Empirical Entropy And Empirical Likelihood Under Censorship

The main research of this paper is applying the empirical entropy (which isalso called exponential titling method,”ET” for short) method and the empiricallikelihood (”EL” for short) method to right censored data and doubly censored data,respectively. EL and ET are both nonparametric statistical method. They havethe almost the same statistical asymptotic property and advantages. Such as forindependent and identically distribution (i.i.d.) complete data, the statistics basedon the two methods asymptotically follow a standard χ~2distribution, which meanscompared with the normal approximation method, both of the two methods don’tneed to estimate the asymptotic variance; the shapes of confdence intervals (CI’s)based on the two methods may be not symmetrical and are automatically decidedby the data. Meanwhile, there is some diference between the two methods. Forexample, EL is Bartlett-Correctable while ET is not; ET is robust while EL is not.In literature, some researchers generalized EL from complete data to right censoreddata. This paper will generalize ET from complete data to right censored data.On the other hand, some authors generalized EL to doubly censored data, but thecorresponding log EL ratio statistics follows a non-standard χ~2distribution. This paper will study how to redefne the EL ratio statistics such that the asymptoticaldistribution is a standard χ~2distribution.This dissertation includes three chapters. In Chapter2, we describe the basicconcept and property of EL and ET, introduce EL and ET under i.i.d. completedata and compare the two methods by simulations on heavy distribution and non-heavy distribution, respectively. Finally, we introduce EL under right censorship,study how to generalize ET to right censored data. In consideration of robustness,we compare the two methods by simulations especially when right censored dataare contaminated. The simulations in this chapter show that:1) for i.i.d. completedata from non-heavy distribution, EL method and ET method have their pros andcons;2) for i.i.d. complete data from heavy distribution, ET method performsbetter than EL method in terms of the coverage of CI and the average length of CI;3) for uncontaminated right censored data, EL method performs a bit better thanET method;4) for contaminated right censored data, ET method performs muchbetter than EL method in terms of the coverage of CI (CCI) and the average lengthof CI (ALCI).In Chapter3, we focus on how to apply EL method to doubly censored data.We will give some real examples and statistical description of doubly censored data.Then we present the existing research works on doubly censored data. Finally wewill study how to get the efective influence function (EIF) of the parameter ofinterest under doubly censorship and how to combine the EIF and EL method so asto make inference. In fact, The log EL ratio based on the new method asymptoticallyfollows a standard χ~2distribution, which maintains the advantage of EL method.And the new method is powerful since it is suitable for both the parameter of linearfunctional and non-linear functional. The simulations show that the new methodperforms better than the weighted EL (WEL) method in terms of CCI, ALCI andthe variance of the length of CI (VLCI).