On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ _n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.     

On the Exponential Decay Rate of the Tail of a Discrete Probability Distribution

Abstract

We give a sufficient condition for the exponential decay of the tail of a discrete probability distribution π = (π _n ) _n≥0 in the sense that lim _n→∞(1/n) log∑ _i>n π _i  = ?θ with 0 < θ < ∞. We focus on analytic properties of the probability generating function of a discrete probability distribution, especially, the radius of convergence and the number of poles on the circle of convergence. Furthermore, we give an example of an M/G/1 type Markov chain such that the tail of its stationary distribution does not decay exponentially.     

Asymptotic properties of a goodness-of-fit test based on maximum correlations

We study the efficiency properties of the goodness-of-fit test based on the Q _n statistic introduced in Fortiana and Grané [Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions, J. R. Stat. Soc. B 65 (2003), pp. 115–126] using the concepts of Bahadur asymptotic relative efficiency and Bahadur asymptotic optimality. We compare the test based on this statistic with those based on the Kolmogorov–Smirnov, the Cramér-von Mises criterion and the Anderson–Darling statistics. We also describe the distribution families for which the test based on Q _n is locally asymptotically optimal in the Bahadur sense and, as an application, we use this test to detect the presence of hidden periodicities in a stationary time series.     

Asymptotic approach for a renewal-reward process with a general interference of chance

ABSTRACT

In this study, a renewal-reward process with a discrete interference of chance is constructed and considered. Under weak conditions, the ergodicity of the process X(t) is proved and exact formulas for the ergodic distribution and its moments are found. Within some assumptions for the discrete interference of chance in general form, two-term asymptotic expansions for all moments of the ergodic distribution are obtained. Additionally, kurtosis coefficient, skewness coefficient, and coefficient of variation of the ergodic distribution are computed. As a special case, a semi-Markovian inventory model of type (s, S) is investigated.     

The ruin probabilities of a discrete time risk model with one-sided linear claim sizes and dependent risks

This article investigates the ruin probabilities of a discrete time risk model with dependent claim sizes and dependent relation between insurance risks and financial risks. The risk-free and risky investments of an insurer lead to stochastic discount factors {θ_n}_{n ? 1}. The claim sizes are assumed to follow a one-sided linear process with independent and identically distributed (i.i.d.) innovations {?_n}_{n ? 1}. The i.i.d. random pairs {(?_n, θ_n)}_{n ? 1} follow a common bivariate Sarmanov-dependent distribution. When the common distribution of the innovations is heavy tailed, we establish some asymptotic estimates for the ruin probabilities of this discrete time risk model.     

Theoretical results on the discrete Weibull distribution of Nakagawa and Osaki

In this paper, we discuss some theoretical results and properties of the discrete Weibull distribution, which was introduced by Nakagawa and Osaki [The discrete Weibull distribution. IEEE Trans Reliab. 1975;24:300–301]. We study the monotonicity of the probability mass, survival and hazard functions. Moreover, reliability, moments, p-quantiles, entropies and order statistics are also studied. We consider likelihood-based methods to estimate the model parameters based on complete and censored samples, and to derive confidence intervals. We also consider two additional methods to estimate the model parameters. The uniqueness of the maximum likelihood estimate of one of the parameters that index the discrete Weibull model is discussed. Numerical evaluation of the considered model is performed by Monte Carlo simulations. For illustrative purposes, two real data sets are analyzed.     

Nonparametric estimation of regression models with mixed discrete and continuous covariates by the K-nn method

In this article we consider the problem of estimating a nonparametric conditional mean function with mixed discrete and continuous covariates by the nonparametric k-nearest-neighbor (k-nn) method. We derive the asymptotic normality result of the proposed estimator and use Monte Carlo simulations to demonstrate its finite sample performance. We also provide an illustrative empirical example of our method.     

In Memoriam: Martin Taitel,Francis E. McLntyre

A discrete uniform distribution is defined over the integers 1, 2, …, k. Following Tenenbein, who investigated the point estimation of k for a sample drawn without replacement, this paper considers a number of point estimators of k for sampling with replacement and presents conservative confidence sets for k for both methods of sampling.     

Parameter Estimation for the Discrete Stable Family

The discrete stable family constitutes an interesting two-parameter model of distributions on the non-negative integers with a Paretian tail. The practical use of the discrete stable distribution is inhibited by the lack of an explicit expression for its probability function. Moreover, the distribution does not possess moments of any order. Therefore, the usual tools—such as the maximum-likelihood method or even the moment method—are not feasible for parameter estimation. However, the probability generating function of the discrete stable distribution is available in a simple form. Hence, we initially explore the application of some existing estimation procedures based on the empirical probability generating function. Subsequently, we propose a new estimation method by minimizing a suitable weighted L ²-distance between the empirical and the theoretical probability generating functions. In addition, we provide a goodness-of-fit statistic based on the same distance.     

