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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]

Several generalizations of the logistic distribution and certain related models are proposed by many authors for modeling various random phenomena such as those encountered in data engineering, pattern recognition, and reliability assessment studies. In this paper, we study generalized q-logistic (GqL) distribution, in which the additional parameters offered increase flexibility of the distribution for modeling purposes. Since the parameter estimation of GqL model is not explored yet, in the present study, we propose different methods for estimating the GqL parameters. In particular, we have made a comprehensive comparison through simulation study, the performance of the maximum likelihood estimation (both complete and censored), maximum product spacing estimation and ordinary least squares estimation methods. Some characterization theorems are obtained and some properties of GqL are established. Finally, we analyze a real data to illustrate the potentiality of the model and also present some graphical illustrations. [ABSTRACT FROM AUTHOR]

In this paper, we establish normalized and self-normalized Cramér-type moderate deviations for the Euler-Maruyama scheme for SDE. Due to our results, Berry-Esseen's bounds and moderate deviation principles are also obtained. Our normalized Cramér-type moderate deviations refine the recent work of Lu et al. (2022). [ABSTRACT FROM AUTHOR]

Characterizations of finite sequences β 1 < ⋯ < β n representing expected values of order statistics from a random sample of size n are given. As a by-product, a characterization of binomial mixtures, when the mixing random variable is supported in the open interval (0, 1), is presented; this enables the exact description of the convex hull of the open binomial curve, as well as the open moment curve. [ABSTRACT FROM AUTHOR]

Abstract.Frailty and resilience models provide a way to introduce random effects in hazard and reversed hazard rate modeling by random variables, called frailty and resilience random variables, respectively, to account for unobserved or unexplained heterogeneity among experimental units. This article investigates the effects of frailty and resilience random variables on the baseline random variables using some shifted stochastic orders based on some ageing properties of the baseline random variables. Relevant examples are presented to illustrate the findings, and the applicability of the results is demonstrated with real-world data, highlighting their relevance in practical scenarios. [ABSTRACT FROM AUTHOR]

Abstract.This study delves into the conditional mode estimation of a randomly censored scalar response variable operating within the framework of strong mixing conditions. We introduce a kernel-based estimator for the conditional mode function. The principal contribution of this investigation lies in the derivation of the asymptotic distribution and the strong rate of convergence of the newly proposed estimators. These findings are established under a set of fairly comprehensive structural assumptions governing the underlying models. Additionally, we conduct a series of simulation studies to showcase the finite sample performance characteristics of the proposed estimator. [ABSTRACT FROM AUTHOR]

We introduce the bivariate unit-log-symmetric model based on the bivariate log-symmetric distribution (BLS) defined in Vila et al. [25] as a flexible family of bivariate distributions over the unit square. We then study its mathematical properties such as stochastic representations, quantiles, conditional distributions, independence of the marginal distributions and marginal moments. Maximum likelihood estimation method is discussed and examined through Monte Carlo simulation. Finally, the proposed model is used to analyze some soccer data sets. [ABSTRACT FROM AUTHOR]

The paper reviews the scientific and pedagogical activities of Julius Kruopis, the pioneer of Lithuanian applied statistics. His contribution to statistics and cooperation with Lithuanian companies in the application of statistical methods in various fields of human activity is described in the most detail, especially in the quality control area and industrial process optimization. His works in probability theory are also mentioned, emphasizing important contributions to approximations of distributions. Peculiarities of his pedagogical activity, textbooks and monographs, and supervision of students' theses and dissertations are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we compare the entropy of the original distribution and its corresponding compound distribution. Several results are established based on convex order and relative log-concave order. The necessary and sufficient condition for a compound distribution to be log-concave is also discussed, including compound geometric distribution, compound negative binomial distribution and compound binomial distribution. [ABSTRACT FROM AUTHOR]

Two novel proofs show that the sum of a specific pair of normal random variables is not normal are established in this note. This is one of the most often misunderstood facts by first-year students in probability theory and statistics. The first proof is concise using the moment generating function. The second proof checks whether the moments of the sum have the property of normal distribution. [ABSTRACT FROM AUTHOR]

