Series Approximation Methods in StatisticsThis book was originally compiled for a course I taught at the University of Rochester in the fall of 1991, and is intended to give advanced graduate students in statistics an introduction to Edgeworth and saddlepoint approximations, and related techniques. Many other authors have also written monographs on this s- ject, and so this work is narrowly focused on two areas not recently discussed in theoretical text books. These areas are, ?rst, a rigorous consideration of Edgeworth and saddlepoint expansion limit theorems, and second, a survey of the more recent developments in the ?eld. In presenting expansion limit theorems I have drawn heavily on notation of McCullagh (1987) and on the theorems presented by Feller (1971) on Edgeworth expansions. For saddlepoint notation and results I relied most heavily on the many papers of Daniels, and a review paper by Reid (1988). Throughout this book I have tried to maintain consistent notation and to present theorems in such a way as to make a few theoretical results useful in as many contexts as possible. This was not only in order to present as many results with as few proofs as possible, but more importantly to show the interconnections between the various facets of asymptotic theory. Special attention is paid to regularity conditions. The reasons they are needed and the parts they play in the proofs are both highlighted. |
Contents
5 | 73 |
6 | 80 |
Characteristic Functions and the BerryEsseen Theorem Edgeworth Series Saddlepoint Series for Densities | 87 |
Multivariate | 130 |
Conditional Distribution Approximations | 156 |
Likelihood | 170 |
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analytic ancillarity ancillary applied asymptotic expansion Barndorff-Nielsen Biometrika bounded calculated characteristic function coefficients components conditional distribution constructed convergence cumulant generating function cumulative distribution function defined derivatives differentiable distributed random variables distribution function approximations double saddlepoint Edgeworth series evaluated exists exponential family expressed factor finite Fourier function approximations fx(x Hence Hermite polynomials heuristic identically distributed random implies independent and identically integrand inversion integral Kolassa lattice distributions likelihood ratio statistic linear matrix maximum likelihood estimator McCullagh mean methods multivariate normal approximation parameter path power series Proof pseudo-moments Qx(x random vector range of integration regularity conditions relative error result saddlepoint approximation saddlepoint density approximation sample series expansion Skovgaard steepest descent sufficient statistic summands Suppose tail probabilities Theorem transform univariate variance zero