The nonparametric bootstrap for the current status model

Abstract

It has been proved that direct bootstrapping of the nonparametric maximum likelihood estimator (MLE) of the distribution function in the current status model leads to inconsistent confidence intervals. We show that bootstrapping of functionals of the MLE can however be used to produce valid intervals. To this end, we prove that the bootstrapped MLE converges at the right rate in the Lp-distance. We also discuss applications of this result to the current status regression model.

Confidence Intervals for the Current Status Model

“…Recently, it was however proved in Groeneboom & Hendrickx () that the naive bootstrap of resampling with replacement from the data does work in case the underlying distribution function is estimated by the SMLE or in case interest is in other functionals than the values of the distribution function. The validity of the naive bootstrap for constructing pointwise CIs around the SMLE and for doing inferences in the current status linear regression model (Groeneboom & Hendrickx, ) is illustrated in Groeneboom & Hendrickx (). Although Durot & Reboul () conjecture that the naive bootstrap fails in their setting, the result of Groeneboom & Hendrickx () suggests that this conjecture might be incorrect and that applications of the naive bootstrap involving the Grenander estimator are worthy of study in further research.…”

Some Developments in the Theory of Shape Constrained Inference

“…The latter method is implemented in the R package curstatCI Groeneboom and Hendrickx (2017a). That this also give a consistent bootstrap is proved in Groeneboom and Hendrickx (2017b). Figure 12 shows the result for a sample of size n = 1000, where F 0 is the standard truncated exponential distribution function on [0, 2] and the T i are uniform on [0, 2].…”

“…It is suggested in Groeneboom and Jongbloed (2015) however, that one can use the nonparametric bootstrap with a smoothed Grenander estimator. Results of this type for the nonparametric bootstrap are proved in Groeneboom and Hendrickx (2017b) for the current status model.…”

“…wheref nh andf * nh are the SMLEs in the original sample and the bootstrap sample, respectively. That this procedure will work is nontrivial, but can be argued along the lines of Groeneboom and Hendrickx (2017b) (see also Section 6). The bandwidths were chosen by least squares cross validation, both in the original and in the bootstrap samples.…”

Score estimation of monotone partially linear index model

“…A1 and A2, which are similar to the assumptions A1 and A2 in BGH, impose boundedness on the support of covariates and the monotone function ψ 0 . These conditions are used to control the entropy of the function classes that characterize (6). We note that Xia and Härdle (2006) and Wang et al (2010) imposed similar conditions.…”