Local empirical likelihood inference for varying-coefficient density-ratio models based on case-control data

In this article, we develop a varying-coefficient density-ratio model for case-control studies. The case and control samples come from two different distributions. Under the model assumption, the ratio of the two densities is related to the linear combination of covariates with varying coefficients through a known function. A special case is the exponential tilt model where the log ratio of the two densities is a linear function of covariates. We propose a local empirical likelihood (EL) approach to estimate the nonparametric coefficient functions. Under some regularity assumptions, the proposed estimators are shown to be consistent and asymptotically normally distributed. The sieve empirical likelihood ratio (SELR) test statistic for detecting whether the varying-coefficients are really constant and other related hypotheses is constructed and it follows approximately a chi-squared distribution. We introduce a modified bootstrap procedure to estimate the null distribution of the SELR when sample size is small. We also examine the performance of proposed method for finite sample sizes through simulation studies and illustrate it with a real dataset. Supplementary materials for this article are available online.