Characterization of probability distributions by conditional expectation of function of record statistics

In this paper, two general classes of distributions have been characterized through conditional expectation of power of difference of two record statistics. Further, some particular cases and examples are also discussed.     

Inequalities for probability contents of convex sets via geometric average

It is shown that: If (X₁, X₂) is a permutation invariant central convex unimodal random vector and if A is a symmetric (about 0) permutation invariant convex set then P{(aX₁, X₂/a) A} is nondecreasing as a varies from )+ to 1 and is non-increasing as a varies from 1 to ∞ (that is, P{(a₁X₁, a₂X₂) ε A} is a Schur-concave function of (log a₁, log a₂). Some extensions of this result for the n-dimensional case are discussed. Applications are given for elliptically contoured distributions and scale parameter families.     

Identifiability of mixtures of power-series distributions and related characterizations

The concept of the identifiability of mixtures of distributions is discussed and a sufficient condition for the identifiability of the mixture of a large class of discrete distributions, namely that of the power-series distributions, is given. Specifically, by using probabilistic arguments, an elementary and shorter proof of the Lüxmann-Ellinghaus's (1987,Statist. Probab. Lett.,5, 375–378) result is obtained. Moreover, it is shown that this result is a special case of a stronger result connected with the Stieltjes moment problem. Some recent observations due to Singh and Vasudeva (1984,J. Indian Statist. Assoc.,22, 93–96) and Johnson and Kotz (1989,Ann. Inst. Statist. Math.,41, 13–17) concerning characterizations based on conditional distributions are also revealed as special cases of this latter result. Exploiting the notion of the identifiability of power-series mixtures, characterizations based on regression functions (posterior expectations) are obtained. Finally, multivariate generalizations of the preceding results have also been addressed.     

Stationarity and almost sure divergence of time averages in interval-valued probability

Our work is motivated by the study of empirical processes (such as flicker noise) that occur in stable systems yet give rise to observations with seemingly divergent time averages. Stationary models for such processes do not exist in the domain of numerical probability, as the ergodic theorems dictate the convergence of time averages of stationary and bounded processes. This has led us to investigate such models in the wider framework of interval-valued probability. In this paper we construct interval-valued probabilities on the space of infinite binary sequences that combine properties of (i) strict stationarity, (ii) unicity of extension from the algebra of cylinder sets to a wider collection containing salient asymptotic events, and (iii) almost sure support of divergence of time averages. These properties are not shared by conventional stochastic models.     

Imprecise stochastic processes in discrete time: global models,imprecise Markov chains,and ergodic theorems

We justify and discuss expressions for joint lower and upper expectations in imprecise probability trees, in terms of the sub- and supermartingales that can be associated with such trees. These imprecise probability trees can be seen as discrete-time stochastic processes with finite state sets and transition probabilities that are imprecise, in the sense that they are only known to belong to some convex closed set of probability measures. We derive various properties for their joint lower and upper expectations, and in particular a law of iterated expectations. We then focus on the special case of imprecise Markov chains, investigate their Markov and stationarity properties, and use these, by way of an example, to derive a system of non-linear equations for lower and upper expected transition and return times. Most importantly, we prove a game-theoretic version of the strong law of large numbers for submartingale differences in imprecise probability trees, and use this to derive point-wise ergodic theorems for imprecise Markov chains.     

Ruin probability and joint distributions of some actuarial random vectors in the compound pascal model

The compound negative binomial model,introduced in this paper,is a discrete time version.We discuss the Markov properties of the surplus process,and study the ruin probability and the joint distributions of actuarial random vectors in this model.By the strong Markov property and the mass function of a defective renewal sequence,we obtain the explicit expressions of the ruin probability,the finite-horizon ruin probability,the joint distributions of T,U(T-1),|U(T)| and 0≤inn相似文献   

Quantization of probability distributions and gradient flows in space dimension 2

In this paper we study a perturbative approach to the problem of quantization of probability distributions in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view [10], [12], [15], we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strict minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a new mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations.     

Characterization and statistical test using truncated expectations for a class of skew distributions

The expectation of left truncated Waring and Pareto distributions is a linear function of the point of truncation. Based on this property, a characterization theorem and statistical tests can be constructed.     

Point and Interval Estimation of P(X < Y): The Normal Case with Common Coefficient of Variation

The problem of estimating R = P(X < Y) originated in the context of reliability where Y represents the strength subjected to a stress X. In this paper we consider the problem of estimating R when X and Y have independent normal distributions with equal coefficient of variation. The maximum likelihood estimation of R when the coefficient of variation is known and when it is unknown is studied. The asymptotic variance of the estimators are obtained and asymptotic confidence intervals are provided. An example is presented to illustrate the procedure. Finally some simulation studies are carried out to study the coverage probability and the lengths of the confidence interval. In particular, lengths of the confidence intervals are compared with and without the assumption of common coefficient of variation. It is observed that the assumption of common coefficient of variation results in considerably tighter intervals.     

