Jackknife empirical likelihood of error variance in partially linear varying-coefficient errors-in-variables models

Abstract

For the partially linear varying-coefficient model when the parametric covariates are measured with additive errors, the estimator of the error variance is defined based on residuals of the model. At the same time, we construct Jackknife estimator as well as Jackknife empirical likelihood statistic of the error variance. Under both the response variables and their associated covariates form a stationary α-mixing sequence, we prove that the proposed estimators and Jackknife empirical likelihood statistic are asymptotic normality and asymptotic χ 2 distribution, respectively. Numerical simulations are carried out to assess the performance of the proposed method.

Jackknifing for partially linear varying-coefficient errors-in-variables model with missing response at random

“…Remark 3.1 Assumptions (A1)-(A4) are standard conditions, which are commonly used in the literature; see Fan and Huang [4], Liu and Liang [12]. Assumption (A5) is always applied on missing data analysis; see Wang et al [20].…”

“…Following Lemma 6.11 in Liu and Liang [12], it follows that β nβ n,-i = O p (n -1 ). We combine this with the fact max 1≤i≤n ξ i = o(n 1/(2s) ) for s > 2.…”

“…From the proof of (6.16) in Liu and Liang [12], it can be checked that (1). According to Lemma A.5, one can write…”

“…For example, Fan et al [2] considered the penalized empirical likelihood and variable selection for high-dimension data. Liu and Liang [12] constructed the asymptotical normality of jackknife estimator for error variance and standard chi-square distribution of jackknife empirical log-likelihood statistic. Fan et al [3] established penalized profile least squares estimation of parameter and non-parameter in the model.…”

“…Thanks to its advantages, the jackknife empirical likelihood approach has been applied by many researchers. See Gong et al [5], Peng et al [15], Yang and Zhao [26], Liu and Liang [12], Yu and Zhao [30] and so on. However, there is a little literature considering the jackknife method for response mean with missing response at random.…”