Order statistics from discrete distributions

Several recurrence relations and identities available for single and product moments of order1 statistics in a sample size n from an arbitrary continuous distribution are extended for the discrete case,, Making use of these recurrence relations it is shown that it is sufficient to evaluate just two single moments and (n-l)/2 product moments when n is odd and two single moments and {n-2)/2 product moments when n is even, in order to evaluate the first, second and product moments of order statistics in a sample of size n drawn from an arbitrary discrete distribution, given these moments in samples of sizes n-1 and less.. A series representation for the product moments of order statistics is derived.. Besides enabling us to obtain an exact and explicit expression for the product moments of order statistics from the geometric distribution, it. makes the computation of the product moments of order statistics from other discrete distributions easy too.     

Moments of the sampled autocovariances and autocorrelations for a Gaussian white-noise process

Cumulants, moments about zero, and central moments are obtained for the mean-corrected serial covariances and serial correlations for series realizations of length n from a white-noise Gaussian process. All first and second moments (and some third, fourth, and higher moments) are given explicitly for the serial covariances; and the corresponding moments for the serial correlations are derived either explicitly or implicitly.     

Generalization of Fisher's z-transformation

We deal with the asymptotic expansions of the means and the variances of the correlation coefficients in truncated bivariate normal populations. The Fisher's z-transformation is generalized for stabilizing variance in a truncated normal population. The Hermite moments are introduced, and the relationship among cross moments, central cross moments, and Hermite moments are discussed.     

An approximation of the inverse moments of the positive hypergeometric distribution

The negative moments of the positive hyper geometric distribution are often approximated by the inverse of the positive moments of this distribution. In this paper, a suitable approximation to the positive hypergeometric distribution is used to obtain the negative moments.     

Relations and identities for the moments of order statistics from a sample containing a single outlier

In this paper, we derive several new recurrence relations and indentities satisfied by the single and the product moments of order statistics from a sample of size n in the presence of an outlier. These recurrence relations involve the first two single moments and the product moments in samples of sized n?1 and less. By making use of these recurrence relations we show that it is sufficient to evaluate at most two single rerents and (n?2)/2 product moment when n is even and two single moments and (n?2)/2 product moments when n is odd, in order to evaluate the first and second single moments and product moments of all order statistics in a sample of size n comprising an outlier, given these moments for the all sample of size less than n. These generalize the results of Govindarajulu (1963), Joshi (1971), and Joshi and Balakrishnan (1982) to the case when the sample includes a single outlier. We also establish some simple identitites involving linear combination of convariances of order statistics. These results can be effectively used to reduce the amount of numerical computation considerably and also to check the accuracy of the computations while evaluating means, variances and covariances of order statistics from an outlier model.     

On fractional moments of quadratic expressions in normal variables1

Fractional moments, product cumulants and product moments of general quadratic expressions in singular and nonsingular normal variables are explicitly evaluated. A general method of deriving such moments is also indicated. Particular cases art; shown to agree with known results.     

Moments of scale mixtures of skew-normal distributions and their quadratic forms

We obtain the first four moments of scale mixtures of skew-normal distributions allowing for scale parameters. The first two moments of their quadratic forms are obtained using those moments. Previous studies derived the moments, but all relevant results do not allow for scale parameters. In particular, it is shown that the mean squared error becomes an unbiased estimator of σ² with skewed and heavy-tailed errors. Two measures of multivariate skewness are calculated.     

Incomplete moments of modified power series districutions with applications

The recurrence relations between the incomplete moments and the factorial incomplete moments of the modified power series distributions (MPSD) are derived. These relations are employed to obtain the experessions for the incomplete moments and the incomplete factorial moments of some particular members of the MPSD class such as the generalized negative binomial, the generalized Poisson, the generalized logrithmic series, the lost game distribution and the distribution of the number of customers served in a busy period. An application of the incomplete moments of the generalized Poisson distribution is provided in the economic selection of a manufactured product. A numerical example is provided using the Poisson distribution and the Generalized Poisson distribution. The example illustrates the difference in results using the two models     

On Order Statistics from Sine Distribution

We derive closed form expressions for the first two moments of order statistics from the sine distribution. For the higher moments, a recurrence relation is given. We also give a recurrence relation for the product moments. These relations will be useful for moment computations based on ordered data.     

A Proposal for a New Bound for Discrete Distributions

In this work we re-examine some classical bounds for non negative integer-valued random variables by means of information theoretic or maxentropic techniques using fractional moments as constraints. The proposed new bound, no more analytically expressible in terms of moments or moment generating function (mgf), is built by mixing classical bounds and the Maximum Entropy (ME) approximant of the underlying distribution; such a new bound is able to exploit optimally all the information content provided by the sequence of given moments or by the mgf. Particular care will be devoted to obtain fractional moments from the available information given in terms of integer moments and/or moment generating function. Numerical examples show clearly that the bound improvement involving the ME approximant based on fractional moments is not trivial.     

