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A Problem of Dimensionality in Normal Mixture Analysis
Author(s) -
BEE MARCO,
FLURY BERNARD
Publication year - 2002
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/1467-9469.00302
Subject(s) - mathematics , mahalanobis distance , statistics , univariate distribution , multivariate normal distribution , random variate , normal distribution , unimodality , distribution (mathematics) , curse of dimensionality , combinatorics , random variable , multivariate statistics , mathematical analysis
Suppose the p ‐variate random vector W , partitioned into q variables W 1 and p ‐ q variables W 2 , follows a multivariate normal mixture distribution. If the investigator is mainly interested in estimation of the parameters of the distribution of W 1 , there are two possibilities: (1) use only the data on W 1 for estimation, and (2) estimate the parameters of the p ‐variate mixture distribution, and then extract the estimates of the marginal distribution of W 1 . In this article we study the choice between these two possibilities mainly for the case of two mixture components with identical covariance matrices. We find the asymptotic distribution of the linear discriminant function coefficients using the work of Efron (1975) and O'Neill (1978), and give a Wald–test for redundancy of W 2 . A simulation study gives further insights into conditions under which W 2 should be used in the analysis: in summary, the inclusion of W 2 seems justified if Δ 2.1, the Mahalanobis distance between the two component distributions based on the conditional distribution of W 2 given W 1 , is at least 2.