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We consider a weighted sum of a series of independent Poisson randomvariables and show that it results in a new compound Poisson distribution whichincludes the Poisson distribution and Poisson distribution of order k. Anexplicit representation for its distribution is obtained in terms of Bellpolynomials. We then extend it to a compound Poisson process and timefractional compound Poisson process (TFCPP). It is shown that theone-dimensional distributions of the TFCPP exhibit over-dispersion property,are not infinitely divisible and possess the long-range dependence property.Also, their moments and factorial moments are derived. Finally, the fractionaldifferential equation associated with the TFCPP is also obtained.

This study aims to explore the issues of evaluating the parameters and the accelerating factor based on constant stress for partially accelerating life tests when the potential failure times have an exponentiated Rayleigh distribution. Within the framework of progressive Type-II censoring schemes, we employ the Newton-Raphson algorithm as an iterative methodology to gain the maximum likelihood estimates, accompanied by proof of the existence of these point estimators. We also construct asymptotic confidence intervals for interested parameters and acceleration factors by utilizing the asymptotical characteristics of the maximum likelihood estimators. The Bayesian estimations of unknown parameters are derived by using the independent gamma priors and dependent Gamma-Dirichlet prior on the basis of square error and relatively smooth LINEX loss functions, respectively. Furthermore, we adopt the importance sampling method to compute Bayesian point estimates and the credible intervals with the highest posterior density. To validate the effectiveness of the suggested approaches, a series of simulated experiments are carried out. Lastly, we conduct analyzes on two actual datasets to show the applicability of the suggested techniques. [ABSTRACT FROM AUTHOR]

Here we study some important properties of the two-tailed Lindley distribution (TLD) and propose a location-scale extension of the TLD. Several properties of the extended TLD are also obtained and an attempt has been made for estimating its parameters by the method of maximum likelihood, along with brief discussion on the existence of the estimators. Further, the distribution is fitted to certain real life data sets for illustrating the utility of the model. A simulation study is carried out for assessing the performance of likelihood estimators of the parameters of the distribution. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new distribution over the unit interval which can be characterized as a ratio of the type Z = d Y / (X + Y) where X and Y are two correlated Birnbaum-Saunders random variables. The density of Z may be unimodal or bimodal. Simple expressions for the cumulative distribution function, moment-generating function and moments are obtained. Moreover, the stress-strength probability between X and Y is calculated explicitly in the symmetric case, that is, when the respective scale parameters are equal. Two applications of the ratio distribution are discussed. [ABSTRACT FROM AUTHOR]

AbstractThe moment generating function and moments of a real-valued random variable can be expressed in terms of the survival function or inverse survival function. In this article, we give the representation of characteristic function via the generalized inverse survival function or generalized inverse distribution function for arbitrary distributions. We extend existing studies, both univariate and multivariate, by giving formulae of the expectation of a more general function of a random variable. These results have potential applications in a variety of fields. To illustrate their applications, we calculate the characteristic functions of several distributions appear in distortion riskmetrics and weighted entropies. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we introduce generalized version of aging functions based on arithmetic, geometric and harmonic means of generalized aging intensity (AI) function. The importance of these functions arise while solving equations related to proportionality of each of the functions, viz., arithmetic, geometric and harmonic mean of AI function with the AI function. The routine work of applications in real life data and simulation with regard to the said aging functions are presented. [ABSTRACT FROM AUTHOR]

Stimulated by the need of describing useful notions related to information measures, we introduce the 'pdf-related distributions'. These are defined in terms of transformation of absolutely continuous random variables through their own probability density functions. We investigate their main characteristics, with reference to the general form of the distribution, the quantiles, and some related notions of reliability theory. This allows us to obtain a characterization of the pdf-related distribution being uniform for distributions of exponential and Laplace type as well. We also face the problem of stochastic comparing the pdf-related distributions by resorting to suitable stochastic orders. Finally, the given results are used to analyse properties and to compare some useful information measures, such as the differential entropy and the varentropy. [ABSTRACT FROM AUTHOR]

As cloud computing continues to evolve, optimizing resource utilization and enhancing system efficiency have become preeminent objectives. Efficient VM migration and dynamic virtual machine consolidation strategies stand as pivotal solutions in achieving these goals. However, the success of these approaches hinges on a thorough understanding of the intricate dependencies among virtual machine (VM) components, spanning software, hardware, services, and resources, which are used in deciding competent VM migration strategies. This paper presents an innovative approach focusing on a novel integrated correlation coefficient for conducting dependency analysis of VM components within a virtual cloud infrastructure. Unlike traditional methodologies, our proposed model, efficient migration through dependency analysis (EMDA), not only accounts for resource utilization metrics but also delves into the complex inter-component relationships of VMs that govern computation in cloud environments. By considering various significant factors, our proposed framework offers a comprehensive technique for deciphering VM dependencies. We have developed a novel system architecture with the necessary functional blocks to streamline the analysis and deployed a sophisticated algorithm to implement our proposed model. Rigorous experiments have been conducted in a simulated virtualized cloud environment to precisely scrutinize the performance of EMDA and compare it with four cutting-edge frameworks. We have used diversified and dynamic scientific workloads to conduct simulations that prove the novelty of our proposed model. In essence, this paper outlines how dependency analysis of VM components empowers cloud infrastructure management, enhancing migration efficiency and cost-effectiveness. [ABSTRACT FROM AUTHOR]

Zero-inflated regression models are generally accepted for modeling data with inflated amounts of zeros. The principle used in constructing the distribution for these models has been extended to modeling certain types of data with inflation at the upper bound of the support. An excellent example of this is the zero and κ-inflated truncated Poisson regression model which utilizes the Poisson distribution. However, the equi-dispersion nature of the Poisson distribution makes the model less suitable for data that are under-dispersed and over-dispersed. Models based on exponentiated exponential geometric distribution (EEGD) are presented as alternatives for modeling any count data with an upper bound, irrespective of the nature of dispersion. The models are obtained by truncating the EEGD at the top and inflating at either or both zero and κ. A simulation study is conducted to examine the performances of the models under certain conditions. The result shows that the closer the dispersion parameter c is from the equi-dispersion line, the better the estimates for over-dispersed data, while the farther the dispersion parameter from the equi-dispersed line, the better the estimates for an under-dispersed model. The application of the models is illustrated using data from the Behavioral Risk Factor Surveillance System 2011. [ABSTRACT FROM AUTHOR]

In recent years, modern statistical process control has established higher concentration, and this approach is reinforced by new control charts to efficiently monitor the process. The goal of this article is to create a control chart based on a modified chain sampling plan (MchSP) when product quality dimensions are followed by a generalized exponential distribution. There was a brief overview of the generalized exponential distribution (GED) and the modified chain sampling strategy. The chart parameters were presented for several sample sizes at various average run length (ARL) values. A variety of tables were provided for the actual application of the produced control chart in industrial applications at various shift values of scale and form parameters. The real data set was used to demonstrate the application of the created MchSP control chart process. In addition, the proposed MchSP control chart process was developed using simulated data. The average run length (ARL) results of the established control chart for shifting both scale and shape parameters were computed for various sample sizes. The results show that when the shift constant (both scale and form) increases, the ARL values decrease. Also, it was discovered that ARL1 values tend to decrease as the shape parameter value increases. [ABSTRACT FROM AUTHOR]

