Product Moments of Multivariate Random Vectors

Abstract

We derive a formula for the product moment EX m 1 1 Á Á Á X m p p , m 1 ! 1, . . . , m n ! 1 in terms of the joint survival function when ðX 1 , . . . , X p Þ is a non-negative random vector. In the course of the derivation, we present an independent approach for deriving the formula for EX m 1 1 X m 2 2 (which is already known in the literature). These formulas will be useful tools for deriving explicit expressions for the product moments for the various bivariate and multivariate distributions in statistics specified by the joint tails of the cumulative distribution function. We illustrate their use by deriving product

Alternative approach to moments of order statistics from one-parameter Weibull distribution

“…This note utilizes survival function to find the moments of order statistics from one-parameter Weibull distribution. Recently, this technique of finding moments has gained more importance in practical situation and Hong called it an alternative expectation formula (Feller, 1966;Nadarajah and Mitov, 2003;Hong, 2012). For a continuous non-negative random variable X, the mean or the first moment can be expressed as…”

Letter to the Editor

“…The Alternative Expectation Formulas of Hong (2015) are due to Nadarajah and Mitov (2003) Dear Editor, Let (X, Y ) denote a nonnegative random vector with joint survival function S(x, y) = Pr(X > x, Y > y). Hong (2015) proved that…”

Computation of the Multivariate Normal Integral over a Complex Subspace

“…Therefore, they are not equal to zero [18]. A lot of works are dedicated to the problem of computation of product moments [see, for example, [22][23][24][25][26][27][28].…”