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Journal of the Indian Society for Probability and Statistics
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ISSN (Online) 2364-9569
Published by Springer Publishing Company

[24 journals]

A Review of Nonparametric Research on Cumulative Distribution Function
Estimation
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  Abstract: Abstract This paper intends to review non-parametric studies on estimating the cumulative distribution function (CDF) of a random variable. Research in this field has mostly utilized kernel-type methods. Generally, studies on estimating CDF by kernel methods can be grouped into three categories, including the studies on the conditions and convergence rate of the estimator, studies on how to select the estimator’s smoothing parameter, and studies on boundary problem-solving. We also consider the CDF estimation for the multivariate and the conditional cases as well as CDF for time series and dependent data. Our approach, however, is not merely a review of these studies. Whenever possible, we have analyzed and compared the strengths and weaknesses of various methods proposed by the researchers.
  PubDate: 2024-07-23
The M/M/S/M+ Queuing System with Clients Abandonment
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  Abstract: Abstract This paper develops a comprehensive analytical model for the M/M/S/M+ queueing system with customer abandonment. The transition rate diagram and associated transition probabilities are derived to characterize the system dynamics. Key performance metrics are obtained, including the mean number of customers in the system, the mean waiting time in the system, the probability of an empty system, the probability of finding all servers busy, and the abandonment probability. Specific focus is given to characterizing the abandonment behavior, deriving expressions for the mean abandonment time and the mean number of abandoned customers. The mathematical formulation provides insights into the impacts of arrival rates, service rates, customer impatience levels and system capacity on performance. Potential applications span various domains where customer abandonment significantly affects system design and resource provisioning, such as call centers, healthcare facilities, transportation services and e-commerce platforms. The analytical queueing model enables quantifying fundamental tradeoffs between service quality metrics like customer waiting times and the potential revenue losses from abandonments. The results can guide capacity planning, staffing optimization, and decision support for design and control of queueing systems with impatient customers.
  PubDate: 2024-07-18
Two-Part Mixed Effects Mixture Model for Zero-Inflated Longitudinal
Compositional Data
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  Abstract: Abstract Compositional data (CD) is mostly analyzed using ratios of components and log-ratio transformations to apply known multivariable statistical methods. Therefore, CD where some components equal zero represents a problem. Furthermore, when the data is measured longitudinally, and appear to come from different sub-populations, the analysis becomes highly complex. Our objective is to build a statistical model addressing structural zeros in longitudinal CD and apply it to the analysis of radiation-induced lung damage (RILD) over time. We propose a two-part mixed-effects model extended to the case where the non-zero components of the vector might come from a two-component mixture population. Maximum likelihood estimates for fixed effects and variance components were calculated by an approximate Fisher scoring procedure base on sixth-order Laplace approximation. The expectation-maximization (EM) algorithm estimates the mixture model’s probability. This model was used to analyze the radiation therapy effect on tissue change in one patient with non-small cell lung cancer (NSCLC), utilizing five CT scans over 24 months. Instead of using voxel-level data, voxels were grouped into larger subvolumes called patches. Each patch’s data is a CD vector showing proportions of dense, hazy, or normal tissue. Proposed method performed reasonably for estimation of the fixed effects, and their variability. However, the model produced biased estimates of the nuisance parameters in the model.
  PubDate: 2024-07-03
A New Regression Model for Over-Dispersed Count Responses Based on Poisson
and Geometric Convolution
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  Abstract: Abstract This article presents an alternative generalized linear regression model specifically designed for count responses that exhibit over-dispersion. The recently developed PoiG distribution features a closed-form expression of the mean unlike many over-dispersed count data models such as the popular COM-Poisson (CMP) distribution. A reparametrized version of the PoiG distribution is proposed in the current work to demonstrate its flexible properties in modelling over-dispersed counts with covariates. The parameters of the proposed regression model are estimated using the method of maximum likelihood estimation and the respective confidence intervals are computed using bootstrap routine. Three benchmark real-world datasets are used to demonstrate the application of the proposed modelling approach. The proposed model is found to be more suitable for modelling over-dispersed count data compared to its closest competitors.
