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In this paper, we prove an extension of the Jajte weak law of large numbers for exchangeable random variables. We make a simulation to illustrate the asymptotic behavior in the sense of convergence in probability for weighted sums of exchangeable weighted random variables. [ABSTRACT FROM AUTHOR]

In this paper, we prove an extension of the Jajte strong law of large numbers for exchangeable random variables, we make a simulation study for the asymptotic behavior in the sense of convergence almost surly for weighted sums of exchangeable weighted random variables. [ABSTRACT FROM AUTHOR]

To strengthen inferences meta‐analyses are commonly used to summarize information from a set of independent studies. In some cases, though, the data may not satisfy the assumptions underlying the meta‐analysis. Using three Bayesian methods that have a more general structure than the common meta‐analytic ones, we can show the extent and nature of the pooling that is justified statistically. In this article, we reanalyze data from several reviews whose objective is to make inference about the COVID‐19 asymptomatic infection rate. When it is unlikely that all of the true effect sizes come from a single source researchers should be cautious about pooling the data from all of the studies. Our findings and methodology are applicable to other COVID‐19 outcome variables, and more generally. [ABSTRACT FROM AUTHOR]

This paper investigates a discrete‐time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First, a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of any function of a surplus sequence in a finite‐time interval. Specifically, the distribution of the maximum surplus is obtained under nonruin conditions. Based on this distribution, the computation of the minimum surplus distribution is given. Asset and risk management–oriented implications are discussed for the obtained distributions based on numerical evaluations. In addition, comparisons are made involving the corresponding results of the classical discrete‐time compound binomial risk model, for which claim occurrences are independent and identically distributed. [ABSTRACT FROM AUTHOR]

Abstract Necessary and sufficient conditions for multivariate total positivity of order 2 (MTP 2) for density functions of some class of exchangeable random variables are obtained. The considered densities occur via symmetrization of particular ordered random variables. As an example, a characterization of the MTP 2 property for the Freund–Weinman multivariate exponential distribution is given. Furthermore, the results are applied to general failure-dependent component lifetimes in systems based on sequential order statistics. In the latter setting, the hazard rate increasing upon failure (HIF) property is also characterized. In particular, the case of underlying distributions satisfying the proportional hazards assumption is considered. The results are supplemented by an analysis of the covariances of the above multivariate exponential distribution. [ABSTRACT FROM AUTHOR]

This paper is concerned with Schur-constant survival models for discrete random variables. Our main purpose is to prove that the associated partial sum process is a non-homogeneous Markov chain. This is shown in three different situations where the random variables considered take values in the sets 0, {0,1} or {0,…,m}, m ≥ 2. The property of Schur-constancy is also compared for these three cases. [ABSTRACT FROM AUTHOR]

In 1986, R. J. Simes proposed a modified Bonferroni test procedure for testing an overall null hypothesis in multiple testing problems, nowadays referred to as the Simes test. The paper of Simes may be considered as a basic step in the development of many new test procedures and new error rate criteria as for example control of the false discovery rate. A key issue is the validity of the Simes test and the underlying Simes inequality under dependence. Although it has been proved that the Simes inequality is valid under suitable assumptions on dependence structures, important cases are not covered yet. In this note we investigate p-values based on exchangeable test statistics in order to explore reasons for the validity or failure of the Simes inequality. We provide sufficient conditions for the asymptotic validity of the Simes inequality and its possible strictness. We also show by means of an easy-to-compute counterexample that exchangeability by itself is not sufficient for the validity of the Simes inequality. [ABSTRACT FROM AUTHOR]

In this paper, robust M-estimation of parameters in a linear system is studied. Some strict theoretical analysis and algorithm analysis about the linear system are given. The results show that the algorithm is simple, and the amount of computation is affected by the number of data is not very large. The results are much better than the least square estimation. [ABSTRACT FROM AUTHOR]

