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- W2048008001 abstract "If $x$ is a random variable with mean zero and variance $sigma^2$, then, according to Chebyshev's inequality, $P{|x| geqq 1} leqq sigma^2$. The corresponding one-sided inequality $P{x geqq 1} leqq sigma^2/(sigma^2 + 1)$ is also known (see e.g. [2, p. 198]). Both inequalities are sharp. A generalization of Chebyshev's inequality was obtained by Olkin and Pratt [1] for $P{|x_1| geqq 1 or cdots or |x_k| geqq 1}$, where $Ex_i = 0, Ex_i^2 = sigma^2$, $Ex_ix_j = sigma^2 rho (i neq j), i, j = 1, cdots, k;$ we give here the corresponding generalization of the one-sided inequality, and we consider also the case where only means and variances are known. To obtain an upper bound for $P{x varepsilon T} equiv P{x_1 geqq 1 text{or} cdots text{or} x_k geqq 1}$, we consider a nonnegative function, $f(x) equiv f(x_1, cdots, x_k)$, such that $f(x) geqq 1$ for $x varepsilon T$. Then $Ef(x) geqq int_{{x varepsilon T}} f(x) dP geqq P{x varepsilon T}$. Since the bound is to be a function of the covariance matrix, $Sigma, f(x)$ must be of the form $(x - a)A(x - a)'$, where $a = (a_1, cdots, a_k), A = (a_{ij}): k times k$. A best bound is one which minimizes $Ef(x) = operatorname{tr} A(Sigma + a'a)$, subject to $f(x) geqq 0, f(x) geqq 1$ on $T$." @default.
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- W2048008001 date "1960-06-01" @default.
- W2048008001 modified "2023-09-28" @default.
- W2048008001 title "A One-Sided Inequality of the Chebyshev Type" @default.
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- W2048008001 doi "https://doi.org/10.1214/aoms/1177705913" @default.
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