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

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 Rd. Given smooth functional f:Θ↦R1, goal to estimate f(θ) based on X1,…,Xn. Θ(a;d):=Rd×{Σ∈C+d:σ(Σ)⊂[1/a,a]},a≥1, σ(Σ) spectrum Σ. θˆ:=(μˆ,Σˆ), μˆ sample mean and Σˆ covariance, observations For an arbitrary f∈Cs(Θ), s=k+1+ρ,k≥0,ρ∈(0,1], we define 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], β=1 for k=0 β>s−1 k≥1. This error rate minimax optimal similar bounds hold more general loss functions. If d=dn≤nα some α∈(0,1) s≥11−α, becomes O(n−1/2). Moreover, s>11−α, estimator fk(θˆ) shown asymptotically efficient. The crucial part construction bias reduction method studied paper statistical models than normal.