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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]

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]