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Dissertation Information for Satish Iyengar

NAME:
- Satish Iyengar

DEGREE:
- Ph.D.

DISCIPLINE:
- Statistics

SCHOOL:
- Stanford University (USA) (1982)

ADVISORS:
- None

COMMITTEE MEMBERS:
- None

MPACT Status: Incomplete - Not_Inspected

Title: ON THE EVALUATION OF CERTAIN MULTIVARIATE NORMAL PROBABILITIES

Abstract: Consider the following problem: if X is an n-dimensional normally distributed random vector with mean zero and covariance (SUMM), evaluate the probability, p(,k), that k components of X exceed a given constant. We call (p(,0),...,p(,n)) the exceedance distribution and study its behavior as (SUMM) varies. The p(,k)'s can be expressed as multidimensional integrals; these expressions, however, are not helpful, for simulation and numerical integration in high dimensions are very expensive.

When (SUMM) is an equicorrelation matrix or has single-factor structure, the probabilities can be written as single integrals. In this dissertation, we propose some methods for approximating the above multidimensional integrals by such single integrals. Ample numerical evidence is given to show that the approximations are quite good. We also prove a theorem which gives conditions for the variance of the exceedance distribution to be greater than that of the approximation. We use this inequality to improve upon earlier approximations and inequalities.

A closely related problem is that of evaluating "vanishing" orthant probabilities: i.e., P{X(,1) > a(,1),...,X(,n) > a(,n)} for large a(,i). A natural approach here is to use generalizations of Mills' ratio. Several authors have studied this approach and stated some conjectures. Studying Mills' ratio from the viewpoint of exponential families, we provide proofs for some of the conjectures and some extensions.

Finally, we state and prove some interesting special results for the equicorrelation case. This case lends itself to much analysis because of its symmetry. These results are new and shed some light on other results that are in the literature.

MPACT Scores for Satish Iyengar

A = 0
C = 1
A+C = 1
T = 0
G = 0
W = 0
TD = 0
TA = 0
calculated 2010-09-28 00:15:44

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