  PubDate: 2024-07-02
On Average Modulus of Random Polynomials over a Unit Circle and Disc
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  Abstract: Abstract This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new results concerning the bounds of the maximum modulus of random polynomials with coefficients distributed as independently as Gaussian and uniform variates, utilizing probability principles to derive findings about the likelihood of the maximum modulus exceeding a specific threshold, using Markov inequality as the primary probabilistic tool. These findings and the approach can potentially initiate the study of a rich class of problems concerning the norms of random polynomials.
  PubDate: 2024-06-24
Parameter Estimation for Some Discretely Observed Class of Stable Driven
Stochastic Differential Equations
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  Abstract: Abstract In this paper, we consider the problem of parameter estimation for a real stochastic model observed at some discrete times, that is a solution of a stochastic differential equation driven by \(\alpha\) -stable processes, \(\alpha \in (1,2)\) . After recalling the non-parametric estimation framework of the drift function namely the Nadaraya-Watson estimation, we provide explicit estimators for the diffusion parameters (the scaling and the driving stable process parameters) based on the Euler-Maruyama scheme. We apply the estimation results to stable driven Ornstein-Uhlenbeck (OU), Cox-Ingersoll-Ross (CIR) and Lotka-Volterra processes. We also consider the estimation of the drift coefficients in the linear case namely, the stable driven OU and CIR processes. The novelty of this paper which is our baseline is the combination of a characteristic sample function method, the least squares or linear statistical regression methods and the Itô formula. We also established under certain conditions, the consistency of their drift coefficient estimators of the stable driven OU and CIR processes. We efficiently discuss our result with numerical simulations using synthetic data. A real data in finance, such as exchange rates is used to fit the parameters of a justified model among the above stable driven processes. As a forthcoming work, we intend to study the rate of convergence of the estimators and to create a package on R software to handle this kind of estimation problem. We are also currently interested in ergodicity properties for a class of stochastic differential equations driven by stable processes.
  PubDate: 2024-06-17
Predictive Root Based Bootstrap Prediction Intervals in Neural Network
Models for Time Series Forecasting
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  Abstract: Abstract Time series (TS) modelling is an important area in the domain of statistics, as it enables us to comprehend the dynamics underlying a particular phenomenon. In the spectrum of non-linear TS data analysis, neural network (NN) models are one of the dominant methods due to their several advantages over statistical methods. However, NN models are unable to provide prediction intervals (PIs) which is an important part of forecasting to capture uncertainties. The predictive root concept earlier used by researchers for both linear and non-linear autoregression models has been extended to ANN models for constructing PIs. Two bootstrap approaches (with and without rescaling) for constructing PIs in ANN models for non-linear TS have been proposed. The performances of the proposed methods have also been evaluated by comparing them with the existing methods using both simulated and real datasets. The proposed methods can be considered as a viable alternative for computing PIs in TS.
  PubDate: 2024-06-15
On Estimation of Stress-Strength Reliability with Zero-Inflated Poisson
Distribution
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  Abstract: Abstract Many real-world phenomena generate count data with inflated number of zeroes. To model such datasets, the zero-inflated Poisson model has been used quite extensively and effectively. Examples of such datasets include the number of defects in manufacturing, the number of visits to a specific location and the number of times suffered a major health issue. Besides the engineering applications of the stress-strength reliability, its use as an index of stochastic comparison is well established. Stress-strength reliability studies for the power series distributions such as the binomial, the Poisson, the geometric and the negative binomial are recent additions in the literature. Realizing their shortcomings in modelling inflated count datasets, we consider two distinct independent marginal zero-inflated Poisson distributions, derive the structure of the corresponding stress-strength reliability parameter and study its nature. Then the stress-strength reliability is estimated in frequentist as well as in the Bayesian approach. Stan, a probabilistic programming language for Bayesian inference, is used for the computation of the Bayes’ estimate. Both the approaches of estimation have almost equivalent performance in terms of bias and mean squared error. Simulation experiments are performed to assess the performance of the estimators. We also present two important real-life applications for demonstrating the utility of the proposed estimators. The applications involve benchmark zero-inflated count datasets related to railway accidents and fishing by groups of people on a camping trip.
