Abstract: The concept of $\phi$-mixing has been used extensively as measures of the weak depen-dence, and the phenomenon of missing data often occurs in various application fields. In existing literatures, the statistical inference under the dependence and missing data, has been deeply studied. However, there are few studies on the case of the dependent and missing data simultaneously. This paper is concerned with the statistical inference simultaneously under the dependence and missing data. In other words, this paper discusses the asymptotic distributions of empirical likelihood ratio statistics for regression coefficients in a linear model under $\phi$-mixing samples with missing data. The regression imputation method is applied to impute the missing data of the response variables, and thus `complete' data for regression coefficients in the linear model are obtained. Furthermore, we employ the score functions to establish the empirical likelihood ratio statistics for the regression vector in the linear model. Under some conditions, it is proved that the empirical likelihood ratio statistics are asymptotically Chi square distributed. This conclusion provides a theoretical basis for the confidence region of the regression coefficients of a linear model under $\phi$-mixing samples with missing data.

Key words: $\phi$-mixing sample, missing data, empirical likelihood, $\chi^2$ distribution