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A general methodology is developed forapproximatingthe distribution of a random variable on the basis of itsexactmoments. More specifically, a probability density function is approximated by the product of a suitable weight function and a linear combination of its associated orthogonal polynomials. A technique for generating a sequence of orthogonal polynomials from a given weight function is provided and the coefficients of the linear combination are explicitly expressed in terms of the moments of the target distribution. On applying this approach to several test statistics, we observed that the resulting percentiles are consistently in excellent agreement with the tabulated values. As well, it is explained that the same moment-matching technique can be utilized to produce densityestimateson the basis of thesamplemoments obtained from a given set of observations. An example involving a well-known data set illustrates the density estimation methodology advocated herein. [ABSTRACT FROM AUTHOR]

moment-based methodology is proposed for approximating the distribution of the distance between two random points belonging to sets that are composed of rectangles. The resulting density approximants are expressed in terms of polynomially adjusted beta density functions. Two norms are being considered: the L¹ norm referred to as the Manhattan distance and the L² norm which corresponds to the Euclidean distance. A few illustrative examples will be presented and certain applications to transportation and routing problems will be pointed out. [ABSTRACT FROM AUTHOR]

Peña and Rodríguez (2002) introduced a portmanteau test for time series which turns out to be more powerful than those proposed by Ljung and Box (1986) and Monti (1994), and approximated its distribution by means of a two-parameter gamma random variable. A polynomially adjusted beta approximation is proposed in this paper. This approximant is based on the moments of the statistic, which can be estimated by simulation or determined by symbolic computations or numerical integration. Various types of time series processes such as AR (1) , MA (1) , ARMA (2, 2) are being considered. The proposed approximation turns out to be nearly exact. [ABSTRACT FROM AUTHOR]

Abstract: An explicit representation of the elements of the inverses of certain patterned matrices involving the moments of nonnegative weight functions is derived in this paper. It is shown that a sequence of monic orthogonal polynomials can be generated from a given weight function in terms of Hankel-type determinants and that the corresponding matrix inverse can be expressed in terms of their associated coefficients and orthogonality factors. This result enables one to obtain an explicit representation of a certain type of approximants which apply to a wide class of positive continuous functions. Convenient expressions for the coefficients of standard classical orthogonal polynomials such as Legendre, Jacobi, Laguerre and Hermite polynomials are also provided. Several examples illustrate the results. [Copyright &y& Elsevier]

Nous obtenons dans cet article la distribution exacte d'une statistique dont le numérateur et le dénominates sont indépendants et sont respectivement constitués d'une combinaison linéaire de k variables khi-deux indépendantes et de la racine k-ième d'un produit de k variables khi-deux indépendantes. Une telle structure se presente lors de l'étude de relations fonctionnelles dans certains modèles linéaires multidimensionnels. Nous dérivons la fonction de densité exacte de cette statistique en termes de séries infinies en évaluant sa transformée inverse de Mellin. [ABSTRACT FROM AUTHOR]