Students: Dissertation: Victoria Viorela Neagoe

Victoria Viorela Neagoe

ABSTRACT
The results obtained by James and Stein in the 1950’s led to the introduction of biased estimators. In the area of simultaneous estimation they have defined a class of biased estimators of several unknown normal means called shrinkage estimators.

The study presents first an interpretation of the shrinkage estimators and the adaptive type estimators in the context of the hyperbolic space of negative constant sectional curvature. These estimators are described geometrically using some isometries of the hyperbolic space, namely the inversions with respect to spheres.

The study continues with the consideration of a Riemannian manifold of positive constant sectional curvature, the n-dimensional sphere. The map inverse to stereographic projection onto the equatorial hyperplane is used to construct a new class of shrinkage estimators called spherical shrinkage estimators. For the new class of estimators a risk function is defined using the mean squared error loss function. Since the problem of finding a radius value minimizing the risk is very difficult, an approximate risk function is proposed. A spherical shrinkage estimator minimizing the approximate risk is defined.

For the cases where the quadratic loss is not appropriate, Pitman’s Measure of Closeness (PMC) is an alternative criterion in comparing estimators. The estimators from the new class of spherical shrinkage are compared next using PMC. Conditions leading to a region of preference of one estimator over another competing estimator are described, and a theoretical optimum radius is computed. Areas of application and open research questions related to the study are mentioned.

Tuesday, June 1, 1999
DISSERTATION COMMITTEE
Robert L. Fountain, Chairman
Marek Elzanowski
Eugene A. Enneking
Andrew M. Fraser
James Pratt, Graduate Studies Rep.