Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.     

A bayesian approach to detect informative observations in an experiment

In this article Lindley's (1956) measure of average information is used to measure the loss of information due to the unavailability of a set of observations in an experiment. This measure of loss of information may be used to detect a set of most informative observations in a given design.     

A WEIGHTED RANK MEASURE OF CORRELATION

Spearman's rank correlation coefficient is not entirely suitable for measuring the correlation between two rankings in some applications because it treats all ranks equally. In 2000, Blest proposed an alternative measure of correlation that gives more importance to higher ranks but has some drawbacks. This paper proposes a weighted rank measure of correlation that weights the distance between two ranks using a linear function of those ranks, giving more importance to higher ranks than lower ones. It analyses its distribution and provides a table of critical values to test whether a given value of the coefficient is significantly different from zero. The paper also summarizes a number of applications for which the new measure is more suitable than Spearman's.     

On Transition and First Hitting Time Densities and Moments of the Ornstein–Uhlenbeck Process

This paper derives transition and first hitting time densities and moments for the Ornstein–Uhlenbeck Process (OUP) between exponential thresholds. The densities are obtained by simplifying the process via Doob’s representation into Brownian motion between affine thresholds. The densities in this paper also offer easy-to-use and fast small-time approximations for the densities of OUP between constant thresholds given that exponential thresholds are virtually constant for a small time. This is of interest for estimation with high-frequency data given that extant approaches for constant thresholds impose a large demand on computing power. The moments of the transition distribution up to order n are derived within a closed-form recursive formula that offers valuable information for management. Expressions for the moments of the first hitting time distribution are also obtained in closed form by simplifying integrals via series expansions.     

Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

Undergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.     

CONDITIONING AND CONCENTRATION OF PRINCIPAL COMPONENTS

This paper uses various gauges to construct principal variables that satisfy criteria of maximal scatter. The solutions coincide with Hotelling's (1933) principal components in structured ensembles and mixtures, including heavy-tailed distributions not having moments. Thus, normal-theory tests are exact in level and power under nonstandard models allowing for correlated vector observations and for certain mixtures having neither moments nor unimodal marginals.     

On Efficiency of Multilevel Splitting

This article focuses on estimating rare events using multilevel splitting schemes. The event of interest is that a Markov process enters some rare set before another (“tabu”) set. It is known that in this setting a large deviations analysis is not always sufficient for constructing asymptotically efficient importance sampling schemes; additional modifications to the change of measure suggested by large deviations are needed. As an alternative, we design an asymptotically efficient multilevel splitting scheme that relies on the large deviations analysis only. This property makes it more flexible and easier to implement than corresponding importance sampling schemes.     

Inverse moments of discrete distributions

In this paper an expression for the inverse moment of order r is given for the truncated binomial and Poisson distributions. This enables one to obtain inverse moments in a finite series. Some applications and multivariate generalizations are also given. The method also enables one to obtain relations between inverse moments and factorial moments and distributions of sums of variables.     

Tests for multivariate normality based on canonical correlations

We propose new affine invariant tests for multivariate normality, based on independence characterizations of the sample moments of the normal distribution. The test statistics are obtained using canonical correlations between sets of sample moments in a way that resembles the construction of Mardia’s skewness measure and generalizes the Lin–Mudholkar test for univariate normality. The tests are compared to some popular tests based on Mardia’s skewness and kurtosis measures in an extensive simulation power study and are found to offer higher power against many of the alternatives.     

Yule–Walker type estimators in periodic bilinear models: strong consistency and asymptotic normality

This paper studies the covariance structure and the asymptotic properties of Yule–Walker (YW) type estimators for a bilinear time series model with periodically time-varying coefficients. We give necessary and sufficient conditions ensuring the existence of moments up to eighth order. Expressions of second and third order joint moments, as well as the limiting covariance matrix of the sample moments are given. Strong consistency and asymptotic normality of the YW estimator as well as hypotheses testing via Wald’s procedure are derived. We use a residual bootstrap version to construct bootstrap estimators of the YW estimates. Some simulation results will demonstrate the large sample behavior of the bootstrap procedure.     

MOMENT ESTIMATION IN THE CLASS OF BISEXUAL BRANCHING PROCESSES WITH POPULATION–SIZE DEPENDENT MATING

This paper concerns the estimation of the offspring mean vector, the covariance matrix and the growth rate in the class of bisexual branching processes with population‐size dependent mating. For the proposed estimators, some unconditional moments and some conditioned to non‐extinction are determined and asymptotic properties are established. Confidence intervals are obtained and, as illustration, a simulation example is given.     

