Abstract
We study the problem of density estimation on [0,1] under Lp norm. We carry out a new piecewise polynomial estimator and prove that it is simultaneously (near)-minimax over a very wide range of Besov classes Bπ,∞α(R). In particular, we may deal with unbounded densities and shed light on the minimax rates of convergence when π<p and α∈(1∕π−1∕p,1∕π].
Highlights
We consider n independent and identically distributed random variablesX1, . . . , Xn defined on an abstract probability space (Ω, E, P )
The minimax rate of convergence, that is the rate at which Rp(F) converges to 0, is the best possible for procedures based solely on assumptions modelled by F
Our estimation strategy is based on projection estimators and on a new estimator selection rule. This procedure may be thought of as a mix between a Lespki-type procedure [Lep92] and the one of [Sar14]. It leads to a piecewise polynomial estimator of degree r that is minimax and adaptive over the full scale of Besov classes π ∈ (0, p) and α ∈ (1/π − 1/p, r + 1)
Summary
X1, . . . , Xn defined on an abstract probability space (Ω, E , P ). This procedure may be thought of as a mix between a Lespki-type procedure [Lep92] and the one of [Sar14] It leads to a piecewise polynomial estimator of degree r that is (near) minimax and adaptive over the full scale of Besov classes π ∈ (0, p) and α ∈ (1/π − 1/p, r + 1). In other terms, it achieves the rates given above, up to logarithmic factors, without the prior knowledge of α and π.
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