  PubDate: 2024-06-05
Survival Function Estimation for Multiple Sequential Time-to-Events
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  Abstract: Abstract Data on multiple time-to-events occurring in a natural sequence are observed in many prospective clinical studies. The estimation of survival functions for such time-to-events gained the attention of many researchers, and the related literature has recorded many developments. This study revisits the problem of survival function estimation for multiple sequentially observed time-to-events and discusses two very easy to straightforward and easy to implement methods. One of these approaches, being non-parametric, assumes independence between the consecutive event times, and the estimators are derived based on well known Kaplan–Meier estimates. The other method is fully parametric, accounts for the dependence between consecutive event times, and assumes the event times to be log-normally distributed. We have studied the finite sample properties of both the non-parametric and parametric approaches at different sample sizes and censoring rates through statistical simulations. Applications of the methods are demonstrated using data from a sample of HIV/AIDS patients in South Carolina. For the patients diagnosed with detectable viral load, we have analyzed their viral load rebound behavior in 24 months post-diagnosis. The marginal and conditional survival curves of viral suppression following diagnosis and viral rebound following viral suppression are estimated and discussed.
  PubDate: 2024-04-30
A Discrete Version of the Half-Logistic Distribution Based on the
Mimicking of the Probability Density Function
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  Abstract: Abstract We introduce a count distribution obtained as a discrete analogue of the continuous half-logistic distribution. It is derived by assigning to each non-negative integer value a probability proportional to the corresponding value of the density function of the parent model. The main features of this new distribution, in particular related to its shape, moments, and reliability properties, are described. Parameter estimation, which can be carried out resorting to different methods including maximum likelihood, is discussed, and a numerical comparison of their performances, based on Monte Carlo simulations, is presented. The applicability of the proposed distribution is proved on two real datasets, which have been already fitted by other well-established count distributions. In order to increase the flexibility of this counting model, a generalization is finally suggested, which is obtained by adding a shape parameter to the continuous one-parameter half-logistic and then applying the same discretization technique, based on the mimicking of the density function.
  PubDate: 2024-04-06
Bioinformatics Analysis in the Identification of Prognostic Signatures for
ER-Negative Breast Cancer Data
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  Abstract: Abstract Breast cancer (BRCA) is the most widespread malignant tumor and the leading cause of death in women. BRCA treatments vary based on the presence of estrogen receptors (ER). Generally, cancer preventive and therapy options for ER-negative BRCA are limited compared to ER-positive BRCA. Therefore, this study investigates the key genes predicting poor prognosis for ER-negative BRCA through the application of LASSO (Least Absolute Shrinkage and Selection Operator) and bioinformatics analysis. These methods are analyzed using a dataset GSE7390 that contains 198 untreated lymph node-negative BRCA patients with gene expressions and clinical information. Differentially Expressed Genes (DEGs) between ER-negative and ER-positive BRCA are found using GEO2R. Here, the regularized regression reduces the dimensionality of DEGs by selecting necessary genes. Later, Support Vector Machines for Survival and Random Survival Forest methods were implemented to construct survival predictive models. Comparatively, LASSO holds higher C-index and AUC values with better prediction accuracy and it selects 28 DEGs consisting of 14 upregulated and 14 downregulated genes. Among them, LCN2 expression is significantly downregulated and ZMYND10 expression is significantly upregulated in tumor tissues compared to normal tissue using GEPIA2. Also, both genes are statistically significant in survival rates for TCGA BRCA patients (p < 0.05). Hence, this study highlights higher LCN2 and low ZMYND10 expression in ER-negative BRCA associated with a poor prognosis. Consequently, LCN2 and ZMYND10 genes can be used to predict and serve as potential biomarkers for future diagnosis and treatment in ER-negative BRCA.
