Empirical Likelihood Inference For Rank Regression With Doubly Truncated Data

Doubly truncated data refers to the data with both left and right truncation,which are mentioned in the literature of astronomy,econometrics and survival analysis.Ying et al.proposed a method for estimating regression parameters when the dependent variable is affected by double truncation in 2020.They extended the Mann-Whitney-type rank estimation and proved that the estimator is consistent and asymptotically normal.A random weighting method is proposed to obtain the confidence interval of any specified component of the parameter.The random weighting method needs to generate a large number of weighted rank estimates by repeatedly generating random weights,which is very time-consuming.The advantage of the empirical likelihood method is that it can avoid the variance estimation and obtain an effective confidence interval.Therefore,we will reduce the computational cost of the random weighting method by applying the empirical likelihood method.This paper introduces the non-smooth empirical likelihood and Jackknife empirical likelihood methods into the rank regression model under doubly truncated data.The article is mainly divided into two parts.The first part of the article mainly studies the Empirical Likelihood(EL)problem of the rank regression model with doubly truncated data.First,construct an estimation function suitable for the rank regression model with doubly truncated data to prove the effectiveness of the empirical likelihood method.After that,14 lemmas are introduced to prove that the non-smooth empirical likelihood ratios statistics of the unknown parameters under the empirical likelihood method closely follows the 4-fold asymptotic chi-square distribution,and the confidence interval formula of the position parameter is obtained.Through numerical simulation,the confidence interval and length are compared with the weighted rank estimation(Random Weighting,RW)method proposed in Ying et al.,and it is concluded that for the empirical coverage probabilities,EL is closer to the nominal level of 95% than the RW method.,RW has a shorter interval length.In addition,when the PLT is fixed and the PRT becomes larger,the average length of the RW and EL confidence intervals becomes longer.The properties of all confidence intervals improve as the sample size increases.As the sample size becomes larger,the difference between the RW and EL confidence intervals gradually decreases.The Q-Q chart clearly shows that the non-smooth EL ratios statistics always closely follows the 4-fold asymptotic chi-square distribution.The second part of the article mainly studies the Jackknife Empirical likelihood(JEL)inference problem of the rank regression model with doubly truncated data.First,construct the estimation function of the rank regression model with doubly truncated data under the JEL method,and then derive it with the proof process similar to the empirical likelihood method,introduce 7 lemmas,and prove that the non-smooth JEL ratios statistics of unknown parameters follows the asymptotic chi-square distribution;Finally,the effectiveness of the JEL method under the rank regression model is verified by numerical simulation,and compared with the RW and EL methods,it is found that the coverage probability of JEL is closer to the nominal level of 95% than the coverage probability of the EL and RW methods.The Q-Q chart clearly shows that the non-smooth JEL ratios statistics always closely follows the asymptotic chi-square distribution.This article applies the three methods of RW,EL,and JEL to AIDS incubation data,and finds that the three confidence intervals and their interval lengths at different confidence levels indicate that there is a significant positive correlation between the incubation period and age,and the EL and JEL methods are much faster than the RW method.