Computational Algorithms for Moments of Accumulated Markov and Semi-Markov Rewards

Power moments for accumulated rewards defined on Markov and semi-Markov chains are studied. A model with mixed time-space termination of reward accumulation is considered for inhomogeneous in time rewards and Markov chains. Characterization of power moments as minimal solutions of recurrence system of linear equations, sufficient conditions for finiteness of these moments and upper bounds for them, expressed in terms of so-called test functions, are given. Backward recurrence algorithms for funding of power moments of accumulated rewards and various time-space truncation approximations reducing dimension of the corresponding recurrence relations are described.     

Relationships Between Central Moments and Cumulants,with Formulae for the Central Moments of Gamma Distributions

Abstract

Simple expressions are presented that relate cumulants to central moments without involving moments about the origin. These expressions are used to obtain recursive formulae for the central moments of the gamma distribution, with exponential and chi-square distributions as special cases.     

Estimating Moments in Linear Mixed Models

In this article, we investigate estimating moments, up to fourth order, in linear mixed models. For this estimation, we only assume the existence of moments. The obtained estimators of the model parameters and the third and fourth moments of the errors and random effects are proved to be consistent or asymptotically normal. The estimation provides a base for further statistical inference such as confidence region construction and hypothesis testing for the parameters of interest. Moreover, the method is readily extended to estimate higher moments. A simulation is carried out to examine the performance of this estimating method.     

Moment density estimation for positive random variables

An unknown moment-determinate cumulative distribution function or its density function can be recovered from corresponding moments and estimated from the empirical moments. This method of estimating an unknown density is natural in certain inverse estimation models like multiplicative censoring or biased sampling when the moments of unobserved distribution can be estimated via the transformed moments of the observed distribution. In this paper, we introduce a new nonparametric estimator of a probability density function defined on the positive real line, motivated by the above. Some fundamental properties of proposed estimator are studied. The comparison with traditional kernel density estimator is discussed.     

A new bivariate beta distribution

A new bivariate beta distribution capable of providing better fits than all its competitors is introduced. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities and conditional moments. The method of maximum likelihood is used to derive the associated estimation procedure. Applications to six bivariate data sets are illustrated.     

Order statistics from the linear-exponential distribution,part I: increasing hazard rate case

For the linear-exponential distribution with increasing hazard rate, exact and explicit expressions for means, product moments and percentage points of order statistics are obtained. Some recurrence relations for both single and product moments of order statistics are also derived. These recurrence relations would enable one to obtain all the higher order moments of order statistics for all sample sizes from those of the lower order     

Relationships among moments of order statistics in samples from two related outlier models and some applications

Balakrishnan (1987a) has recently shown that the moments of order statistics in samples drawn from a continuous population with pdf f(x) symmetric about zero comprising a single outlier with pdf g(x) also symmetric about zero can be expressed in terms of the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero and the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero comprising a single outlier with pdf obtained by folding g(x) at zero. The cumulative round off error involved in the numerical evaluation of the moments of order statistics from the symmetric outlier model, using a table of the moments of order statistics from the folded population and the moments of order statistics from the folded outlier model, has also been studied by Balakrishnan (1987a) and shown to be not serious. Making use of these results we study here the robustness of some estimators of th location and scale parameters of a double exponential distribution.     

A NOTE ON MOMENTS OF PROGRESSIVELY TYPE II CENSORED ORDER STATISTICS

ABSTRACT

Upper and lower bounds for moments of progressively Type II censored order statistics in terms of moments of (progressively Type II censored) order statistics are derived. In particular, this yields conditions for the existence of moments of progressively Type II censored order statistics based on an absolutely continuous distribution function.     

Estimation of product moments of a stationary stochastic process with application to estimation of cumulants and cumulant spectral densities

Inference concerning the structure of stationary stochastic processes can be investigated by looking at properties of various cumulant spectral densities of order two and higher. However, except for cases when cumulants and product moments are identical, estimation of higher-order cumulant spectral densities has been restricted by the dependence of higher-order cumulants on lower-order product moments. By first estimating product moments and then using an identity between product moments and cumulants, asymptotically unbiased and consistent estimates of cumulants are obtained. This in turn leads to asymptotically unbiased and consistent estimators of higher-order cumulant spectral densities. In addition, asymptotic normality of product-moment estimators is exhibited under weak dependence.     

Moments and generating functions for the absortion distribution and its negative binomial analogue

The moments of the absorption are difficult to obtain. The generating functions are basic hypergeometric functions. This paper shows how to define two shift operators that allow elementary arguments to be used to develop recursions for the expected values of general functions. The exact moments of the distribution follow. The generating function for the negative binomial analogue gives the moments directly.