HISTOGRAM ESTIMATORS OF BIVARIATE DENSITIES (MULTIVARIATE, STATISTICS)
HUSEMANN, JOYCE ANN STEVENS
Doctor of Philosophy
One-dimensional fixed-interval histogram estimators of univariate probability density functions are less efficient than the analogous variable-interval estimators which are constructed from intervals whose lengths are determined by the criterion of integrated mean squared error (IMSE) minimization. Similarly, two-dimensional fixed-cell-size histogram estimators of bivariate probability density functions are less efficient than variable-cell-size estimators whose cell sizes are determined from IMSE minimization. Only estimators whose cell sides are parallel to the coordinate axes are examined. The estimators are classified according to the functional dependence of their cell dimensions upon x and y: each cell dimension of the Minimally Restricted Mesh depends upon both x and y; one cell dimension of the Semi-fixed-dimension Mesh is fixed, and the other depends upon either x alone or y alone; one cell dimensions of the Variable-dimensions Mesh I depends upon x and the other upon y; one cell dimension of the Variable-dimension Mesh II depends upon x alone or y alone and the other depends upon both x and y. The Minimally Restricted Mesh results in the smallest IMSE of the four types, but is not implementable. The other meshes are implementable and are listed above in order of decreasing IMSE. Random vectors from Dirichlet, mixed bivariate and elliptical bivariate normal distributions were generated and used to construct optimal histograms. The Variable-dimension Mesh II produced histograms having IMSEs from 20 to 90 percent smaller than those from histograms based upon optimal fixed-dimension meshes. The most substantial improvements were observed for mixed bivariate normal densities having strongly unequal variances. Modest improvements (20%) were observed for skewed densities and slightly elliptical densities, but no improvements were observed in cases of highly elliptical densities whose axes were rotated 45(DEGREES) from the coordinate axes.