Bootstrap confidence intervals for isotonic estimators in a stereological problem

Abstract

Let $\mathbf{X}=(X_{1},X_{2},X_{3})$ be a spherically symmetric random vector of which only $(X_{1},X_{2})$ can be observed. We focus attention on estimating $F$, the distribution function of the squared radius $Z:=X_{1}^{2}+X_{2}^{2}+X_{3}^{2}$, from a random sample of $(X_{1},X_{2})$. Such a problem arises in astronomy where $(X_{1},X_{2},X_{3})$ denotes the three dimensional position of a star in a galaxy but we can only observe the projected stellar positions $(X_{1},X_{2})$. We consider isotonic estimators of $F$ and derive their limit distributions. The results are nonstandard with a rate of convergence $\sqrt{n/{\log n}}$. The isotonized estimators of $F$ have exactly half the limiting variance when compared to naive estimators, which do not incorporate the shape constraint. We consider the problem of constructing point-wise confidence intervals for $F$, state sufficient conditions for the consistency of a bootstrap procedure, and show that the conditions are met by the conventional bootstrap method (generating samples from the empirical distribution function).