Geometrical, Algebraic, Functional and Correlation Inequalities Applied in Support of James-Stein Estimator for Multidimensional Projections

Michael Fundator

doi:10.11648/j.acm.20180703.14

Abstract

Isoperimetric, Milman reverse, Hilbert, Widder, Fan-Taussky-Todd, Landau, and Fortuin–Kasteleyn–Ginibre (FKG) inequalities in n dimensions in investigations of multidimensional estimators support the use of James-Stein estimator against classical least squares as applied to Cumulant Analysis, Associate Random Variables, and Time Series Analysis.

Highlights

Introduction ofStein’s Estimators for Exponential Family [46-48]
If X is from exponential family of distributions its density is given by functional dependence:
This important identity is very useful for the application of the methods of linear combinations and ratios of random variables of exponential family of distributions that would be discussed in Subsection 2.3.1. and for future development of the theory of Stein estimator and different tests of hypotheses for multidimensional exponential families

Summary

Introduction

“quoting Virgil: At last they landed, where from far your eyes May view the turrets of new Carthage rise; There bought a space of ground, which Byrsa call’d, From the bull’s hide they first inclos’d, and wall’d. (Aeneid, Dryden’s translation) This refers to the legend of Dido. Virgil’s version has it that Dido, daughter of the king of Tyre, fled her home after her brother had killed her husband She ended up on the north coast of Africa, where she bargained to buy as much land as she could enclose with an oxhide. Earthly factors mar the purity of the problem, for surely the clever Dido would have chosen an area by the coast so as to exploit the shore as part of the perimeter. This is essential for the mathematics as well as for the progress of the story. The very good example are James-Stein estimators that depend on the number of dimensions and in dimensions more than 2 is deferent from the classical least squares estimator that almost uniformly reduces every problem to 2-dimensional (2D) consideration

Isoperimetric Inequalities

Another approach for n-polygons the problem becames to maximize

10. After 20 years Carleman published proof based on power series theorem

Properties of Statistics and the Uniformly Most Powerful Invariant Test

Simple Rules for the Reduction of Dimensions