  PubDate: 2024-03-31
On Mathai–Haubold Past Entropy Measure
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  Abstract: Abstract In this paper, Mathai–Haubold past entropy measure is proposed and its properties are studied. Also some generalized inequalities related to Mathai–Haubold entropy measure are discussed. A Kernel based non-parametric estimator for the proposed measure is provided when the underlying sample follows \(\rho\) -mixing dependence condition. The consistency property and asymptotic normality of the proposed estimator are established. A simulation study is conducted to assess the performance of the estimator. A data set is analyzed for illustrative purposes.
  PubDate: 2024-03-26
Simulation Study of Estimators of the Gamma Rate Parameter Using MLE as a
Baseline Estimator
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  Abstract: Abstract Classical estimation methods of the rate parameter of the gamma distribution have shown to have quality issues. In this paper we propose three estimators namely linear shrinkage, preliminary test and linear shrinkage preliminary test for rate parameter of the gamma distribution using maximum likelihood estimation as a baseline estimator. The salient feature of the proposed estimators is the optimality and robustness property that is defined on belief of an uncertain prior information (UPI). Expressions for bias and relative efficiency using Maximum Likelihood Estimation (MLE) as a baseline estimator have been derived with asymptotic properties. A Monte Carlo simulation work is carried out with degree of belief in the UPI. The study results shows that even though the proposed estimators utilizes the UPI from the neighbourhood of the true rate parameter, they are more efficient and minimally biased when prior information is closed to the neighbourhood of the rate parameter compared to the classical Maximum Likelihood Method.
  PubDate: 2024-03-18
Single-Index Mixed-Effects Model for Asymmetric Bivariate Clustered Data
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  Abstract: Abstract Studies/trials assessing status and progression of periodontal disease (PD) usually focus on quantifying the relationship between the clustered (tooth within subjects) bivariate endpoints, such as probed pocket depth (PPD), and clinical attachment level (CAL) with the covariates. Although assumptions of multivariate normality can be invoked for the random terms (random effects and errors) under a linear mixed model (LMM) framework, violations of those assumptions may lead to imprecise inference. Furthermore, the response-covariate relationship may not be linear, as assumed under a LMM fit, and the regression estimates obtained therein do not provide an overall summary of the risk of PD, as obtained from the covariates. Motivated by a PD study on Gullah-speaking African-American Type-2 diabetics, we cast the asymmetric clustered bivariate (PPD and CAL) responses into a non-linear mixed model framework, where both random terms follow the multivariate asymmetric Laplace distribution (ALD). In order to provide a one-number risk summary, the possible non-linearity in the relationship is modeled via a single-index model, powered by polynomial spline approximations for index functions, and the normal mixture expression for ALD. To proceed with a maximum-likelihood inferential setup, we devise an elegant EM-type algorithm. Moreover, the large sample theoretical properties are established under some mild conditions. Simulation studies using synthetic data generated under a variety of scenarios were used to study the finite-sample properties of our estimators, and demonstrate that our proposed model and estimation algorithm can efficiently handle asymmetric, heavy-tailed data, with outliers. Finally, we illustrate our proposed methodology via application to the motivating PD study.
  PubDate: 2024-03-16
A Modified Power Lindley Distribution
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  Abstract: Abstract Many probability distributions have been discussed by various authors to model the lifetime data. However, looking into the variety of data coming in this era, there are more and more requirements for distributions to fit the data. This article introduces a new probability distribution, a modified power Lindley distribution, to fit a greater number of situations and more general applications. Its rich statistical properties are established. The proposed distribution can be used to model the data sets with decreasing, increasing, decreasing-increasing–decreasing and upside-down bathtub-shaped hazard rates. The parameters of the proposed distribution are estimated by the maximum likelihood method. A simulation study is carried out to compute the average estimate, bias, mean square error, and average width of the asymptotic confidence interval. Two real data sets are analyzed to illustrate the flexibility of the proposed model.
