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The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and j th largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes. [ABSTRACT FROM AUTHOR]

This paper proposes several methodologies whose objective consists of securing copula density estimates. More specifically, this aim will be achieved by differentiating bivariate least-squares polynomials fitted to Deheuvels' empirical copulas, by making use of Bernstein's approximating polynomials of appropriately selected orders; by differentiating linearized distribution functions evaluated at optimally spaced grid points; and by implementing the kernel density estimation technique in conjunction with a repositioning of the pseudo-observations and a certain criterion for determining suitable bandwidths. Smoother representations of such density estimates can further be secured by approximating them by means of moment-based bivariate polynomials. The various copula density estimation techniques being advocated herein are successfully applied to an actual dataset as well as a random sample generated from a known distribution. [ABSTRACT FROM AUTHOR]

When a p×p$$ p\times p $$ real positive definite matrix S$$ S $$ follows a Wishart or, more generally, a matrix‐variate gamma distribution with shape parameter α$$ \alpha $$ and positive definite scale parameter matrix B$$ B $$, one can represent S$$ S $$ as XX′$$ X{X}^{\prime } $$ for some matrix X$$ X $$ of dimension p×q$$ p\times q $$. When p>q$$ p>q $$, S$$ S $$ has a singular distribution whose properties can be studied via the density function of X$$ X $$. It will be shown that when X$$ X $$ follows a matrix‐variate extended Gaussian distribution, the density function of the resulting singular gamma distribution can be obtained by making use of successive transformations and their associated Jacobians. The singular Wishart distribution will then be obtained as a particular case. The marginal and conditional density functions resulting from an arbitrary partitioning of X$$ X $$ will be considered as well. The same technique will also be applied to the derivation of the density functions of real and complex singular type‐1 and type‐2 beta‐distributed matrices. It so happens that the proposed approach, which is based on manipulations involving the wedge products of certain differential elements, generally proves more efficient than the intricate procedures that have hitherto been employed in the literature. [ABSTRACT FROM AUTHOR]

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 discusses the fitting of suitable models to rainfall observations. Daily rainfall amounts were aggregated to monthly data using the Thiessen polygons method and multivariate seasonal vector integrated autoregressive moving average models (sVARIMA) were fitted to the monthly cumulative rainfall volume. The data were obtained from the 12 Palestinian meteorological gauge stations located across the 5 governorates of the Gaza Strip and incorporated 42 years (from 1973 to 2014) of irregular daily precipitation. It can be concluded that the use of sVARIMA models in the environmental sciences provide a useful approach for forecasting rainfall data as a preliminary guideline toward short and long-term sustainable water resources management. [ABSTRACT FROM AUTHOR]

The determination of the distribution of aggregate losses is of crucial importance for an insurer. In this paper, we propose a technique for approximating the distribution of univariate and bivariate aggregate losses, which is solely based on their moments. Accordingly, this methodology can be implemented without any specific knowledge of the claim number or size distributions. The numerical examples presented herein indicate that the proposed approach constitutes a viable alternative to the commonly used recursive and FFT methods. [ABSTRACT FROM AUTHOR]

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]

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]

A new generalized extended inverse Gaussian density function involving a confluent hypergeometric function is defined. Closed form representations of a generalized extended inverse Gaussian density function and the corresponding moment generating function are provided in terms of Meijer's G-function. Some other statistical functions which are associated with the proposed distribution are also derived. Many distributions can be obtained as limiting cases of the proposed density function. This density is utilized to model an actual data set. The new distribution provides a better fit than related distributions as measured by the Anderson-Darling and Cramér-von Mises statistics. The proposed distribution could, for instance, find applications in the physical and biological sciences and in particular be helpful to researchers working in special functions as well as reliability analysis, survival models and diffraction theory. [ABSTRACT FROM AUTHOR]

This paper proposes univariate and bivariate density estimation techniques whereby averaged shifted histograms are smoothed by means of polynomials. In the univariate case, the density estimate is obtained as a moment-based polynomial approximation to an averaged shifted histogram. The polynomials are expressed as linear combinations of Legendre polynomials. In the bivariate case, the product of polynomial approximations to the marginal density functions is adjusted by making use of a bivariate polynomial whose coefficients are such that the joint moments of the resulting distribution agree with the sample moments up to a certain order. As well, it is explained that this approach actually gives rise to copula density functions. Alternatively, a density estimate may be obtained by smoothing a bivariate averaged shifted histogram with a least-squares approximating polynomial. Several illustrative examples are presented. [ABSTRACT FROM AUTHOR]

