1960/12/01 by Albert W. Marshall, Ingram Olkin · 7 citations
Mathematics · Decision Sciences · #Mathematical Approximation and Integration #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.1214/aoms/1177705673
If X is a random variable with EX2 = σ2, then by Chebyshev's inequality, P\|X| \geqq ε\ \leqq σ2/ε2. If in addition EX = 0, one obtains a corresponding one-sided inequality P\X \geqq ε\ \leqq σ2/ (ε2 + σ2) (see, e.g., [8] p. 198). In each case a distribution for X is known that results in equality, so that the bounds are sharp. By a change of variable we can take ε = 1. There are many possible multivariate extensions of (1.1) and (1.2). Those providing bounds for P\max1 \leqq j \leqq k |Xj| \geqq 1\ and P\|max1 \leqq j \leqq k Xj \geqq 1\ have been investigated in [3, 5, 9] and [4], respectively. We consider here various inequalities involving (i) the minimum component or (ii) the product of the components of a random vector. Derivations and proofs of sharpness for these two classes of extensions show remarkable similarities. Some of each type occur as special cases of a general theorem in Section 3. Bounds are given under various assumptions concerning variances, covariances and independence. Notation. We denote the vector (1, ⋯, 1) by e and (0, ⋯, 0) by 0; the dimensionality will be clear from the context. If x = (x1, ⋯, xk) and y = (y1, ⋯, yk), we write x \geqq y(x > y) to mean xj \geqq yj(xj > yj), j = 1, 2, ⋯, k. If Σ = (σij): k × k is a moment matrix, for convenience we write σjj = σ2j, j = 1, ⋯, k. Unless otherwise stated, we assume that Σ is positive definite.