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T-calibration in semi-parametric models

Anja Mühlemann, Johanna Ziegel

TL;DR

This note investigates calibration of semi-parametric models for identifiable functionals through the lens of all consistent loss functions. It leverages the Choquet/mixture representation of scoring rules, linking calibration to the existence of a unique parameter $\beta^*$ that minimizes every elementary (or Choquet) loss $L_H$, and defines T-calibrated models via the functional $T$ induced by an identification function $V$. When a single optimal parameter exists, the model is T-calibrated and, under suitable conditions, recovers the targeted function $g(X)$; when multiple Pareto-optimal parameters arise, the paper analyzes their structure and provides explicit examples to illustrate non-uniqueness. The results offer a theoretical framework for understanding calibration under loss-function choice in semi-parametric regression and a diagnostic tool for model misspecification via Pareto sets. Overall, the work clarifies how calibration, loss consistency, and identifiability interact to shape parameter estimation beyond mean-structured models.

Abstract

This note relates the calibration of models to the consistent loss functions for the target functional of the model. We demonstrate that a model is calibrated if and only if there is a parameter value that is optimal under all consistent loss functions.

T-calibration in semi-parametric models

TL;DR

This note investigates calibration of semi-parametric models for identifiable functionals through the lens of all consistent loss functions. It leverages the Choquet/mixture representation of scoring rules, linking calibration to the existence of a unique parameter that minimizes every elementary (or Choquet) loss , and defines T-calibrated models via the functional induced by an identification function . When a single optimal parameter exists, the model is T-calibrated and, under suitable conditions, recovers the targeted function ; when multiple Pareto-optimal parameters arise, the paper analyzes their structure and provides explicit examples to illustrate non-uniqueness. The results offer a theoretical framework for understanding calibration under loss-function choice in semi-parametric regression and a diagnostic tool for model misspecification via Pareto sets. Overall, the work clarifies how calibration, loss consistency, and identifiability interact to shape parameter estimation beyond mean-structured models.

Abstract

This note relates the calibration of models to the consistent loss functions for the target functional of the model. We demonstrate that a model is calibrated if and only if there is a parameter value that is optimal under all consistent loss functions.

Paper Structure

This paper contains 5 sections, 2 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Identifiable functionals and mixture representations
  3. T-calibrated models
  4. Pareto-optimal parameters
  5. Discussion

Key Result

Theorem 3.2

Suppose that Assumptions assu:2 holds, and that there exists $\beta^* \in \Theta$ such that $\inf_{\beta \in \Theta} \mathsf{E} S_\eta(m(X;\beta),Y)= \mathsf{E} S_\eta(m(X;\beta^*),Y)$ for all $\eta \in \mathbb{R}$. Assume that the random variables $V(m(X;\beta^*)\pm \epsilon',Y)$ are integrable for

Figures (1)

  • Figure 1: Pareto optimal parameters in blue for the model in Example \ref{['ex:Anja']} and regression function $g_1$ and $g_2$ in the left and right panel, respectively.

Theorems & Definitions (10)

  • Example 2.1
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['thm:1']}
  • Definition 4.1
  • Example 4.2
  • Example 4.3