  Neutral to the right (NTR) processes were introduced by Doksum in 1974 as
Bayesian priors on the class of distributions on the real line. Since that time
there have been numerous applications to models that arise in survival analysis
subject to possible right censoring. However, unlike the Dirichlet process, the
larger class of NTR processes has not been used in a wider range of more
complex statistical applications. Here, to circumvent some of these
limitations, we describe a natural extension of NTR processes to arbitrary
Polish spaces, which we call spatial neutral to the right processes. Our
construction also leads to a new rich class of random probability measures,
which we call NTR species sampling models. We show that this class contains the
important two parameter extension of the Dirichlet process. We provide a
posterior analysis, which yields tractable NTR analogues of the
Blackwell--MacQueen distribution. Our analysis turns out to be closely related
to the study of regenerative composition structures. A new computational
scheme, which is an ordered variant of the general Chinese restaurant
processes, is developed. This can be used to approximate complex posterior
quantities. We also discuss some relationships to results that appear outside
of Bayesian nonparametrics.
