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 matrix evaluate the probability, pk, that k components of X exceed a given constant. We call (po,...,pn) the exceedance distribution and study its behavior as the covariance matrix varies. The pk's can be expressed as multidimensional integrals; these expressions, however, are not helpful, for simulation and numerical intergration in high dimensions are very expensive. When covariance 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 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 12, 1982
- Accession Number
- ADA119368
Entities
People
- Satish Iyengar
Organizations
- Stanford University