes (GERBIL)) and fully conditional specification (FCS) procedures for integrating multiple outcome variables simultaneously. The Monte Carlo simulation study shows the benefits of GERBIL and FCS with predictive mean matching in terms of balancing the Monte Carlo bias and variance. We further evaluate our proposed method by combining the information from Tribal Behavioral Risk Factor Surveillance System and Behavioral Risk Factor Surveillance System data files. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

11 pages, 303 KiB

Open AccessArticle

Model Selection with Missing Data Embedded in Missing-at-Random Data

by Keiji Takai and Kenichi Hayashi

Stats 2023, 6(2), 495-505; https://doi.org/10.3390/stats6020031 - 11 Apr 2023

Cited by 2 | Viewed by 2022

Abstract

When models are built with missing data, an information criterion is needed to select the best model among the various candidates. Using a conventional information criterion for missing data may lead to the selection of the wrong model when data are not missing [...] Read more.

When models are built with missing data, an information criterion is needed to select the best model among the various candidates. Using a conventional information criterion for missing data may lead to the selection of the wrong model when data are not missing at random. Conventional information criteria implicitly assume that any subset of missing-at-random data is also missing at random, and thus the maximum likelihood estimator is assumed to be consistent; that is, it is assumed that the estimator will converge to the true value. However, this assumption may not be practical. In this paper, we develop an information criterion that works even for not-missing-at-random data, so long as the largest missing data set is missing at random. Simulations are performed to show the superiority of the proposed information criterion over conventional criteria. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

12 pages, 3114 KiB

Open AccessArticle

Consecutive-k₁ and k₂-out-of-n: F Structures with a Single Change Point

by Ioannis S. Triantafyllou and Miltiadis Chalikias

Stats 2023, 6(1), 438-449; https://doi.org/10.3390/stats6010027 - 16 Mar 2023

Viewed by 1988

Abstract

In the present paper, we establish a new consecutive-type reliability model with a single change point. The proposed structure has two common failure criteria and consists of two different types of components. The general framework for constructing the so-called consecutive-k₁ and [...] Read more.

In the present paper, we establish a new consecutive-type reliability model with a single change point. The proposed structure has two common failure criteria and consists of two different types of components. The general framework for constructing the so-called consecutive-k₁ and k₂-out-of-n: F system with a single change point is launched. In addition, the number of path sets of the proposed structure is determined with the aid of a combinatorial approach. Moreover, two crucial performance characteristics of the proposed model are studied. The numerical investigation carried out reveals that the behavior of the new structure is outperforming against its competitors. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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16 pages, 350 KiB

Open AccessArticle

On Weak Convergence of the Bootstrap Copula Empirical Process with Random Resample Size

by Salim Bouzebda

Stats 2023, 6(1), 365-380; https://doi.org/10.3390/stats6010023 - 22 Feb 2023

Viewed by 1598

Abstract

The purpose of this note is to provide a description of the weak convergence of the random resample size bootstrap empirical process. The principal results are used to estimate the sample rank correlation coefficients using Spearman’s and Kendall’s respective methods. In addition to [...] Read more.

The purpose of this note is to provide a description of the weak convergence of the random resample size bootstrap empirical process. The principal results are used to estimate the sample rank correlation coefficients using Spearman’s and Kendall’s respective methods. In addition to this, we discuss how our findings can be applied to statistical testing. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

10 pages, 281 KiB

Open AccessArticle

Panel Data Models for School Evaluation: The Case of High Schools’ Results in University Entrance Examinations

by Manuel Salas-Velasco

Stats 2023, 6(1), 312-321; https://doi.org/10.3390/stats6010019 - 13 Feb 2023

Viewed by 2528

Abstract

To what extent do high school students’ course grades align with their scores on standardized college admission tests? People sometimes make the argument that grades are “inflated”, but many school districts only use outcome-based descriptive methods for school evaluation. In order to answer [...] Read more.

To what extent do high school students’ course grades align with their scores on standardized college admission tests? People sometimes make the argument that grades are “inflated”, but many school districts only use outcome-based descriptive methods for school evaluation. In order to answer that question, this paper proposes econometric models for panel data, which are less well-known in educational evaluation. In particular, fixed-effects and random-effects models are proposed for assessing student performance in university entrance examinations. School-level panel data analysis allows one knowing if results in college admission tests vary more between high schools than within a high school in different academic years. Another advantage of using panel data includes the ability to control for school-specific unobserved heterogeneity. For empirical implementation, official transcript data and university entrance test scores of Spanish secondary schools are used. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

14 pages, 494 KiB

Open AccessArticle

A Class of Enhanced Nonparametric Control Schemes Based on Order Statistics and Runs

by Nikolaos I. Panayiotou and Ioannis S. Triantafyllou

Stats 2023, 6(1), 279-292; https://doi.org/10.3390/stats6010017 - 8 Feb 2023

Cited by 3 | Viewed by 1575

Abstract

In this article, we establish a new class of nonparametric Shewhart-type control charts based on order statistics with signaling runs-type rules. The proposed charts offer to the practitioner the opportunity to reach, as close as possible, a pre-specified level of performance by determining [...] Read more.