In this paper, the strong consistency of M-estimator in the linear models for exchangeable random variable samples is studied. Some suitably sufficient conditions about the strong consistency of M-estimator are given by performing the strict theoretical analysis. The results are much better than the strong consistency of M-estimator in the linear models for the identically random variable samples. [ABSTRACT FROM PUBLISHER]

We develop our previous works concerning the identification of the collection of significant factors determining some, in general, nonbinary random response variable. Such identification is important, e.g., in biological and medical studies. Our approach is to examine the quality of response variable prediction by functions in (certain part of) the factors. The prediction error estimation requires some cross-validation procedure, certain prediction algorithm, and estimation of the penalty function. Using simulated data, we demonstrate the efficiency of our method. We prove a new central limit theorem for introduced regularized estimates under some natural conditions for arrays of exchangeable random variables. [ABSTRACT FROM PUBLISHER]

The interrelation between signs and permutations in maximum inequalities is studied in this paper. The relationship is based on a lemma that reduces a rearrangement problem to a problem of choosing signs. It helps simplify proofs and find new facts and general settings. The following inequality is one of the main results of the paper: Let x1, ..., xn Σnk=1 xk = 0, be a collection of elements of a normed space X. Then for any collection of signs θ=(θ1,...,θn) and any t > 0, P {π: max1≧k≧n ‖ Σk1 xπ(i) ‖>t} ≧CP {π: max 1≧k≧n ‖Σk1 xπ(i)Σ‖> t/c} where π ϵ IIn, Πn is the group of all permutations of {1, ..., n}, P is the uniform distribution on it, and C is an absolute constant. The inequality is unimprovable (the inverse inequality also holds with some other constant); it generalizes well-known results due to Garsia; Maurey, and Pisier; Kashin and Saakyan; Chobanyan and Salehi; and Levental. All the inequalities of this paper can be restated as maximum inequalities for exchangeable random variables. [ABSTRACT FROM AUTHOR]

We consider two different sets of exchangeable samples which are assumed to be dependent. A single set of observations is obtained from these two dependent samples. The distribution of single order statistic, and the joint distribution of the minimum and an arbitrary order statistic are derived. The results are illustrated in the context of reliability problem. [ABSTRACT FROM AUTHOR]

Most of the research, on the study of the reliability properties of technical systems, assume that the components of the systems operate independently. However, in real life situation, it is more reasonable to assume that there is dependency among the components of the system. In this paper, we consider a ( n− k+1)-out-of- n structure in which the component lifetimes are dependent random variables. We investigate stochastic properties of the inactivity time of the failed components of the system, extending some existing results in the literature where the components of the system are assumed to be independent and identically distributed. The results are then extended to the case where the system has an arbitrary coherent structure with exchangeable components. [ABSTRACT FROM AUTHOR]

We consider coherent and mixed systems with exchangeable components whose lifetimes have positive and finite variances. We present sharp lower and upper bounds on the variance of the system lifetime, expressed in terms of the system signature and the variance of a single component. The bounds are attained for the power and Pareto distributions of the component lifetimes. [ABSTRACT FROM AUTHOR]

During the past few decades, substantial research has been carried out on start-up demonstration tests. In this paper, we study the class of binary start-up demonstration tests under a general framework. Assuming that the outcomes of the start-up tests are described by a sequence of exchangeable random variables, we develop a general form for the exact waiting time distribution associated with the length of the test (i.e., number of start-ups required to decide on the acceptance or rejection of the equipment/unit under inspection). Approximations for the tail probabilities of this distribution are also proposed. Moreover, assuming that the probability of a successful start-up follows a beta distribution, we discuss several estimation methods for the parameters of the beta distribution, when several types of observed data have been collected from a series of start-up tests. Finally, the performance of these estimation methods and the accuracy of the suggested approximations for the tail probabilities are illustrated through numerical experimentation. [ABSTRACT FROM AUTHOR]

