Confidence Intervals for the Current Status Model

Abstract

ABSTRACT. We discuss a new way of constructing pointwise confidence intervals for the distribution function in the current status model. The confidence intervals are based on the smoothed maximum likelihood estimator, using local smooth functional theory and normal limit distributions. Bootstrap methods for constructing these intervals are considered. Other methods to construct confidence intervals, using the non-standard limit distribution of the (restricted) maximum likelihood estimator, are compared with our approach via simulations and real data applications.

The nonparametric bootstrap for the current status model

“…The proof uses empirical process theory and results on tail probabilities for √ n(P n − P n ) F for classes F with finite entropy integrals. Similar results are proved using martingale theory in Section 11.2 of [17] for the original sample and in [14] for a smooth bootstrap empirical process. Since…”

“…This raises the question if one should really use confidence intervals based on the MLE instead of on a faster converging estimate. This latter procedure is followed in [14], where the authors consider constructing confidence intervals around the smoothed maximum likelihood estimator (SMLE) of F 0 in the current status model. The SMLE is a kernel estimate based on the MLE with an asymptotic normal distribution, instead of Chernoff's limiting distribution ( [16]).…”

“…The SMLE is a kernel estimate based on the MLE with an asymptotic normal distribution, instead of Chernoff's limiting distribution ( [16]). The bootstrap method proposed in [14] is however still based on the smooth bootstrap procedure described in [32] and not on Efron's nonparametric bootstrap. We show in this paper that the construction of confidence intervals around the SMLE based on the nonparametric bootstrap can also be proved to be valid, where one does not resample from a smooth estimate of F 0 , but just resamples with replacement from the pairs (T i , ∆ i ) in the original sample.…”

“…This method already has been used without proof in [17] and also in [18] and the present manuscript intends to fill the gap of the missing proofs here. An important difference with the smooth bootstrap in [14] is that for the centering of the estimates in the nonparametric bootstrap samples the SMLE of the original sample is used, whereas this will not work for the resampling as proposed in [14]; in the latter case one needs to center the estimates in the bootstrap samples by a kernel convolution of the SMLE in the original sample. It is not clear which method is better, and the most striking fact is the similarity of the results of the two methods in our simulations.…”

Nonparametric Shape-Restricted Regression

“…It is well-known that both the above bootstrap schemes -bootstrapping pairs and bootstrapping residuals withf n =f n -yield inconsistent estimators of H n ; see [1,122,82,125,56]. Intuitively, the inconsistency of the residual bootstrap procedure can be attributed to the lack of smoothness off n .…”