Confidence bands are confidence sets for an unknown function f, containing all functions within some sup-norm distance of an estimator. In the density estimation, regression, and white noise models, we consider the problem of constructing adaptive confidence bands, whose width contracts at an optimal rate over a range of Hölder classes.While adaptive estimators exist, in general adaptive confidence bands do not, and to proceed we must place further conditions on f. We discuss previous approaches to this issue, and show it is necessary to restrict f to fundamentally smaller classes of functions.We then consider the self-similar functions, whose Hölder norm is similar at large and small scales. We show that such functions may be considered typical functions of a given Hölder class, and that the assumption of self-similarity is both necessary and sufficient for the construction of adaptive bands. Finally, we show that this assumption allows us to resolve the problem of undersmoothing, creating bands which are honest simultaneously for functions of any Hölder norm.

The problem of constructing confidence sets that are adaptive in L 2 -loss over a continuous scale of Sobolev classes of probability densities is considered. Adaptation holds, where possible, with respect to both the radius of the Sobolev ball and its smoothness degree, and over maximal parameter spaces for which adaptation is possible. Two key regimes of parameter constellations are identified: one where full adaptation is possible, and one where adaptation requires critical regions be removed. Techniques used to derive these results include a general nonparametric minimax test for infinite-dimensional null-and alternative hypotheses, and new lower bounds for L 2 -adaptive confidence sets. every n ∈ N and some constant M that depends on α, α ′ , r, B. B) If s > 2r and C n is an L 2 -adaptive and honest confidence set forΣ(r, ρ n ) ∪ Σ(s), for every α, α ′ > 0, then necessarily lim inf n ρ n n r/(2r+1/2) > 0.

In quantitative finance, we often model asset prices as semimartingales, with drift, diffusion and jump components. The jump activity index measures the strength of the jumps at high frequencies, and is of interest both in model selection and fitting, and in volatility estimation. In this paper, we give a novel estimate of the jump activity, together with corresponding confidence intervals. Our estimate improves upon previous work, achieving near-optimal rates of convergence, and good finite-sample performance in Monte-Carlo experiments.Comment: Published at http://dx.doi.org/10.1214/15-AOS1349 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

We describe a novel algorithm for noisy global optimisation and continuum-armed bandits, with good convergence properties over any continuous reward function having finitely many polynomial maxima. Over such functions, our algorithm achieves square-root regret in bandits, and inverse-square-root error in optimisation, without prior information.Our algorithm works by reducing these problems to tree-armed bandits, and we also provide new results in this setting. We show it is possible to adaptively combine multiple trees so as to minimise the regret, and also give near-matching lower bounds on the regret in terms of the zooming dimension.

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, . . . , 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases.