Practical Small-Sample Asymptotics for Regression Problems

Abstract Saddlepoint approximations are derived for sums of independent, but not necessarily identically distributed random variables, along with generalizations to estimating equations and multivariate problems. These results are particularly useful for accurately approximating the distribution of regression coefficients. General formulas are given for the distribution of the coefficients arising out of a generalized linear model with both canonical and noncanonical link functions. We illustrate the case of logistic regression with a real data example and show how the Gibbs sampler may be used to obtain confidence sets for each regression parameter based on the saddlepoint approximation.

[1]  J. Simonoff.  Multivariate Density Estimation , 1996 .

[2]  Trevor J. Sweeting,et al.  Approximate Bayesian computation based on signed roots of log-density ratios (with discussion) , 1996 .

[3]  Nancy Reid,et al.  The Roles of Conditioning in Inference , 1995 .

[4]  T. Sweeting.  A Bayesian approach to approximate conditional inference , 1995 .

[5]  Trevor J. Sweeting,et al.  A framework for Bayesian and likelihood approximations in statistics , 1995 .

[6]  Elvezio Ronchetti,et al.  Empirical Saddlepoint Approximations for Multivariate M-estimators , 1994 .

[7]  J. Kolassa.  Series Approximation Methods in Statistics , 1994 .

[8]  Christopher G. Small,et al.  Hilbert Space Methods in Probability and Statistical Inference: Small/Hilbert , 1994 .

[9]  C. Small,et al.  Hilbert Space Methods in Probability and Statistical Inference , 1994 .

[10]  Ricardo Estrada,et al.  Asymptotic Analysis: A Distributional Approach , 1993 .

[11]  Thomas J. DiCiccio,et al.  Simple Modifications for Signed Roots of Likelihood Ratio Statistics , 1993 .

[12]  Nancy Reid,et al.  Statistical Theory and Modeling: In Honour of Sir David Cox, FRS. , 1992 .

[13]  M. Tanner,et al.  Facilitating the Gibbs Sampler: The Gibbs Stopper and the Griddy-Gibbs Sampler , 1992 .

[14]  Donald A. Pierce,et al.  Practical Use of Higher Order Asymptotics for Multiparameter Exponential Families , 1992 .

[15]  L. J. Wei,et al.  A Lack-of-Fit Test for the Mean Function in a Generalized Linear Model , 1991 .

[16]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[17]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[18]  R. Tibshirani.  Noninformative priors for one parameter of many , 1989 .

[19]  D. Cox,et al.  Asymptotic techniques for use in statistics , 1989 .

[20]  Anthony C. Davison,et al.  Approximate conditional inference in generalized linear models , 1988 .

[21]  Nancy Reid,et al.  Saddlepoint Methods and Statistical Inference , 1988 .

[22]  M. Fligner.  Small Sample Asymptotics , 1988 .

[23]  Ib M. Skovgaard,et al.  Saddlepoint expansions for conditional distributions , 1987, Journal of Applied Probability.

[24]  D. Cox,et al.  Parameter Orthogonality and Approximate Conditional Inference , 1987 .

[25]  E. Ronchetti,et al.  General Saddlepoint Approximations with Applications to L Statistics , 1986 .

[26]  Philip Hougaard.  Saddlepoint approximations for curved exponential families , 1985 .

[27]  Charles Stein,et al.  On the coverage probability of confidence sets based on a prior distribution , 1985 .

[28]  J. Lawless.  Statistical Models and Methods for Lifetime Data , 1983 .

[29]  O. Barndorff-Nielsen.  On a formula for the distribution of the maximum likelihood estimator , 1983 .

[30]  H. Daniels.  Saddlepoint approximations for estimating equations , 1983 .

[31]  Chris Field,et al.  Small Sample Asymptotic Expansions for Multivariate $M$-Estimates , 1982 .

[32]  C. Field,et al.  Small-sample asymptotic distributions of M-estimators of location , 1982 .

[33]  Joseph Bak,et al.  Complex Analysis , 1982 .

[34]  Reuven Y. Rubinstein,et al.  Simulation and the Monte Carlo method , 1981, Wiley series in probability and mathematical statistics.

[35]  S. Rice,et al.  Saddle point approximation for the distribution of the sum of independent random variables , 1980, Advances in Applied Probability.

[36]  H. Daniels.  Exact saddlepoint approximations , 1980 .

[37]  David R. Cox,et al.  Edgeworth and Saddle‐Point Approximations with Statistical Applications , 1979 .

[38]  B. L. Welch,et al.  On Formulae for Confidence Points Based on Integrals of Weighted Likelihoods , 1963 .

[39]  H. Daniels.  Saddlepoint Approximations in Statistics , 1954 .