Saddlepoint quantiles and distribution curves with bootstrap applications (Q1965953)

Let \(\kappa(\tau)=\log E\exp(\tau X)\) be a known cumulant generating function of a random variable \(X\). The saddlepoint estimate \(\widehat F\) of the distribution function \(F\) of \(X\) can be defined as follows. Let \[ p(\tau)= \begin{cases} \Phi(\zeta)+\phi(\zeta)(1/\zeta - 1/z),& \text{ if \(\tau\not=0\)}\\ 1/2 +\kappa^{(3)}/(6\sigma^3(2\pi)^{1/2}),& \text{ if \(\tau=0,\)}\end{cases} \] \(q(\tau)=\kappa'(\tau)\), where \(\zeta=\text{sign}(\tau)\sqrt{2(\tau q(\tau)-\kappa(\tau))}\) and \(z=\tau\sqrt{\kappa''(\tau)}\), \(\phi\) and \(\Phi\) are the density and the c.d.f. of the standard normal distribution. Then \(\widehat F(q(\tau))=p(\tau)\). The direct application of these equations for the \(\alpha\)-quantile \(F^{-1}(\alpha)\) estimation requires two nested levels of iteration. The author proposes to tabulate \(\widehat F(\alpha)\) at the points \(\alpha_j=q(\tau_j)\) with appropriately chosen \(\tau_j\) and then interpolate the results by a cubic spline. In this case only one iteration cycle is required to calculate \(\widehat F^{-1}(\alpha)\). Applications of the saddlepoint approximation to bootstrap procedures for three variance reduction techniques, post-stratification and importance bootstrap are considered. Two figures which should demonstrate the accuracy of the method are refereed to but are absent in the paper.