The Estimator Of The Expectation Of The Function And Its Application With Interval-censored Data

The problem of analyzing the time of a given event has become increasingly commonin many applied fields, such as demographic statistics, medicine, biology, epidemiology,economy and sociology, we call the time of a certain event as failure time data. However,because of the limited objective condition the exact time of interest cannot be observed,but a certain interval in which it falls into is known. In statistics we call this type data asinterval-censored data.In this paper, we construct the estimator of the expectation of the function which isdenoted by Eh(T) of interval-censored data by unbiased transformation idea, where h(T)is a continuous function of the target variable T. It is simplified to deal with the problemof T itself when h(T) = T. In many situations, we have to consider the function of h(T)in interval-censored data. For example, in the accelerated failure model h(T) = logT, andwhen h(T) = Tr it becomes the problem of the origin moment.When the target variable T is nonnegative and the function of h(T) is continuous,byuse of unbiased transformation idea we construct the estimators of Eh(T) under caseI and case II interval-censored data, which are denoted by h*(V,δ) and h**(U,V,δ₁,δ₂)respectively. It's worth noting that it is possible that the variance of the estimators donot exist, so it is necessary to give some limits to the estimators and the distributionof the target variable. When h(0) <∞we only need to make a limit at the variableapproach to infinity, which the distributions of target variable T and V . On the otherhand, if h(0) =∞what we should do is to make a restriction at the variable approachto zero of the distribution of target variable's distribution. Focus on the issue, this papermakes a discussion to case I and case II interval-censored data and draws a conclusion.The asymptotic normality and consistency are also established. Later, the conclusion areapplied to the parametric estimator of linear regression model with interval-censored data.At last, we verify the e*ect of the estimator under model (I) and model (II) by sim-ulation.