Bearing in mind an increasing popularity of the fractional calculus the main aim of this paper is to derive several new representation formulae for the cumulative distribution function (cdf) of the McKay I ν Bessel distribution including the Grünwald-Letnikov fractional derivative; also, two connection formulae between cdf of the McKay I ν random variable and the so–called Neumann series of modified Bessel functions of the first kind are established, providing, consequently, a new integral representation for such cdf in terms of a definite integral. Another fashion expression for the given cdf is derived in terms of the Grünwald-Letnikov fractional derivative of the widely applicable Marcum Q–function, which represents a certain simplification of the already existing relationship between McKay I ν random variable and a Marcum Q–functions. The exposition ends with some open questions, drawing the interested reader's attention, among others, to the summation of some Neumann series. [ABSTRACT FROM AUTHOR]

A balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by many uncertainties, it is more natural to induce some sort of randomness in the balancing-like sequences. If, in a balancing-like sequence, the growth rate is assumed to be a random variable, the resulting sequence will be a stochastic process and the sequence of expectations, in many cases, cannot be described as a binary recurrence sequence. In some cases, the growth rate of expectations increases without limit while, in some cases, it remains finite. [ABSTRACT FROM AUTHOR]

We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner's semicircle law is of order n - 1 2 with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order n - 1 , thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalize to the setting of sums of free non-identically distributed bounded self-adjoint random variables leading to a new rate of convergence in the free central limit theorem. [ABSTRACT FROM AUTHOR]

We introduce probabilistic Stirling numbers of the first kind s Y (n , k) associated with a complex-valued random variable Y satisfying appropriate integrability conditions, thus completing the notion of probabilistic Stirling numbers of the second kind S Y (n , k) previously considered by the first author. Combinatorial interpretations, recursion formulas, and connections between s Y (n , k) and S Y (n , k) are given. We show that such numbers describe a large subset of potential polynomials, on the one hand, and the moments of sums of i. i. d. random variables, on the other, establishing their precise asymptotic behavior without appealing to the central limit theorem. We explicitly compute these numbers when Y has a certain familiar distribution, providing at the same time their combinatorial meaning. [ABSTRACT FROM AUTHOR]

The purpose of this study is to construct a mortality model that reasonably explains survival curves and mortality rates in terms of the decline in biological function, which is the phenomenon of ageing. In this model, an individual organism is regarded as a collection of subsystems, and for each subsystem, the model defines human mortality by introducing positive self-repair mechanisms and stochastically generated negative external shocks. The probability density function of the time of death is derived explicitly, and the model parameters are estimated using life tables from Japan and the UK, which demonstrate the existence of multiple parameter sets that fit well with the observed data. [ABSTRACT FROM AUTHOR]

Street lamp is a great asset for human society with a narrow beam spread light. The extensive proliferation of solar power in street lamps causes power outages due to their variable power-generated profiles. Thus Smart Street Lamp Fog Intelligence (SSLFI) framework based on hierarchical learning was proposed for efficient energy management in solar street lamps. Smart Street Lamp (SSL) shifts its brightness at higher and lower light levels with a comforting, energy-efficient gleam of light. The fog intelligence framework forecasts the SSL output power through short-term probabilistic energy consumption forecasts using Q-NARX-BiLSTM (Quantile Regression-Nonlinear Auto-Regressive Neural Networks with exogenous input-Bidirectional Long short-term memory) model. NARX-BiLSTM of two module types: (1) NARXNN (Nonlinear Auto-Regressive Neural Networks with exogenous input) model generates SSL power consumption and (2) BiLSTM (Bidirectional Long short-term memory) model generates SSL power forecasts. The quantile regression with the NARX-BiLSTM (Nonlinear Auto-Regressive Neural Networks with exogenous input-Bidirectional Long short-term memory) model forecasts the seasonal patterns achieving non-parametric interval predictions. The probabilistic predictions of power consumption are determined based on the conditional quantile using an improved kernel density estimation approach. The fuzzy inference system adopts forecasting results to diagnose fault conditions in street lamps. The experiment results show that the proposed framework SSLFI outperformed the state-of-the-art models forecasting under different weather conditions. [ABSTRACT FROM AUTHOR]

