A Construction of Lancaster Probabilities with Margins in the Multidimensional Meixner Class

The well-known Meixner class (Meixner, 1934) of probabilities on R has been extended recently to R ^d (Pommeret, 1996). This generalized Meixner class corresponds to the simple quadratic natural exponential families characterized by Casalis (1996). Following Lancaster (1975), the present paper offers a characterization of the joint probability of a randomvector ( X, Y ) such that the two variables X and Y on R ^d belong to the multidimensional Meixner class and fulfil a bi-orthogonality condition involving orthogonal polynomials. The joint probabilities, called Lancaster probabilities, are characterized by two sequences of orthogonal polynomials with respect to the margins and a sequence of expectations of products. Some multivariate probabilities are studied, namely the Poisson-Gaussian and the gamma-Gaussian.     

A REGRESSION CHARACTERISATION OF THE MEIXNER HYPERGEOMETRIC DISTRIBUTION

The paper describes the Meixner hypergeometric distribution, characterised by properties of the regression of products of linear transformations of random variables with respect to residuals.     

On Stein Identity,Chernoff Inequality,and Orthogonal Polynomials

In this article, we present a general method for deriving Stein-like identity and Chernoff-like inequality based on orthogonal polynomials. In order to illustrate our method, some applications are given with respect to normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution, not only for random variables but also for random vectors, resulting corresponding Stein-like identity and Chernoff-like inequality are obtained consequently. Within our best knowledge, some of our matrix version results are new in the literature. In addition, forward difference formulae of Charlier polynomials, Krawtchouk polynomials and Meixner polynomials, Stein-like identity, and Chernoff-like inequality with respect to Beta distribution, as well as Rodrigues formula of Meixner polynomials are also prepared in the first time within our limited information. Interestingly, as far as normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution are concerned, we found that their Stein-like identity and corresponding Chernoff-like inequality are related closely, by examining their Rodrigues formula.     

EXACT AND APPROXIMATED RELATIONS BETWEEN NEGATIVE HYPERGEOMETRIC AND NEGATIVE BINOMIAL PROBABILITIES

We derive orthogonal expansions in terms of the Meixner polynomials of the first kind for hypergeometric probabilities. We show how these expansions can be used to obtain negative binomial approximations to negative hypergeometric probabilities. Some limit properties of these approximations are studied and also the extension of these results to cumulative probabilities.     

ODD MAN OUT—THE MEIXNER HYPERGEOMETRIC DISTRIBUTION

Two open problems are described for the hyperbolic secant distribution, as a special case of the more general Meixner hypergeomet-ric distribution. The first concerns the completeness of the family of functions sech θ x , θ≥ 0. The second concerns the characterization of the class of canonical correlation sequences in bivariate distributions with marginals sech θ x , sech θ y. In both cases some partial results are put forward.     

On Some Applications of a Symbolic Representation of Non Centered Lévy Processes

By using a symbolic technique known in the literature as the classical umbral calculus, we characterize two classes of polynomials related to Lévy processes: the Kailath-Segall and the time-space harmonic polynomials. We provide the Kailath-Segall formula in terms of cumulants and we recover simple closed-forms for several families of polynomials with respect to not centered Lévy processes, such as the Hermite polynomials with Brownian motion, Poisson-Charlier polynomials with Poisson processes, actuarial polynomials with Gamma processes, first kind Meixner polynomials with Pascal processes, and Bernoulli, Euler, and Krawtchuk polynomials with suitable random walks.     

