Vol 27, No 1 (2018)

We consider parametric exponential families of dimension K on the real line. We study a variant of boundary crossing probabilities coming from the multi-armed bandit literature, in the case when the real-valued distributions form an exponential family of dimension K. Formally, our result is a concentration inequality that bounds the probability that B^ψ(θ̂_n, θ*) ≥ f(t/n)/n, where θ* is the parameter of an unknown target distribution, θ̂_n is the empirical parameter estimate built from n observations, ψ is the log-partition function of the exponential family and B^ψ is the corresponding Bregman divergence. From the perspective of stochastic multi-armed bandits, we pay special attention to the case when the boundary function f is logarithmic, as it is enables to analyze the regret of the state-of-the-art KL-ucb and KL-ucb+ strategies, whose analysis was left open in such generality. Indeed, previous results only hold for the case when K = 1, while we provide results for arbitrary finite dimension K, thus considerably extending the existing results. Perhaps surprisingly, we highlight that the proof techniques to achieve these strong results already existed three decades ago in the work of T. L. Lai, and were apparently forgotten in the bandit community. We provide a modern rewriting of these beautiful techniques that we believe are useful beyond the application to stochastic multi-armed bandits.

The class of limit distribution functions (df’s) of the random bivariate extreme, central and intermediate generalized order statistics (gos) from independent and identically distributed random variables (rv’s) is fully characterized. The cases, when the random sample size is independent of the basic variables and when the interrelation between the random sample size and the basic variables is not restricted, are considered.

For a multivariate normal set-up, it is well known that themaximumlikelihood estimator (MLE) of covariance matrix is neither admissible nor minimax under the Stein loss function. In this paper, we reveal that the MLE based on the Iwasawa parameterization leads to minimaxity with respect to the Stein loss function. Furthermore, a novel class of loss functions is proposed so that the minimum risks of the MLEs are identical in different coordinate systems, Cholesky parameterization and full Iwasawa parameterization. In other words, the MLEs based on these two different parameterizations are characterized by the property of minimaxity, without a Stein paradox. The application of our novel method to the high-dimensional covariance matrix problem is also discussed.