# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I: The equal-means case. (English) Zbl 1098.62530
Summary: We consider the problem of discriminating, on the basis of random ‘training’ samples, between two independent multivariate normal populations, ${N}_{p}\left(\mu ,{{\Sigma }}_{1}\right)$ and ${N}_{p}\left(\mu ,{{\Sigma }}_{2}\right)$, which have a common mean vector $\mu$ and distinct covariance matrices ${{\Sigma }}_{1}$ and ${{\Sigma }}_{2}$. Using the theory of Bessel functions of the second kind of matrix argument developed by C. S. Herz [Ann. Math. 61, 474–523 (1955; Zbl 0066.32002)], we derive stochastic representations for the exact distributions of the ‘plug-in’ quadratic discriminant functions for classifying a newly obtained observation. These stochastic representations involve only chi-squared and F-distributions, hence we obtain an efficient method for simulating the discriminant functions and estimating the corresponding probabilities of misclassification. For some special values of p, ${{\Sigma }}_{1}$ and ${{\Sigma }}_{2}$ we obtain explicit formulas and inequalities for the probabilities of misclassification. We apply these results to data given by Stocks [Ann. Eugen. 5, 1–55 (1933)] in a biometric investigation of the physical characteristics of twins, and to data provided by A. C. Rencher [Methods of Multivariate Analysis. (1995; Zbl 0836.62039)] in a study of the relationship between football helmet design and neck injuries. For each application we estimate the exact probabilities of misclassification, and in the case of Stocks’ data we make extensive comparisons with previously published estimates.

##### MSC:
 62H10 Multivariate distributions of statistics 62H30 Classification and discrimination; cluster analysis (statistics) 62E15 Exact distribution theory in statistics