A measure of slope-rotatability for second order response surface experimental designs

In the design of experiments for estimating the slope of a response surface, slope-rotatability is a desirable property. In this paper, a measure is introduced that enables us to assess the degree of slope-rotatability for a given response surface design. The measure takes the value zero if and only if the design is slope-rotatable, and becomes larger as the design deviates from a slope-rotatable design. Examples of applying this measure to some response surface designs are also given.     

A class of maximum-entropy multivariate distributions

This paper characterizes a class of multivariate distributions that includes the multinormal and is contained in the exponential family. The wide range of possible applications of these distributions is suggested by some of hte characteristics germane to them: First, they maximize Shannon's entropy among all distributions that have finite moments of given orders. As such, they constitute a class of distributions that includes the multinormal and some likely alternatives. Second, they can exhibit several modes, and, further-more, they do so with a relatively small number of parameters (compared to mixtures of multinormals). Third, they are the stationary distributions of certain diffusion processes. Fourth, they approximate, near the multinormal, the multivariate Pearson family. And fifth, the maximum likelihood estimators of their population moments are the sample moments. Two possible methods of estimating the distributions are studied in this paper: maximum likelihood estimation, and a fast procedure that can be used to find consistent estimators of the parameters via sample moments. A FORTTAN subroutine that implements the latter method is also provided.     

A New Measure of Kurtosis Adjusted for Skewness

Studies of kurtosis often concentrate on only symmetric distributions. This paper identifies a process through which the standardized measure of kurtosis based on the fourth moment about the mean can be written in terms of two parts: (i) an irreducible component, about L₄, which can be seen to occur naturally in the analysis of fourth moments; (ii) terms that depend only on moments of lower order, in particular including the effects of asymmetry attached to the third moment about the mean. This separation of the effect of skewness allows definition of an improved measure of kurtosis. This paper calculates and discusses examples of the new measure of kurtosis for a range of standard distributions.     

The distribution of a moment estimator for a parameter of the generalized poisson distribution

The ratio of the sample variance to the sample mean estimates a simple function of the parameter which measures the departure of the Poisson-Poisson from the Poisson distribution. Moment series to order n?²⁴ are given for related estimators. In one case, exact integral formulations are given for the first two moments, enabling a comparison to be made between their asymptotic developments and a computer-oriented extended Taylor series (COETS) algorithm. The integral approach using generating functions is sketched out for the third and fourth moments. Levin's summation algorithm is used on the divergent series and comparative simulation assessments are given.     

Population moments and moments of the sample mean

This paper presents new formulae which simultaneously express and estimate moments of the sample mean and estimate population moments, from a simple random sample drawn without replcement from a finite population. By avoiding the generality of the multivariate case, these two problems are not only unified but are made significantly more tractable. Explicit solution are given up to eighth moments. Asymptotic results for infinite populations are also given.     

BOUNDS ON BIVARIATE DISTRIBUTION FUNCTIONS WITH GIVEN MARGINS AND MEASURES OF ASSOCIATION

We find pointwise best-possible bounds on the bivariate distribution function of continuous random variables with given margins and a given value of the population version of a nonparametric measure of association such as Kendall's tau or Spearman's rho.     

Information,measure shifts and distribution metrics

Reflexive shifting of a given distribution, using its own distribution function, can reveal information. The shifts are changes in measure such that the separation of the resulting left and right unit shifted distributions reveals the binary entropy of position, called locational or partition entropy. This can be used for spread and asymmetry functions. Alternatively, summary metrics for distributional asymmetry and spread can be based on the relative strengths of left- and right-hand shifts. Such metrics are applicable even for long tail densities where distributional moments may not exist.     

A BAYESIAN INTERPRETATION OF MULTIPLE POINT ESTIMATES

Consider a large number of econometric investigations using different estimation techniques and/or different subsets of all available data to estimate a fixed set of parameters. The resulting empirical distribution of point estimates can be shown - under suitable conditions - to coincide with a Bayesian posterior measure on the parameter space induced by a minimum information procedure. This Bayesian interpretation makes it easier to combine the results of various empirical exercises for statistical decision making. The collection of estimators may be generated by one investigator to ensure the satisfaction of our conditions, or they may be collected from published works, where behavioral assumptions need to be made regarding the dependence structure of econometric studies.     

Pareto Poisson–Lindley distribution with applications

A new lifetime distribution is introduced based on compounding Pareto and Poisson–Lindley distributions. Several statistical properties of the distribution are established, including behavior of the probability density function and the failure rate function, heavy- and long-right tailedness, moments, the Laplace transform, quantiles, order statistics, moments of residual lifetime, conditional moments, conditional moment generating function, stress–strength parameter, Rényi entropy and Song's measure. We get maximum-likelihood estimators of the distribution parameters and investigate the asymptotic distribution of the estimators via Fisher's information matrix. Applications of the distribution using three real data sets are presented and it is shown that the distribution fits better than other related distributions in practical uses.