### On Fitting Polynomials to Averaged Shifted Histograms

#### Abstract

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.

#### Keywords

density function estimation; histogram, smoothing; polynomial approximation; bivariate data; copulas

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