Robust Testing of Functionals

Abstract

Given a smooth d-parametric model, let T = T(ψ) be a functional that is Fisher consistent and differentiable at the model with influence curve ψ. In our paper, the multisided testing problem about T,

$${\text{H}}:{\text{ }}T'{\text{Cov}}{(\psi )^{ - 1}}T \leqslant {b^2}{\text{ vs}}{\text{. K}}:T'{\text{Cov}}{(\psi )^{ - 1}}T \geqslant {c^2}$$

which extends the classical multiparameter hypotheses to nonparametrized measures, is studied using a locally asymptotic robust approach. Assuming any full infinitesimal neighborhoods, and i.i.d. observations, we derive a locally asymptotic maximin (LAM) upper bound for the testing power at level α of the form

$$\Pr [{x^2}(d;{c^2}) > {c_\alpha }(d;{b^2})],$$

where χ²(d; c ²) denotes a χ²-variable with d degrees of freedom and noncentrality parameter c2, and c_α(d;b ²) is the upper α-point of a χ²(d;b ²) distribution. Depending on suitable constructions, the LAM bound is attained by the test ϕ(ψ) = (ϕ ₁,ϕ ₂,…),

$${\phi _n} = {\text{J[}}n{S_n}'{\text{Cov}}{(\psi )^{ - 1}}{S_n} > {c_\alpha }(d;{b^2})],{\text{ }}n = 1,2,...,$$

where (S ₁, S ₂,…) = S(ψ) is an asymptotically linear estimator with influence curve ψ, and J denotes the indicator function. This optimality, however, degenerates to some kind of unbiasedness since all other “estimator tests” ϕ(ϱ) with ϱ ≠ ψ achieve asymptotic minimum power zero.

The selection of the functional T(ψ) to be tested leads to nontrivial optimality problems. In the one dimensional, onesided case,

$${\text{H: }}T \leqslant b{\text{ }}vs.{\text{ K}}:T \geqslant c,$$

one may want to test ε-contamination or total variation balls P vs. Q using functionals. Then maximization of the corresponding LAM bound

$$\Pr [N\left( {0,1} \right){u_\alpha } - \left( {c - b} \right)/\sqrt {Var\left( \psi \right)} ]$$

subject to the inclusions P ⊂ H, Q ⊂ K, yields an asymptotic version of the Huber-Strassen maximin test for P vs. Q based on least favorable pairs. In the multidimensional case, this idea leads to the minimization of the maximum eigenvalue of the information-standardized covariance subject to a bound on the self-standardized sensitivity.