The Exponentiated Type Distributions

Gupta et al. [Commun. Stat., Theory Methods 27, 887–904, 1998] introduced the exponentiated exponential distribution as a generalization of the standard exponential distribution. In this paper, we introduce four more exponentiated type distributions that generalize the standard gamma, standard Weibull, standard Gumbel and the standard Fréchet distributions in the same way the exponentiated exponential distribution generalizes the standard exponential distribution. A treatment of the mathematical properties is provided for each distribution.     

A lifetime model with increasing failure rate

This paper deals with a new two-parameter lifetime distribution with increasing failure rate. This distribution is constructed as a distribution of a random sum of independent exponential random variables when the sample size has a zero truncated binomial distribution. Various statistical properties of the distribution are derived. We estimate the parameters by maximum likelihood and obtain the Fisher information matrix. Simulation studies show the performance of the estimators. Also, estimation of the parameters is considered in the presence of censoring. A real data set is analyzed for illustrative purposes and it is noted that the distribution is a good competitor to the gamma, Weibull, exponentiated exponential, weighted exponential and Poisson-exponential distributions for this data set.     

汽车保险损失率期权定价模型及实证研究

随着汽车保险行业的迅速发展,如何通过证券衍生产品来转嫁汽车保险越发引起人们的重视。本文在Taehan Bae等人的研究基础上给出了当索赔额分布服从指数分布、Γ-分布、混合指数分布、对数正态分布时的汽车保险损失率期权的定价公式,并以太平洋保险公司的有关索赔数据作为样本,利用Γ-分布下的汽车保险损失率期权定价公式对其进行实证研究,得到汽车保险损失率期权价格的近似值,具有很好的理论意义和现实意义。     

An extension of Goldie's result and further results in infinite divisibility

Summary It is shown that gamma random variables satisfy a certain factorisation property. Using this property, Goldie's result on mixtures of the exponential distributions is extended. The note also establishes that the generalized inverse Gaussian distribution and the generalized hyperbolic distribution considered by Barndorff-Nielsen and Halgreen (1977) are indeed self-decomposable.     

一般寿命分布和定时截尾的Bayes变量抽样方案

林（1994）研究了指数分布和定时截尾的变量抽样方案．本文将讨论一般寿命分布和定时截尾的一次抽样方案．在多项式损失函数的假设下,我们讨论了Weibull分布、双参数指数分布和-分布三种情形,并着重讨论Weibull分布的情形．本文还提出了一个可用于近似地确定最优抽样方案的有报算法,并且进行了灵敏度分析,还同林较早的模型（1990,1994）做了比较．     

两个广义伽玛分布之间新的Kullback-Leibler距离

在Kullback-Leibler距离的基础上,对Kullback-Leibler距离进行改进,给出了新的Kullback-Leibler距离,并讨论了它的性质.计算了两个不同广义伽玛分布之间新的Kullback-Leibler距离.推导出伽玛分布、Weibull分布、Rayleigh分布、正态分布、指数分布新的Kullback-Leibler距离.另外在新的KullbackLeibler距离下,还得到digamma函数Ψ(x)=(Γ'(x)/(Γ(x))为单调递增函数.     

The bivariate gamma exponential distribution with application to drought data

The exponential and the gamma distributions have been the traditional models for drought duration and drought intensity data, respectively. However, it is often assumed that the drought duration and drought intensity are independent, which is not true in practice. In this paper, an application of the bivariate gamma exponential distribution is provided to drought data from Nebraska. The exact distributions ofR =X +Y,P =XY andW =X/(X +Y) and the corresponding moment properties are derived whenX andY follow this bivariate distribution.     

Can Data Recognize Its Parent Distribution?

This study is concerned with model selection of lifetime and survival distributions arising in engineering reliability or in the medical sciences. We compare various distributions—including the gamma, Weibull, and lognormal—with a new distribution called geometric extreme exponential. Except for the lognormal distribution, the other three distributions all have the exponential distribution as special cases. A Monte Carlo simulation was performed to determine sample sizes for which survival distributions can distinguish data generated by their own families. Two methods for decision are by maximum likelihood and by Kolmogorov distance. Neither method is uniformly best. The probability of correct selection with more than one alternative shows some surprising results when the choices are close to the exponential distribution.     

