SOME RESULTS FOR ESTIMATING BIVARIATE DENSITIES USING KERNEL, ORTHOGONAL SERIES AND PENALIZED LIKELIHOOD PROCEDURES
NEZAMES, DONNA DALANGAUSKAS
Doctor of Philosophy
In this work, three extensions of univariate nonparametric probability density estimators into two dimensions are analyzed in terms of statistical and numerical properties. The asymptotically optimal smoothing parameters of the bivariate kernel estimate are derived. Since the optimal smoothing parameters depend on prior knowledge of the underlying density function, an objective method based on functional iteration is investigated. The second nonparametric estimator is formulated from a weighted orthogonal series. A data-based procedure that selects the smoothing parameters by minimizing an approximate expression of the integrated mean square error is investigated. The estimate is proven to be consistent with the asymptotic rate of convergence equivalent to that of the kernel method. A hybrid procedure is proposed that estimates the asymptotically optimal smoothing parameters in the kernel estimate by using the orthogonal series estimate in place of the density function. The third estimate, a discrete maximum penalized-likelihood estimate, is proven to exist, be unique and be consistent pointwise almost surely. A procedure to numerically implement the scheme is presented. To compare the integrated mean square error with that of the kernel method, a preliminary simulation investigation is studied.