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A moment-based density approximation technique that is based on a generalization of Pearson's system of frequency curves is introduced in this paper. More specifically, the derivative of the logarithm of a continuous density function is expressed as a ratio of polynomials whose coefficients are determined by solving a linear system, and a simple representation of the resulting density function is provided. Additionally, a result relating a sample to its moments is stated and derived. It is then explained that, when used in conjunction with sample moments, the methodology being herein advocated can be utilized for the purpose of modeling data sets, irrespective of their size. Several illustrative examples are presented. [ABSTRACT FROM AUTHOR]

This paper provides a simplified representation of the exact density function of R, the sample correlation coefficient. The odd and even moments of R are also obtained in closed forms. Being expressed in terms of generalized hypergeometric functions, the resulting representations are readily computable. Some numerical examples corroborate the validity of the results derived 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]

Abstract: A bivariate gamma-type density function involving a confluent hypergeometric function of two variables is being introduced. The inverse Mellin transform technique is employed in conjunction with the transformation of variable technique to obtain its moment generating function, which is expressed in terms of generalized hypergeometric functions. Its cumulative distribution function is given in closed form as well. Many distributions such as the bivariate Weibull, Rayleigh, half-normal and Maxwell distributions can be obtained as limiting cases of the proposed gamma-type density function. Computable representations of the moment generating functions of these distributions are also provided. [Copyright &y& Elsevier]

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]

An extension of the Weibull distribution which involves an additional shape parameter is being proposed. Interestingly, the additional parameter acts somewhat as a location parameter while the support of the distribution remains the positive half-line. Since the gamma distribution is a particular case of this distribution, the latter is referred to as a gamma-Weibull distribution. The gamma-Weibull distribution is in fact a reparameterization of the generalized gamma distribution, which has received little attention in recent years. Some parameters of the gamma-Weibull model have a more straightforward interpretation than those associated with the generalized gamma distribution. Moreover, the gamma-Weibull distribution does not contain a threshold parameter. Accordingly, it readily lends itself to various estimation methodologies and exhibits regular asymptotics. Numerous distributions such as the Rayleigh, half-normal and Maxwell distributions can also be obtained as special cases. The moment generating function of a gamma -Weibull random variable is derived by making use of the inverse Mellin transform technique and expressed in terms of generalized hypergeometric functions. This provides computable representations of the moment generating functions of several of the distributions that were identified as particular cases. Other statistical functions such as the cumulative distribution function of a gamma-Weibull random variable, its moments, hazard rate and associated entropy are also given in closed form. The proposed reparametrization is utilized to model two data sets. The gamma-Weibull distribution provides a better fit than the two parameter Weibull model or its shifted counterpart, as measured by the Anderson-Darling and Cramer-von Mises statistics. [ABSTRACT FROM AUTHOR]

This article proposes a five-parameter exponential-type distribution whose exact moments and normalizing constant are obtained as inverse Mellin transforms and expressed as generalized hypergeometric functions. Several commonly used statistical distributions such as the gamma, Weibull and half-normal turn out to be particular cases of this generalized distribution. It is shown in a numerical example that the proposed distribution produces a better fit than the Weibull or inverse Gaussian distributions when used for modeling a data set consisting of maximum flood levels over a certain time period. [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]

A moment based methodology is proposed for approximating the distribution of the distance between two random points belonging to sets that are composed of rectangles or rectangular parallelepipeds. The resulting density approximants are expressed in terms of Jacobi orthogonal polynomials. Two norms are being considered: the L 1 norm referred to as the Manhattan distance and the L 2 norm corresponding to the Euclidean distance. Several illustrative examples will be presented and certain applications to transportation and routing problems will be pointed out. The results can be readily extended to determine the distribution of the distance between a fixed point or a line segment and sets in ℜ or ℜ 3 , and to sets belonging to higher dimensions as well. [ABSTRACT FROM AUTHOR]

Several estimators of the coefficient of an AR(1) process can be expressed as the ratio of two quadratic forms. In this article, we are considering the ordinary least-squares, a modified least-squares, the Yule–Walker, and Burg's estimators. It will be shown that the modified least-squares estimator is the least biased and that the ordinary least-squares and Burg's estimators share very similar distributional properties. An integral representation of the moments of these estimators is provided and a methodology is proposed for correcting their bias. Bounds for the supports of the Yule–Walker and Burg's estimators are determined and the density functions of those estimators are then approximated in terms of Jacobi polynomials. Finally, confidence intervals for the autoregressive coefficient are determined from replicated series. [ABSTRACT FROM AUTHOR]