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In this paper, a varying-coefficient density-ratio model for case-control studies is developed. We investigate the local empirical likelihood diagnosis of varying coefficient density-ratio model for case-control data. The local empirical log-likelihood ratios for the nonparametric coefficient functions are introduced. First, the estimation equations based on empirical likelihood method are established. Then, a few of diagnostic statistics are proposed. At last, we also examine the performance of proposed method for finite sample sizes through simulation studies.

Varying coefficient models are often used as extensions of classical linear models (e.g. Shumway [

In this paper, we consider the following general two-sample varying-coefficient density-ratio model

where

et al. [

where

Various density-ratio models for some conventional density functions were discussed in Kay and Little [

The empirical likelihood method origins from Thomas & Grunkemeier [

Over the last several decades, the diagnosis and influence analysis of linear regression model has been fully developed (R.D. Cook and S. Weisberg [

The remainder of the article is organized as follows. Local empirical likelihood and estimation equation are presented in Section 2. The main results are given in Section 3. An example is given to illustrate our results in Section 4.

Let

group and case group, respectively. Let

the pooled sample. Assume that

where

tion corresponding to

Assume that all components of

Let

denote

where

represents the size of the local neighborhood. The kernel weight is used to give smoother weight to data with

where

Motivated by Zhu and Ibrahim (2008), we regard

Obviously, the maximum empirical likelihood estimates

We consider the local influence method for a case-weight perturbation

in which

We consider two local influence measures based on the normal curvature

The most popular local influence measures include

as

the most influential perturbation to the empirical likelihood function, whereas the

As the discuss of Zhu et al. (2008), for varying-coefficient density-ratio model, we can deduce that

where

are trivariate normal densities with means

Because

We draw 1000 data sets with sample size

choose the Epanechnikov kernel

In order to checkout the validity of our proposed methodology, we change the value of the first, 125th, 374th,

789th and 999th data. For every case, it is easy to obtain

ples, we evaluated their maximum empirical likelihood estimators.

Consequently, it is easy to calculate the value of

It can be seen from the result of

In this paper, we considered the statistical diagnosis for varying-coefficient density-ratio model based on local empirical likelihood. Through simulation study, we illustrate that our proposed method can work fairly well.