On some moving-average processes with exponentially distributed innovations

Summary In this paper we discusse the stationary sequence of random variables which are formed from an independent identically distributed sequence, according to the moving-average model of ordern. Some properties of the process are considered. The joint bivariate exponential distribution is given, as well as the distribution of the sum.     

Modified slashed generalized exponential distribution

Abstract

In this article, we introduce a new distribution for modeling positive data sets with high kurtosis, the modified slashed generalized exponential distribution. The new model can be seen as a modified version of the slashed generalized exponential distribution. It arises as a quotient of two independent random variables, one being a generalized exponential distribution in the numerator and a power of the exponential distribution in the denominator. We studied various structural properties (such as the stochastic representation, density function, hazard rate function and moments) and discuss moment and maximum likelihood estimating approaches. Two real data sets are considered in which the utility of the new model in the analysis with high kurtosis is illustrated.     

Power binomial exponential distribution: Modeling,simulation and application

ABSTRACT

The binomial exponential 2 (BE2) distribution was proposed by Bakouch et al. as a distribution of a random sum of independent exponential random variables, when the sample size has a zero truncated binomial distribution. In this article, we introduce a generalization of BE2 distribution which offers a more flexible model for lifetime data than the BE2 distribution. The hazard rate function of the proposed distribution can be decreasing, increasing, decreasing–increasing–decreasing and unimodal, so it turns out to be quite flexible for analyzing non-negative real life data. Some statistical properties and parameters estimation of the distribution are investigated. Three different algorithms are proposed for generating random data from the new distribution. Two real data applications regarding the strength data and Proschan's air-conditioner data are used to show that the new distribution is better than the BE2 distribution and some other well-known distributions in modeling lifetime data.     

A new multivariate model involving geometric sums and maxima of exponentials

We study the joint distribution of (X, Y, N), where N has a geometric distribution while X and Y are, respectively, the sum and the maximum of N i.i.d. exponential random variables. We present fundamental properties of this class of mixed trivariate distributions, and discuss their applications. Our results include explicit formulas for the marginal and conditional distributions, joint integral transforms, moments and related parameters, stability properties, and stochastic representations. We also derive maximum likelihood estimators for the parameters of this distribution, along with their asymptotic properties, and briefly discuss certain generalizations of this model. An example from finance, where N represents the duration of the growth period of the daily log-returns of currency exchange rates, illustrates the modeling potential of this model.     

The Exponentiated G Poisson Model

We study a new family of distributions defined by the minimum of the Poisson random number of independent identically distributed random variables having a general exponentiated G distribution. Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics and their moments, reliability, and Shannon entropy are derived. Maximum likelihood estimation of the model parameters is investigated. Two special models of the new family are discussed. We perform an application to a real data set to show the potentiality of the proposed family.     

On linear combinations of independent exponential variables

The exact distribution of a linear combination of n indepedent negative exponential random variables , when the coefficients cf the linear combination are distinct and positive , is well-known. Recently Ali and Obaidullah (1982) extended this result by taking the coeff icients to be arbitrary real numbers. They used a lengthy geometric.

al approach to arrive at the result . This article gives a simple derivation of the result with the help of a generalized partial fraction technique. This technique also works when the variables involved are gamma variables with certain types of parameters. Results are presented in a form which can easily be programmed for computational purposes. Connection of this problem t o various problems in different fields is also pointed out.     

On order statistics from non-identical right-truncated exponential random variables and some applications

By considering order statistics arising from n independent non-identically distributed right-truncated exponential random variables, we derive in this paper several recurrence relations for the single and the product moments of order statistics. These recurrence relations are simple in nature and could be used systematically in order to compute all the single and the product moments of order statistics for all sample sizes in a simple recursive manner. The results for order statistics from a multiple-outlier model (with a slippage of p observations) from a right-truncated exponential population are deduced as special cases. These results will be useful in assessing robustness properties of any linear estimator of the unknown parameter of the right-truncated exponential distribution, in the presence of one or more outliers in the sample. These results generalize those for the order statistics arising from an i.i.d. sample from a right-truncated exponential population established by Joshi (1978, 1982).     

Higher order moments of order statistics from INID symmetric random variables

In this paper we present analogues of Balakrishnan's (1989) relations that relate the triple and quadruple moments of order statistics from independent and nonidentically distributed (I.NI.D.) random variables from a symmetric distribution to those of the folded distribution. We then apply these results, along with the corresponding recurrence relations for the exponential distribution derived recently by Childs (2003), to study the robustness of the Winsorized variance.     

Large Deviation Principles for Sequences of Maxima and Minima

In this article, we consider sequences of i.i.d. random variables and, under suitable conditions on the (common) distribution function, we prove large deviation principles for sequences of maxima, minima and pairs formed by maxima and minima. The i.i.d. random variables can be either unbounded or bounded; in the first case maxima and minima have to be suitably normalized.     

