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

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. Copyright Sociedad de Estadística e Investigación Operativa 2014