12 research outputs found
A Geometric Reduction Approach for Identity Testing of Reversible Markov Chains
We consider the problem of testing the identity of a reversible Markov chain
against a reference from a single trajectory of observations. Employing the
recently introduced notion of a lumping-congruent Markov embedding, we show
that, at least in a mildly restricted setting, testing identity to a reversible
chain reduces to testing to a symmetric chain over a larger state space and
recover state-of-the-art sample complexity for the problem
The minimax risk in testing the histogram of discrete distributions for uniformity under missing ball alternatives
We consider the problem of testing the fit of a discrete sample of items from
many categories to the uniform distribution over the categories. As a class of
alternative hypotheses, we consider the removal of an ball of radius
around the uniform rate sequence for . We deliver a sharp
characterization of the asymptotic minimax risk when as the
number of samples and number of dimensions go to infinity, for testing based on
the occurrences' histogram (number of absent categories, singletons,
collisions, ...). For example, for and in the limit of a small expected
number of samples compared to the number of categories (aka
"sub-linear" regime), the minimax risk asymptotes to , with the
normal survival function. Empirical studies over a range of problem parameters
show that this estimate is accurate in finite samples, and that our test is
significantly better than the chisquared test or a test that only uses
collisions. Our analysis is based on the asymptotic normality of histogram
ordinates, the equivalence between the minimax setting to a Bayesian one, and
the reduction of a multi-dimensional optimization problem to a one-dimensional
problem