The maximum of Brownian motion minus a parabola

We derive a simple integral representation for the distribution of the maximum of Brownian motion minus a parabola, which can be used for computing the density and moments of the distribution, both for one-sided and two-sided Brownian motion.Comment: 7 pages, 4 figures, to appear in the Electronic Journal of Probabilit

The bivariate current status model

For the univariate current status and, more generally, the interval censoring model, distribution theory has been developed for the maximum likelihood estimator (MLE) and smoothed maximum likelihood estimator (SMLE) of the unknown distribution function, see, e.g., [12], [7], [4], [5], [6], [10], [11] and [8]. For the bivariate current status and interval censoring models distribution theory of this type is still absent and even the rate at which we can expect reasonable estimators to converge is unknown. We define a purely discrete plug-in estimator of the distribution function which locally converges at rate n^{1/3} and derive its (normal) limit distribution. Unlike the MLE or SMLE, this estimator is not a proper distribution function. Since the estimator is purely discrete, it demonstrates that the n^{1/3} convergence rate is in principle possible for the MLE, but whether this actually holds for the MLE is still an open problem. If the cube root n rate holds for the MLE, this would mean that the local 1-dimensional rate of the MLE continues to hold in dimension 2, a (perhaps) somewhat surprising result. The simulation results do not seem to be in contradiction with this assumption, however. We compare the behavior of the plug-in estimator with the behavior of the MLE on a sieve and the SMLE in a simulation study. This indicates that the plug-in estimator and the SMLE have a smaller variance but a larger bias than the sieved MLE. The SMLE is conjectured to have a n^{1/3}-rate of convergence if we use bandwidths of order n^{-1/6}. We derive its (normal) limit distribution, using this assumption. Finally, we demonstrate the behavior of the MLE and SMLE for the bivariate interval censored data of [1], which have been discussed by many authors, see e.g., [18], [3], [2] and [15].Comment: 18 pages, 7 figures, 4 table

The remaining area of the convex hull of a Poisson process

In Cabo and Groeneboom (1994) the remaining area of the left-lower convex hull of a Poisson point process with intensity one in the first quadrant of the plane was analyzed, using the methods of Groeneboom (1988), giving formulas for the expectation and variance of the remaining area for a finite interval of slopes of the boundary of the convex hull. However, the time inversion argument of Groeneboom (1988) was not correctly applied in Cabo and Groeneboom (1994), leading to an incorrect scaling constant for the variance. The purpose of this note is to show how the correct application of the time inversion argument gives the right expression, which is in accordance with results in Nagaev and Khamdamov (1991) and Buchta (2003).Comment: 7 pages, 3 figure

Maximum smoothed likelihood estimators for the interval censoring model

We study the maximum smoothed likelihood estimator (MSLE) for interval censoring, case 2, in the so-called separated case. Characterizations in terms of convex duality conditions are given and strong consistency is proved. Moreover, we show that, under smoothness conditions on the underlying distributions and using the usual bandwidth choice in density estimation, the local convergence rate is $n^{-2/5}$ and the limit distribution is normal, in contrast with the rate $n^{-1/3}$ of the ordinary maximum likelihood estimator.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1256 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric confidence intervals for monotone functions

We study nonparametric isotonic confidence intervals for monotone functions. In Banerjee and Wellner (2001) pointwise confidence intervals, based on likelihood ratio tests for the restricted and unrestricted MLE in the current status model, are introduced. We extend the method to the treatment of other models with monotone functions, and demonstrate our method by a new proof of the results in Banerjee and Wellner (2001) and also by constructing confidence intervals for monotone densities, for which still theory had to be developed. For the latter model we prove that the limit distribution of the LR test under the null hypothesis is the same as in the current status model. We compare the confidence intervals, so obtained, with confidence intervals using the smoothed maximum likelihood estimator (SMLE), using bootstrap methods. The `Lagrange-modified' cusum diagrams, developed here, are an essential tool both for the computation of the restricted MLEs and for the development of the theory for the confidence intervals, based on the LR tests.Comment: 31 pages, 13 figure

Current status linear regression

We construct $\sqrt{n}$-consistent and asymptotically normal estimates for the finite dimensional regression parameter in the current status linear regression model, which do not require any smoothing device and are based on maximum likelihood estimates (MLEs) of the infinite dimensional parameter. We also construct estimates, again only based on these MLEs, which are arbitrarily close to efficient estimates, if the generalized Fisher information is finite. This type of efficiency is also derived under minimal conditions for estimates based on smooth non-monotone plug-in estimates of the distribution function. Algorithms for computing the estimates and for selecting the bandwidth of the smooth estimates with a bootstrap method are provided. The connection with results in the econometric literature is also pointed out.Comment: 64 pages, 6 figure