We introduce a certain discrete probability distribution P n , m , k , l ; q having non-negative integer parameters n, m, k, l and quantum parameter q which arises from a zonal spherical function of the Grassmannian over the finite field F q with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single q-Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating 4 ϕ 3 -hypergeometric series. [ABSTRACT FROM AUTHOR]

A framework of two-sided densities is presented for asymmetric continuous distributions consisting of two branches each with its own generating density. The framework supports the construction of distributions with positive support and a specified mode. Examples thereof shall be constructed using the beta and Burr Type XII distributions for their left and right branch densities. The examples are parameterized via left and right branch scale parameters, a mode parameter and two parameters determining the heaviness of its right tail. Keeping one of the tail parameters fixed, a procedure solving for their parameters is presented given a lower and upper quantile, a mode and a conditional-value-at-risk, popular in risk management of insurance losses. While valuable on its own right, that solution may be used as a starting point for a maximum likelihood routine. The estimation of the parameters is demonstrated using the classical insurance Danish fire loss data set and a French business loss interruption data set. Both data sets are publicly available. Developed models compare favorably with prior models fitted to the Danish fire loss data in the literature. [ABSTRACT FROM AUTHOR]

We introduce a new unified extension of the integral form of Euler's beta function with a MacDonald function in the integrand and establish functional upper bounds for it. We use this definition to extend as well the Gaussian and Kummer's confluent hypergeometric functions, for which we provide bounding inequalities. Moreover, we use our extension of the beta function to define a new probability distribution, for which we establish raw moments and moment inequalities and, as by-products, Turán inequalities for the initially defined extended beta function. [ABSTRACT FROM AUTHOR]

Quantum field theories can be applied to compute various statistical properties of random tensors. In particular, signed distributions of tensor eigenvalues/vectors are the easiest, which can be computed as partition functions of four-fermi theories. Though signed distributions are different from genuine ones because of extra signs of weights, they are expected to coincide in vicinities of ends of distributions. In this paper, we perform a case study of the signed eigenvalue/vector distribution of the real symmetric order-three random tensor. The correct critical point and the correct end in the large N limit are obtained from the four-fermi theory, for which a method using the Schwinger-Dyson equation is very efficient. Since locations of ends are particularly important in applications, such as the largest eigenvalues and the best rank-one tensor approximations, signed distributions are the easiest and highly useful through the Schwinger-Dyson method. [ABSTRACT FROM AUTHOR]

The main concern of this paper is providing a flexible discrete model that captures every kind of dispersion (equi-, over- and under-dispersion). Based on the balanced discretization method, a new discrete version of Burr–Hatke distribution is introduced with the partial moment-preserving property. Some statistical properties of the new distribution are introduced, and the applicability of proposed model is evaluated by considering counting series. A new integer-valued autoregressive (INAR) process based on the mixing Pegram and binomial thinning operators with discrete Burr–Hatke innovations is introduced, which can model contagious data properly. The different estimation approaches of parameters of the new process are provided and compared through the Monte Carlo simulation scheme. The performance of the proposed process is evaluated by four data sets of the daily death counts of the COVID-19 in Austria, Switzerland, Nigeria and Slovenia in comparison with some competitor INAR(1) models, along with the Pearson residual analysis of the assessing model. The goodness of fit measures affirm the adequacy of the proposed process in modeling all COVID-19 data sets. The fundamental prediction procedures are considered for new process by classic, modified Sieve bootstrap and Bayesian forecasting methods for all COVID-19 data sets, which is concluded that the Bayesian forecasting approach provides more reliable results. [ABSTRACT FROM AUTHOR]

The class of log-symmetric distributions is a generalization of log-normal distribution and includes some well-known distributions such as log-normal, log-logistic, log-Laplace, log-Cauchy, log-power-exponential, log-student-t, log-slash, and Birnbaum-Saunders distributions. In this paper, several characterization results are obtained for log-symmetric distributions based on moments of some functions of the parent distribution and also on the basis of some properties of order statistics. Specifically, when X is identical in distribution with a decreasing continuous function $ h(X) $ h (X) , then a relationship is established between upper and lower order statistics which is then utilized to construct characterization results for log-symmetric distributions in terms of functions of order statistics. The established results can be used for constructing a goodness-of-fit test for log-symmetric distributions. [ABSTRACT FROM AUTHOR]