Log-concavity of stirling numbers and unimodality of stirling distributions

A series of inequalities involving Stirling numbers of the first and second kinds with adjacent indices are obtained. Some of them show log-concavity of Stirling numbers in three different directions. The inequalities are used to prove unimodality or strong unimodality of all the subfamilies of Stirling probability functions. Some additional applications are also presented.     

Average behavior of greedy algorithms for the minimization knapsack problem: General coefficient distributions

For the minimization knapsack problem with Boolean variables, primal and dual greedy algorithms are formally described. Their relations to the corresponding algorithms for the maximization knapsack problem are shown. The average behavior of primal and dual algorithms for the minimization problem is analyzed. It is assumed that the coefficients of the objective function and the constraint are independent identically distributed random variables on [0, 1] with an arbitrary distribution having a density and that the right-hand side d is deterministic and proportional to the number of variables (i.e., d = μn). A condition on μ is found under which the primal and dual greedy algorithms have an asymptotic error of t.     

Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions

In the present paper, our aim is to establish several formulas involving integral transforms, fractional derivatives, and a certain family of extended generalized hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results. A probability density function involving the extended generalized hypergeometric function is introduced, and its properties are studied. The corresponding properties of some of the classical probability distributions and their associated probability density functions are easily derivable as special cases of our general results. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonnull distributions of two test criteria for independence under local alternatives

For testing the independence of q-sets in a p-variate normal population, the asymptotic distributions of the likelihood ratio test, and the test proposed by the author under local alternatives are derived in terms of noncentral χ² variates.     

The Fourier-series method for inverting transforms of probability distributions

This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this purpose, we also describe two methods for inverting Laplace transforms based on the Post-Widder inversion formula. The overall procedure is illustrated by several queueing examples.     

Concentration inequalities and laws of large numbers under epistemic and regular irrelevance

his paper presents concentration inequalities and laws of large numbers under weak assumptions of irrelevance that are expressed using lower and upper expectations. The results build upon De Cooman and Miranda’s recent inequalities and laws of large numbers. The proofs indicate connections between the theory of martingales and concepts of epistemic and regular irrelevance.     

Jessen–Wintner type random variables and fractal properties of their distributions

Formulas are given for the Lebesgue measure and the Hausdorff–Besicovitch dimension of the minimal closed set S_ξ supporting the distribution of the random variable ξ = 2^–k τ_k, where τ_k are independent random variables taking the values 0, 1, 2 with probabilities p _0k , p _1k , p _2k , respectively. A classification of the distributions of the r.v. ξ via the metric‐topological properties of S_ξ is given. Necessary and sufficient conditions for superfractality and anomalous fractality of S_ξ are found. It is also proven that for any real number a ₀ ∈ [0, 1] there exists a distribution of the r.v. ξ such that the Hausdorff–Besicovitch dimension of S_ξ is equal to a ₀. The results are applied to the study of the metric‐topological properties of the convolutions of random variables with independent binary digits, i.e., random variables ξⁱ = , where η_k are independent random variables taking the values 0 and 1. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplication of distributions in any dimension: Applications to δ-function and its derivatives

In two previous papers the author introduced a multiplication of distributions in one dimension and he proved that two one-dimensional Dirac delta functions and their derivatives can be multiplied, at least under certain conditions. Here, mainly motivated by some engineering applications in the analysis of the structures, we propose a different definition of multiplication of distributions which can be easily extended to any spatial dimension. In particular we prove that with this new definition delta functions and their derivatives can still be multiplied.     

Sets of desirable gambles: Conditioning, representation, and precise probabilities

The theory of sets of desirable gambles is a very general model which covers most of the existing theories for imprecise probability as special cases; it has a clear and simple axiomatic justification; and mathematical definitions are natural and intuitive. However, much work remains to be done until the theory of desirable gambles can be considered as generally applicable to reasoning tasks as other approaches to imprecise probability are. This paper gives an overview of some of the fundamental concepts for reasoning with uncertainty expressed in terms of desirable gambles in the finite case, provides a characterization of regular extension, and studies the nature of maximally coherent sets of desirable gambles, which correspond to finite sequences of probability distributions, each one of them defined on the set where the previous one assigns probability zero.     

�������������������������������ʼ���ǿ��������

Strong laws of large numbers play key role in nonadditive probability theory. Recently, there are many research papers about strong laws of large numbers for independently and identically distributed (or negatively dependent) random variables in the framework of nonadditive probabilities (or nonlinear expectations). This paper introduces a concept of weakly negatively dependent random variables and investigates the properties of such kind of random variables under aframework of nonadditive probabilities and sublinear expectations. A strong law of large numbers is also proved for weakly negatively dependent random variables under a kind of sublinear expectation as an application