Evaluating the lifetime performance index for the exponential lifetime products

Process capability indices (PCIs) are used to measure process potential and performance. Since the lifetime of products generally may possess an exponential, gamma or Weibull distribution, etc., so under a two-parameter exponential distribution, this study constructs a uniformly minimum variance unbiased estimator (UMVUE) of the lifetime performance index based on the right type II censored sample. Then the UMVUE of the lifetime performance index is utilized to develop the new hypothesis testing procedure in the condition of known L. Finally, a practical example is illustrated to employ the testing procedure to determine whether the process is capable.     

Estimates for distributions of the Voronoi polygon's geometric characteristics

An exponential upper bound on the distribution of the Voronoi polygon having n hyperfaces is proved. Using a new integral formula for the Poisson process, the conditional distribution of volume of the fundamental region given n is found to be gamma distribution. This yields an upper bound on distribution of the polygon's volume.     

A Comparison of Two Sequential Metropolis-Hastings Algorithms with Standard Simulation Techniques in Bayesian Inference in Reliability Models Involving a Generalized Gamma Distribution

In this paper we consider the generalized gamma distribution as introduced in Gåsemyr and Natvig (1998). This distribution enters naturally in Bayesian inference in exponential survival models with left censoring. In the paper mentioned above it is shown that the weighted sum of products of generalized gamma distributions is a conjugate prior for the parameters of component lifetimes, having autopsy data in a Marshall-Olkin shock model. A corresponding result is shown in Gåsemyr and Natvig (1999) for independent, exponentially distributed component lifetimes in a model with partial monitoring of components with applications to preventive system maintenance. A discussion in the present paper strongly indicates that expressing the posterior distribution in terms of the generalized gamma distribution is computationally efficient compared to using the ordinary gamma distribution in such models. Furthermore, we present two types of sequential Metropolis-Hastings algorithms that may be used in Bayesian inference in situations where exact methods are intractable. Finally these types of algorithms are compared with standard simulation techniques and analytical results in arriving at the posterior distribution of the parameters of component lifetimes in special cases of the mentioned models. It seems that one of these types of algorithms may be very favorable when prior assessments are updated by several data sets and when there are significant discrepancies between the prior assessments and the data.     

基于先验信息的社会碳成本贝叶斯不确定性分析

针对气候变化及经济影响存在的巨大不确定性,研究了气候变化不确定性以及先验信息对社会碳成本的影响.在贝叶斯理论框架下,采用指数分布刻画气候变化的分布特征,假设尾部变化率是一个随机变量,给出其伽玛先验分布,推导了气候变化分布的贝叶斯先验预测分布.并分别基于指数分布以及帕累托先验预测分布计算了社会碳排放成本.模拟分析发现,在未融合先验信息的情况下,由于尾部概率很小,不管是否修正消费与气候变化之间的关系,截尾社会碳成本和未截尾社会碳成本几乎重合.然而,在利用贝叶斯方法融合先验信息的情况下,社会碳成本容易受到先验信息的影响.但是,通过修正消费与气候变化之间的关系后,发现社会碳成本受先验信息的影响比较少.     

Series representations for densities functions of a family of distributions—Application to sums of independent random variables

Series representations for several density functions are obtained as mixtures of generalized gamma distributions with discrete mass probability weights, by using the exponential expansion and the binomial theorem. Based on these results, approximations based on mixtures of generalized gamma distributions are proposed to approximate the distribution of the sum of independent random variables, which may not be identically distributed. The applicability of the proposed approximations are illustrated for the sum of independent Rayleigh random variables, the sum of independent gamma random variables, and the sum of independent Weibull random variables. Numerical studies are presented to assess the precision of these approximations.     

Classical and Bayesian inference in two parameter exponential distribution with randomly censored data

This paper deal with the classical and Bayesian estimation for two parameter exponential distribution having scale and location parameters with randomly censored data. The censoring time is also assumed to follow a two parameter exponential distribution with different scale but same location parameter. The main stress is on the location parameter in this paper. This parameter has not yet been studied with random censoring in literature. Fitting and using exponential distribution on the range \((0, \infty )\), specially when the minimum observation in the data set is significantly large, will give estimates far from accurate. First we obtain the maximum likelihood estimates of the unknown parameters with their variances and asymptotic confidence intervals. Some other classical methods of estimation such as method of moment, L-moments and least squares are also employed. Next, we discuss the Bayesian estimation of the unknown parameters using Gibbs sampling procedures under generalized entropy loss function with inverted gamma priors and Highest Posterior Density credible intervals. We also consider some reliability and experimental characteristics and their estimates. A Monte Carlo simulation study is performed to compare the proposed estimates. Two real data examples are given to illustrate the importance of the location parameter.     

