Empirical likelihood for nonlinear models with missing responses

Abstract

In this paper, a nonlinear model with response variables missing at random is studied. In order to improve the coverage accuracy for model parameters, the empirical likelihood (EL) ratio method is considered. On the complete data, the EL statistic for the parameters and its approximation have a χ 2 asymptotic distribution. When the responses are reconstituted using a semi-parametric method, the empirical loglikelihood on the response variables associated with the imputed data is also asymptotically χ 2 . The Wilks theorem for EL on the parameters, based on reconstituted data, is also satisfied. These results can be used to construct the confidence region for the model parameters and the response variables. It is shown via Monte Carlo simulations that the EL methods outperform the normal approximation-based method in terms of coverage probability for the unknown parameter, including on the reconstituted data. The advantages of the proposed method are exemplified on real data.

Empirical likelihood confidence regions for the parameters of a two phases nonlinear model with and without missing response data

“…for j ∈ J. We suppose that π 1 (x i ) > 0 and π 2 (x j ) > 0, which are a common suppositions in the literature, see for example the papers of Sun et al (2009), Xue (2009) and Ciuperca (2011). Moreover, lim n→∞ i∈I δ i = lim n→∞ j∈J δ j = ∞, which means that, we have a significant number of values for Y non missing.…”

“…We consider the following nonlinear function , 20]. The same function was considered in Ciuperca (2011) where the model was estimated by penalized least absolute method and in Ciuperca and Salloum (2013) to test the change in the regression parameters of the nonlinear model. There are three different cases for the error terms :…”

“…for all 1 ≤ i ≤ n. The MAR assumption is a common condition for statistical analysis with missing data and is reasonable in many practical situations, see Little and Rubin (1987), Qin et al (2009) and Ciuperca (2011).…”

“…To construct the confidence regions for the coefficients in the linear regression model, Chen (1994) proposed a nonparametric method based on empirical likelihood. Qin et al (2009), Xue (2009) and Ciuperca (2011) considered this same problem but for the models with missing response data. Kolaczyk (1994) shows that empirical likelihood is justified as a method of inference for a class of linear models, and shows in particular how empirical likelihood may be used with generalized linear models.…”

Empirical likelihood test in a posteriori change-point nonlinear model

“…(8) In view of Theorem 2 of [22], using assumptions (A1), (A2) and (A3), each sum of the right-hand side of (8) converges in law to χ 2 (d). Taking into account that the two terms of relation (8) involved two independent sets of random vectors we obtain the theorem.…”