The Least Upper Bound on the Poisson-Negative Binomial Relative Error

In this article, basic mathematical computations are used to determine the least upper bound on the relative error between the negative binomial cumulative distribution function with parameters n and p and the Poisson cumulative distribution function with mean λ =nq = n(1 ? p). Following this bound, it is indicated that the negative binomial cumulative distribution function can be properly approximated by the Poisson cumulative distribution function whenever q is sufficiently small. Five numerical examples are presented to illustrate the obtained result.     

On the error term in bahadur's representation of an order statistic

Bahadur (1966) presented a representation of an order statistic, giving its asymptotic distribution and the rate of convergence, under weak assumptions on the density function of the parent distribution. In this paper we consider the mean(squared) deviation of the error term in Bahadur’s approximation of the q th sample quantile (q_n ). We derive a uniform bound on the mean (squared) deviation of q_n , not depending on the value of q. An application of the given result provides the corresponding result for a kernel type estimator of the q th quantile.     

Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n^?1), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.     

The 3 F 2 with Complex Parameters as Generating Function of Discrete Distribution

A new discrete family of probability distributions that are generated by the ₃ F ₂ function with complex parameters is presented. Some of the properties of this new family are studied as well as methods of estimation for its parameters. It affords considerable flexibility of shape which turns the distribution into an appropriate candidate for modeling data that cannot be adequately fitted by classical families with fewer parameters. Finally, three examples in the fields of Agriculture and Education are included in order to show the versatility and utility of this distribution.     

Limit laws for the cumulative number of ties for the maximum in a random sequence

Let {_Xn,n?1}$\{X_n,n?1\}$ be a sequence of independent identically distributed random variables, taking nonnegative integer values. An observation _Xn$X_n$ is a tie for the maximum if _Xn=max{_X1,…,_Xn-1}$X_n=\max \{X_1,\dots ,X_{n-1}\}$. In this paper, we obtain weak and strong laws of large numbers and central limit theorems for the cumulative number of ties for the maximum among the first n$n$ observations.     

Some characterizations of discrete distributions based on weak records

Let X ₁, X ₂,... be iid random variables (rv's) with the support on nonnegative integers and let (W _n , n≥0) denote the corresponding sequence of weak record values. We obtain new characterization of geometric and some other discrete distributions based on different forms of partial independence of rv's W _n and W _n+r —W _n for some fixed n≥0 and r≥1. We also prove that rv's W ₀ and W _n+1 —W _n have identical distribution if and only if (iff) the underlying distribution is geometric.     

A discrete distribution associated with a pure birth process

A discrete distribution associated with a pure birth process starting with no individuals, with birth rates λ ⁿ =λ forn=0, 2, …,m−1 and λ ⁿ forn≥m is considered in this paper. The probability mass function is expressed in terms of an integral that is very convenient for computing probabilities, moments, generating functions and others. Using this representation, the mean and the k-th factorial moments of the distribution are obtained. Some nice characterizations of this distribution are also given.     

A note on LeCam’s bound for the distance between the Poisson binomial and the Poisson distribution

n possibly different success probabilities p ₁, p ₂, ..., p _n is frequently approximated by a Poisson distribution with parameter λ = p ₁ + p ₂ + ... + p _n . LeCam's bound p ² ₁ + p ² ₂ + ... + p _n ² for the total variation distance between both distributions is particularly useful provided the success probabilities are small. The paper presents an improved version of LeCam's bound if a generalized d-dimensional Poisson binomial distribution is to be approximated by a compound Poisson distribution. Received: May 10, 2000; revised version: January 15, 2001     

Approximate maximum likelihood estimation in the presence of extra-poisson variation

A two-stage hierarchical model for analysis of discrete data with extra-Poisson variation is examined. The model consists of a Poisson distribution with a mixing lognormal distribution for the mean. A method of approximate maximum likelihood estimation of the parameters is proposed. The method uses the EM algorithm and approximations to facilitate its implementation are derived. Approximate standard errors of the estimates are provided and a numerical example is used to illustrate the method.     

On the Markov property of order statistics

The order statistics from a sample of size n≥3 from a discrete distribution form a Markov chain if and only if the parent distribution is supported by one or two points. More generally, a necessary and sufficient condition for the order statistics to form a Markov chain for (n≥3) is that there does not exist any atom x₀ of the parent distribution F satisfying F(x₀-)>0 and F(x₀)<1. To derive this result a formula for the joint distribution of order statistics is proved, which is of an interest on its own. Many exponential characterizations implicitly assume the Markov property. The corresponding putative geometric characterizations cannot then be reasonably expected to obtain. Some illustrative geometric characterizations are discussed.