This paper proposes univariate and bivariate density estimation techniques whereby averaged shifted histograms are smoothed by means of polynomials. In the univariate case, the density estimate is obtained as a moment-based polynomial approximation to an averaged shifted histogram. The polynomials are expressed as linear combinations of Legendre polynomials. In the bivariate case, the product of polynomial approximations to the marginal density functions is adjusted by making use of a bivariate polynomial whose coefficients are such that the joint moments of the resulting distribution agree with the sample moments up to a certain order. As well, it is explained that this approach actually gives rise to copula density functions. Alternatively, a density estimate may be obtained by smoothing a bivariate averaged shifted histogram with a least-squares approximating polynomial. Several illustrative examples are presented. [ABSTRACT FROM AUTHOR]

A bandwidth selection method that combines the concept of least-squares cross-validation and the plug-in approach is being introduced in connection with kernel density estimation. A simulation study reveals that this hybrid methodology outperforms some commonly used bandwidth selection rules. It is shown that the proposed approach can also be readily employed in the context of variable kernel density estimation. We conclude with two illustrative examples. [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: The inverse Mellin transform technique is utilized to obtain closed form representations of the generalized Boltzmann factors associated with several Weibull-type models such as the generalized gamma, Maxwell, Rayleigh and half-normal distributions. The results complement those already available in the Physics literature in connection with the distribution of certain variables affecting the behavior of nonequilibrium systems subject to complex dynamics, which include for instance computable expressions for the generalized Boltzmann factors induced by the gamma, , uniform and lognormal statistical models. [Copyright &y& Elsevier]

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]

Density estimates that are expressible as the product of a base density function and a linear combination of orthogonal polynomials are considered in this paper. More specifically, two criteria are proposed for determining the number of terms to be included in the polynomial adjustment component and guidelines are suggested for the selection of a suitable base density function. A simulation study reveals that these stopping rules produce density estimates that are generally more accurate than kernel density estimates or those resulting from the application of the Kronmal–Tarter criterion. Additionally, it is explained that the same approach can be utilized to obtain multivariate density estimates. The proposed orthogonal polynomial density estimation methodology is applied to several univariate and bivariate data sets, some of which have served as benchmarks in the statistical literature on density estimation. [ABSTRACT FROM PUBLISHER]

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]

Numerous types of asymmetrical distributions such as the logistic, Weibull, gamma, and beta distributions have been used for modeling various random phenomena such as those encountered in data engineering, pattern recognition, and reliability assessment studies. Several generalizations of the logistic distribution, and certain related models, are proposed in this paper. The corresponding density functions involve an additional parameter, denoted by q, which allows for increased flexibility for modeling purposes; in fact, the larger this parameter is, the lower the mode of the resulting distribution will be. Generalizations of the type-1 and type-2 beta distributions are introduced, along with their logistic-type counterparts; the moments and cumulants of the latter are also derived. Other extensions are discussed including a q-analog of the generalized type-2 beta model, a q-extended generalized logistic distribution, and q-analogs of generalizations of the Dirichlet distribution. As is shown graphically, the proposed univariate distributions can generate a wide array of unimodal or symmetric bimodal curves. [ABSTRACT FROM AUTHOR]

Abstract: In a recent survey, Olkin [The 70th anniversary of the distribution of random matrices: a survey, Linear Algebra Appl. 354 (2002) 231–243] has looked into 70 years of development on random matrices, giving special attention to rectangular matrices. Gupta and Richards [Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233–256] have looked into generalizations of the Dirichlet family of distributions for the multivariate case. Mathai [An Introduction to Geometrical Probability: Distributional Aspects with Applications, Gordon and Breach Scientific Publishers, New York, 1999] developed a gamma type distribution on rectangular matrices in connection with the distributional aspects of random volumes. This idea is developed further in the present article to define rectangular matrix-variate gamma type, beta type and Dirichlet type distributions with location and scale matrices; connections among these distributions and various properties are pointed out. This paper discusses the complex matrix-variate cases and the corresponding real cases are also listed along with each result. The complex rectangular matrix-variate t and Cauchy distributions are obtained as particular cases. [Copyright &y& Elsevier]

