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July, 1974 Probability Inequalities and Errors in Classification
Somesh Das Gupta
Ann. Statist. 2(4): 751-762 (July, 1974). DOI: 10.1214/aos/1176342762


Let $X$ and $Y$ be two $p \times 1$ random vectors distributed according to a normal distribution with respective mean vectors $\mu$ and $a\mu$ and covariance matrix $\begin{pmatrix}I_p & \rho I_p \\ \rho I_p & I_p\end{pmatrix}.$ Let $S$ be a random $p \times p$ matrix distributed as the Wishart distribution $W_p(I_p, r)$, independently of $X$ and $Y$. For fixed $a, \rho$, and $c$, some sufficient conditions are obtained for which $P\lbrack X'Y < c\rbrack$ and $P\lbrack X'S^{-1}Y < c\rbrack$ increase with $\mu'\mu$. These results are used to show a monotonicity property of the probabilities of correct classification of a class of rules for classifying an observation into one of two normal distributions. For the classification problem, some estimates of the probability of correct classification of the minimum distance rule are studied.


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Somesh Das Gupta. "Probability Inequalities and Errors in Classification." Ann. Statist. 2 (4) 751 - 762, July, 1974.


Published: July, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0285.62032
MathSciNet: MR365914
Digital Object Identifier: 10.1214/aos/1176342762

Primary: 62H30
Secondary: 60E05

Keywords: ‎classification‎ , estimates of probability or correct classification , Monotonicity , multivariate normal distribution , Probability inequalities , probability of correct classification , two populations

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 4 • July, 1974
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