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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]

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]

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]

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]

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]

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]