Empirical Likelihood for a Varying Coefficient Model With Longitudinal Data

Abstract

In this article local empirical likelihood-based inference for a varying coefficient model with longitudinal data is investigated. First, we show that the naive empirical likelihood ratio is asymptotically standard chi-squared when undersmoothing is employed. The ratio is self-scale invariant and the plug-in estimate of the limiting variance is not needed. Second, to enhance the performance of the ratio, mean-corrected and residual-adjusted empirical likelihood ratios are recommended. The merit of these two bias corrections is that without undersmoothing, both also have standard chi-squared limits. Third, a maximum empirical likelihood estimator (MELE) of the time-varying coefficient is defined, the asymptotic equivalence to the weighted least-squares estimator (WLSE) is provided, and the asymptotic normality is shown. By the empirical likelihood ratios and the normal approximation of the MELE/WLSE, the confidence regions of the time-varying coefficients are constructed. Fourth, when some components are of particular interest, we suggest using mean-corrected and residual-adjusted partial empirical likelihood ratios to construct the confidence regions/intervals. In addition, we also consider the construction of the simultaneous and bootstrap confidence bands. A simulation study is undertaken to compare the empirical likelihood, the normal approximation, and the bootstrap methods in terms of coverage accuracies and average areas/widths of confidence regions/bands. An example in epidemiology is used for illustration.KEY WORDS: Confidence band; Maximum empirical likelihood estimator.

Asymptotics of SIMEX-based variance estimation

“…Motivated by Xue and Zhu (2007), we introduce a similar approach to give the simultaneous confidence bands for g ps (X, −1) over a given subinterval [a, b] of S(g). In the following content of this subsection, we use g ps (X ) to represent g ps (X, −1) for brevity.…”

Block Empirical Likelihood for Semiparametric Varying-Coefficient Partially Linear Errors-in-Variables Models with Longitudinal Data

“…, into account as a whole. Hence, similar to Xue and Zhu [13], we introduce the auxiliary random vector…”

“…Owen [11] applied empirical likelihood to linear regression models and Kolaczyk [12] made further extensions to generalized linear models. Recently, Xue and Zhu [13] considered the varying coefficient models. You and Zhou [4], Huang and Zhang [5], and Zhao and Xue [14] investigated the empirical likelihood confidence regions for varying-coefficient partially linear models.…”

A simultaneous confidence corridor for varying coefficient regression with sparse functional data

“…Construction of computationally simple SCCs with exact coverage probability is known to be difficult even with independent cross-sectional data; see, Wang and Yang (2009) Xue and Zhu (2007) proposed maximum empirical likelihood estimators and constructed SCCs for the coefficient functions. All these SCCs are Bonferroni-type variability bands according to Hall and Titterington (1988).…”