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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]

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]