Statistical Inference Of Partially Linear Single-indicator Cox Proportional Hazards Models Under Length-biased Data

Cox Proportional Hazard model is a widely used tool in survival analysis.However,in practical application,the model is easily affected by the assumption when the covariates are linear,then we will get the wrong estimates.When the dimension of covariates is high,the nonparametric function will be affected by the"curse of dimensionality”.The single-index model can solve the problem of"curse of dimensionality”very well.In practical conditions,we often meet the survival time is left truncated data type,when the truncation variables are uniformly distributed,we call the data with this property length deviation data.When the survival time accords with the data type of length deviation,some theories and methods of Cox model are no longer applicable.Therefore,this paper will study the statistical inference of partial linear single-index Cox proportional hazard model with length bias data.Firstly,this paper studies the estimation of partial linear single-parameter Cox proportional hazard model with length bias data.By using B-spline fitting connection function and combining the idea of composite partial likelihood function,we give the estimation method of regression parameters,and prove the consistency and asymptotic normality of the regression parameters.At the same time,Breslow proposed estimator （?）₀（t）for cumulative benchmark risk function under length bias data.Under different sample size and different deletion rate,we have done a lot of simulation calculations,the results show that our estimation method has a good estimation effect.As an application,we analyze the Oscar dataset using the method proposed in our paper,and give the estimation results of regression parameters,fitting curves and 95%confidence intervals of nonlinear connection functions.Secondly,this paper studies the hypothesis testing problem of a partial linear single-index Cox proportional hazard model with length bias data.Using the idea of likelihood ratio test,we give a test scheme for the hypothesis of whether the connection function is linear or not.The simulation results show that the proposed method is effective when the connection function is assumed to have different curvatures.The test results of the Oscar data set show that the number of movies with more than four stars（X₁）and the proportion of the total number of movies after the nomination（X₂）have nonlinear effects on the life span of the actors.