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).