In this paper, in order to test whether changes have occurred in a nonlinear parametric regression, we propose a nonparametric method based on the empirical likelihood. Firstly, we test the null hypothesis of no-change against the alternative of one change in the regression parameters. Under null hypothesis, the consistency and the convergence rate of the regression parameter estimators are proved. The asymptotic distribution of the test statistic under the null hypothesis is obtained, which allows to find the asymptotic critical value. On the other hand, we prove that the proposed test statistic has the asymptotic power equal to 1. These theoretical results allows find a simple test statistic, very useful for applications. The epidemic model, a particular model with two change-points under the alternative hypothesis, is also studied. Numerical studies by Monte-Carlo simulations show the performance of the proposed test statistic, compared to an existing method in literature.
A non parametric method based on the empirical likelihood is proposed for detecting the change in the coefficients of high-dimensional linear model where the number of model variables may increase as the sample size increases. This amounts to testing the null hypothesis of no change against the alternative of one change in the regression coefficients. Based on the theoretical asymptotic behaviour of the empirical likelihood ratio statistic, we propose, for a fixed design, a simpler test statistic, easier to use in practice. The asymptotic normality of the proposed test statistic under the null hypothesis is proved, a result which is different from the χ 2 law for a model with a fixed variable number. Under alternative hypothesis, the test statistic diverges. We can then find the asymptotic confidence region for the difference of parameters of the two phases. Some Monte-Carlo simulations study the behaviour of the proposed test statistic.Keywords Two-sample · high-dimension · linear model · empirical likelihood test. IntroductionThe technology development and fast numerical techniques make possible to consider and study statistical models with a large number of variables. High-dimensional model refers to a model whose the number p of explanatory variables increases to infinity as the number n of observations converges to infinity. When p diverges, traditional statistical methods may not work with this kind of growth dimensionality.Most of the literature works on high-dimensional model utilize the LASSO (Least Absolute Shrinkage and Selection Operator) type methods, in order to automatically select the significant variables. The principle of these methods, introduced by Tibshirani (1996), is to optimize a penalized process, more precisely, a process with a L 1 -type penalty. If the model contains outliers, the parameter estimators by the least squares method with LASSO penalty have a large error. An alternative method is then the penalized quantile method. Thereby, Dicker et al. (2013) consider a quantile model with seamless-L 0 penalty when the number p of explanatory variables is such that p → ∞, p/n → 0 as n → ∞. For a general quantile regression, Wu and Liu (2009) propose the SCAD penalty, while, in Zou andYuan (2008), a composite quantile regression is considered with an adaptive LASSO penalty. The case p → ∞ is also considered in Fan and Peng (2004) for a non-concave penalized likelihood method, when p 5 /n → ∞. Concerning the group selection methods for high-dimensional models, the readers find in Huang et al. (2012) a review of methods. All these methods are based first on the principle of selecting (automatically) the significant variables. Then, the dependent variable is modeled only as a function of the significant variables, in order to have more accurate parameter estimators and a better adjustment for the dependent variable.
In this paper, we use the empirical likelihood method to construct the confidence regions for the difference between the parameters of a two-phases nonlinear model with random design. We show that the empirical likelihood ratio has an asymptotic chi-squared distribution. The result is a nonparametric version of Wilk's theorem. Empirical likelihood method is also used to construct the confidence regions for the difference between the parameters of a two-phases nonlinear model with response variables missing at randoms (MAR). In order to construct the confidence regions of the parameter in question, we propose three empirical likelihood statistics : Empirical likelihood based on complete-case data, weighted empirical likelihood and empirical likelihood with imputed values. We prove that all three empirical likelihood ratios have asymptotically chi-squared distributions. The effectiveness of the proposed approaches in aspects of coverage probability and interval length is demonstrated by a Monte-Carlo simulations.
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