Saddlepoint‐Based Bootstrap Inference for Quadratic Estimating Equations

Abstract

We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half-century. Copyright (c) 2008 Board of the Foundation of the Scandinavian Journal of Statistics.

Examining the Drivers for Franchised Chains Performance through the Lens of the Dynamic Capabilities Approach

“…Research has proven that bootstrapping produces better results than simply applying regression analysis to an original sample (Efron 1979;Efron and Tibshirani 1993). The bootstrap procedure to estimate the statistical significance of regression coefficients may also mitigate potential problems of nonlinearity and non-normality of the quadratic term-curvilinear relationship between the proportion of franchised units and chain performance (Efron 1979(Efron , 1987Efron and Tibshirani 1993;Paige, Trindade, and Fernando 2009). We created a sampling distribution with 5,000 bootstrap resamples (from the original sample of 189 franchised chains), using a stratified procedure to maintain the integrity of the original data.…”

“…Results had similar significance levels and negligible changes in coefficient values. Based on the recommendations of Efron and Tibshirani (1993), and Paige, Trindade, and Fernando (2009) concerning the non-normality of the quadratic terms, we decided to retain results of bootstrapping.We thank an anonymous reviewer for pointing this out.…”

Exact and Approximate Statistical Inference for Nonlinear Regression and the Estimating Equation Approach

“…Fourth, the exact (or improved) statistical inference can be extended to statistical models when variance–covariance matrices are subject to estimation using the EE approach (Paige & Trindade, ).…”

Fast and Accurate Inference for the Smoothing Parameter in Semiparametric Models

“…In Paige et al . () we proposed an easy to implement parametric bootstrap percentile method of confidence interval construction for a generic model parameter , by indirectly saddlepoint approximating the distribution of an estimator constructed from a vector of observations, . We termed this the saddlepoint‐based bootstrap (SPBB).…”

“…Another goal of the present paper is to demonstrate that our proposed saddlepoint bootstrap method is in fact applicable in a quite general setting. The essential elements of this methodology were recently pioneered by us in Paige, Trindade & Fernando (), and can be used whenever the estimator in question can be expressed as the root of an equivalent estimating equation that is a quadratic form in normal random variables. We adapt this for penalised spline models, and show that it can be used for inference on the smoothing parameter estimated under a variety of criteria, such as maximum and restricted maximum likelihood, generalised cross‐validation and Akaike's information criterion.…”