Hájek–Le Cam convolution theorem

Let ℘ = {P_θ | θ ∈ Θ ⊂ ℝ^k} be a regular parametric model, and q(θ): Θ → ℝ^m be a parameter in this model (typically a parameter is just one of the components of vector θ). Assume that function q is differentiable on Θ, with the m × k matrix of derivatives denoted as q̇_θ. Define

${\displaystyle I_{q(\theta )}^{-1}={\dot {q}}(\theta )I^{-1}(\theta ){\dot {q}}(\theta )'}$ — the information bound for q,

${\displaystyle \psi _{q(\theta )}={\dot {q}}(\theta )I^{-1}(\theta ){\dot {\ell }}(\theta )}$ — the efficient influence function for q,

where I(θ) is the Fisher information matrix for model ℘, ${\displaystyle \scriptstyle {\dot {\ell }}(\theta )}$ is the score function, and ′ denotes matrix transpose.

Theorem (Bickel 1998, Th.2.3.1). Suppose T_n is a uniformly (locally) regular estimator of the parameter q. Then

1. There exist independent random m-vectors ${\displaystyle \scriptstyle Z_{\theta }\,\sim \,{\mathcal {N}}(0,\,I_{q(\theta )}^{-1})}$ and Δ_θ such that

   ${\displaystyle {\sqrt {n}}(T_{n}-q(\theta ))\ {\xrightarrow {d}}\ Z_{\theta }+\Delta _{\theta },}$

   where ^d denotes convergence in distribution. More specifically,

   ${\displaystyle {\begin{pmatrix}{\sqrt {n}}(T_{n}-q(\theta ))-{\tfrac {1}{\sqrt {n}}}\sum _{i=1}^{n}\psi _{q(\theta )}(x_{i})\\{\tfrac {1}{\sqrt {n}}}\sum _{i=1}^{n}\psi _{q(\theta )}(x_{i})\end{pmatrix}}\ {\xrightarrow {d}}\ {\begin{pmatrix}\Delta _{\theta }\\Z_{\theta }\end{pmatrix}}.}$

2. If the map θ → q̇_θ is continuous, then the convergence in (A) holds uniformly on compact subsets of Θ. Moreover, in that case Δ_θ = 0 for all θ if and only if T_n is uniformly (locally) asymptotically linear with influence function ψ_q(θ)

• Bickel, Peter J.; Klaassen, Chris A.J.; Ritov, Ya’acov; Wellner Jon A. (1998). Efficient and adaptive estimation for semiparametric models. New York: Springer. ISBN 0-387-98473-9.CS1 maint: uses authors parameter (link)