W.-D. Richter, J. Schumacher
Asymptotic Expansions for Large Deviation Probabilities of Noncentral Generalized Chi-Square Distributions
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.), See also {30E15}

41A63 Multidimensional problems (should also be assigned at least one other classification number in this section)

60F10 Large deviations

62E17 Approximations to distributions (nonasymptotic)

62E20 Asymptotic distribution theory

Abstract: Asymptotic expansions for large deviation probabilities are used to approximate
the cumulative distribution functions of noncentral generalized chi-square distributions,
preferably in the far tails. The basic idea of how to deal with the tail
probabilities consist in first rewriting these probabilities as large parameter values
of the Laplace transform of a suitably defined function f_k, second making a series
expansion of this function and third applying a certain modification of Watsons
lemma.
The function f_k is deduced by applying a geometric representation formula for
spherical measures to the domain of large deviations under consideration. At the so
called dominating point, the largest main curvature of the boundary of this domain
tends to one as the large deviation parameter approaches infinity. Therefore, the
dominating point degenerates asymptotically. For this reason the recent asymptotic
expansion for large deviations in Breitung and Richter (1996) does not
apply.
Assuming a suitable parameterized expansion for the inverse g^{-1} of the negative
logarithm of the density generating function we derive a series expansion for the
function f_k. Note that low order coefficients from the expansion of g^{-1} influence
practically all coefficients in the expansion of the tail probabilities.
Keywords: Asymptotic expansion, approximation tail probabilities, noncentral distribution, spherical distribution, generalized chi-square distribution, geometric representation, large deviations, Watson's lemma