This study presents a piecewise non-linear binary subdivision scheme based on a four-point linear binary subdivision scheme. This scheme is first transformed into a piecewise linear subdivision scheme and then into a piecewise non-linear subdivision scheme. The non-linear subdivision approach removes artifacts and distortions from limit curves, producing better continuous curves. The study demonstrates that this technique is a workable technique for producing smooth curves, especially when avoiding bends, distortion, and oscillation. [ABSTRACT FROM AUTHOR]

We propose a new generator which has been used as a generalized class, and can also be helpful in generating new flexible generalized classes of distributions for continuous random variable. The newly proposed generator-cum-generalized class does not involve any extra parameter, and its functional form plays an important role along with baseline models to develop flexible models. In literature, such classes had been reported as Marshall-Olkin G-class, exponentiated G-class, Transmuted G-class, exponentiated generalized G-class, and Flexible G-class. So, any parent (or baseline) model can be substituted in the proposed class which has no extra burden on the parameters of the class. Some needful characteristics of the newly proposed class are obtained. Furthermore, a special model of the class, that is, the flexible Kumaraswamy distribution is considered and its properties are reported. The parameters estimation is dealt through the method of maximum likelihood, and a simulation is carried out to assess the performance of model's parameters. Four real-life data sets are analyzed to show usefulness of the proposed model in comparison to some well-established competitive models. It is found that the proposed model yields low values of the goodness-of-fit statistics as compared to the other models, and hence our proposed model performed better as compared to others on these four data sets. [ABSTRACT FROM AUTHOR]

In this paper, a new member of the Poisson family of distributions called the Poisson-Exponential-Gamma (PEG) distribution for modelling count data is proposed by compounding the Poisson with Exponential-Gamma distribution. The first four moments about the origin and the mean of the new PEG distribution were obtained. The expressions for its coefficient of variation, skewness, kurtosis, and index of dispersion were equally derived. The parameters of the PEG distribution were estimated using the Maximum Likelihood Method. Its relative performance based on the Goodness-of-Fit (GoF) criteria was compared with those provided by seven of the existing related distributions (Poisson, Negative-Binomial, Poisson-Exponential, Poisson-Lindley, Poisson-Shanker, Poisson-Shukla, and Poisson Entropy-Based Weighted Exponential distributions) in the literature on three different published real-life count data sets. The GoF assessment of all these distributions was performed based on the values of their loglikelihoods ( - 2 logLik ), Akaike Information Criteria, Akaike Information Criteria Corrected, and Bayesian Information Criteria. The results showed that the new PEG distribution was relatively more efficient for modelling (over-dispersed) count data than any of the seven existing distributions considered. The new PEG distribution is therefore recommended as a credible alternative for modelling count data whenever relative gain in the model's efficiency is desired. [ABSTRACT FROM AUTHOR]

We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons–Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton–Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency benefits to using FFT, Newton–Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method’s performance and support our theoretical findings. [ABSTRACT FROM AUTHOR]

We propose a new generator which has been used as a generalized class, and can also be helpful in generating new flexible generalized classes of distributions for continuous random variable. The newly proposed generator-cum-generalized class does not involve any extra parameter, and its functional form plays an important role along with baseline models to develop flexible models. In literature, such classes had been reported as Marshall-Olkin G-class, exponentiated G-class, Transmuted G-class, exponentiated generalized G-class, and Flexible G-class. So, any parent (or baseline) model can be substituted in the proposed class which has no extra burden on the parameters of the class. Some needful characteristics of the newly proposed class are obtained. Furthermore, a special model of the class, that is, the flexible Kumaraswamy distribution is considered and its properties are reported. The parameters estimation is dealt through the method of maximum likelihood, and a simulation is carried out to assess the performance of model’s parameters. Four real-life data sets are analyzed to show usefulness of the proposed model in comparison to some well-established competitive models. It is found that the proposed model yields low values of the goodness-of-fit statistics as compared to the other models, and hence our proposed model performed better as compared to others on these four data sets.

