Estimation of smooth functionals in normal models: Bias reduction and asymptotic efficiency
Vladimir Koltchinskii, Mayya Zhilova
The Annals of Statistics · 2021 · Volume 49 · Issue 5
Abstract
Let X1,…,Xn be i.i.d. random variables sampled from a normal distribution N(μ,Σ) in Rd with unknown parameter θ=(μ,Σ)∈Θ:=Rd×C+d, where C+d is the cone of positively definite covariance operators in Rd. Given a smooth functional f:Θ↦R1, the goal is to estimate f(θ) based on X1,…,Xn. Let Θ(a;d):=Rd×{Σ∈C+d:σ(Σ)⊂[1/a,a]},a≥1, where σ(Σ) is the spectrum of covariance Σ. Let θˆ:=(μˆ,Σˆ), where μˆ is the sample mean and Σˆ is the sample covariance, based on the observations X1,…,Xn. For an arbitrary functional f∈Cs(Θ), s=k+1+ρ,k≥0,ρ∈(0,1], we define a functional fk:Θ↦R such that supθ∈Θ(a;d)‖fk(θˆ)−f(θ)‖L2(Pθ)≲s,β‖f‖Cs(Θ)[(a n∨aβs( d n)s)∧1], where β=1 for k=0 and β>s−1 is arbitrary for k≥1. This error rate is minimax optimal and similar bounds hold for more general loss functions. If d=dn≤nα for some α∈(0,1) and s≥11−α, the rate becomes O(n−1/2). Moreover, for s>11−α, the estimator fk(θˆ) is shown to be asymptotically efficient. The crucial part of the construction of estimator fk(θˆ) is a bias reduction method studied in the paper for more general statistical models than normal.
Keywords
60B20, 62G20, 62H12, 62H25, bias reduction, bootstrap chain, Concentration, efficiency, random homotopy, smooth functionals
Enable JavaScript for citation tools and additional paper interactions.