2. Preliminaries and model assumptions

In classical insurance analysis, the treatment of insurance claims typically revolves around modeling them as random variables, denoted as $X$ , with corresponding density functions represented by $f(x|\theta )$ . Here, $\theta$ signifies a fixed parameter associated with the risk, though often obscured by uncertainty. However, the Bayesian approach to risk analysis introduces a fundamental shift in perspective: $\theta$ is no longer regarded as a fixed value but rather as a random variable itself. This paradigmatic change necessitates the adoption of prior distributions for $\Theta$ within heterogeneous insurance portfolios.

The Bayesian framework offers considerable advantages to actuaries. First, it liberates them from the constraints of specific models, fostering adaptability in tackling a wide array of problems. Once adept at analyzing one scenario, actuaries find themselves equipped to handle analogous, albeit more intricate, situations with minimal additional complexity. This inherent flexibility is invaluable in navigating the dynamic landscape of insurance and risk management.

Second, the Bayesian approach simplifies the process of obtaining estimates. Notably, it provides a seamless transition from obtaining point estimates to deriving interval estimates. Also, the key point in the Bayesian approach is it allows actuaries to assess the uncertainty of their inference in terms of probabilities. This characteristic is particularly pertinent in contemporary actuarial practice, where stakeholders increasingly demand not only precise estimates but also insights into the uncertainty surrounding them.

By embracing the Bayesian framework, actuaries can navigate the complexities of risk analysis with greater confidence and efficiency. It empowers them to provide robust evidence regarding the quality and reliability of their estimates, thereby enhancing decision-making processes within the insurance and financial industries. As the landscape of risk management continues to evolve, the Bayesian approach stands as a beacon of innovation and adaptability in the pursuit of informed decision-making and risk mitigation strategies.

Consider a scenario where $X_1, X_2, \ldots , X_n$ represent a vector of insurance claims, while $Z_i = (Z_{i1}, Z_{i2}, \ldots , Z_{im})$ denotes the covariate vector associated with individual risk $\theta _i$ , for $i = 1, 2, \ldots , n$ . Each individual risk is characterized by a unique risk profile, encapsulated by a scalar $\theta _i$ . Importantly, $\theta _i$ is not a fixed parameter but rather a realization of a random element denoted as $\Theta _i$ , reflecting the inherent uncertainty associated with each individual’s risk profile.

Continuing from the above, let us use the following to elaborate on the division of observations into homogeneous subpopulations:

Classification of the nonhomogeneous random sample $X_1, X_2, \ldots ,X_n,$ into $k$ homogeneous subpopulation, suggested by Model Assumption 1, plays a crucial role in insurance pricing, leveraging observable characteristics to group insured individuals with similar expected claims. This classification enables the development of premium rating systems, which express a priori information about new policyholders or insured individuals lacking claims experience. However, a priori classification schemes may not capture all relevant factors for premium rating, as some factors are unmeasurable or unobservable.

Classification of the nonhomogeneous random sample $X_1,X_2,\ldots ,X_n$ into $k$ homogeneous subpopulations, as suggested by Model Assumption 1, plays a crucial role in insurance pricing. This process leverages observable characteristics to group insured individuals with similar expected claims. Such classification enables the development of premium rating systems, which provide initial information about new policyholders or insured individuals lacking claims experience. However, these a priori classification schemes might overlook certain relevant factors in premium rating, as some factors are unmeasurable or unobservable.

To address these limitations, a posteriori classification, also known as experience rating, becomes essential. This system re-rates risks by incorporating claims experience into the rating process, resulting in a more equitable and rational price discrimination scheme. By integrating actual claims data, insurers can refine their pricing strategies, better align premiums with individual risk profiles, and promote fairness in insurance pricing.

Diao and Weng (Reference Diao and Weng2019) introduced the RTC model, which employs statistical techniques like regression trees to partition the measurable space $X$ into smaller regions where simple models provide accurate fits. In the subsequent step, for each region, they apply the Bühlmann-Straub credibility premium formula to predict the credibility premium. Specifically, given observed data $X_i$ and associated information $Z_{i,1}, \ldots , Z_{i,m}$ for $i=1,\ldots,n$ , a statistical model such as a regression tree determines the probability that the claim experience $X_1,X_2,\ldots,X_{n}$ arises from Population $l$ . If this probability exceeds $0.5$ , the credibility premium is predicted using the model developed for Population 1; otherwise, the model developed for Population 2 is used. As the RTC method employs the Bühlmann-Straub credibility premium formula, its credibility premium is given by $P^{{\rm RTC}}_l = \alpha _l \bar {X}_l + (1-\alpha _l) \mu _l$ when Population $l=1,2,\ldots,k$ is chosen. However, without any classification, the credibility premium is $P^{{\rm Credibility}}_{{\rm Total}}=\alpha \bar {X}+(1-\alpha ) \mu$ , where $\alpha ={n}/{(n + \frac {\sigma ^2}{\tau ^2})}$ .

