Multivariate probability density estimation: Some statistical properties

Univariate statistical theories have always inspired similar development for multivariate statistical theories. For years the lack of computational capabilities hindered the progress of developing multivariate statistical methods. In the last decade or so, this obstacle has been removed by high speed computers and excellent software including S-Plus, Matlab, and libraries like IMSL and NAG.; Following the univariate methods, many methods were generalized to develop multivariate density estimators. In two dimensions, histogram-type estimators developed on hexagonal bins were proved to give slightly better estimates in terms of Asymptotic Mean Integrated Squared Error (AMISE)—Scott (1988). But, this poses a question: What will be the counterpart of a hexagon in a dimension higher than two? For example, the hexagonal concept in 3-d would lead to a truncated octahedron, and so forth. The rectangular bins grow exponentially in higher dimensions and pose the problem of the “Empty Space” phenomenon, Scott and Thompson (1983) or “Curse of Dimensionality,” Bellman (1961).; This problem inspired a different kind of binning: the Delaunay Tessellation. In this dissertation, complete algorithms and computational aspects are developed. Following this, it is proved that the estimator (1) is consistent; (2) is conditionally maximum likelihood; (3) has asymptotic distribution. Also, (4) The tiles grow much slower than the rectangular binning.; In addition to this, it is also observed that the number of tiles will grow proportionately as the data grows. This will remove the above dilemma.