High-order asymptotic approximations for improved inference under exceptionally low false positive error rates

A methodology for obtaining accurate approximations to p-values of likelihood-based test statistics for a scalar parameter is proposed. The procedure hinges on the well-established technique of stochastic expansion for computing high-order asymptotic approximations to the respective cumulants, which are subsequently fed into Edgeworth and saddlepoint approximations. The novel contribution of the paper is threefold. The first is the derivation of fifth-order accurate formulas for the cumulant and Edgeworth expansions of the classical triad of statistics: Likelihood Ratio, Score, and Wald. The second is the derivation of a result that synchronizes a particular saddlepoint approximation’s order of accuracy with that of the cumulants, while ensuring convexity of the respective (truncated cumulant) generating function. Third, the potential of the methodology is illustrated in an area of recent widespread interest involving the search for rare signals where control of exceptionally low false positive error rates is of paramount importance, while simultaneously dealing with nuisance parameters. The high-order expansions provide a poignant illustration of how a daunting analytical bookkeeping exercise can be automated via the twenty-first-century capabilities of a symbolic mathematics package.