The Moments Do Not Determine a Distribution

Table 1 corrects the critical values for testing normality reported by Lilliefors (1967). The corrected table allows us to derive a simple analytic approximation to the upper tail probabilities of his test statistic for probabilities less than .10. With few exceptions, the approximation is more accurate than Lilliefors's original table.     

Two Moments of the Logitnormal Distribution

We display the first two moment functions of the Logitnormal(μ, σ²) family of distributions, conveniently described in terms of the Normal mean, μ, and the Normal signal-to-noise ratio, μ/σ, parameters that generate the family. Long neglected on account of the numerical integrations required to compute them, awareness of these moment functions should aid the sensible interpretation of logistic regression statistics and the specification of “diffuse” prior distributions in hierarchical models, which can be deceiving. We also use numerical integration to compare the correlation between bivariate Logitnormal variables with the correlation between the bivariate Normal variables from which they are transformed.     

Matching Three Moments with Minimal Acyclic Phase Type Distributions

Abstract

A number of approximate analysis techniques are based on matching moments of continuous time phase type (PH) distributions. This paper presents an explicit method to compose minimal order continuous time acyclic phase type (APH) distributions with a given first three moments. To this end we also evaluate the bounds for the first three moments of order n APH distributions (APH(n)). The investigations of these properties are based on a basic transformation, which extends the APH(n ? 1) class with an additional phase in order to describe the APH(n) class.     

All Invariant Moments of the Wishart Distribution

Abstract.  In this paper, we compute moments of a Wishart matrix variate U of the form E ( Q ( U )) where Q ( u ) is a polynomial with respect to the entries of the symmetric matrix u , invariant in the sense that it depends only on the eigenvalues of the matrix u . This gives us in particular the expected value of any power of the Wishart matrix U or its inverse U ^{− 1}. For our proofs, we do not rely on traditional combinatorial methods but rather on the interplay between two bases of the space of invariant polynomials in U . This means that all moments can be obtained through the multiplication of three matrices with known entries. Practically, the moments are obtained by computer with an extremely simple Maple program.     

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ _n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.     

Moments Determine the Tail of a Distribution (But Not Much Else)

How much information does a finite collection of moments carry about the underlying distribution? We revive an old bound, give a new, simple formula for its calculation, and demonstrate that although very little can be said about the central part of the distribution, the tail is much more sharply defined.     

Estimation of the Lognormal-Pareto Distribution Using Probability Weighted Moments and Maximum Likelihood

This article deals with the estimation of the lognormal-Pareto and the lognormal-generalized Pareto distributions, for which a general result concerning asymptotic optimality of maximum likelihood estimation cannot be proved. We develop a method based on probability weighted moments, showing that it can be applied straightforwardly to the first distribution only. In the lognormal-generalized Pareto case, we propose a mixed approach combining maximum likelihood and probability weighted moments. Extensive simulations analyze the relative efficiencies of the methods in various setups. Finally, the techniques are applied to two real datasets in the actuarial and operational risk management fields.     

Moments of Order Statistics from Weibull Distribution in the Presence of Multiple Outliers

In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X₁, X₂, …, X_{n ? p} are independent with density function f(x) while the remaining, p observations (outliers) X_{n ? p + 1}, …, X_n are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.     

A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution

In this paper we give a class of row-column designs with the property that the i-th row and the j-th column have precisely r treatments in common. A conjecture that such designs are quasi-factorial is disproved by showing that the designs given in this paper are not quasi-factorial. It is also shown that the designs given here are nearly optimal.     

Moments of a ratio of two quadratic forms

In this paper, we give the exact moments of a ratio of qua- dratic forms in normal variables, where the quadratic forms are not assumed to be independent. This problem was tackled by other authors who gave approximations and partial results. Here we will give the exact moments for the general case.     

Moments for a ratio of correlated gamma variates

This paper considers the finite integral moments for the ratio, R = X/Y, where X and Y re correlated gamma distributed variables. An analytical and numerical comparison is given for two classes of underlying bivariate gamma distributions. It is shown that the two bivariate gamma structures provide indentical experessions for the mth unadjussted moment, E(R^m), if and only if either of the following conditions hold : 1) X and Y are uncorrelated of 2) m=1. A numerical evaluation is performed to determine the extent that the two methods differ whenever the variables are correlated     

The Exact General Formulas for the Moments of a Ridge Regression Estimator when the Regression Error Terms Follow a Multivariate t Distribution