Distributions of gaps between neighboring local maxima in sequences of i.i.d. random variables are studied. The sequence of these gaps is shown to be a hidden Markov chain. Explicit formulas for the joint distributions of neighboring gaps are obtained. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we obtain an expression between the sums of the marginal distributions of the order statistics and the common marginal distribution of an exchangeable random sequence. We also derive an expression between the sums of the joint distribution of two order statistics and the two dimensional joint distribution of an exchangeable random sequence. [Copyright &y& Elsevier]

We investigate several aspects of the recently introduced consecutive successes distance failures start-up demonstration test under an exchangeable model. By assuming that the probability of a successful start-up attempt p is a random variable, instead of a fixed value, the outcomes of the successive start-ups become s-dependent exchangeable binary random variables. From a practical point of view, this is a more realistic model than the ordinary model of s-independent start-ups. Critical quantities of the test such as the expected length of the test, the probability of acceptance of the equipment under test, as well as the distribution of the length of the test, are derived. Illustrative numerical examples based on a Beta-mixing distribution for p, and comparisons with the corresponding i.i.d. model, are presented. Furthermore, a comparison study is performed between the consecutive successes distance failures and other competitive start-up demonstration tests. Finally, inferential procedures for the estimation of the unknown parameter(s) of the Beta-mixing distribution are also discussed. [ABSTRACT FROM PUBLISHER]

Abstract: Huang and Kotz (1999) [17] considered a modification of the Farlie–Gumbel–Morgenstern (FGM) distribution, introducing additional parameters, and paved the way for many research papers on modifications of FGM distributions allowing high correlation. The first part of the present paper is a review of recent developments on bivariate Huang–Kotz FGM distributions and their extensions. In the second part a class of new bivariate distributions based on Baker’s system of bivariate distributions is considered. It is shown that for a model of a given order, this class of distributions with fixed marginals which are based on pairs of order statistics constructed from the bivariate sample observations of dependent random variables allows higher correlation than Baker’s system. It also follows that under certain conditions determined by Lin and Huang (2010) [21], the correlation for these systems converges to the maximum Fréchet–Hoeffding upper bound as the sample size tends to infinity. [Copyright &y& Elsevier]

Abstract: Generalized Leibniz triangles have been used in nonextensive statistical mechanics as theoretical models that yield q-Gaussians () as attractors. We study such triangles from a probability point of view. Our results show that one can get any distribution on (or any distribution that has a compact support, after a linear transform) from such triangles, including q-Gaussians with . Next we propose conceptual models that are triangular arrays of row-wise exchangeable random variables and yield q-Gaussians for and as attractors, via laws of large numbers and central limit theorems, respectively. [Copyright &y& Elsevier]

For practical applications in reliability analysis, the assumption of dependence among lifetimes of components of the system is more realistic than the assumption of independence. This paper investigates the residual lifetimes of coherent systems in situation where there exists a stochastic dependence among the components of the system. A stochastic comparison among residual lives of k-out-of-n systems with exchangeable components is conducted. The mean residual life function of a k-out-of-n system with exchangeable components is investigated. Special examples in the case of a system consisting of two dependent components with a joint Gumbel's bivariate exponential distribution and for a system having n components with Marshall and Olkin's multivariate exponential distribution, illustrating the behavior of the mean residual life function are provided. The extension to general coherent systems with exchangeable components using properties of Samaniego's signature are given. [ABSTRACT FROM AUTHOR]

G. Edelman, O. Sporns and G. Tononi have introduced the neural complexity of a family of random variables, defining it as a specific average of mutual information over subfamilies. We show that their choice of weights satisfies two natural properties, namely invariance under permutations and additivity, and we call any functional satisfying these two properties an intricacy. We classify all intricacies in terms of probability laws on the unit interval and study the growth rate of maximal intricacies when the size of the system goes to infinity. For systems of a fixed size, we show that maximizers have small support and exchangeable systems have small intricacy. In particular, maximizing intricacy leads to spontaneous symmetry breaking and lack of uniqueness. [ABSTRACT FROM AUTHOR]

Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and L bounds on the ensemble then give L convergence. [ABSTRACT FROM AUTHOR]