Comprehensive families of copulas including the three basic copulas (at least as limit cases) are useful tools to model countermonotonicity, independence, and comonotonicity of pairs of random variables on the same probability space. In this contribution, we study how the transition from a (basic) copula to a copula modeling a different dependence behavior can be realized by means of ordinal sums based on one of the three basic copulas, perturbing one of the three basic copulas (considering some appropriate parameterized transformations) and truncating the results using the Fréchet-Hoeffding bounds. We provide results and examples showing the flexibility and the restrictions for obtaining new copulas or comprehensive families and illustrate the development of their dependence parameters.

This article establishes the functional average as an important estimand for causal inference. The significance of the estimand lies in its robustness against traditional issues of confounding. We prove that this robustness holds even when the probability distribution of the outcome, conditional on treatment or some other vector of adjusting variables, differs almost arbitrarily from its counterfactual analogue. This article also examines possible estimators of the functional average, including the sample mid-range, and proposes a new type of bootstrap for robust statistical inference: the Hoeffding bootstrap. After this, the article explores a new class of variables, the U{\mathcal{U}} class, that simplifies the estimation of functional averages. This class of variables is also used to establish mean exchangeability in some cases and to provide the results of elementary statistical procedures, such as linear regression and the analysis of variance, with causal interpretations. Simulation evidence is provided. The methods of this article are also applied to a National Health and Nutrition Survey data set to investigate the causal effect of exercise on the blood pressure of adult smokers.

A novel approach to the construction of absolutely continuous distributions over the unit interval is proposed. Considering two absolutely continuous random variables with positive support, this method conditions on their convolution to generate a new random variable in the unit interval. This approach is demonstrated using some popular choices of positive random variables, such as the exponential, Lindley, and gamma. Some existing distributions, like the uniform, the beta, and the Kummer-beta, are formulated with this method. Several new structures of density functions having potential for future applications in real-life problems are also provided. One of the new distributions, namely the LCG, is considered for detailed study along with a related distribution, namely the GCL. The moments, hazard rate, cumulative distribution function, stress-strength reliability, random sample generation using the quantile function, method of moments along with maximum likelihood estimation, and regression modeling are discussed for both the distributions. Real-life applications of the proposed models and the corresponding regression models show promising results. [ABSTRACT FROM AUTHOR]

This paper introduces a new probability distribution called the mixture symmetric gamma (MSG) distribution, which is defined as a mixture of two symmetric gamma distributions. Its statistical properties and applications are explored. We first examine its mathematical properties, including the possible shapes of the corresponding probability density function, as well as the moments, and the moment‐generating function. We then look at parameter estimation using various frequentist and Bayesian methods, such as moment estimation, maximum likelihood method, least‐squares method, and Bayesian approaches. In addition, the prediction of future observations under the MSG model is extensively covered, considering both frequentist and Bayesian perspectives, including median prediction, best unbiased prediction, and Bayesian prediction. A comprehensive simulation study is conducted to evaluate the performance of the proposed estimation and prediction techniques. Finally, the practical applicability of the MSG model is demonstrated through the analysis of four real‐world datasets. It is shown to outperform several well‐known competing models in terms of goodness‐of‐fit. The results highlight the inherent simplicity, efficiency, robustness, and intuitive interpretability of the MSG distribution, making it a compelling choice for modeling data with a symmetric pattern, with potential applications in diverse domains. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of mean-field stochastic differential equations driven by both Brownian motion and fractional Brownian motion. Using the Banach fixed point theorem and inequality technique, we obtain sufficient conditions for the existence and uniqueness of Weyl almost periodic solutions in distribution of the equations under consideration. [ABSTRACT FROM AUTHOR]