In this article, we establish a new class of nonparametric Shewhart-type control charts based on order statistics with signaling runs-type rules. The proposed charts offer to the practitioner the opportunity to reach, as close as possible, a pre-specified level of performance by determining appropriately their design parameters. Special monitoring schemes, already established in the literature, are ascertained to be members of the proposed class. In addition, several new nonparametric control charts that belong to the family are introduced and studied in some detail. Exact formulae for the variance of the run length distribution and the average run length (ARL) for the proposed monitoring schemes are also derived. A numerical investigation is carried out and demonstrates that the proposed schemes acquire competitive performance in detecting the shift of the underlying distribution. Although the large number of design parameters is quite hard to handle, the numerical results presented throughout the lines of the present manuscript provide practical guidance for the implementation of the proposed charts. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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21 pages, 5159 KiB

Open AccessArticle

A New Class of Alternative Bivariate Kumaraswamy-Type Models: Properties and Applications

by Indranil Ghosh

Stats 2023, 6(1), 232-252; https://doi.org/10.3390/stats6010014 - 30 Jan 2023

Cited by 1 | Viewed by 1758

Abstract

In this article, we introduce two new bivariate Kumaraswamy (KW)-type distributions with univariate Kumaraswamy marginals (under certain parametric restrictions) that are less restrictive in nature compared with several other existing bivariate beta and beta-type distributions. Mathematical expressions for the joint and marginal density [...] Read more.

In this article, we introduce two new bivariate Kumaraswamy (KW)-type distributions with univariate Kumaraswamy marginals (under certain parametric restrictions) that are less restrictive in nature compared with several other existing bivariate beta and beta-type distributions. Mathematical expressions for the joint and marginal density functions are presented, and properties such as the marginal and conditional distributions, product moments and conditional moments are obtained. Additionally, we show that both the proposed bivariate probability models have positive likelihood ratios dependent on a potential model for fitting positively dependent data in the bivariate domain. The method of maximum likelihood and the method of moments are used to derive the associated estimation procedure. An acceptance and rejection sampling plan to draw random samples from one of the proposed models along with a simulation study are also provided. For illustrative purposes, two real data sets are reanalyzed from different domains to exhibit the applicability of the proposed models in comparison with several other bivariate probability distributions, which are defined on

[0, 1] \times [0, 1] .

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(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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20 pages, 373 KiB

Open AccessArticle

Estimating Smoothness and Optimal Bandwidth for Probability Density Functions

by Dimitris N. Politis, Peter F. Tarassenko and Vyacheslav A. Vasiliev

Stats 2023, 6(1), 30-49; https://doi.org/10.3390/stats6010003 - 27 Dec 2022

Viewed by 1885

Abstract

The properties of non-parametric kernel estimators for probability density function from two special classes are investigated. Each class is parametrized with distribution smoothness parameter. One of the classes was introduced by Rosenblatt, another one is introduced in this paper. For the case of [...] Read more.

The properties of non-parametric kernel estimators for probability density function from two special classes are investigated. Each class is parametrized with distribution smoothness parameter. One of the classes was introduced by Rosenblatt, another one is introduced in this paper. For the case of the known smoothness parameter, the rates of mean square convergence of optimal (on the bandwidth) density estimators are found. For the case of unknown smoothness parameter, the estimation procedure of the parameter is developed and almost surely convergency is proved. The convergence rates in the almost sure sense of these estimators are obtained. Adaptive estimators of densities from the given class on the basis of the constructed smoothness parameter estimators are presented. It is shown in examples how parameters of the adaptive density estimation procedures can be chosen. Non-asymptotic and asymptotic properties of these estimators are investigated. Specifically, the upper bounds for the mean square error of the adaptive density estimators for a fixed sample size are found and their strong consistency is proved. The convergence of these estimators in the almost sure sense is established. Simulation results illustrate the realization of the asymptotic behavior when the sample size grows large. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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10 pages, 794 KiB

Open AccessBrief Report

On the Vector Representation of Characteristic Functions

by Wolf-Dieter Richter

Stats 2023, 6(4), 1072-1081; https://doi.org/10.3390/stats6040067 - 10 Oct 2023

Cited by 1 | Viewed by 1644

Abstract

Based upon the vector representation of complex numbers and the vector exponential function, we introduce the vector representation of characteristic functions and consider some of its elementary properties such as its polar representation and a vector power expansion. Full article

(This article belongs to the Special Issue Advances in Probability Theory and Statistics)

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