Efficiency bounds for semiparametric models with singular score functions

Abstract

This paper is concerned with asymptotic efficiency bounds for the estimation of the finite dimension parameter $\theta \in {\mathbb{R}}^p$ of semiparametric models that have singular score function for θ at the true value ${\theta }_{\star }.$ The resulting singularity of the matrix of Fisher information means that the standard bound for $\theta -{\theta }_{\star }$ 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 ${\kappa }_j(\theta )=({\theta }_1-{\theta }_{\star 1},{({\theta }_2-{\theta }_{\star 2})}^j),$ with j = 2 and 3, respectively and argue that an estimator $\widehat{\theta }$ is efficient if ${\kappa }_j(\widehat{\theta })$ 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 ${\kappa }_2(\theta ).$ This transformation effectively gives the efficiency bound for the estimation of ${\kappa }_2(\theta )$ 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 $V_{\star }^{-1}$ as weighting matrix, where $V_{\star }$ 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.

Keywords:

• Efficient estimation
• semiparametric models
• singular score
• moment condition models
• under-identification

Notes

1 We refer to Powell (1994) for a more precise characterization of semiparametric models.

2 The score function in the direction of θ is given by ${\nabla }_{\theta }\hspace{0.17em}\text{ln}\hspace{0.17em}(f_{(\theta ,h)}^2(·))$ which amounts to $2{\nabla }_{\theta }{f}_{(\theta ,h)}(·)/f_{(\theta ,h)}(·).$

3 The sets ${\Theta }_1,{\mathcal{C}}_1$ and H₁ are defined as in Section 2 but with the spaces introduced in the current section.

4 We refer the reader to, e.g., Dovonon and Renault (2019) for a more general specification of the second-order local identification condition and its characterizations.

Additional information

Funding

The paper has benefited from the comments of the editor and two anonymous referees. This research is partially supported by the Fonds de Recherche du Québec - Société et Culture (FRQSC), and by the National Science Foundation grant DMS 1228164 and SES 1229261.