General Saddlepoint Approximations with Applications to L Statistics (original) (raw)

The Multivariate Saddlepoint Approximation to the Distribution of Estimators: A General Approach

Journal of Mathematics and System Science, 2016

We develop the theory of multivariate saddlepoint approximations. Our treatment differs from the one in Barndorff-Nielsen and Cox (1979, equation (4.7)) in two aspects: 1) our results are satisfied for random vectors that are not necessarily sums of independent and identically distributed random vectors, and 2) we consider that the sample is taken from any distribution, not necessarily a member of the exponential family of densities. We also show the relationship with the corresponding multivariate Edgeworth approximations whose general treatment was developed by Durbin in 1980, emphasizing that the basic assumptions that support the validity of both approaches are essentially similar.

Statistical Inference and Exact Saddle Point Approximations

2018 IEEE International Symposium on Information Theory (ISIT), 2018

Statistical inference may follow a frequentist approach or it may follow a Bayesian approach or it may use the minimum description length principle (MDL). Our goal is to identify situations in which these different approaches to statistical inference coincide. It is proved that for exponential families MDL and Bayesian inference coincide if and only if the renormalized saddle point approximation for the conjugated exponential family is exact. For 1-dimensional exponential families the only families with exact renormalized saddle point approximations are the Gaussian location family, the Gamma family and the inverse Gaussian family. They are conjugated families of the Gaussian location family, the Gamma family and the Poisson-exponential family. The first two families are self-conjugated implying that only for the two first families the Bayesian approach is consistent with the frequentist approach. In higher dimensions there are more examples.

Saddlepoint approximations and tests based on multivariate M-estimates

The Annals of Statistics, 2003

We consider multidimensional M-functional parameters defined by expectations of score functions associated with multivariate M-estimators and tests for hypotheses concerning multidimensional smooth functions of these parameters. We propose a test statistic suggested by the exponent in the saddlepoint approximation to the density of the function of the M-estimates. This statistic is analogous to the log likelihood ratio in the parametric case. We show that this statistic is approximately distributed as a chi-squared variate and obtain a Lugannani-Rice style adjustment giving a relative error of order n −1. We propose an empirical exponential likelihood statistic and consider a test based on this statistic. Finally we present numerical results for three examples including one in robust regression.

Conditional saddlepoint approximations for non–continuous and non–lattice distributions

Journal of Statistical Planning and Inference, 2007

This manuscript presents an approximation to the distribution function of a smooth transformation of a random vector, conditional on the event that values of other smooth transformations of the same random vector lie in a small rectangle. This approximation is used to extend the application of standard saddlepoint conditional tail area approximations in circumstances beyond continuous and lattice cases currently justified in the literature. We consider application to two examples, finite sampling and score testing in logistic regression, where conditioning on a rectangle is essential.

Empirical saddlepoint approximations of the Studentized ratio and regression estimates for finite populations

Statistics & Probability Letters, 2005

We obtain saddlepoint approximations for tail probabilities of the Studentized ratio and regression estimates of the population mean for a simple random sample taken without replacement from a finite population. This is only possible if we know the entire population, so we also obtain empirical saddlepoint approximations based on the sample alone. These empirical approximations can be used for tests of significance and confidence intervals for the population mean. We compare the empirical approximation to the true saddlepoint approximation, both theoretically and numerically. The empirical saddlepoint is related to a bootstrap method for finite populations and we give numerical comparisons of these. We show that for data which contains extreme outliers, poor approximations can be obtained in the case of regression estimates, both for the saddlepoint and empirical saddlepoint, but for less extreme data the saddlepoint and empirical saddlepoint approximations are extremely close to the corresponding Monte Carlo and bootstrap approximations.

General Saddlepoint Approximations of Marginal Densities and Tail Probabilities

Journal of the American Statistical Association, 1996

Saddlepoint approximations of marginal densities and tail probabilities of general nonlinear statistics are derived. They are based on the expansion of the statistic up to the second order. Their accuracy is shown in a variety of examples, including logit and probit models and rank estimators for regression.

Empirical Saddlepoint Approximations for Multivariate M-Estimators

Journal of the Royal Statistical Society: Series B (Methodological), 1994

In this paper. we investigate the use of the empirical distribution function in place of the underlying distribution function F to construct an empirical saddlepoint approximation to the density In of a general multivariate M-estimator. We obtain an explicit form for the error term in the approximation, investigate the effect of renormalizing the estimator, carry out some numerical comparisons and discuss the regression problem.