Composite likelihood inference by nonparametric saddlepoint tests (original) (raw)
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Nonparametric saddlepoint test and pairwise likelihood inference
The asymptotic distribution of the pairwise log likelihood ratio is a linear combination of independent chi-square random variables with coefficients depending on the elements of the Godambe information. Adjusted versions of the pairwise log likelihood statistic have been proposed, but they still depend on the Godambe information matrix. Approximated p-values for testing a composite hypothesis may be obtained by refferring the observed value of such statistics to a critical value. The asymptotic theory can be used to approximate the desired quantile, but the approximation may be inaccurate. In this work we provide a nonparametric saddlepoint statistic derived from the pairwise score function. This statistic enjoys some desirable properties: it is asymptotically chi-square distributed and the approximation has a relative error of second order. Thereby our proposal claims a high level of accuracy with no need to estimate the Godambe information.
Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2008
In testing that a given distribution P belongs to a parameterized family P, one is often led to compare a nonparametric estimate A n of some functional A of P with an element A θ n corresponding to an estimate θ n of θ. In many cases, the asymptotic distribution of goodness-of-fit statistics derived from the process n 1/2 (A n − A θ n ) depends on the unknown distribution P . It is shown here that if the sequences A n and θ n of estimators are regular in some sense, a parametric bootstrap approach yields valid approximations for the P -values of the tests. In other words if A * n and θ * n are analogs of A n and θ n computed from a sample from P θ n , the empirical processes n 1/2 (A n − A θ n ) and n 1/2 (A * n − A θ * n ) then converge jointly in distribution to independent copies of the same limit. This result is used to establish the validity of the parametric bootstrap method when testing the goodnessof-fit of families of multivariate distributions and copulas. Two types of tests are considered: certain procedures compare the empirical version of a distribution function or copula and its parametric estimation under the null hypothesis; others measure the distance between a parametric and a nonparametric estimation of the distribution associated with the classical probability integral transform. The validity of a two-level bootstrap is also proved in cases where the parametric estimate cannot be computed easily. The methodology is illustrated using a new goodness-of-fit test statistic for copulas based on a Cramér-von Mises functional of the empirical copula process.
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.
Improved likelihood inference in generalized linear models
Computational Statistics & Data Analysis, 2014
We address the issue of performing testing inference in generalized linear models when the sample size is small. This class of models provides a straightforward way of modeling normal and non-normal data and has been widely used in several practical situations. The likelihood ratio, Wald and score statistics, and the recently proposed gradient statistic provide the basis for testing inference on the parameters in these models. We focus on the small-sample case, where the reference chi-squared distribution gives a poor approximation to the true null distribution of these test statistics. We derive a general Bartlett-type correction factor in matrix notation for the gradient test which reduces the size distortion of the test, and numerically compare the proposed test with the usual likelihood ratio, Wald, score and gradient tests, and with the Bartlett-corrected likelihood ratio and score tests. Our simulation results suggest that the corrected test we propose can be an interesting alternative to the other tests since it leads to very accurate inference even for very small samples. We also present an empirical application for illustrative purposes. 1
Bootstrapping modified goodness-of-fit statistics with estimated parameters
Statistics & Probability Letters, 2005
Goodness-of-fit tests are proposed for testing a composite null hypothesis that is a general parametric family of distribution functions. They are distribution-free under the null hypothesis and have a limiting normal distribution under the null and the alternative hypothesis. To avoid the estimation of the asymptotic variance under the alternative hypothesis, we propose consistent bootstrap estimators. r 2004 Elsevier B.V. All rights reserved. MSC: primary 62G09; 62G10
Biometrika, 1993
By comparing the expansions of the empirical log-likelihood ratio and the empirical cumulant generating function calculated at the saddlepoint, we investigate the relationship between empirical likelihood and empirical saddlepoint approximations. This leads to a nonparametric approximation of the density of a multivariate M-estimator based on the empirical likelihood and, on the other hand, it provides nonparametric confidence regions based on the empirical cumulant generating function. Some examples illustrate the use of the empirical likelihood in saddlepoint approximations and vice versa.
Comparison of Extended Empirical Likelihood Methods: Size and Shape of Test Based Confidence Regions
Statistica Sinica, 2018
Empirical likelihood is a general non-parametric inference methodology. It uses likelihood principle in a way that is analogous to that of parametric likelihoods. In a wide range of applications the methodology was shown to provide likelihood ratio statistics that have limiting chi-square distributions and observe a nonparametric version of Wilks theorem. Amongst recent extensions of the empirical likelihood are the analysis of censored data, longitudinal data and semi-parametric regression model. However, this property of Wilks theorem only remained true in some but not in others. This motivates our discussion of relative optimality of extended empirical likelihood methods. We compare extended empirical likelihood methods and evaluate their relative optimality by comparing the confidence regions provided by inverting the likelihood ratio tests. We show that those extension methods with its likelihood ratio statistic observing the Wilks theorem provides the smallest confidence region. Specific examples are provided for the case of censored data analysis and estimating equations involving nuisance parameters.
Gradient statistic: Higher-order asymptotics and Bartlett-type correction
Electronic Journal of Statistics, 2013
We obtain an asymptotic expansion for the null distribution function of the gradient statistic for testing composite null hypotheses in the presence of nuisance parameters. The expansion is derived using a Bayesian route based on the shrinkage argument described in [10]. Using this expansion, we propose a Bartlett-type corrected gradient statistic with chi-square distribution up to an error of order o(n −1) under the null hypothesis. Further, we also use the expansion to modify the percentage points of the large sample reference chi-square distribution. Monte Carlo simulation experiments and various examples are presented and discussed.