Adaptive confidence sets in $$L^2$$

Abstract

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.

Locally adaptive confidence bands

“…Their condition is shown to be necessary and sufficient. Similar important conclusions concerning adaptivity in terms of confidence statements are obtained under Hilbert space geometry with corresponding L 2 -loss, see Juditsky and Lambert-Lacroix (2003), Baraud (2004), Genovese and Wasserman (2005), Cai and Low (2006), Robins and van der Vaart (2006), Bull and Nickl (2013), and Nickl and Szabó (2016). Concerning L p -loss, we also draw attention to Carpentier (2013).…”

Adaptive confidence sets for matrix completion

“…and then apply Theorem 2.2. As test statistic, we propose an infimum-test which has previously been used by Bull and Nickl [6] and Nickl and van de Geer [33] in density estimation and high-dimensional regression, respectively (see also Section 6.2.4. in [20]). Since σ 2 = Eǫ 2 ij is known we can define the statistic…”

Frequentist coverage of adaptive nonparametric Bayesian credible sets

“…It is known that ℓ 2 -confidence balls can be honest over a model of regularity α and possess a radius that adapts to the minimax rate whenever the true parameter is of smoothness contained in the interval [α, 2α]. Thus, these balls can adapt to double a coarsest smoothness level [Juditsky and Lambert-Lacroix (2003), Cai and Low (2006), Robins and van der Vaart (2006), Bull and Nickl (2013)]. The fact that the coarsest level α and the radius of the ball must be known [as shown in Bull and Nickl (2013)] makes this type of adaptation somewhat theoretical.…”

“…Thus, these balls can adapt to double a coarsest smoothness level [Juditsky and Lambert-Lacroix (2003), Cai and Low (2006), Robins and van der Vaart (2006), Bull and Nickl (2013)]. The fact that the coarsest level α and the radius of the ball must be known [as shown in Bull and Nickl (2013)] makes this type of adaptation somewhat theoretical. This type of adaptation is not considered in the present paper (in fact, we do not have a coarsest regularity level α).…”