Higher-order approximations to the quantile of the distribution for a class of statistics in the first-order autoregression

摘要

Higher-order approximations for quantiles can be derived upon inversions of the Edgeworth and saddlepoint approximations to the distribution function of a statistic. The inversion of the Edgeworth expansion leads to the well known Cornish-Fisher expansion. This paper deals with the inversions of Esscher's, Lugannani-Rice and r~* saddlepoint approximations for a class of statistics in the first-order autoregression. Such inversions provide analytically explicit approximations to the quantile, alternative to the Cornish-Fisher expansion. We assess the accuracy of the new approximations both theoretically and numerically and compare them with the normal approximation and the second and third-order Cornish-Fisher expansions.