\begin{algorithmic}[1]
    \State Simulate $N$ random variables from $\text{Uniform}(0,1)$; Denote $\tilde{U}=(\tilde{U}_{(1)},...,\tilde{U}_{(N)})$ as the vector of $N$ simulated Uniformly distributed random variables
    \State Denote $\tilde{Y}^{*}_{ij}=(\tilde{Y}^{*}_{ij,(1)},...,\tilde{Y}^{*}_{ij,(N)})$ as the vector of $N$ simulated variables for cell $(i,j)$ from the ensemble
    \For{$n=1,..,N$}                    
        \If{$\tilde{U}_{(n)} \in (\sum_{m=1}^{l-1}w_m,\sum_{m=1}^{l}w_m]$}
            \State simulate a random variable from the $l^{th}$ component distribution, denoted as $\tilde{Y}_{ij}^{(l)}$
            \State Set $\tilde{Y}^{*}_{ij,(n)}=\tilde{Y}_{ij}^{(l)}$
        \EndIf 
    \EndFor
    \State Repeat the above two steps for each cell $(i,j)$
    \State Calculate the simulated reserve based on the simulated random variables for each cell $(i,j)$: $\tilde{R}^{*}=(\tilde{R}^{*}_{(1)},\tilde{R}^{*}_{(2)},...,\tilde{R}^{*}_{(N)})=(\sum_{i,j\in D^{out}}\tilde{Y}_{ij,(1)}^{*},\sum_{i,j\in D^{out}}\tilde{Y}_{ij,(2)}^{*},...,\sum_{i,j\in D^{out}}\tilde{Y}_{ij,(N)}^{*})$
    \State Calculate the empirical $75^{th}$ quantile of the $N$ simulated reserves for the ensemble: $R^{*}_{75}$
\end{algorithmic}

\begin{algorithmic}
[1]
\State Simulate $N$ random variables from $\text{Uniform}(0,1)$; Denote $\tilde{U}=(\tilde{U}_{(1)},...,\tilde{U}_{(N)})$ as the vector of $N$ simulated Uniformly distributed random variables
\State Denote $\tilde{Y}^{*}_{ij}=(\tilde{Y}^{*}_{ij,(1)},...,\tilde{Y}^{*}_{ij,(N)})$ as the vector of $N$ simulated variables for cell $(i,j)$ from the ensemble
\For{$n=1,..,N$}
\If{$\tilde{U}_{(n)} \in (\sum_{m=1}^{l-1}w_m,\sum_{m=1}^{l}w_m]$}
\State simulate a random variable from the $l^{th}$ component distribution, denoted as $\tilde{Y}_{ij}^{(l)}$
\State Set $\tilde{Y}^{*}_{ij,(n)}=\tilde{Y}_{ij}^{(l)}$
\EndIf
\EndFor
\State Repeat the above two steps for each cell $(i,j)$
\State Calculate the simulated reserve based on the simulated random variables for each cell $(i,j)$: $\tilde{R}^{*}=(\tilde{R}^{*}_{(1)},\tilde{R}^{*}_{(2)},...,\tilde{R}^{*}_{(N)})=(\sum_{i,j\in D^{out}}\tilde{Y}_{ij,(1)}^{*},\sum_{i,j\in D^{out}}\tilde{Y}_{ij,(2)}^{*},...,\sum_{i,j\in D^{out}}\tilde{Y}_{ij,(N)}^{*})$
\State Calculate the empirical $75^{th}$ quantile of the $N$ simulated reserves for the ensemble: $R^{*}_{75}$
\end{algorithmic}
{
    "title": "ensemble distributional forecasting for insurance loss reserving",
    "id": "2206.08541",
    "abstract": "loss reserving generally focuses on identifying a single model that can generate superior predictive performance. however, different loss reserving models specialise in capturing different aspects of loss data. this is recognised in practice in the sense that results from different models are often considered, and sometimes combined. for instance, actuaries may take a weighted average of the prediction outcomes from various loss reserving models, often based on subjective assessments.   in this paper, we propose a systematic framework to objectively combine (i.e. ensemble) multiple _stochastic_ loss reserving models such that the strengths offered by different models can be utilised effectively. our framework contains two main innovations compared to existing literature and practice. firstly, our criteria model combination considers the full distributional properties of the ensemble and not just the central estimate - which is of particular importance in the reserving context. secondly, our framework is that it is tailored for the features inherent to reserving data. these include, for instance, accident, development, calendar, and claim maturity effects. crucially, the relative importance and scarcity of data across accident periods renders the problem distinct from the traditional ensembling techniques in statistical learning.   our framework is illustrated with a complex synthetic dataset. in the results, the optimised ensemble outperforms both (i) traditional model selection strategies, and (ii) an equally weighted ensemble. in particular, the improvement occurs not only with central estimates but also relevant quantiles, such as the 75th percentile of reserves (typically of interest to both insurers and regulators).",
    "categories": "stat.me q-fin.rm stat.ap",
    "doi": "",
    "created": "2022-06-17",
    "updated": "2024-02-19",
    "authors": [
        "benjamin avanzi",
        "yanfeng li",
        "bernard wong",
        "alan xian"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2206.08541"
}
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