Asymptotic expansions for large deviation probabilities of noncentral generalized chi-square distributions (Q1840777)

Several asymptotic representations for large deviation probabilities are known in the literaure for the one-dimensional as well as the multi-dimensional case [cf. \textit{H. E. Daniels}, Int. Stat. Rev. 55, 37-48 (1987; Zbl 0614.62016), \textit{C. Ittrich, D. Krause} and \textit{W.-D. Richter}, Statistics 34, No. 1, 53-101 (2000), \textit{K. Breitung} and \textit{W.-D. Richter}, J. Multivariate Anal. 58, No. 1, 1-20 (1996; Zbl 0864.41026)]. The authors solve the problem of evaluating the tail probabilities \(1-\)CQ\((k,\delta^2;g)\) where CQ\((k,\delta^2;g)\) denotes the noncentral \(g\)-generalized chi-square distribution with \(k\) d.f. and non-centrality parameter \(\delta^2\) for shperically distributed random vectors. Using a geometric representation formula for the measure, \(1-\)CQ\((k,\delta^2;g)\) gives a Laplace type integral representation which can be expanded in a series. A modified version of Watson's lemma finally leads to the desired asymptotic expansion for the tail probabilities \(1-\)CQ\((k,\delta^2;g)\). The authors apply their result for Kotz type densities and present some numerical calculations.