In this article, we consider functions having the properties of both the cumulative distribution function and the quantile function. Due to these characteristics, we term such functions quantile cumulative distribution functions. The comparison distribution function defined by Parzen (1982) for comparing the probability distributions of two populations is an example of such a function. We discuss some properties of the quantile cumulative distribution function and its usefulness in generating new continuous probability distributions on the unit interval. An overview of some of their applications is also presented to emphasize the importance of the quantile cumulative distribution function. [ABSTRACT FROM AUTHOR]

Aging Intensity (AI) function is a quantitative measure of hazard function (hazard rate/failure rate), which is used for evaluating the aging behaviour of a component/system. Although variety of research are now available on various properties such as modelling and analysis of AI function; however, a detailed theoretical study on the estimation of the same has not been considered. Accordingly, in the present study, we propose two nonparametric estimators for aging intensity function based on right-censored dependent data scheme and study their properties. Asymptotic properties of the estimators are established under suitable regularity conditions. A simulation study and real data analysis have been carried out to illustrate the performance of the estimators. [ABSTRACT FROM AUTHOR]

In this paper, the Bahadur representation of the empirical and Bernstein polynomials estimators of the quantile function based on associated sequences are investigated. The rate of approximation depends on the rate of decay in covariances, and it converges to the optimal rate observed under independence when the covariances quickly approach zero. As an application, we establish a Berry-Esseen bound with the rate O (n - 1 / 3) assuming polynomial decay of covariances. All these results are established under fairly general conditions on the underlying distributions. Additionally, we perform Monte Carlo simulations to evaluate the finite sample performance of the suggested estimators, utilizing an associated and non-mixing model. [ABSTRACT FROM AUTHOR]

We address the problem of testing for the invariance of a probability measure under the action of a group of linear transformations. We propose a procedure based on consideration of one-dimensional projections, justified using a variant of the Cramér–Wold theorem. Our test procedure is powerful, computationally efficient, and dimension-independent, extending even to the case of infinite-dimensional spaces (multivariate functional data). It includes, as special cases, tests for exchangeability and sign-invariant exchangeability. We compare our procedure with some previous proposals in these cases, in a small simulation study. The paper concludes with two real-data examples. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as standard normal variates. Additionally, the paper provides a clear exposition of the developed methodology. To establish our results, we develop a novel approach utilizing the classical Cauchy's bounds for the zeros of a deterministic polynomial with complex coefficients. We also corroborate our analytical results with extensive simulations. The methodology developed in the paper can potentially be applied to a broad class of problems regarding bounds and the distribution of zeros in the theory of random polynomials. [ABSTRACT FROM AUTHOR]

In this study, a new generalization of Samade distribution has been proposed. The weighted Samade distribution (WSD) is a new distribution designed for modeling real-life data. The survival function, hazard function, moments, moment-generating function, order statistics, and entropies of this distribution have all been determined. The parameters of the suggested model are estimated using the maximum likelihood estimation technique. Finally, the helpfulness and application of the distribution have been demonstrated by lung cancer data. [ABSTRACT FROM AUTHOR]

In this article, we proposed a new probability distribution called length-biased Chris-Jerry distribution in which length-biased distribution is a specific case of weighted distribution. Also, we determined the structural properties of length-biased Chris-Jerry distribution and the unknown parameters were estimated by maximum likelihood estimator and tested by likelihood ratio test. Finally, we emphasized a real lifetime data set to reveal how the proposed distribution worked in it. [ABSTRACT FROM AUTHOR]

