Nonparametric estimation of bivariate mean residual life function
Ghebremichael, Musie S.
Doctor of Philosophy
In survival analysis the additional lifetime that an object survives past a time t is called the residual life function of the object. Mathematically speaking if the lifetime of the object is described by a random variable T then the random variable R(t) = [T - t| T > t] is called the residual life random variable. The quantity e(t) = E( R(t)) = E[T - t|T > t] is called the mean residual lifetime (mrl) function or the life expectancy at age t. There are numerous situations where the bivariate mrl function is important. Times to death or times to initial contraction of a disease may be of interest for litter mate pairs of rats or for twin studies in humans. The time to a deterioration level or the time to reaction of a treatment may be of interest in pairs of lungs, kidneys, breasts, eyes or ears of humans. In reliability, the distribution of the lifelengths of a particular pair of components in a system may be of interest. Because of the dependence among the event times, we can not get reliable results by using the univariate mrl function on each event times in order to study the aging process. The bivariate mrl function is useful in analyzing the joint distribution of two event times where these times are dependent. In recent years, though considerable attention has been paid to the univariate mrl function, relatively little research has been devoted to the analysis of the bivariate mrl function. The specific contribution of this dissertation consists in proposing, and examining the properties of, nonparametric estimators of the bivariate mean residual life function when a certain order among such functions exists. That is, we consider the problem of nonparametric estimation of a bivariate mrl function when it is bounded from above by another known or unknown mrl function. The estimators under such an order constraint are shown to perform better than the empirical mrl function in terms of mean squared error. Moreover, they are shown to be projections, onto an appropriate space, of the empirical mean residual life function. Under suitable technical conditions, the asymptotic theory of these estimators is derived. Finally, the procedures are applied to a data set on bivariate survival. More specifically, we have used the Diabetic Retinopathy Study (DRS) data to illustrate our estimators. In this data set, the survival times of both left and right eyes are given for two groups of patients: juvenile and adult diabetics. Thus, it seems natural to assume that the mrl for the juvenile diabetics be longer than the mrl of the adult diabetics. Under this assumption, we calculated the estimators of the mrl function for each group. We have also calculated the empirical mrl functions of the two groups and compared them with the estimators of the mrl function obtained under the above assumption.