Empirical Likelihood for Right Censored Lifetime Data

Abstract

This paper considers the empirical likelihood (EL) construction of confidence intervals for a linear functional θ based on right censored lifetime data. Many of the results in literature show that −2 log(empirical likelihood ratio) has a limiting scaled-χ 2 1 distribution, where the scale parameter is a function of the unknown asymptotic variance.The scale parameter has to be estimated for the construction. Additional estimation would reduce the coverage accuracy for θ. This diminishes a main advantage of the EL method for censored data. By utilizing certain influence functions in an estimating equation, it is shown that under very general conditions, −2 log(EL ratio) converges weakly to a standard χ 2 1 distribution and thereby eliminates the need for estimating the scale parameter. Moreover, a special way of employing influence functions eases the otherwise very demanding computations of the EL method. Our approach yields smaller asymptotic variance of the influence function than those comparable ones con-

Two-step semiparametric empirical likelihood inference

“…Since then, this approach has been used in other semiparametric settings, including in [87] for semiparametric regressions with longitudinal data, in [83,85,70] for models with missing data, and in [90,44,69,82] for other semiparametric problems. Explicit recognition of the benefits of using influence functions as estimating equations to obtain chi-squared limiting distributions for EL ratio tests is given in [89,30]. [89] considered finite dimensional nuisance parameters, and although they discussed two applications in semiparametric models, no theoretical results were given for infinite dimensional nuisance parameters.…”

“…[89] considered finite dimensional nuisance parameters, and although they discussed two applications in semiparametric models, no theoretical results were given for infinite dimensional nuisance parameters. [30] proposed using a special influence function for a scalar parameter defined through an estimating equation with right censored data. Relative to this literature, the main contribution of this article is to provide a general theory of bias-corrected EL in semiparametric models.…”

“…For semiparametric models the most popular method uses a two-step (plug-in) procedure in which the first-step estimator replaces the infinitedimensional nuisance parameter, while in the second step the plug-in EL ratio is used to obtain inferences for the finite-dimensional parameter of interest. This two-step semiparametric EL approach has been considered by a number of authors, including [75] for partially linear models, [86] for single-index models, [89,81,30] for various censored regression problems, [77,78,79,76,74,70] for various missing data problems, and [7,48,49,12,11] for other semiparametric problems. [18,88] provide recent surveys on EL inference in the context of semiparametric regression models.…”

Empirical likelihood based on synthetic right censored data

“…Then, based on the simulated data, we can compare the performance of ICconfidence interval (He et al, 2016), ScaledEL-confidence interval (Wang and Jing, 2001) and MeanEL-confidence intervals I A , I B proposed in the previous section. In our simulation, the parameter of interests, θ, is the mean of T , therefore the estimating equation is g(T, θ) = T − θ.…”

“…We use this dataset to illustrate our proposed method described in Section 2. Figure 2 presents the 95% confidence intervals for the mean survival time base on four different methods, Mean-A, Mean-B described in Section 2 and the methods in Wang and Jing (2001) and He et al (2016). All confidence intervals produced by Mean-A, IC and Scale methods contain the Maximum Empirical Likelihood point estimate value 3286.…”