Estimation of Smooth Functionals in Normal Models: Bias Reduction and Asymptotic Efficiency
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Let X_1,..., X_n be i.i.d. random variables sampled from a normal distribution N(μ,Σ) in R^d with unknown parameter θ=(μ,Σ)∈Θ:=R^d× C_+^d, where C_+^d is the cone of positively definite covariance operators in R^d. Given a smooth functional f:ΘR^1, the goal is to estimate f(θ) based on X_1,..., X_n. Let Θ(a;d):=R^d×{Σ∈ C_+^d: σ(Σ)⊂ [1/a, a]}, a≥ 1, where σ(Σ) is the spectrum of covariance Σ. Let θ̂:=(μ̂, Σ̂), where μ̂ is the sample mean and Σ̂ is the sample covariance, based on the observations X_1,..., X_n. For an arbitrary functional f∈ C^s(Θ),s=k+1+ρ, k≥ 0, ρ∈ (0,1], we define a functional f_k:ΘR such that sup_θ∈Θ(a;d)f_k(θ̂)-f(θ)_L_2(P_θ)≲_s, βf_C^s(Θ)[(a/√(n) a^β s(√(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=d_n≤ n^α for some α∈ (0,1) and s≥1/1-α, the rate becomes O(n^-1/2). Moreover, for s>1/1-α, the estimators f_k(θ̂) is shown to be asymptotically efficient. The crucial part of the construction of estimator f_k(θ̂) is a bias reduction method studied in the paper for more general statistical models than normal.
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