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In this paper, we investigate the empirical likelihood diagnosis of modal linear regression models. The empirical likelihood ratio function based on modal regression estimation method for the regression coefficient is introduced. First, the estimation equation based on empirical likelihood method is established. Then, some diagnostic statistics are proposed. At last, we also examine the performance of proposed method for finite sample sizes through simulation study.

The mode of a distribution is regarded as an important feature of data. Several authors have made efforts to identify the modes of population distributions for low-dimensional data. See, for example, Muller and Sawitzki [

Given a random sample

Lee [

The empirical likelihood method originates 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 rest of the paper is organized as follows. In Section 2, we review the modal regression. In Section 3, empirical likelihood and estimation equation are presented. The main results are given in Section 4. Simulation study is given to illustrate our results in Section 5.

Suppose a response variable

The idea of modal linear regression can be easily generalized to other models such as nonlinear regression, nonparametric regression, and varying coefficient partially linear regression. To include the intercept term in (1), we assume that the first element of

Yao and Li [

where

In this section, we review empirical likelihood based on modal regression for regression coefficients, then establish the estimation equations.

Similarly to Zhao, Zhang and Liu [

Note that

By the method of Lagrange multipliers, similar to that used in Owen (2001),

where

Motivated by Zhu and Ibrahim [

Obviously, the maximum empirical likelihood estimates

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

likelihood function

1 vector with all elements equal to 1, represents no perturbation to the empirical likelihood, because

where

fluence graph

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. [

where

We generate data-sets from following model

where the covariates

In order to check out 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

Consequently, it is easy to calculate the value of

From the figure, we can see that in most cases, the value of

In this paper, we considered the statistical diagnosis for modal linear regression models based on empirical likelihood. Through simulation study, we illustrate that our proposed method can work fairly well.