The following provides a more general version of Diao and Weng (Reference Diao and Weng2019)’s finding for the RTC model under the square error loss function:

Proof. To drive an induction argumentation, consider the case of $k=2$ . Using the concave function $f(x,y) = \frac {1}{1/{x}+1/{y}}$ , we may conclude that

\begin{align*} \underset {\sim }{R}^{{\rm RTC}} \Big(P_1^{{\rm RTC}}, P_2^{{\rm RTC}}\Big)&= \sum _{l=1}^{2}\omega _lf\bigg(\frac {\sigma _l^2}{n},\tau _l^2\bigg)\\ &\leq f\bigg(\omega _1\frac {\sigma _1^2}{n} + \omega _2\frac {\sigma _2^2}{n}, \omega _1 \tau _1^2 + \omega _2 \tau _2^2 \bigg)\\ &=R_{{\rm Total}}^{{\rm Credibility}}(P^{{\rm Credibility}}). \end{align*}

The rest of the proof arrives under an induction argumentation.

Theorem 1 provides key insights into the risk management implications of utilizing the RTC premium within insurance pricing frameworks. By considering the square error loss function, it underscores the significance of balancing within-subpopulation variance $\sigma _l^2$ with between-subpopulation variance $\tau _l^2$ . This balance is essential for optimizing the mixing proportions $\omega$ to minimize overall risk, highlighting the importance of strategic decision-making in risk assessment and pricing strategies.

Furthermore, the theorem establishes that incorporating subclassifications based on $\omega$ into the MCF premium does not increase the overall risk. In fact, it suggests that the resulting risk is no greater than that associated with $P^{{\rm Bayes}}$ (Bayes premium) without subclassifications. This implies that subclassification strategies improve risk management efficacy without elevating overall risk levels, emphasizing the effectiveness of the RTC premium in refining pricing strategies and enhancing risk assessment accuracy.

Additionally, Theorem 1 elucidates that partitioning a collective of individual risks does not compromise the prediction accuracy of the RTC formula premium, provided that structural parameters of resulting sub-collectives can be accurately computed. While this theoretical foundation supports the use of partitioning-based premium prediction methods, it also acknowledges the inevitability of statistical estimation errors. The challenge lies in balancing the benefits of partitioning against the adverse effects of estimation errors, necessitating a judicious trade-off in decision-making processes.

Model Assumption 2. Additional to Model Assumption 1 , suppose that for each random variable $X_i$ , for $i=1,2,\ldots,n$ , there exists additional information $Z_{i,1}, Z_{i,2}, \ldots , Z_{i,m}$ such that using the following logistic regression, one can evaluate the probability that observation belongs to the $l{{\rm th}}$ subpopulation, that is,

(1) \begin{align} \omega _l = P(X_i \in PoP_l|z_{i,1}, z_{i,2}, \ldots , z_{i,m}) = \frac {1} {1+\exp {\{-\beta _{0} - \sum _{j=1}^{m}\beta _{l}z_{j,l}}\}}, \end{align}

for $l=1,2,\ldots,k.$ Moreover, suppose that the claim experience $X_1,X_2,\ldots,X_n$ , given parameter vector $\boldsymbol{\Psi }=(\theta _1,\theta _2,\ldots,\theta _k)^\prime$ , follows a $k$ -component normal mixture distribution $\sum _{l=1}^{k}\omega _l N(\theta _l,\sigma _l^2)$ , where $\sigma _l^2$ are given, and for $l=1,2,\ldots,k$ , $\Theta _l$ has a conjugate prior distribution $N(\mu _l,\tau _l^2)$ .