  PubDate: 2024-03-09
On Some Characterizations of Probability Distributions Based on Maxima or
Minima of Some Families of Dependent Random Variables
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  Abstract: Abstract Most of the characterizations of probability distributions are based on properties of functions of possibly independent random variables. We investigate characterizations of probability distributions through properties of minima or maxima of max-independent, min-independent and quasi-independent random variables generalizing the results from independent random variables of Kotlarski (Aequ Math 17:77–82, 1978), Prakasa Rao (Identifiability in stochastic models: characterization of probability distributions, Academic Press, New York, 1992) and Klebanov (Math Notes 13:71–72, 1973).
  PubDate: 2024-03-04
On Relative Cumulative Extropy, Its Residual (Past) Measures and their
Applications in Estimation and Testing
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  Abstract: Abstract This article introduces a new measure of distance between two probability distributions based on the notion of extropy, called the relative cumulative extropy. A test for goodness-of-fit for standard uniform distribution is developed using relative cumulative extropy and compared its power with some existing well known tests. We extend the measure to its dynamic forms, namely the dynamic relative survival and failure extropies, respectively for the residual and past lifetime random variables. Some characterization theorems using the proposed measures for the additive hazards model, additive dynamic survival extropy model, and exponential distributions are derived. Non-parametric kernel estimators are proposed for the estimation of the dynamic measures and their performance is validated through simulation studies. The application and scope of the relative cumulative extropy measure in the image analysis are further investigated. A real data application to examine the usefulness of the relative cumulative residual extropy is also carried out.
  PubDate: 2024-02-25
A Study on Quantile based Cumulative Residual Extropy of Order Statistics
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  Abstract: Abstract In recent times, there has been a growing interest among researchers in utilizing quantile-based approaches for assessing the uncertainty associated with random variables. Distinct from traditional distribution function methods, quantile-based measurements offer unique perspectives. This paper investigates the extropy of order statistics by introducing a novel approach based on quantiles and explores its properties. Additionally, we present a nonparametric estimator and its application to this new measure using distributions commonly employed in lifetime data analysis.
  PubDate: 2024-02-16
A New Algorithm for the Partition of Pearson’s Chi-Squared Statistic for
Multiway Contingency Table
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  Abstract: Abstract Pearson’s chi-squared statistic is one of the most common statistical tools used to assess the association between two or more categorical variables that have been cross-classified to form a contingency table. In many practical settings, multiple categorical variables are “paired-off” and analysed by identifying association structures between two variables only. However, there are less well-known tools that allow the analyst to explore the association structure of categorical variables that form a multi-way contingency table. This paper presents an ANOVA-like decomposition of the chi-squared statistic for four-way and five-way contingency tables and can be extended for the analysis of higher-way contingency tables. Furthermore, we propose an efficient algorithm for partitioning the statistic that leads to two-way and higher-way terms. The proposed algorithm reduces the complexity involved in the calculation of the terms of the partition and will be demonstrated by way of a simulation and practical example.
  PubDate: 2024-02-12
Statistical Estimation of Exponential Power Distribution on Different
Progressive Type-II Censoring Schemes
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  Abstract: Abstract In recent life-testing experiments, Progressive Type-II (PT2) censoring has been used by several researchers in various problems. However, one drawback of PT2 is that the length of the experiment might be fairly long. Progressive Type-II hybrid (PT2H) and later adaptive progressive Type-II (APT2) censorings were used to overcome this disadvantage. This study investigates the advantages and disadvantages of these censoring techniques using the Exponential Power (EP) distribution. The EP distribution is known for its ability to model increasing and bathtub-shaped failure rates, and it has gained acceptance in several domains, especially in the areas of reliability-related decision-making and cost analysis. This makes the EP model a useful alternative to the most widely used Weibull distribution in some instances. To assess the efficiency of both Maximum Likelihood Estimators and Bayesian estimators generated through PT2, PT2H, and APT2, we have conducted a comprehensive simulation study. Additionally, we compare the performance of PT2, PT2H, and APT2 when applied to the EP distribution with popular time models using real data.
  PubDate: 2023-12-15

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