Nonparametric Mean Residual Life And Semiparametric Models Under Length-biased Data And Their Applications

The main tasks of survival analysis are:(1)statistical analysis and inference of the distribution of survival time;(2)analysis of the relationship between survival time and potential influence factors.Its theory and methods can be properly used to deal with common censored data in practice,such as right censored data,left-truncated data and interval-type censored data,etc.In this thesis,the length-biased data we study is a special case of left-truncated data.In left-truncated sampling,if chances that the initial event happens at any calendar time ? are equal,or the left truncation variable follows a uniform distribution,then this kind of collected data is also called length-biased data,which has been introduced in detail by a lot of literature such as Vardi(1982),Gupta and Keating(1986),Luo and Tsai(2009),etc.Length-biased data is widespread in life test work.Especially recently,a lot of research work on different models under length-biased data has been done by statistical researchers,such as Huang and Qin(2011,2012),Bai et al.(2016),Shi et al.(2018)and so on.Inspired by their research work,in this thesis,we not only study the moment-type estimators and empirical likelihood confidence interval of mean residual life function with length-biased data,but also analyze and study coefficients' estimations of Aalen additive hazards model and additive mean residual life model.The main research content and innovations of this thesis is further introduced in the following.In Chapter 1,definitions of length-biased data and mean residual life function(M-RL)are firstly introduced and their research statuses are also summarized;in addition,several semiparametric models referred in this thesis,such as proportional likelihood ratio model,semiparametric hazards model and mean residual life regression model,are also introduced and their development situations are summarized,too.In Chapter 2,several nonparametric estimations of the MRL are developed with length-biased data.Mean residual life function,which describes the remaining mean life of a unit or an individual that has been alive for t years(or other units of time),is an important biological statistical function in survival analysis and life table studies.It is widely used in many life tests,for example:in medicine,people with incurable diseases,such as cancer patients,may want to know their remaining mean life very much;in clinical treatment,doctors need to calculate the residual life to advise patients an appropriate treatment plan.There are two main research motivations in this chapter:(1)In general,nonparametric estimators of the MRL are usually constructed based on nonparametric estimations of survival distribution,such as Gill(1983),Zhao et al.(2013)and so on,but they need to calculate the integral of an estimated survival function,which may be sometimes very troublesome to deal with;(2)To the best of our knowledge,for nonparametric statistical inference of length-biased data,most of existing research work focuses on nonparametric inference of survival function of interest and little research has been done on the MIRL.So,in this chapter,we study moment-type estimators of the MRL with length-biased complete data and censored length-biased data,respectively,and prove large sample properties of our proposed estimators through the Hadamard derivative principle.In order to assess the proposed methods,a series of numerical simulations are carried out.In addition,the MRL of the older in Channing House data is analyzed by proposed methods in this chapter.The main innovations of this chapter are:(1)With complete length-biased data,we firstly derive the relationship between the MRL and survival functions of length-biased variable and left truncation variable,and then construct an unbiased moment-type estimation of distribution function of the length-biased variable and a composite unbiased moment-type estimation of survival function of the left truncation variable,respectively.Based on above referred work,the moment-type estimation of the MRL is constructed under complete length-biased data;(2)With censored length-biased data,we firstly derive moment-type estimations of survival function and its function of length-biased variable,and then propose three nonparametric estimations of the MRL by utilizing the relationship of The MRL and survival function of length-biased variable;(3)Asymptotical normalities of our proposed estimators are proved through the central limit theorem and Hadamard derivative principle.In Chapter 3,empirical likelihood(EL)confidence interval of the MRL is stud-ied with length-biased right-censored data.As a nonparametric statistical inference method,empirical likelihood has been widely used to construct confidence intervals of some unknown quantities in statistical problems.When the adjusted empirical likeli-hood method(see wang and Jing,2001;Qin and Zhao,2007)developed under random censored data is employed to construct a confidence interval,an additional scale pa-rameter needs to be estimated,and this no doubt increases the difficulty of calculation.Fortunately,He et al.