Under Model Assumption 2, Jahanbani et al. (2024) introduced the LRC premium as $\sum _{l=1}^{k}\omega _l[\xi _1\bar {X}_l+(1-\xi _l)\mu _l] .$ They also showed that, under the squared error loss function, its corresponding risk function is:

\begin{eqnarray*} R^{{\rm LRC}}(P^{{\rm LRC}}) &=&\sum _{l=1}^{k}\omega _l^2 \bigg [ \xi _l^2 \frac {\sigma _l^2}{n} + (1-\xi _l)^2 \tau _l^2 \bigg ], \end{eqnarray*}

where $\xi _l=\sum _{i=0}^{n} \omega ^i_l (1-\omega _l)^{n-i}\binom {n}{i}\frac {i\tau _l^2}{i\tau _l^2 + \sigma _l^2}.$

Also, it has been shown that in the case of $k=2,$ the LRC premium dominates. Jahanbani et al. (2024) demonstrated that under certain conditions, the risk function of LRC dominates the RTC premium whenever $\omega$ is around $0.5$ . Additionally, see Jahanbani et al. (2022) for practical applications of the RTC model.

3. Mixture credibility formula

This section serves as the main contribution of this article, presenting the introduction of an MCF model. This model integrates the $k$ -means technique into credibility theory, aiming to improve premium prediction accuracy as measured by the risk function under the square error loss function. Through the incorporation of machine learning methods, the MCF model demonstrates enhanced performance compared to traditional approaches in credibility theory.

Our MCF model offers a versatile approach to leverage covariate information for premium prediction, consisting of three key steps. First, we introduce a $k$ -means-based algorithm to utilize covariate information, effectively partitioning a collective of risks into distinct sub-collectives. This segmentation ensures homogeneity within each sub-collective while promoting heterogeneity across sub-collectives in terms of risk profiles. Second, the credibility premium formula is applied to each sub-collective, enabling precise estimation of premiums tailored to the characteristics of each segment. Lastly, we aggregate these segment-specific credibility premiums to derive the overall insurance premium, employing logistic regression to estimate mixture weights for each class. This comprehensive approach maximizes the utilization of covariate information while enhancing the accuracy and flexibility of premium prediction.

The following theorem evaluates the risk function for the MCF when the square error loss function is employed.

Theorem 2. Under Model Assumption 1 , and probabilities of belonging to each class $(\omega _1, \omega _2, \ldots , \omega _k)$ , the total premium is computed as $P^{{\rm MCF}} = \sum _{l=1}^{k}\omega _l [\alpha _l \bar {X_l} + (1-\alpha _l) \mu _l]$ . Consequently, the total risk function under the square error loss function can be expressed as follows:

\begin{eqnarray*} R^{{\rm MCF}}(P^{{\rm MCF}})&=&\sum _{l=1}^{k} \omega _l^2\bigg [\alpha _l^2 \frac {\sigma _l^2}{n_l} + (1-\alpha _l)^2 \tau _l^2\bigg ]. \end{eqnarray*}

Proof. Using the Model Assumption 1, one may write

\begin{eqnarray*} R^{{\rm MCF}}(P^{{\rm MCF}})&=& E\bigg [(P^{{\rm MCF}}- \mu (\Theta ))^2\bigg ]=E\bigg [\bigg(P^{{\rm MCF}}- \sum _{l=1}^{k}\omega _l\mu (\Theta _l) \bigg)^2\bigg ]\\ &=&E\bigg [\bigg (\sum _{l=1}^{k}\bigg [\omega _l[\alpha _l \bar {X_l} + (1-\alpha _l) \mu _l] - \omega _l\mu (\Theta _l)\bigg ]\bigg )^2\bigg ]\\ &=&E\bigg [\bigg (\sum _{l=1}^{k}\bigg [\omega _l[\alpha _l \bar {X_l} + (1-\alpha _l) \mu _l] - \omega _l\mu (\Theta _l)\pm \omega _l\alpha _l \mu (\Theta _l)\bigg ]\bigg )^2\bigg ]\\ &=&\sum _{l=1}^{k}\omega _l^2\alpha _l^2 E\bigg [\bigg ( \bar {X_l}-\mu (\Theta _l)\bigg )^2\bigg ] + \sum _{l=1}^{k}\omega _l^2(1-\alpha _l)^2 E\bigg [\bigg ( \mu _l-\mu (\Theta _l)\bigg )^2\bigg ] \\ &=& \sum _{l=1}^{k}\omega _l^2\alpha _l^2 \frac {\sigma _l^2}{n_l} + \sum _{l=1}^{k}\omega _l^2(1-\alpha _l)^2 Var\left [\mu (\Theta _l)\right ] \\ &=&\sum _{l=1}^{k} \omega _l^2\bigg [\alpha _l^2 \frac {\sigma _l^2}{n_l} + (1-\alpha _l)^2 \tau _l^2\bigg ]. \end{eqnarray*}

The following theorem demonstrates that the MCF under the square error loss function outperforms the RTC method introduced by Diao and Weng (Reference Diao and Weng2019).

