Comments on: Nonparametric inference with generalized likelihood ratio tests

This is a very interesting paper reviewing the technique for testing semiparametric hypotheses using GLR tests. I’d like to supplement Fan and Jiang’s review with some cautions and a somewhat different point of view. 1 The Wilks phenomenon Rigorous results for smooth parametric models, see, for example, Bickel and Doksum (2005, Chap. 6) do say that, if θ̂ , η̂ or, equivalently, (θ̂η̂, η̂) are MLE’s, then 2( (θ̂ , η̂) − (θ0, η̂θ0)) ⇒ χ2 d , where d is the dimension of Θ . But if η̂ is not the MLE this result may fail to hold. In particular it will fail if, in the case of θ, η real, E(θ0,η0) ∂ ∂θ (X1, θ0, η0) · ∂ ∂η (X1, θ0, η0) = 0. More generally, if θ and η are infinite dimensional, the requirement is that the tangent spaces at (θ0, η0) of the models with θ = θ0 kept fixed, ◦ P η , and η = η0 kept fixed, ◦ P θ , are orthogonal in L2(P(θ0,η0))— see Bickel et al. (1993). All of Fan and Jiang’s examples satisfy this condition— appropriately generalized to the general dependent case—see Bickel and Kwon (2001). Murphy’s does not but the estimator (θ̂ , η̂) that she uses is efficient, i.e., behaves like the MLE in nice parametric situations. We give a heuristic argument below why the Wilks phenomenon can only be expected if η̂ is efficient or the tangent spaces are orthogonal. This comment refers to the invited paper available at: http://dx.doi.org/10.1007/s11749-007-0080-8. P.J. Bickel ( ) Department of Statistics, University of California at Berkeley, 367 Evans Hall, Berkeley, CA 94710-3860, USA e-mail: [email protected]