The probability density function in Rice–Middleton model, which describes the behavior of the single sinusoidal random signal combined with Gaussian noise is expressed in three mutually independent ways: firstly, with the aid of an integral representation of the modified Bessel function of the first kind of integer order; secondly, by a hyperbolic cosine differential operator and thirdly, applying the Grünwald–Letnikov fractional derivative. The cumulative distribution functions are also described in all these cases, and also using the Nuttall Q–function. An associated, seemingly new, probability distribution is introduced which cumulative distribution function and the raw moments of general real order are obtained whilst the characteristic function's power series form is inferred. The exposition ends with a discussion in which by–product summations are given for the considered Neumann series of the second type built by modified Bessel functions of the second kind having integer order. [ABSTRACT FROM AUTHOR]

In this paper, we consider one-parameter regularly varying discrete distribution generated by Levy probability (RDLP). Some useful plots of the models are illustrated. Mathematically, to propose the RDLP model as a discrete distribution in bioinformatics, some common statistical facts such as unimodality, skewness to the right, upward/downward convexity, regular variation at infinity, and asymptotically constant slowly varying component are established for the model. Based on the Monte Carlo method and limited-memory Broyden–Fletcher–Goldfarb–Shanno for bound-constrained optimization, simulation studies are done to estimate bias, mean square error, and maximum likelihood estimation for the unknown parameter of the model. Some asymptotic properties and asymptotic expansions are given for useful functions. Moreover, we shall attempt to examine the RDLP model with a real count data set to show its application, and also compare it with two rival models. [ABSTRACT FROM AUTHOR]

We analyze the influence of the random effect transformation on four regularity classes of distribution functions. We consider regularly varying, dominatedly varying, long-tailed, and generalized long-tailed distribution functions. We apply the random effect transformation to the primary distribution functions from these classes and investigate whether the resulting distribution function remains in the same class. We find that the random effect transformation has the greatest impact on regularly varying distribution functions. We illustrate the obtained theoretical results with several examples. [ABSTRACT FROM AUTHOR]

In this paper, we establish some stochastic comparison results for largest claim amounts of two sets of independent and also for interdependent portfolios under the setup of the proportional odds model. We also establish stochastic comparison results for aggregate claim amounts of two sets of independent portfolios. Further, stochastic comparisons for largest claim amounts from two sets of independent multiple-outlier claims have also been studied. The results we obtained apply to the whole family of extended distributions, also known as the Marshall–Olkin family of distributions. We have given many numerical examples to illustrate the results obtained. [ABSTRACT FROM AUTHOR]

The fractional Lindley distribution is used to model the distribution of perturbations in count data regressions, which allow for dealing with widely dispersed data. It is obtained from the non-fractional Lindley distribution by replacing the support $\mathbb{T} = {\mathbb{R}^ + }$ T = R + by ${\mathbb{T}} = {\mathbb{N}}\backslash \{ 0\} $ T = N ∖ { 0 } and applying time scale theory, whose ambition is to unify the theories of difference equations and differential equations, integral and differential calculus, and the calculus of finite differences. It thus provides a framework for the study of dynamical systems in discrete-continuous time. Delta moments are discrete-time Laplace transforms of the frequency function of the fractional Lindley distribution. The parameter of the fractional Lindley distribution is estimated by least squares, weighted least squares, maximum likelihood, moments, and proportions. The moment estimator always exists, so that delta moments result from the nabla Laplace transform of the frequency function of the fractional Lindley distribution. The maximum likelihood estimates have the least mean-square errors. The proportion method works satisfactorily only when the mode of the distribution is null and the proportion of zeros is high. A simulation allows for quantifying the mean-square errors associated with the estimators. A count regression based on the fractional Lindley distribution with data on the total number of stays after hospital admission among U.S. residents aged 65 and over shows that the Akaike information criteria is significantly lower than with the uniform Poisson and Poisson regressions. [ABSTRACT FROM AUTHOR]

There has been much work on the distribution of independent or dependent random variables. But we are not aware of any work giving exact results for the distribution of the sum of randomly weighted random variables. In this paper, we derive exact results for the randomly weighted sum of two dependent random variables. The derived expressions are for the cumulative distribution function, conditional expectation, moment generating function, value at risk, expected shortfall and the limiting tail behavior of the randomly weighted sum of two dependent random variables. Two numerical illustrations are given. [ABSTRACT FROM AUTHOR]

