An identity for the product moments of order statistics

A general identity for the product moments of successive order statistics is given, which is valid in a class of probability distributions including Weibull, Pareto, exponential and Burr distributions.     

On the nearness of record values to order statistics from Pitman’s measure of closeness

In the two-sample prediction problem, record values from the present sample may be used as predictors of order statistics from a future sample. In this paper, we investigate the nearness of record statistics (upper and lower) to order statistics from a location-scale family of distributions in the sense of Pitman closeness and discuss the corresponding monotonicity properties. We then determine the closest record value to a specific order statistic from a future sample. Even though in general it depends on the parent distribution, exact and explicit expressions are derived for the required probabilities in the case of exponential and uniform distributions, and some computational results are presented as well. Finally, we consider the mean squared error criterion and examine the corresponding results in the exponential case.     

A general recurrence relation for moments of order statistics in a class of probability distributions and characterizations

Summary In a class of distribution functions, including exponential, power function, Pareto, Lomax, and logistic distributions, a general recurrence relation for moments of order statistics is given. The validity of this identity for certain constants and some sequence of order statistics leads to characterizations of probability distributions. Several recurrence relations and characterization results known from the literature are particular cases of the theorems stated.     

L-statistics from multivariate unified skew-elliptical distributions

We study here the distributions of order statistics and linear combinations of order statistics from a multivariate unified skew-elliptical distribution. We show that these distributions can be expressed as mixtures of unified skew-elliptical distributions, and then use these mixture forms to study some distributional properties and moments.     

Characterizations of the Pareto and related distributions

In this note, a class of Pareto distributions is characterized based on the Shannon entropy of k-record statistics. As a consequence of that characterizations of the uniform and exponential distributions are given. Received: October 1999     

Moment inequalities for order statistics with applications to characterizations of distributions

General inequalities of Hölder type between moments of order statistics and moments of record values respectively are derived. Special choices of the involved sample sizes and ranks and discussions of when equality is attained in these inequalities yield several characterizations of well known distributions, such as the uniform, polynomial, Pareto, reflected Pareto, exponential, Weibull distribution and some others.     

Exact inference for the difference of Laplace location parameters

We consider exact procedures for testing the equality of means (location parameters) of two Laplace populations with equal scale parameters based on corresponding independent random samples. The test statistics are based on either the maximum likelihood estimators or the best linear unbiased estimators of the Laplace parameters. By conditioning on certain quantities we manage to express their exact distributions as mixtures of ratios of linear combinations of standard exponential random variables. This allows us to find their exact quantiles and tabulate them for several sample sizes. The powers of the tests are compared either numerically or by simulation. Exact confidence intervals for the difference of the means corresponding to those tests are also constructed. The exact procedures are illustrated via a real data example.     

Sharp exponential and entropy bounds on expectations of generalized order statistics

Sharp lower and upper bounds on expected values of generalized order statistics are proven by the use of Moriguti's inequality combined with the Young inequality. The bounds are expressed in terms of exponential moments or entropy. They are attainable providing new characterizations of some nontrivial distributions. Received October 2001/Revised May 2002     

Testing outliers from an exponential population

Summary Likes obtains the distributions ofDixon’s statistics for an exponential population and tabulates upper 100α% points (α=0.1, 0.05, 0.01) of some of these distributions. The distribution of these statistics can be expressed in terms of finite series of beta functions and hence the probabilities of the rejection of suspected observed outliers may be easily calculated on a desk calculator. Thus we may avoid the difficult task of tabulating 100α% values of these statistics.     

Top-<Emphasis Type="Italic">k</Emphasis>-lists

Top-k-lists are introduced as sequences of k-dimensional random vectors with ordered components being k largest observations from a sequence of independent identically distributed random variables. Such lists changing in time are natural stochastic models of ranking tables which appear in many situations in real life, when one wants to keep a track of several best results in a given field. Here we study basic properties of top-k-lists as joint distributions, conditional structures, representations, driving examples of top-k-lists from exponential and uniform distributions, asymptotics and a relation to generalized order statistics.     

Statistical inference in simplicially contoured sample distributions

For generalizations of the n-dimensional two parameter exponential distribution with identical marginals with threshold and dispersion parameters the exact distributions of estimators and test statistics are given. Under certain conditions the consistency of the estimators and the rate of convergence is shown. Therefore generalized Gamma- and F-distributions are defined.     

