| Abstract: | Let U and V be two independent positive random variables having continuous distribution functions F and G, respectively. Under left truncation, both U and V are observable only when U>=V. Hence, one observe n iid pairs (X_1,Y_1),....,(X_n,Y_n) from H*(x,y)=P(U<=x,V<=y|U>=V), the conditional distribution of (U, V) given that U>=V . Let a_f=inf{x:F(x)>0} and b_f=sup{x:F(x)<1} denote the lower and upper boundaries of U, respectively. Similarly, define a_g and b_g for V. The nonparametric maximum likelihood estimate of \bar{F(x)}=1-F(x) , \hat\bar{F_n{(x)}=\ {Pi{z<=x}[1-d\hat{lambda_n(z)}, was derived by Lynden-Bell (1971), where \hat\{lambda_n(z)} is the estimated cumulative hazard function of F. Let k={(F,G):a_g<=a_f,b_g<=b_f,F(0)=G(0)=0}. Suppose that (F,G) belong to k. For the estimation of the cumulative hazard function of F (denoted by lambda), Lemma 2 of Woodroofe (1985) computed the bias of the estimator \hat\lambda_n. In this note, it is pointed out that the assertion of Lemma 2 is incorrect. The estimator \hat\lambda_n can seriously underestimate lambda when the n^-1 th quantile of G is larger than that of F. In that case,\hat\bar{F_n{(x)} would seriously overestimate \bar{F(x)} . One way, which would reduce the bias of the estimator, is to consider instead the estimation of \bar{F(x)}/\bar{F(Y_(1))}, where Y_(1) denotes the smallest ordered values of {Y_i}'s .Simulation results support our argument. |