劳动收入份额的决定与波动机制研究

本文通过数理经济模型的理论分析和计量经济模型的实证分析发现,决定社会产品初次分配中劳动收入份额高低的因素,除了政府的生产税税率之外,主要是劳动者的实际工资水平和生产技术水平,且二者的作用方向完全相反,都取决于要素替代弹性的大小。当资本和劳动的替代弹性小于1时,劳动者的实际工资水平越高,劳动收入份额就越高,劳动者的生产技术水平越高,劳动收入份额就越低;而当资本和劳动的替代弹性大于1时,则劳动者的实际工资水平越高,劳动收入份额就越低;劳动者的生产技术水平越高,劳动收入份额就越高。理论和实证的分析还表明,劳动者货币工资水平和实际工资水平的增长随着经济周期的波动而波动,实际工资水平增长率的波动与经济周期的波动方向相反,导致劳动收入份额也随经济周期反向波动。     

Some practical advice on polynomial regression analysis from blocked response surface designs

It is often necessary to run response surface designs in blocks. In this paper the analysis of data from such experiments, using polynomial regression models, is discussed. The definition and estimation of pure error in blocked designs are considered. It is recommended that pure error is estimated by assuming additive block and treatment effects, as this is more consistent with designs without blocking. The recovery of inter-block information using REML analysis is discussed, although it is shown that it has very little impact if the design is nearly orthogonally blocked. Finally prediction from blocked designs is considered and it is shown that prediction of many quantities of interest is much simpler than prediction of the response itself.     

A statistical test for the hypothesis of Gaussian random function

A Gaussian random function is a functional version of the normal distribution. This paper proposes a statistical hypothesis test to test whether or not a random function is a Gaussian random function. A parameter that is equal to 0 under Gaussian random function is considered, and its unbiased estimator is given. The asymptotic distribution of the estimator is studied, which is used for constructing a test statistic and discussing its asymptotic power. The performance of the proposed test is investigated through several numerical simulations. An illustrative example is also presented.     

不完全信息下纯交换帕累托最优的实现研究

现有的理论中，纯交换帕累托最优状况不是唯一的，且认为只要进行交换就必然能实现帕累托最优。其原因是纯交换帕累托最优条件不充分和交换原则存有缺陷。根据交换是一个讨价还价的过程，采用博弈分析方法，在不完全信息条件下，只要遵循理性人交换原则，交换必能实现，但不一定是帕累托最优。如果信息不对称方所判断的效用函数没有改变对方真实效用函数的偏好性质，交换必然能达到唯一的帕累托最优状况，但利益分割不公平，存在交换剥削；如果效用函数的偏好性质改变，帕累托最优不能实现，而是帕累托改进，但其消费状态也是唯一的。     

OPTIMAL TESTS OF SIGNIFICANCE

To perform a test of significance of a null hypothesis, a test statistic is chosen which is expected to be small if the hypothesis is false. Then the significance level of the test for an observed sample is the probability that the test statistic, under the assumptions of the hypothesis, is as small, or smaller than, its observed value. A "good" test statistic is taken to be one which is stochastically small when the null hypothesis is false. Optimal test statistics are defined using this criterion and the relationship of these methods to the Neyman-Pearson theory of hypothesis testing is considered.     

Locally best invariant test for outliers in a gamma type distribution

It is shown that the locally best invariant test for the existence of outliers for scale parameters of the gamma distribution is given by Bartholomew's test for exponentiality which is the ratio of the sum of squares of the data to the square of the sample mean. The optimality robustness, including null and nonnull robustness of the test is shown. A small simulation study to compare the power among the other eight competitive tests for testing exponentiality is performed. It is seen that the locally best invariant test is not always best but is reasonably good. It is slightly better than Cochran's test and suffers less from the limiting masking effect.     

Setup adjustment under unknown process parameters and fixed adjustment cost

Consider a machine that can start production off-target where the initial offset is unknown and unobservable. The goal is to determine the optimal series of machine adjustments that minimize the expected value of the sum of quadratic off-target costs and fixed adjustment costs. Apart of the unknown initial offset, the process is supposed to be in a state of statistical control, so the process model is applicable to discrete-part production processes. The process variance is also assumed unknown. We show, using a dynamic programming formulation based on the Bayesian estimation of all unknown process parameters, how the optimal process adjustment policy is of a deadband form where the width of the deadband is time-varying and U-shaped. Computational results and implementation details are presented. The simpler case of a known process variance is also solved using a dynamic programming approach. It is shown that the solution to this case is a good approximation to the first case, when the variance is actually unknown. The unknown process variance solution, however, is the most robust with respect to variation in the process parameters.     

