  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).