We propose to introduce a new class of bivariate probability distributions, which we believe is of great interest to statisticians and data scientists. However different from the conventional Weibull it might be, the density function posited herein allows to generalize its properties in two dimensions (2D). This new function, essentially, has structure characteristics and properties different from those of the various bivariate Weibull-type functions found in the literature. The main features, such as the marginal distributions, moments, characteristic functions of this bivariate density are defined. Two related maximum likelihood estimation algorithms are also explicated, tested, and compared. Numerical simulations show the practicality of these algorithms as well as the interest of the new density in several areas of data analysis and extreme values modeling. [ABSTRACT FROM AUTHOR]

We consider the Hotelling T 2 test in low sample size, high dimensional setting. We partition the p variables into b > 1 blocks of p/b variables and use the union-intersection principle to propose a testing procedure that computes the T 2 test in each block. We show that the proposed method is more powerful than Hotelling T 2 test. We also consider Wilks method of outlier detection and use the union-intersection principle to search for outliers in blocks of variables. The significance level and the power function of the new test are investigated. We show that the new outlier detection method produces more power compared to Wilks test. [ABSTRACT FROM AUTHOR]

Similar to usual lower records, Bottom- k -lists (Kozan and Tanil, İstatistik J Turk Stat Assoc 13:73–79, 2013) have a wide range of practical applications in meteorology, hydrology, sports, etc. Also, exceedance statistics can be viewed as a close relative to tolerance limits—an important field of statistical science. In this study, an idea of combining these two important subjects together is studied and an exceedance statistic is defined based on bottom- k -lists in an independent and identically distributed (iid) continuous random sequence. Probability mass function (pmf) of a selected exceedance statistic is obtained. Also, an illustrative application of the exceedance statistic is given. [ABSTRACT FROM AUTHOR]

A random variable X has a standard normal distribution if and only if E [ f ′ (X) ] = E [ X f (X) ] for any continuous and piecewise continuously differentiable function f such that the expectations exist. This first-order characterizing equation, called the Stein identity, has been extended to other univariate distributions. For the multivariate normal distribution, a number of Stein identities have already been developed, all of them second order equations. In this study, we developed a new Stein characterization for the bivariate normal distribution. Unlike many existing multivariate versions in the literature, ours is a system of first-order equations which has the univariate Stein identity as a special case. We also constructed a generalized Stein characterization for other absolutely continuous bivariate distributions. Finally, we illustrated how this Stein characterization looks like for some known absolutely continuous bivariate distributions. [ABSTRACT FROM AUTHOR]

This paper considers parallel and series systems with heterogeneous components having dependent exponential lifetimes. The underlying dependence is assumed to be Archimedean and the component lifetimes are supposed to be connected according to an Archimedean copula. Sufficient conditions are found to dominate a parallel system with heterogenous exponential components, with respect to the dispersive order, by another parallel system with homogenous exponential components where the dependence structure between lifetimes of components is the same. We also compare two series systems (and two parallel systems) with general one-parameter dependent components and with respect to the usual stochastic ordering. Examples are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper studies some stochastic properties of random quantiles according to the newly defined reliability measure called reversed aging intensity function. Preservation property of reversed aging intensity order under random quantile is obtained and using it, a lower bound and an upper bound for the reversed aging intensity function of a random quantile are derived. Preservation of two related monotonic reliability classes under random quantiles is also studied. We finally apply our results for reliability analysis of series systems with heterogeneous component lifetimes. Examples are included to examine and analyze the obtained results. [ABSTRACT FROM AUTHOR]

Recently, a lot of works have been done on the study of the failure rate (FR), the aging intensity function (AI) and their properties. In the article, we analyze the aging classes based on monotonicity of new functions viz., arithmetic mean, geometric mean, and harmonic mean of AI function, respectively. Also, the relationships between these functions and the classical AI function are discussed. Moreover, some characterizations of distributions based on these functions are established. Some characterizations of proportional hazard rate and AI family of distributions are discussed. Investigations are illustrated by some examples and applications of aging functions in data analysis. [ABSTRACT FROM AUTHOR]

