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Semiparametric Fisher Information in Models parametrized by a Normed Space

Telmo Pérez-Izquierdo

Abstract

This paper studies semiparametric Fisher information in models parametrized by general normed spaces. The main contribution is to establish that positive semiparametric Fisher information is equivalent to the gradient of the parameter of interest lying in the range of the adjoint score operator. This result generalizes a key theorem Van Der Vaart (1991) and provides a unified framework linking differentiability and information, beyond Hilbert spaces. The paper develops a normed-space mean-square-differentiable models for two canonical problems: estimation of the average of a known transformation and estimation of a density at a point. In these applications, it shows that positive information holds if and only if the transformation has finite variance and if and only if the density has positive mass at the evaluation point, respectively. These findings offer a novel information-theoretic perspective on known minimax results and clarify the conditions under which root-n estimation is possible.

Semiparametric Fisher Information in Models parametrized by a Normed Space

Abstract

This paper studies semiparametric Fisher information in models parametrized by general normed spaces. The main contribution is to establish that positive semiparametric Fisher information is equivalent to the gradient of the parameter of interest lying in the range of the adjoint score operator. This result generalizes a key theorem Van Der Vaart (1991) and provides a unified framework linking differentiability and information, beyond Hilbert spaces. The paper develops a normed-space mean-square-differentiable models for two canonical problems: estimation of the average of a known transformation and estimation of a density at a point. In these applications, it shows that positive information holds if and only if the transformation has finite variance and if and only if the density has positive mass at the evaluation point, respectively. These findings offer a novel information-theoretic perspective on known minimax results and clarify the conditions under which root-n estimation is possible.

Paper Structure

This paper contains 10 sections, 9 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Setup and motivating examples
  3. Overview of the results
  4. Differentiability and positive information
  5. One-to-one score operators
  6. General score operators
  7. Closed range and local identifiability
  8. Applications
  9. Conclusion
  10. Proofs

Key Result

Proposition 4.1

Let $A$ be one-to-one. Then, $\mathcal{I}>0$ if and only if $\dot\psi A^{-1}$ is continuous. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (14)

  • Example 1: Average of a known transformation
  • Example 2: Density at a point
  • Proposition 4.1
  • Proposition 4.2
  • Lemma 4.1
  • Theorem 4.1
  • Example 3: continues=ex:mean
  • Proposition 4.3
  • Example 4: continues=ex_mean
  • Proposition 5.1
  • ...and 4 more