Slashed generalized exponential distribution

In this paper, we introduce an extension of the generalized exponential (GE) distribution, making it more robust against possible influential observations. The new model is defined as the quotient between a GE random variable and a beta-distributed random variable with one unknown parameter. The resulting distribution is a distribution with greater kurtosis than the GE distribution. Probability properties of the distribution such as moments and asymmetry and kurtosis are studied. Likewise, statistical properties are investigated using the method of moments and the maximum likelihood approach. Two real data analyses are reported illustrating better performance of the new model over the GE model.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

Regression mean residual life of a parallel system of dependent components

In this paper, we consider a system consisting of two dependent components and we are interested in the average remaining life of the component that fails last when (i) the first failure occurs at time t and (ii) the first failure occurs after time t. For both the cases, expressions are derived in the case of general bivariate normal distribution and a class of bivariate exponential distribution including bivariate exponential distribution of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution.     

指数族分布中参数的经验贝叶斯估计

指数族分布是一类应用广泛的分布类,包括了泊松分布、Gamma分布、Beta分布、二项分布等常见分布.在非寿险中,索赔额或索赔次数过程常常被假定服从指数族分布,由于风险的非齐次性,指数族分布中的参数θ也为随机变量,假定服从指数族共轭先验分布.此时风险参数的估计落入了Bayes框架,风险参数θ的Bayes估计被表达“信度”形式.然而,在实际运用中,由于先验分布与样本分布中仍然含有结构参数,根据样本的边际分布的似然函数估计结构参数,从而获得风险参数的经验Bayes估计,最后证明了该经验Bayes估计是渐近最优的.     

A Mixed Bivariate Distribution Connected with Geometric Maxima of Exponential Variables

We study the joint distribution of X and N, where N has a geometric distribution and X is the maximum of N i.i.d. exponential variables, independent of N. We present basic properties of these mixed bivariate distributions and discuss parameter estimation for this model. An example from finance, where N represents the number of consecutive positive daily log-returns of currency exchange rates, illustrates stochastic modeling potential of these laws.     

Bayesian D-optimal design for logistic regression model with exponential distribution for random intercept

ABSTRACT

Nowadays, generalized linear models have many applications. Some of these models which have more applications in the real world are the models with random effects; that is, some of the unknown parameters are considered random variables. In this article, this situation is considered in logistic regression models with a random intercept having exponential distribution. The aim is to obtain the Bayesian D-optimal design; thus, the method is to maximize the Bayesian D-optimal criterion. For the model was considered here, this criterion is a function of the quasi-information matrix that depends on the unknown parameters of the model. In the Bayesian D-optimal criterion, the expectation is acquired in respect of the prior distributions that are considered for the unknown parameters. Thus, it will only be a function of experimental settings (support points) and their weights. The prior distribution of the fixed parameters is considered uniform and normal. The Bayesian D-optimal design is finally calculated numerically by R3.1.1 software.     

The Joint Distribution of the Sum and the Maximum of IID Exponential Random Variables

The authors establish the joint distribution of the sum X and the maximum Y of IID exponential random variables. They derive exact formuli describing the random vector (X, Y), including its joint PDF, CDF, and other characteristics; marginal and conditional distributions; moments and related parameters; and stochastic representations leading to further properties of infinite divisibility and self-decomposability. The authors also discuss parameter estimation and include an example from climatology that illustrates the modeling potential of this new bivariate model.     

On the power of goodness-of-fit tests for the exponential distribution under progressive Type-II censoring

The aim of this article is to review existing goodness-of-fit tests for the exponential distribution under progressive Type-II censoring and to provide some new ideas and adjustments. In particular, we consider two-parameter exponentially distributed random variables and adapt the proposed test procedures to our scenario if necessary. Then, we compare their power by an extensive simulation study. Furthermore, we propose five new test procedures that provide reasonable alternatives to those already known.     

Exact likelihood inference for Laplace distribution based on Type-II censored samples

We develop exact inference for the location and scale parameters of the Laplace (double exponential) distribution based on their maximum likelihood estimators from a Type-II censored sample. Based on some pivotal quantities, exact confidence intervals and tests of hypotheses are constructed. Upon conditioning first on the number of observations that are below the population median, exact distributions of the pivotal quantities are expressed as mixtures of linear combinations and of ratios of linear combinations of standard exponential random variables, which facilitates the computation of quantiles of these pivotal quantities. Tables of quantiles are presented for the complete sample case.     

Note on sums of a random number of random variables with exponential family distributions

With the help of the result that exponential-type families are determined by their mean value functions it is shown that stochastic independence of the random variables S_N and N-S_N characterizes the Poisson and Bernoulli distributions simultaneously.     

Positive dependence of random variables with a common marginal distribution

In this paper we review some notions of positive dependence of random variables with a common univariate marginal distribution and describe the related moment and probability inequalities. We first present a comparison between i.i.d. random variables and exchangeable random variables via an application of de Finetti's theorem, then describe some useful probability inequalities via partial orderings of the strength of their positive dependence. Finally, we state a result for random variables which are not necessarily exchangeable. Special applications to the multivariate normal distribution will be discussed, and the results involve only the correlation matrix of the distribution.