Using recent results concerning non-uniqueness of the center of the mix for completely mixable probability distributions, we obtain the following result: For each d ∈ N and each non-empty bounded Borel set B ⊂ R d , there exists a d-dimensional probability distribution μ satisfying the following: For each n ≥ 3 and each probability distribution ν on B, there exist d-dimensional random vectors X ν , 1 , X ν , 2 , ⋯ , X ν , n such that 1 n (X ν , 1 + X ν , 2 + ⋯ + X ν , n) ∼ ν and X ν , i ∼ μ for i = 1 , 2 , ⋯ , n . We also show that the assumption regarding the boundedness of the set B cannot be completely omitted, but it can be substantially weakened. [ABSTRACT FROM AUTHOR]

The closeness or similarity between two populations is usually assessed by means of overlap measures based on their probability distributions. Matusita's measure, Morisita's measure and Weitzman's measure are three commonly used overlap measures. These measures are defined based on the probability density functions of the two distributions. There are several probability distributions that do not have a tractable probability density function, though the corresponding quantile density function has an explicit form. In such cases, the traditional approach fails and the overlap measures need to be defined in terms of quantile functions. The present paper considers a quantile-based study of the overlap measures. We also illustrate the usefulness of the proposed quantile-based overlap measures using several examples. [ABSTRACT FROM AUTHOR]

This paper introduces a bounded probability distribution which is derived from the Muth distribution. The main statistical properties are studied and analytical expressions are provided for the moments, incomplete moments, inverse of the cumulative distribution function, extropy, Lorentz and Bonferroni curves, among others. Moreover, it possesses both monotone and non-monotone hazard rate functions so the new distribution is rich enough to model real data. Different estimation methods are applied to estimate the parameters of the model and a Monte Carlo simulation study assesses their performances. The usefulness in practical applications is illustrated using two real data sets and the results show that the proposed distribution provides better fits than other competing distributions commonly used to model data with bounded support. [ABSTRACT FROM AUTHOR]

Effective risk exposure descriptions can be made using continuous distributions. To illustrate the level of exposure to a certain danger, it is better to use a single number, or at the very least, a small set of numbers. These risk exposure numbers, which are commonly referred to as significant risk indicators, are unquestionably the output of a particular model. In this regard, five key indicators are utilized to define the risk exposure in the reinsurance revenues data. For this specific purpose we introduce a new distribution called odd log-logistic XGamma model. We estimated the parameters using maximum-likelihood method, least squares method and Bayesian method. Monte Carlo simulation study is performed under a set of conditions and controls. The risk exposure under the reinsurance revenue data was also described using five important risk indicators, including value-at-risk, tail-value-at-risk, tail variance, tail mean-variance, and mean excess loss function. These statistical measures were developed for the proposed new model. [ABSTRACT FROM AUTHOR]

Here we consider beta and/or gamma mixture of the bivariate generalized hypergeometric factorial moment distribution (BGHFMD) of Kumar (Metrika, 2008). Further, it is establised that the BGHFMD can be obtained as a limit of another BGHFMD. [ABSTRACT FROM AUTHOR]

In this paper, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, nonnegative random variables is studied for two cases: (i) continuous and (ii) discrete random variables. First, a recursive formula of the joint cumulative distribution function (CDF) is derived in both cases. Then recurrence relations of the joint probability density function (PDF) and the joint probability mass function (PMF) are given in the former and the latter case, respectively. Interestingly, there is a fundamental difference between the joint PDF and PMF. The proofs are simple and mainly based on the following tools from calculus and discrete mathematics: differentiation under the integral sign (also known as Leibniz’s integral rule), the law of total probability, and mathematical induction. In addition, this work generalizes previous results in the literature, and finally presents several extensions of the methodology. [ABSTRACT FROM AUTHOR]