AbstractA new two-parameter discrete distribution, namely the PoiG distribution, is derived by the convolution of a Poisson variate and an independently distributed geometric random variable. This distribution generalizes both the Poisson and geometric distributions and can be used for modeling over-dispersed as well as equi-dispersed count data. A number of important statistical properties of the proposed count model, such as the probability generating function, the moment generating function, the moments, the survival function and the hazard rate function have been studied in detail. Monotonic properties such as the log concavity are also studied. Stochastic ordering for the proposed model is also investigated in detail. Method of moments and the maximum-likelihood estimators of the parameters of the proposed model are found to work well in simulation experiments as well as in the real-life data analysis. Asymptotic confidence intervals and the coverage probabilities of the parameters are also studied through extensive simulation experiments. Two real life modelling applications demonstrate that the proposed distribution may prove to be useful for the practitioners for modelling over-dispersed count data compared to its closest competitors. [ABSTRACT FROM AUTHOR]

Let (Ω , Σ , p) be a probability measure space and let X : Ω → R k be a (vector valued) random variable. We suppose that the probability pX induced by X is absolutely continuous with respect to the Lebesgue measure on R k and set fX as its density function. Let ϕ : R k → R n be a C1-map and let us consider the new random variable Y = ϕ (X) : Ω → R n. Setting m : = max { rank (J ϕ (x)) : x ∈ R k } , we prove that the probability pY induced by Y has a density function fY with respect to the Hausdorff measure H m on ϕ (R k) which satisfies f Y (y) = ∫ ϕ − 1 (y) f X (x) 1 J m ϕ (x) d H k − m (x) , for H m − a. e. y ∈ ϕ (R k). Here J m ϕ is the m-dimensional Jacobian of ϕ. When J ϕ has maximum rank we allow the map ϕ to be only locally Lipschitz. We also consider the case of X having probability concentrated on some m-dimensional sub-manifold E ⊆ R k and provide, besides, several examples including algebra of random variables, order statistics, degenerate normal distributions, Chi-squared and "Student's t" distributions. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the tail behavior of randomly weighted sums and their maxima of dependent subexponential random variables, in which the primary random variables X 1 , ... , X n are real-valued and dependent following two general dependence structures, respectively, and the random weights θ 1 , ... , θ n are another n positive and arbitrarily dependent random variables, but independent of X 1 , ... , X n.. Under some technical conditions, we derive some asymptotic formulas for the tail probability of the randomly weighted sums and their maxima, which coincide with some existing ones in the literature. The merit of our results is that unbounded supports for the random weights are allowed and the distributions of primary random variables can be different. [ABSTRACT FROM AUTHOR]

In this paper, the asmptotics is considered for the distribution tail of a randomly stopped sum ${S_{\nu }}={X_{1}}+\cdots +{X_{\nu }}$ of independent identically distributed consistently varying random variables with zero mean, where ν is a counting random variable independent of $\{{X_{1}},{X_{2}},\dots \}$. The conditions are provided for the relation $\mathbb{P}({S_{\nu }}\gt x)\sim \mathbb{E}\nu \hspace{0.1667em}\mathbb{P}({X_{1}}\gt x)$ to hold, as $x\to \infty $, involving the finiteness of $\mathbb{E}|{X_{1}}|$. The result improves that of Olvera-Cravioto [14], where the finiteness of a moment $\mathbb{E}|{X_{1}}{|^{r}}$ for some $r\gt 1$ was assumed.

Functions ff on [0,1]m{\left[0,1]}^{m} such that every composition f∘(g1,…,gm)f\circ \left({g}_{1},\ldots ,{g}_{m}) with dd-dimensional distribution functions g1,…,gm{g}_{1},\ldots ,{g}_{m} is again a distribution function, turn out to be characterized by a very natural monotonicity condition, which for d=2d=2 means ultramodularity. For m=1m=1 (and d=2d=2), this is equivalent with increasing convexity.