Efficiency bounds for semiparametric models with singular score functions

This paper is concerned with asymptotic efficiency bounds for the estimation of the finite dimension parameter θ∈Rp of semiparametric models that have singular score function for θ at the true value θ⋆. The resulting singularity of the matrix of Fisher information means that the standard bound for θ−θ⋆ is not defined. We study the case of single rank deficiency of the score and focus on the case where the derivative of the root density in the direction of the last parameter component, θ2, is nil while the derivatives in the p – 1 other directions, θ1, are linearly independent. We then distinguish two cases: (i) The second derivative of the root density in the direction of θ2 and the first derivative in the direction of θ1 are linearly independent and (ii) The second derivative of the root density in the direction of θ2 is also nil but the third derivative in θ2 is linearly independent of the first derivative in the direction of θ1. We show that in both cases, efficiency bounds can be obtained for the estimation of κj(θ)=(θ1−θ⋆1,(θ2−θ⋆2)j), with j = 2 and 3, respectively and argue that an estimator θ̂ is efficient if κj(θ̂) reaches its bound. We provide the bounds in form of convolution and asymptotic minimax theorems. For case (i), we propose a transformation of the Gaussian variable that appears in our convolution theorem to account for the restricted set of values of κ2(θ). This transformation effectively gives the efficiency bound for the estimation of κ2(θ) in the model configuration (i). We apply these results to locally under-identified moment condition models and show that the generalized method of moments (GMM) estimator using V⋆−1 as weighting matrix, where V⋆ is the variance of the estimating function, is optimal even in these non standard settings. Examples of models are provided that fit the two configurations explored.