## Valid model-free prediction of future insurance claims.(English)Zbl 07469929

The authors consider conformal prediction as a model-free method to provide provably valid predictions for the size of future insurance claims uniformly over all sample sizes and all distributions.
Let $$X_{n}$$ $$(n \in \mathbb{N})$$ be real-valued random variables such that the joint distribution of any finite subsequence is permutation invariant. For $$M\colon [\mathbb{R}]^{<\infty} \times \mathbb{R} \rightarrow \mathbb{R}$$, called nonconformity measure, given a set of observations $$\mathcal{X}_{n} = \{x_{0},x_{1},\dots, x_{n-1}\}$$, the plausibility function $$\textrm{pl}_{\mathcal{X}_{n}}\colon \mathbb{R} \rightarrow \mathbb{R}$$ is calculated as follows:
1.
Let $$x \in \mathbb{R}$$ and set $$x_{n} = x$$, $$\mathcal{X}_{n+1} = \mathcal{X}_{n}\cup \{x\}$$.
2.
For every $$i\leq n$$, set $$m_{i} = M(\mathcal{X}_{n+1}\setminus \{x_{i}\},x_{i})$$.
3.
Set $$\textrm{pl}_{\mathcal{X}_{n}}(x) = \frac{1}{n+1}\sum_{i\leq n} \mathbb{I}_{[m_{i}\geq m_{n}]}$$.
The authors note that, in non-degenerate cases, $$\textrm{pl}_{\mathcal{X}_{n}}(X_{n+1})$$ is uniformly distributed on $$\{1/(n+1),2/(n+1),\dots,1\}$$. Hence, for every $$\alpha \in (0,1)$$, $C_{\alpha}(\mathcal{X}_{n}) = \{x\colon \textrm{pl}_{\mathcal{X}_{n}}(x) > \lfloor(n+1)\alpha\rfloor/(n+1) \}$ satisfies that the probability of $$X_{n+1}\notin C_{\alpha}(\mathcal{X}_{n})$$ does not exceed $$\alpha$$, which makes $$C_{\alpha}(\mathcal{X}_{n})$$ the $$100(1-\alpha)\%$$ conformal prediction region.
The authors calculate $$m_{i}$$ explicitly in the case $$M$$ is the sample cumulative density function of the form $$M(\mathcal{X}_{n},x) = \frac{1}{n}\sum_{i<n}K(x,x_{i},h)$$ where $$K(\cdot,\theta,h)$$ is a kernel distribution function with parameter $$\theta \in \mathbb{R}$$ and bandwidth $$h$$, and observe that if $$x \mapsto K(x,x,h)$$ is constant then $$\textrm{pl}_{\mathcal{X}_{n}}(x) = \frac{1}{n}\sum_{i<n}\mathbb{I}_{[x_{i}\geq x]}$$ and $$C_{\alpha}(\mathcal{X}_{n}) = [0,x_{(k)})$$ where $$x_{(k)}$$ is the $$k^{\textrm{th}}$$ largest element of $$\mathcal{X}_{n}$$ with $$k = \min\{n,\lfloor (n+1)(1-\alpha)\rfloor+1\}$$.
The authors use fire claims data and car injury claims data to calculate conformal prediction intervals and compare them to prediction intervals based on other non-parametric methods. Possible extensions of the method to conditional predictions are also considered.

### MSC:

 91G05 Actuarial mathematics 62G30 Order statistics; empirical distribution functions 60G25 Prediction theory (aspects of stochastic processes) 62M20 Inference from stochastic processes and prediction 62G05 Nonparametric estimation 62G07 Density estimation

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### References:

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