Estimation of smooth functionals in normal models: Bias reduction and asymptotic efficiency

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.