Missing data and time-dependent covariates often arise simultaneously in longitudinal studies,and directly applying classical approaches may result in a loss of efficiency and biased estimates.To deal with this problem,we propose weighted corrected estimating equations under the missing at random mechanism,followed by developing a shrinkage empirical likelihood estimation approach for the parameters of interest when time-dependent covariates are present.Such procedure improves efficiency over generalized estimation equations approach with working independent assumption,via combining the independent estimating equations and the extracted additional information from the estimating equations that are excluded by the independence assumption.The contribution from the remaining estimating equations is weighted according to the likelihood of each equation being a consistent estimating equation and the information it carries.We show that the estimators are asymptotically normally distributed and the empirical likelihood ratio statistic and its profile counterpart follow central chi-square distributions asymptotically when evaluated at the true parameter.The practical performance of our approach is demonstrated through numerical simulations and data analysis.
A new expectation-maximization（EM） algorithm is proposed to estimate the parameters of the truncated multinormal distribution with linear restriction on the variables. Compared with the generalized method of moments（GMM） estimation and the maximum likelihood estimation（MLE） for the truncated multivariate normal distribution, the EM algorithm features in fast calculation and high accuracy which are shown in the simulation results. For the real data of the national college entrance exams（NCEE）, we estimate the distribution of the NCEE examinees＇ scores in Anhui, 2003, who were admitted to the university of science and technology of China（USTC）. Based on our analysis, we have also given the ratio truncated by the NCEE admission line of USTC in Anhui, 2003.
We investigate the precise large deviations of random sums of negatively dependent random variables with consistently varying tails. We find out the asymptotic behavior of precise large deviations of random sums is insensitive to the negative dependence. We also consider the generalized dependent compound renewal risk model with consistent variation, which including premium process and claim process, and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.
