Practical Use of Higher Order Asymptotics for Multiparameter Exponential Families

Abstract

SUMMARY Recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families, especially in generalized linear models. The aim here is to clarify and explore these. Attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family. Excellent approximations to exact conditional inferences are often available, in terms of simple adjustments to the signed square root of the likelihood ratio statistic. The focus is on distinguishing between two aspects of the adjustments: one reducing effects of nuisance parameter estimation and the other adjusting for little information regarding the parameter of interest. Numerical results are given for some Poisson and multinomial models.

Higher-order approximations for interval estimation in binomial settings

“…It should be noted first that the same inconvenience for the boundaries y 1 = 0 or n 1 or y 2 = 0 or n 2 , noted in the one-binomial setting, extends to this two binomial situation. Pierce and Peters (1992) refer to these data points as uninformative and ignore them. Unfortunately, these cases are needed in the calculation of the actual coverage probability, when the purpose is to compare the coverage probability of the confidence intervals with the corresponding nominal level.…”

“…For a complete derivation of r * see Barndorff-Nielsen and Cox (1994, Chapter 6.6). In multiparameter exponential models with a scalar parameter of interest, the statistic r * is derived from a saddlepoint approximation to the conditional distribution of the maximum likelihood estimator ; see Pierce and Peters (1992). As in the one-parameter case, r * is invariant to interest respecting reparameterization and it has a density function which agrees with the standard normal density function with relative error O(n −3/2 ), for continuous models.…”

“…In discrete settings, various works have commented on the accuracy of the r * confidence intervals. There are some references which adjust for the discrete nature of the problem: for instance, Pierce and Peters (1992) discuss the accuracy of the r * confidence limits, obtained by considerations of continuity corrections, in approximating the exact confidence limits for a scalar component of interest. Davison and Wang (2002) and Brazzale et al (2007, Chapter 2) examine numerically the r * confidence limits without continuity corrections, which accurately approximate the limits derived from a continuous embedding model of the discrete problem.…”

Comment

“…It gives very poor approximation, especially when the sample size is small or when the underlying distribution is far away from normal distribution (see Pierce and Peters 1992 for some examples). Barndorff-Nielsen (1991) proposed a modified likelihood ratio statistic.…”

Canonical regression models for exponential families

“…These results are useful in settings where more than one coordinate of the canonical parameter follows a regression model, especially in multi-parameter exponential families where exact methods are well defined but computationally difficult. For other applications, we refer the readers to Pierce and Peters (1992).…”