Conditional distributions of generalized order statistics and some characterizations

Generalized order statistics have been introduced in Kamps (1995a). They enable a unified approach to several models of ordered random variables, e.g. (ordinary) order statistics, record values, sequential order statistics, record values from non-identical distributions. The purpose of this paper is to develop conditional distributions of one generalized order statistic given another and to characterize the underlying continuous distribution by different conditional expectations. Well-known results for ordinary order statistics and record values are extended to generalized order statistics. Received: July 1997     

Specification diagnostics based on Laguerre alternatives for econometric models of duration

Score tests of the null hypothesis of exponentially distributed durations (conditional on regressors) against alternatives in a family of approximations to arbitrary distributions for non-negative random variables are developed. The test statistics take a simple, easily calculated and interpreted form. The alternatives considered are expansions of various orders in Laguerre polynomials. An economic model generating exponential unemployment duration is presented. The statistics are applied in a look at unemployment durations in the Denver Income Maintenance Experiment data.     

Distribution of Dixon’s statistics in the case of an exponential population

Summary The distributions of Dixon’s statistics for the case of the sample from an exponential population are found. For some of these statistics the upper 100 α percentage points (α=0,1; 0,05 and 0,01) are tabulated.

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Zusammenfassung Die Verteilungen der Variablen von Dixon werden für den Fall der Stichproben aus einer exponentialen Grundgesamtheit gefunden. Für einige dieser Variablen werden die oberen 100 α Prozentpunkte (α=0,1; 0,05 und 0,01) tabelliert.

     

Generalized densities of order statistics

Let X ₁, ... , X _n be independent identically distributed random variables with distribution F . We derive expressions for generalized joint 'densities' of order statistics of X ₁, ... , X _n , for arbitrary distributions F , in terms of Radon–Nikodym derivatives with respect to product measures based on F . We then give formulae for conditional distributions of order statistics and use them to derive results concerning Markov properties of order statistics, formulae for distributions of trimmed sums, and other useful representations. Our approach leads to simple and natural expressions which appear not to have been given before.     

Optimal weighted average power similar tests for the covariance structure in the linear regression model

This paper suggests a procedure for the construction of optimal weighted average power similar tests for the error covariance matrix of a Gaussian linear regression model when the alternative model belongs to the exponential family. The paper uses a saddlepoint approximation to construct simple test statistics for a large class of problems and overcomes the computational burden of evaluating the complicated integrals arising in the derivation of optimal weighted average power tests. Extensions to panel data models are considered. Applications are given to tests for error autocorrelation in the linear regression model and in a panel data framework.     

面板数据非线性阈值协整的检验方法

非线性阈值协整是线性协整的后续发展。本文使用两机制TR模型对Westerlund和Edgerton(2005)的面板数据协整向量结构突变模型进行扩展,提出截距项具有阈值效应、截距项和斜率系数都具有阈值效应的面板数据非线性阈值协整模型。在此基础上,本文进而分别构造Zc、Ztc、Zr、Ztr统计量检验阈值协整,并对上述统计量的极限分布进行了数学推导,发现它们都收敛于随机泛函。仿真实验结果表明,有限样本下上述检验统计量具有较小的水平扭曲和较高的检验势。     

中国宏观经济变量的时间趋势属性研究

本文提出了一种新的时间趋势属性的检验方法,该方法融合了非线性模型与线性模型。本文构建了三个Wald类检验统计量及一个稳健检验统计量,推导出了这些统计量的极限分布并分析了其有限样本下的统计性质。应用该检验程序,本文分析了我国24个重要宏观经济变量的时间趋势属性,结果表明,其中22个经济变量具有非线性平滑转移特征,其时间趋势属性表现为确定性。     

Reparameterized inverse gamma regression models with varying precision

In this article, we propose a mean linear regression model where the response variable is inverse gamma distributed using a new parameterization of this distribution that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive response variable, as usual in the context of generalized linear models. The variance function of the proposed model has a quadratic form. The inverse gamma distribution is a member of the exponential family of distributions and has some distributions commonly used for parametric models in survival analysis as special cases. We compare the proposed model to several alternatives and illustrate its advantages and usefulness. With a generalized linear model approach that takes advantage of exponential family properties, we discuss model estimation (by maximum likelihood), black further inferential quantities and diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the obtained results. A real application using minerals data set collected by Department of Mines of the University of Atacama, Chile, is considered to demonstrate the practical potential of the proposed model.