Extensive photometric studies of the globular clusters located toward the center of the Milky Way have been historically neglected. The presence of patchy differential reddening in front of these clusters has proven to be a significant obstacle to their detailed study. We present here a well defined and reasonably homogeneous photometric database for 25 of the brightest Galactic globular clusters located in the direction of the inner Galaxy. These data were obtained in the B, V, and I bands using the Magellan 6.5 m Telescope and the Hubble Space Telescope. A new technique is extensively used in this paper to map the differential reddening in the individual cluster fields, and to produce cleaner, dereddened color-magnitude diagrams for all the clusters in the database. Subsequent papers will detail the astrophysical analysis of the cluster populations, and the properties of the obscuring material along the clusters' lines of sight.

ABSTRACT. The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.

Abstract\Ve study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function. The likelihood ratio test for this hypothesis is derived in the particular case of interval censoring (or current status data) and its limiting distribution is obtained. The limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving Brownian motion + t 2 and greatest convex minorants thereof.

The behavior of maximum likelihood estimates (MLEs) and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied. This class of problems differs radically from the usual parametric or semiparametric situations in that the MLE of the monotone function at a point converges to the truth at rate $n^{1/3}$ (slower than the usual $\sqrt{n}$ rate) with a non-Gaussian limit distribution. A framework for likelihood based estimation of monotone functions is developed and limit theorems describing the behavior of the MLEs and the likelihood ratio statistic are established. In particular, the likelihood ratio statistic is found to be asymptotically pivotal with a limit distribution that is no longer $\chi^2$ but can be explicitly characterized in terms of a functional of Brownian motion. Applications of the main results are presented and potential extensions discussed.Comment: Published at http://dx.doi.org/10.1214/009053606000001578 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

In this paper, we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate $n^{1/3}$. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function $f$ on $[0,\infty)$, is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for $f(t_0)$, where $t_0\in(0,\infty)$ is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical distribution function $\mathbb{F}_n$ or its least concave majorant $\tilde{F}_n$, does not have any weak limit in probability. We provide a set of sufficient conditions for the consistency of any bootstrap method in this example and show that bootstrapping from a smoothed version of $\tilde{F}_n$ leads to strongly consistent estimators. The $m$ out of $n$ bootstrap method is also shown to be consistent while generating samples from $\mathbb{F}_n$ and $\tilde{F}_n$.Comment: Published in at http://dx.doi.org/10.1214/09-AOS777 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org