Inventory control using a service constraint on the expected customer order waiting time

In this paper an analytical model is presented which permits the determination of the minimal inventory reorder point subject to (a) a maximum specified expected customer order waiting time or (b) a maximum specified probability of a customer order waiting more than a predetermined time span. The probability distribution of the customer order waiting time is determined for an arbitrary demand distribution (e.g. normal, gamma or exponential) and under stochastic replenishment lead time conditions by use of arguments from renewal theory. The results established can be used for the customer-oriented control of inventory policy as well as for the analysis of multi-echelon inventory systems.     

Distributional Properties of Exceedance Statistics

Behaviour of a sequence of independent identically distributed random variables with respect to a random threshold is investigated. Three statistics connected with exceeding the threshold are introduced, their exact and asymptotic distributions are derived. Also distribution-free properties, leading to some common and some new discrete distributions, are considered. Identification of equidistribution of observations and the threshold are discussed. In this context relations between the exponential and gamma distributions are studied and a new derivation of the celebrated Laplace expansion for the standard normal distribution function is given.     

Generalized variance estimators in the multivariate gamma models

It is already known that the uniformly minimum variance unbiased (UMVU) estimator of the generalized variance always exists for any natural exponential family. However, in practice, this estimator is often difficult to obtain. This paper provides explicit forms of the UMVU estimators for the bivariate and symmetric multivariate gamma models, which are diagonal quadratic exponential families. For the non-independent multivariate gamma models, it is shown that the UMVU and the maximum likelihood estimators are not proportional.      

Distributional analysis of a generalization of the Polya process

A nonhomogeneous birth process generalizing the Polya process is analyzed, and the distribution of the transition probabilities is shown to be the convolution of a negative binomial distribution and a compound Poisson distribution, whose secondary distribution is a mixture of zero-truncated geometric distributions. A simplified form of the secondary distribution is obtained when the transition intensities have a particular structure, and may sometimes be expressed in terms of Stirling numbers and special functions such as the incomplete gamma function, the incomplete beta function, and the exponential integral. Conditions under which the compound Poisson form of the marginal distributions may be improved to a geometric mixture are also given.     

Bayesian inference and life testing plans for generalized exponential distribution

Recently generalized exponential distribution has received considerable attentions. In this paper, we deal with the Bayesian inference of the unknown parameters of the progressively censored generalized exponential distribution. It is assumed that the scale and the shape parameters have independent gamma priors. The Bayes estimates of the unknown parameters cannot be obtained in the closed form. Lindley’s approximation and importance sampling technique have been suggested to compute the approximate Bayes estimates. Markov Chain Monte Carlo method has been used to compute the approximate Bayes estimates and also to construct the highest posterior density credible intervals. We also provide different criteria to compare two different sampling schemes and hence to find the optimal sampling schemes. It is observed that finding the optimum censoring procedure is a computationally expensive process. And we have recommended to use the sub-optimal censoring procedure, which can be obtained very easily. Monte Carlo simulations are performed to compare the performances of the different methods and one data analysis has been performed for illustrative purposes. This work was partially supported by a grant from the Department of Science and Technology, Government of India     

Characterizations of natural exponential families with power variance functions by zero regression properties

Summary Series of new characterizations by zero regression properties are derived for the distributions in the class of natural exponential families with power variance functions. Such a class of distributions has been introduced in Bar-Lev and Enis (1986) in the context of an investigation of reproductible exponential families. This class is broad and includes the following families: normal, Poisson-type, gamma, all families generated by stable distributions with characteristic exponent an element of the unit interval (among these are the inverse Gaussian, Modified Bessel-type, and Whittaker-type distributions), and families of compound Poisson distributions generated by gamma variates. The characterizations by zero regression properties are obtained in a unified approach and are based on certain relations which hold among the cumulants of the distributions in this class. Some remarks are made indicating how the techniques used here can be extended to obtain characterizations of general exponential families.The work of this author was performed while he was a visitor in the Department of Statistics, State University of New York at Buffalo