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The inverse moment of the noncentral chi-squared variable is approximated in simple forms based on its asymptotic expansions. The inverse moment is expanded as the noncentrality parameter tends to infinity proportionally to degrees of freedom. Accuracies of our approximations can be examined through numerical evaluation. It is observed that our approximations perform well in a wide range of values of the noncentrality parameter or degrees of freedom. The noncentral chi-squared distribution has been used in various as- pects of statistical theory. Inverse moments of the distribution often ap- pear in evaluating risks of estimators and powers of test statistics. In the inverse regression problem, Oman (1985) gave an exact formula for the mean squared error of Krutchkoff's inverse estimator by use of inverse mo- ments of the noncentral chi-squared variable. In its extension to multi- variate problem, approximations of the inverse moments were discussed in Lieftinck-Koeijers (1988). The 1st inverse moment of the noncentral chi- squared variable was also used in evaluating the risk of the James-Stein estimator introduced by James and Stein (1961) under the quadratic loss. Stein (1966) gave an approximation of the 1st inverse moment. Egerton and Laycock (1982) gave a simple form of the 1st inverse moment and an exact evaluation of the risk of the James-Stein estimator. An exact expres- sion of the n-thinverse moment as a finite sum was given by Bock et al. (1984). The characteristics and fundamental properties of the noncentral chi-squared distribution are summarized with other related distributions in Johnson et al. (1995). Let χ 2p,λ) denote a noncentral chi-squared variable with p degrees of freedom and the noncentrality parameter λ. We present an asymptotic expansion of the n-thinverse moment, E {(1/χ 2p,λ) ) n } ,a sp tends to infin- ity under the assumption that λ/p is constant. An advantage is that our

ASYMPTOTIC APPROXIMATIONS OF THE INVERSE MOMENT OF THE NONCENTRAL CHI-SQUARED VARIABLE