Combined Likelihood Of Bi-variable

In this paper, we consider combined empirical likelihood and maximum like-lihood for bi-variable inference problems. Firstly, given the constraint of double auxiliary information equations, a semiparametric method is proposed for the parameter inference of conditional probability models. With a newly defined combined likelihood function, the parameter estimation problem comes down to a constrained optimization model, which can be efficiently solved with the KKT condition and the method of lagrange multipliers. In addition, we have shown that the proposed method produces valid inferences with sound asymp-totic properties such as congruence and asymptotic normality. Secondly, we have introduced a new likelihood ratio function R(μ), which generalized the one pro-posed by Qin. With the new likelihood ratio function, the trust region for the underlying parameter can be easily constructed. Moreover, we show that the asymptotic distribution of -21ogR(μ) is χ2.Finally, we generalize above re-sults that are based on independent assumption to the cases of multivariate and a-mixing sequences. The corresponding asymptotic properties are also proven. Simulation results demonstrate that the point estimations with the proposed methods have both small biases(less than 0.001) and small variances(less than 0.001). The constructed trust region is sound and has good symmetry. These results provide further evidence to the validity and feasibility of our methods.