Abstract: Two different exchangeable samples are considered and these two samples are assumed to be independent of each other. From these two samples a new sample is combined and treated as a single set of observations. The distribution of a single order statistic and the joint distribution of two order statistics for a new mixed sample are derived and expressed in terms of joint distribution functions. As a special case the distribution of a single order statistic and the joint distribution of two nonadjacent order statistics from exchangeable random variables are obtained. The results presented in this paper allows widespread applications in modelling of various lifetime data, biomedical sciences, reliability and survival analysis, actuarial sciences etc., where the assumption of independence of data cannot be accepted and the exchangeability is a more realistic assumption. [Copyright &y& Elsevier]

We consider semiparametric multivariate data models based on copula representation of the common distribution function. A copula is characterized by a parameter of association and marginal distribution functions. This parameter and the marginal distributions are unknown. In this article, we study the estimator of the parameter of association in copulas with the marginal distribution functions assumed as nuisance parameters restricted by the assumption that the components are identically distributed. Results of this work could be used to construct special kinds of tests of homogeneity for random vectors having dependent components. [ABSTRACT FROM AUTHOR]

Consecutive successes total failures (CSTF) is a well known start-up demonstration test procedure in which a unit under test is accepted when successive start-up attempts produce a specified number of consecutive successes before a specified number of failures; otherwise the unit is rejected, and in both cases testing (the experiment) is terminated. The CSTF procedure is studied here assuming that the probability of a successful start-up is a random variable. Under this assumption, the outcomes of the attempted start-ups are dependent random variables following an exchangeable model. The unconditional, and the conditional probability distributions of the waiting time (length of the test) are derived and studied in this situation. The first two moments of unconditional waiting time distribution are calculated. Numerical and graphical illustrations are provided, and comparisons are made with the corresponding results for the i.i.d. model. [ABSTRACT FROM AUTHOR]

This article proposes nonparametric inference procedures for analyzing microarray gene expression data that are reliable, robust, and simple to implement. They are conceptually transparent and require no special-purpose software. The analysis begins by normalizing gene expression data in a unique way. The resulting adjusted observations consist of gene-treatment interaction terms (representing differential expression) and error terms. The error terms are considered to be exchangeable, which is the only substantial assumption. Thus, under a family null hypothesis of no differential expression, the adjusted observations are exchangeable and all permutations of the observations are equally probable. The investigator may use the adjusted observations directly in a distribution-free test method or use their ranks in a rank-based method, where the ranking is taken over the whole data set. For the latter, the essential steps are as follows: 1. Calculate a Wilcoxon rank-sum difference or a corresponding Kruskal-Wallis rank statistic for each gene. 2. Randomly permute the observations and repeat the previous step. 3. Independently repeat the random permutation a suitable number of times. Under the exchangeability assumption, the permutation statistics are independent random draws from a null cumulative distribution function (c.d.f.) approximated by the empirical c.d.f. Reference to the empirical c.d.f. tells if the test statistic for a gene is outlying and, hence, shows differential expression. This feature is judged by using an appropriate rejection region or computing a p-value for each test statistic, taking into account multiple testing. The distribution-free analog of the rank-based approach is also available and has parallel steps which are described in the article. The proposed nonparametric analysis tends to give good results with no additional refinement, although a few refinements are presented that may interest some investigators. The implementation is illustrated with a case application involving differential gene expression in wild-type and knockout mice of an E.coli lipopoly-saccharide (LPS) endotoxin treatment, relative to a baseline untreated condition. [ABSTRACT FROM AUTHOR]

Central limit theorems for exchangeable random variables are studied when limits are scale mixtures of normals. First, necessary and sufficient conditions are given under the asymptotic tail probability condition for the mixands:Second, when the weak limits have a particular form, i.e., the mixing measure comes directly from de Finetti's Theorem, necessary and sufficient conditions are given. Finally, some applications are discussed. [ABSTRACT FROM AUTHOR]