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A new strategy for finite-sample valid prediction of future insurance claims in the regression setting

Liang Hong

TL;DR

This work addresses the lack of finite-sample valid prediction intervals for future insurance claims in the regression setting by proposing a general strategy that converts an unsupervised iid predictive method into regression predictions. The core idea writes $Y = h(X) + W$ with $W = Y - h(X)$, constructs a finite-sample valid interval for $W_{n+1}$ using conformal/prediction techniques, and translates it back to a $Y$-interval via $L(W^n) + h(X_{n+1})$ and $U(W^n) + h(X_{n+1})$, yielding infinite families of finite-sample valid intervals for $Y_{n+1}$. The approach, demonstrated through simulations and automobile-claims data, achieves nominal coverage with efficiency that depends on the chosen transformation $h$, offering actuaries a model-free, information-rich mechanism for interval prediction that leverages predictors while preserving finite-sample guarantees. It extends the applicability of conformal prediction into regression for insurance contexts and provides practical guidance on selecting transformations to balance interval length and coverage. $Y = f^*(X) + \varepsilon$ with $W = Y - h(X)$ underpins the theoretical framing, and the method remains broadly applicable beyond insurance to any regression setting requiring finite-sample predictive intervals.

Abstract

The extant insurance literature demonstrates a paucity of finite-sample valid prediction intervals of future insurance claims in the regression setting. To address this challenge, this article proposes a new strategy that converts a predictive method in the unsupervised iid (independent identically distributed) setting to a predictive method in the regression setting. In particular, it enables an actuary to obtain infinitely many finite-sample valid prediction intervals in the regression setting.

A new strategy for finite-sample valid prediction of future insurance claims in the regression setting

TL;DR

This work addresses the lack of finite-sample valid prediction intervals for future insurance claims in the regression setting by proposing a general strategy that converts an unsupervised iid predictive method into regression predictions. The core idea writes with , constructs a finite-sample valid interval for using conformal/prediction techniques, and translates it back to a -interval via and , yielding infinite families of finite-sample valid intervals for . The approach, demonstrated through simulations and automobile-claims data, achieves nominal coverage with efficiency that depends on the chosen transformation , offering actuaries a model-free, information-rich mechanism for interval prediction that leverages predictors while preserving finite-sample guarantees. It extends the applicability of conformal prediction into regression for insurance contexts and provides practical guidance on selecting transformations to balance interval length and coverage. with underpins the theoretical framing, and the method remains broadly applicable beyond insurance to any regression setting requiring finite-sample predictive intervals.

Abstract

The extant insurance literature demonstrates a paucity of finite-sample valid prediction intervals of future insurance claims in the regression setting. To address this challenge, this article proposes a new strategy that converts a predictive method in the unsupervised iid (independent identically distributed) setting to a predictive method in the regression setting. In particular, it enables an actuary to obtain infinitely many finite-sample valid prediction intervals in the regression setting.
Paper Structure (12 sections, 1 theorem, 33 equations, 6 tables, 2 algorithms)

This paper contains 12 sections, 1 theorem, 33 equations, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The problem at hand
  4. Parametric models, non-parametric models, and model-free methods
  5. Two types of validity
  6. New strategy
  7. Proposed strategy
  8. Application to prediction of insurance claims
  9. Illustration
  10. Simulated data
  11. Automobile bodily injury claims data
  12. Concluding remarks

Key Result

Theorem 1

Let $\mathsf{P}$ denote the distribution of an exchangeable sequence $Z_1,Z_2,\ldots$, and let $\mathsf{P}^{n+1}$ be the corresponding joint distribution of $Z^{n+1}=\{Z_1,\ldots,Z_n,Z_{n+1}\}$. For $0<\alpha<1$, put $t_n(\alpha) = (n+1)^{-1}\lfloor (n+1)\alpha \rfloor$, where $\lfloor a \rfloor$ de where the supremum is over all distributions $\mathsf{P}$ for the exchangeable sequence.

Theorems & Definitions (1)

  • Theorem 1