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]

Computable representations of the probability density function and the cumulative distribution function of Hermitian quadratic forms in spherically distributed vectors are obtained from a stochastic decomposition of central quadratic forms into the product of two independent quantities: the square of the norm of the vectors and a linear combination of the components of a Dirichlet random vector. The expressions derived for the Gaussian case appear to lend themselves more readily to numerical evaluation than those currently available in the literature. The distribution of noncentral Hermitian quadratic forms is obtained via a certain mixture representation of the density function of elliptically contoured random vectors. A numerical application to the periodogram is proposed for the case of spherically distributed residuals. [Copyright &y& Elsevier]

Les auteurs déduisent d'un résultat de Imhof une représentation intégrate de la fonction de répartition d'une combinaison Iinéaire quelconque des composantes d'un vecteur aléatoire de Dirichlet. Comme IIs le font valoir, les lois de plusieurs statistiques peuvent ětre déterminées de façon similaire, notamment celles des rapports du type F, des indices de Moran et de Geary, de la statistique de corrélation spatiale de Cliff-Ord, ou des coefficients de détermination et de corrélation sérielle expérimentaux. Les auteurs expliquent en outre le rǒle clef joué par les combinaisons linéaires des composantes d'un vecteur aléatoire de Dirichlet dans la décomposition de formes quadratiques construites à partir de vecteurs obèissant à une loi sphèrique. IIs déterminent, à titre illustratif, la loi du spectre échantillonnal associé à une chronique générée par un processus de type ARMA. [ABSTRACT FROM AUTHOR]

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]

We introduce a three-parameter extension of the Lindley distribution, which has as sub-models the Lindley and Marshall-Olkin Lindley distributions. The proposed model turns out to be quite flexible: its probability density function can be decreasing or unimodal and its associated hazard rate may be increasing, decreasing, unimodal or bathtub-shaped. Since this new distribution has a survival function and a hazard rate that can be expressed in closed form, it can readily be simulated and used to analyze censored data. Computable expressions are obtained for certain statistical functions such as its quantile function, ordinary and incomplete moments, moment generating function, order statistics and reliability function. The maximum likelihood method is utilized to obtain estimates of the model parameters and a simulation study is carried out to assess the performance of the corresponding maximum likelihood estimators. Two illustrative examples involving hydrological data sets are presented. [ABSTRACT FROM AUTHOR]

We introduce a three-parameter extension of the Lindley distribution, which has as sub-models the Lindley and Marshall-Olkin Lindley distributions. The proposed model turns out to be quite flexible: its probability density function can be decreasing or unimodal and its associated hazard rate may be increasing, decreasing, unimodal or bathtub-shaped. Since this new distribution has a survival function and a hazard rate that can be expressed in closed form, it can readily be simulated and used to analyze censored data. Computable expressions are obtained for certain statistical functions such as its quantile function, ordinary and incomplete moments, moment generating function, order statistics and reliability function. The maximum likelihood method is utilized to obtain estimates of the model parameters and a simulation study is carried out to assess the performance of the corresponding maximum likelihood estimators. Two illustrative examples involving hydrological data sets are presented. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive the predictive distributions of one and several future responses including their average, on the basis of a type II doubly censored sample from a two parameter exponential life testing model. We also determine the highest predictive density interval for a future response. A numerical example is provided to illustrate the results. [Copyright &y& Elsevier]

We introduce a three-parameter extension of the exponential distribution which contains as sub-models the exponential, logistic-exponential and Marshall-Olkin exponential distributions. The new model is very flexible and its associated density function can be decreasing or unimodal. Further, it can produce all of the four major shapes of the hazard rate, that is, increasing, decreasing, bathtub and upside-down bathtub. Given that closed-form expressions are available for the survival and hazard rate functions, the new distribution is quite tractable. It can be used to analyze various types of observations including censored data. Computable representations of the quantile function, ordinary and incomplete moments, generating function and probability density function of order statistics are obtained. The maximum likelihood method is utilized to estimate the model parameters. A simulation study is carried out to assess the performance of the maximum likelihood estimators. Two actual data sets are used to illustrate the applicability of the proposed model. [ABSTRACT FROM AUTHOR]