(2015)developed a special influence function to be the estimat-ing equation with random censored data.In their work,the limiting distribution of-2log(empirical likelihood ratio)is proved to be a standard ?12 distribution under gen-eral condition and can be used to construct confidence interval without any additional scale parameters.Inspired by their work,in this chapter we aim to develop an empirical log-likelihood function for the MRL with length-biased right-censored data,where the limiting distribution of corresponding-2 log(empirical likelihood ratio)is a standard ?12 distribution under some general conditions.In addition,we also derive an asymptotic normal confidence interval of the MRL with length-biased right-censored data and com-pare it with our empirical likelihood method as well as the "simple" bootstrap method(Bilker and Wang,1997)through a simulation study.Results show that the mean width of the EL-based confidence interval is much shorter than that derived by the other meth-ods.The main innovations of this chapter are:(1)By utilizing the relationship between the MRL and density function of the length-biased variable,an estimating equation of the MRL is constructed through the inverse probability of weighting technique and an empirical log-likelihood ratio of the MRL is defined under censored length-biased data;(2)The limiting distribution of our proposed empirical log-likelihood ratio is proved to be a standard ?12 distribution under some general conditions;(3)An asymptotically normal confidence interval of the MRL is derived under some general conditions.In Chapter 4,we study composite estimating equations of regression coefficients in Aalen additive hazards model with length-biased right-censored data.Aalen additive hazards model plays an important role in survival analysis.It describes the instanta-neous risk of an individual as the sum of a basic hazards function and a specific attribute of an individual.Huang and Qin(2013)found that the density function of the left trun-cation variable in left-truncated sampling owns a structure of proportional likelihood ratio model(Luo and Tsai,2012).In length-biased sampling the left truncation variable and remaining life time share the same distribution.Therefore,the density function of remaining life time owns a structure of proportional likelihood ratio model,too.In this chapter,we propose a pairwise pseudo-likelihood estimating equation only relying on re-maining life observations based on the invariance property of parameters in proportional likelihood ratio model and the unique structure of length-biased sample.In order to ab-sorb information contained in the sample as more as possible,two composite estimating equations are also developed based on the pairwise pseudo-likelihood estimating equation proposed by Huang and Qin(2013)and composite conditional estimating equation pro-posed by Ma et.al.(2015).Finally,we analyze the survival difference between Male and Female in the Channing House data through the proposed pairwise pseudo-likelihood estimating equation,which only relies on observations of residual life variable.The main innovations of this chapter are:(1)The conditional density function of complete remaining residual life variable in censored length-biased sampling is firstly derived,and then,based on the proportional likelihood ratio model structure of this density function,the pairwise pseudo-likelihood estimating equation relying only on complete remaining life observations is constructed;(2)By taking observations of the left truncation variable and remaining life time as bivariate survival data,a composite pairwise pseudo-likelihood estimating equation is constructed.In addition,by combining the composite pairwise pseudo-likelihood es-timating equation with the composite conditional estimating equation proposed by Ma et al.(2015),a composite conditional-pairwise pseudo-likelihood estimating equation is also constructed.(3)According to the asymptotic properties of U-statistics,we derive asymptot-ic normalities of our composite pairwise pseudo-likelihood estimation and composite conditional-pairwise pseudo-likelihood estimation.In Chapter 5,we develop estimating equations of additive mean residual life model under censored length-biased data.Additive mean residual life model is one of the most common model in survival analysis,and has wide applications in many practice fields such as clinical medical test,actuarial science,etc.Existing statistical inference on this model is usually discussed under right censored data and cannot be directly used to analyze length-biased data.In order to overcome this problem,in this chapter we propose a martingale-type estimation and an inverse probability of weighting technique estimation for the baseline mean residual life function and covariate coefficients under censored length-biased data,and discuss large properties of these estimations.In addition,the numerical simulation results show that empirical biases,empirical standard deviations and empirical mean square errors derived by inverse probability of weighting technique estimation of coefficients are significantly smaller than that derived by martingale-type estimation when the sample capacity and censoring rate are the same.However,it is worth noting that the martingale-type estimation of coefficients can be applied to a variety of samples,such as standard length-biased data,observations of left-truncation variable,observations of censored remaining life time,etc.In addition,proportional mean residual life model and transformed mean residual life model(Sun and Zhang,2009)can also be discussed in the similar way we introduce in this chapter.The main innovations of this chapter are:(1)The relationship between cumulative hazards function of the left truncation variable and mean residual life model of the population of interest is firstly derived,and then the estimating equation for additive mean residual life function is proposed through martingale technique;(2)The inverse probability of weighting technique is adopted to construct an es-timating equation of additive mean residual life model,and a weighting function is introduced into the equation to explore information contained in the sample as more as possible to improve the accuracy of estimation;(3)Large properties of two type of estimations are studied under some general conditions.