Saddlepoint Approximations in Statistics

摘要: It is often required to approximate the distribution of some statistic whose exact cannot be conveniently obtained. When first few moments are known, a common procedure fit law Pearson or Edgeworth type having same as far they given. Both these methods satisfactory in practice, but have drawback that errors "tail" regions sometimes comparable with frequencies themselves. The approximation particular notoriously can assume negative values such regions. characteristic function may and difficulty then analytical one inverting Fourier transform explicitly. In this paper we show for mean sample size $n$, ratio two means, its probability density, when it exists, obtained nearly always by method steepest descents. This gives an asymptotic expansion powers $n^{-1}$ dominant term, called saddlepoint approximation, has number desirable features. error incurred use $O(n^{-1})$ against more usual $O(n^{-1/2})$ associated normal approximation. Moreover shown important class cases relative uniformly over whole admissible range variable. descents was used systematically Debye Bessel functions large order (Watson [17]) introduced Darwin Fowler (Fowler [9]) into statistical mechanics, where remained indispensable tool. Apart from work Jeffreys [12] occasional isolated applications other writers (e.g. Cox [2]), technique been largely ignored on theory. present paper, distributions densities discussed first, being density $\bar{x}$ $n$. how related alternative Khinchin [14] and, slightly different context, Cramer [5]. General conditions established under which all $\bar{x}$, corresponding result expansion. case discrete variables briefly discussed, finally approximating ratios.