Edgeworth and saddlepoint expansions also have analogues for distributions of random vectors. As in the univariate case these expansions will be derived with reference to characteristic functions and cumulant generating functions, and hence these will be defined first. Subsequently Edgeworth density approximations will be defined. Just as in the univariate case, the Edgeworth approximation to probabilities that a random vector lies in a set is the integral of the Edgeworth density over that set; however, since sets of interest are usually not rectangular, theorems for the asymptotic accuracy of these approximations are difficult to prove. These proofs are not presented here. Approximation for variables on a multivariate lattice are discussed. Multivariate saddlepoint approximations are also defined, by a multivariate extension of steepest descent methods. These methods are also used to approximate conditional probabilities.
KeywordsCumulative Distribution Function Random Vector Univariate Case Hermite Polynomial Saddlepoint Approximation
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