Huang (1999 Huang , J. C. ( 1999 ). Improving the estimation precision for a selected parameter in multiple regression analysis: an algebraic approach . Econ. Lett. 62 : 261 – 264 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed a feasible ridge regression (FRR) estimator to estimate a specific regression coefficient. Assuming that the error terms follow a normal distribution, Huang (1999 Huang , J. C. ( 1999 ). Improving the estimation precision for a selected parameter in multiple regression analysis: an algebraic approach . Econ. Lett. 62 : 261 – 264 .[Crossref], [Web of Science ®] , [Google Scholar]) examined the small sample properties of the FRR estimator. In this article, assuming that the error terms follow a multivariate t distribution, we derive an exact general formula for the moments of the FRR estimator to estimate a specific regression coefficient. Using the exact general formula, we obtain exact formulas for the bias, mean squared error (MSE), skewness, and kurtosis of the FRR estimator. Since these formulas are very complex, we compare the bias, MSE, skewness, and kurtosis of the FRR estimator with those of ordinary least square (OLS) estimator by numerical evaluations. Our numerical results show that the range of MSE dominance of the FRR estimator over the OLS estimator is widen under a fat tail distributional assumption.     

Moments of the trace of a noncentral wishart matrix

Several articles have appeared on the moments of the trace of a noncentral Wishart matrix. Partial results are given in these papers but the representations of these partial results are often too cumbersome for any practical use. In this paper general results are given in compact form which are readily usable in practical problems.     

Moments of a product Pearson type VII density distribution

A new Pearson type VII distribution is introduced by taking the product of two Pearson type VII pdfs. Detailed moment properties and particular cases are derived.     

Moments of sample moments

The problem offinding expressions for sampling moments of sample moments has been ahistorically old one. This problem is treated here, with the use of partitions and multi partitions , for the univariate as well as the multivariate case. The systematic combinatorial approach minimizes the chance of omitting any

contributions and making errors in their computation. Componentwise identification is made possible , soerrors can be located. From the complete set of general moment formulae, s pecial cases may be obtained by identifying identical variables.     

Relations for Moments of Progressively Type-II Censored Order Statistics from Log-Logistic Distribution with Applications to Inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a log-logistic distribution. The use of these relations in a systematic recursive manner would enable the computation of all the means, variances and covariances of progressively Type-II right censored order statistics from the log-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁,…, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Malik (1987 Balakrishnan , N. , Malik , H. J. ( 1987 ). Moments of order statistics from truncated log-logistic distribution . J. Statist. Plann. Infer. 17 : 251 – 267 .[Crossref], [Web of Science ®] , [Google Scholar]) and Balakrishnan et al. (1987 Balakrishnan , N. , Malik , H. J. , Puthenpura , S. ( 1987 ). Best linear unbiased estimation of location and scale parameters of the log-logistic distribution . Commun. Statist. Theor. Meth. 16 : 3477 – 3495 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The moments so determined are then utilized to derive best linear unbiased estimators for the scale- and location-scale log-logistic distributions. A comparison of these estimates with the maximum likelihood estimates is made through Monte Carlo simulation. The best linear unbiased predictors of progressively censored failure times is then discussed briefly. Finally, a numerical example is presented to illustrate all the methods of inference developed here.     

Estimation of the Length Distribution of Marine Populations in the Gaussian-multinomial Setting using the Method of Moments

In this paper, the problem of estimation of the length distribution of marine populations in the Gaussian-multinomial model is considered. For the purpose of the mean and covariance parameter estimation, the method of moments estimators are developed. That is, minimum variance linear unbiased estimator for the mean frequency vector is derived and a consistent estimator for the covariance matrix of the length observations is presented. The usefulness of the proposed estimators is illustrated with an analysis of real cod length measurement data.     

The Sample Selection Model from a Method of Moments Perspective

It is shown how the usual two-step estimator for the standard sample selection model can be seen as a method of moments estimator. Standard GMM theory can be brought to bear on this model, greatly simplifying the derivation of the asymptotic properties of this model. Using this setup, the asymptotic variance is derived in detail and a consistent estimator of it is obtained that is guaranteed to be positive definite, in contrast with the estimator given in the literature. It is demonstrated how the MM approach easily accommodates variations on the estimator, like the two-step IV estimator that handles endogenous regressors, and a two-step GLS estimator. Furthermore, it is shown that from the MM formulation, it is straightforward to derive various specification tests, in particular tests for selection bias, equivalence with the censored regression model, normality, homoskedasticity, and exogeneity.