A simple and asymptotically optimal test for the equality of k(≥2) exponential distributions based on type II censored samples

A test for the equality of two or more two-parameter exponential distributions is suggested. It is developed on an intuitive basis and is obtained by combining two independent tests by the Fisher method (1950, pp. 99-101). The test is simple for application and is optimal asymptotically in the sense of Bahadur efficiency (1960). A numerical example is discussed to illustrate its application in a real-world situation. The Monte Carlo simulation is used for calculating its power which is compared with that of the test suggested by Singh and Narayan (1983). The suggested test is found oftener more powerful.     

Category Distinguishability and Observer Agreement

It is common in the medical, biological, and social sciences for the categories into which an object is classified not to have a fully objective definition. Theoretically speaking the categories are therefore not completely distinguishable. The practical extent of their distinguishability can be measured when two expert observers classify the same sample of objects. It is shown, under reasonable assumptions, that the matrix of joint classification probabilities is quasi-symmetric, and that the symmetric matrix component is non-negative definite. The degree of distinguishability between two categories is defined and is used to give a measure of overall category distinguishability. It is argued that the kappa measure of observer agreement is unsatisfactory as a measure of overall category distinguishability.     

Partitioned cross-validation

Partitioned cross-validation is proposed as a method for overcoming the large amounts of across sample variability to which ordinary cross-validation is subject. The price for cutting down on the sample noise is that a type of bias is intriduced. A theory is presented for optimal trade-off of this variance and bias. Comparison with other bandwidth selection methods is given.     

On choosing between fixed and random block effects in some no-interaction models

The purpose of this article is to strengthen the understanding of the relationship between a fixed-blocks and random-blocks analysis in models that do not include interactions between treatments and blocks. Treating the block effects as random has been recommended in the literature for balanced incomplete block designs (BIBD) because it results in smaller variances of treatment contrasts. This reduction in variance is large if the block-to-block variation relative to the total variation is small. However, this analysis is also more complicated because it results in a subjective interpretation of results if the block variance component is non-positive. The probability of a non-positive variance component is large precisely in those situations where a random-blocks analysis is useful – that is, when the block-to-block variation, relative to the total variation, is small. In contrast, the analysis in which the block effects are fixed is computationally simpler and less subjective. The loss in power for some BIBD with a fixed effects analysis is trivial. In such cases, we recommend treating the block effects as fixed. For response surface experiments designed in blocks, however, an opposite recommendation is made. When block effects are fixed, the variance of the estimated response surface is not uniquely estimated, and in practice this variance is obtained by ignoring the block effect. It is argued that a more reasonable approach is to treat the block effects to be random than to ignore it.     

Sequential search using question-sets with bounded intersections

The paper is concerned with sequential search on a finite set: an unknown element of the finite set is to be found testing its subsets, and getting the information that the unknown element is or is not an element of the tested subset. The optimum of the strategy lengths is found under the condition that the intersection of any two different test-sets is bounded. This condition is generalized taking the intersection of any m different test-sets instead of two, and the general problem is also solved.     

Book Reviews

This article uses a Bayesian unit-root test in stochastic volatility models. The time series of interest is the volatility that is unobservable. The unit-root testing is based on the posterior odds ratio, which is approximated by Markov-chain Monte Carlo methods. Simulations show that the testing procedure is efficient for moderate sample size. The unit-root hypothesis is rejected in seven market indexes, and some evidence of nonstationarity is observed in the TWSI of Taiwan.     

A Modified Knox Test of Space-Time Clustering

Tests of space-time clustering such as the Knox test are used by epidemiologists in the preliminary analysis of datasets where an infectious aetiology is suspected. The Knox test statistic is the number of cases close in both space and time to another case. The test statistic proposed here is the excess number of such cases over that expected under H₀ of no infection. It is argued that this modified test is more powerful than the Knox test, because the test statistic is not heavily tied as is the Knox test statistic. The use of the test is illustrated with examples.