## Bernoulli

- Bernoulli
- Volume 9, Number 1 (2003), 137-165.

### Semi-parametric efficiency, distribution-freeness and invariance

Marc Hallin and Bas J.M. Werker

#### Abstract

Semi-parametric models typically involve a finite-dimensional parameter of interest ${\pmb\theta}\in{\pmb\Theta}\subseteq\mathbb{R}^k$, along with an infinite-dimensional nuisance parameter~$f$. Quite often, the submodels corresponding to a fixed value of ${\pmb\theta}$ possess a group structure that induces a maximal invariant $\sigma$-field ${\cal B}({\pmb\theta})$. In classical examples, where $f$ denotes the density of some independ\-ent and identically distributed innovations, ${\cal B}({\pmb\theta})$ is the $\sigma$-field generated by the ranks of the residuals associated with the parameter value ${\pmb\theta}$. It is shown that semi-parametrically efficient distribution-free inference procedures can generally be constructed from parametrically optimal ones by conditioning on ${\cal B}({\pmb\theta})$; this implies, for instance, that semi-parametric efficiency (at given $\pmb\theta$ and $f$) can be attained by means of rank-based methods. The same procedures, when combined with a consistent estimation of the underlying nuis\-ance density $f$, yield conditionally distribution-free semi-parametrically efficient inference methods, for example, semi-parametrically efficient permutation tests. Remarkably, this is achieved without any explicit tangent space or efficient score computations, and without any sample-splitting device. By means of several examples, including both i.i.d. and time-series models, we show how these results apply in models for which rank-based inference or permutation tests have so far seldom been considered.

#### Article information

**Source**

Bernoulli, Volume 9, Number 1 (2003), 137-165.

**Dates**

First available in Project Euclid: 6 November 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1068129013

**Digital Object Identifier**

doi:10.3150/bj/1068129013

**Mathematical Reviews number (MathSciNet)**

MR1963675

**Zentralblatt MATH identifier**

1020.62042

**Keywords**

adaptiveness distribution-freeness local asymptotic normality ranks

#### Citation

Hallin, Marc; Werker, Bas J.M. Semi-parametric efficiency, distribution-freeness and invariance. Bernoulli 9 (2003), no. 1, 137--165. doi:10.3150/bj/1068129013. https://projecteuclid.org/euclid.bj/1068129013