Estimating Population Size for a Continuous Time Frailty Model with Covariates in a Capture-recapture Study Ying Xu 1 , Liping Liu 1,* , Na You 2 , Hungyu Pan 3 and Paul Yip 4 1 LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China 2 School of Public Health, Peking University, Beijing 100083, China 3 National Tsing-Hua University, HsinChu, Taiwan 300 4 Dept. of Statistics and Actuarial Science, University of Hong Kong, Hong Kong, China ∗ email: liping@math.pku.edu.cn Summary A continuous time frailty capture-recapture model is proposed for estimating population size of a closed population with the use of observed covariates to explain individuals’ heterogeneity in presence of a random effect. A conditional likelihood approach is used to derive the estimate of parameters, and the Horvitz-Thompson estimator is adopted to estimate the unknown population size. Asymptotic normality of the estimates is obtained. Simulation results and a real example show that the proposed method works satisfactorily. Key words: Conditional likelihood; Counting process; Covariate; Frailty; Intensity function. 1 Introduction Capture-recapture sampling and models have been widely used to estimate sizes of biological populations. Nowadays, in addition to biological and ecolog- 1