Proof. Using the result of the above two theorems along with the fact that for all $l=1,2,\ldots, k$ $0 \leq \omega _l\leq 1$ , one may have:

\begin{align*} &R^{{\rm MCF}}(P^{{\rm MCF}}) - \underset {\sim }{R}^{{\rm RTC}}\Big(P_1^{{\rm RTC}},\ldots, P_k^{{\rm RTC}}\Big) \\&\quad = \sum _{l=1}^{k}\omega _l^2\bigg [\alpha _l^2 \frac {\sigma _l^2}{n_l} + (1-\alpha _l)^2 \tau _l^2\bigg ] -\sum _{l=1}^{k} \omega _l\bigg [\alpha _l^2 \frac {\sigma _l^2}{n_l} + (1-\alpha _l)^2 \tau _l^2\bigg ]\\ &\quad = \sum _{l=1}^{k} \bigg [\alpha _l^2 \frac {\sigma _l^2}{n_l} + (1-\alpha _l)^2 \tau _l^2\bigg ]\omega _l(\omega _l-1)\leq 0. \end{align*}

We proceed to compare the LRC model with the MCF. Specifically, we demonstrate, under the assumption of normality, that the MCF dominates the LRC model.

Proof. In the case of $k=2,$ the difference between the risk functions of two premiums can be restated as

\begin{align*} &R^{{\rm MCF}}(P^{{\rm MCF}}) - R^{{\rm LRC}}(P^{{\rm LRC}})\\&\quad =\sum _{l=1}^{2} \omega _l^2 \bigg [ \alpha _l^2 \frac {\sigma _l^2}{n_l} + (1-\alpha _l)^2 \tau _l^2 \bigg ] - \sum _{l=1}^{2}\omega _l^2\bigg [ \xi _l^2 \frac {\sigma _l^2}{n_l} + (1-\xi _l)^2 \tau _l^2 \bigg ] \\ &\quad=\sum _{l=1}^{2} \omega _l^2\frac {\sigma _l^2}{n_l}\bigg [ \alpha _l^2 -\xi _l^2\bigg ] +\sum _{l=1}^{2} \omega _l^2 \tau _l^2 \bigg [(1-\alpha _l)^2 - (1-\xi _l)^2\bigg ]\\ &\quad =\sum _{l=1}^{2} \omega _l^2\frac {\sigma _l^2}{n_l} \sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i}\bigg [\frac {n_l\tau _l^2}{n_l\tau _l^2 + \sigma _l^2} +\frac {i\tau _l^2}{i\tau _l^2 + \sigma _l^2}\bigg ]\bigg [\frac {n_l\tau _l^2}{n_l\tau _l^2 + \sigma _l^2} -\frac {i\tau _l^2}{i\tau _l^2 + \sigma _l^2}\bigg ]\\ &\qquad +\sum _{l=1}^{2} \omega _l^2 \tau _l^2 \sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i}\bigg [\frac {\sigma _l^2 }{n_l\tau _l^2 + \sigma _l^2} + \frac {\sigma _l^2}{i\tau _l^2+\sigma _l^2}\bigg ]\bigg [\frac {\sigma _l^2}{n_l\tau _l^2 + \sigma _l^2} -\frac {\sigma _l^2}{i\tau _l^2+\sigma _l^2}\bigg ] \\ &\quad =\sum _{l=1}^{2} \omega _l^2\frac {\sigma _l^2 \tau _l^2}{n_l} \sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i}\tau _l^2\bigg [\frac {n_l}{n_l\tau _l^2 + \sigma _l^2} +\frac {i}{i\tau _l^2 + \sigma _l^2}\bigg ]\bigg [\frac {n_l}{n_l\tau _l^2 + \sigma _l^2} -\frac {i}{i\tau _l^2 + \sigma _l^2}\bigg ] \\ &\qquad \!\!+\sum _{l=1}^{2} \omega _l^2\frac {\sigma _l^2 \tau _l^2}{n_l} \sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i} n_l\sigma _l^2\bigg [\frac {1}{n_l\tau _l^2 + \sigma _l^2} + \frac {1}{i\tau _l^2+\sigma _l^2}\bigg ]\bigg [\frac {1}{n_l\tau _l^2 + \sigma _l^2} -\frac {1}{i\tau _l^2+\sigma _l^2}\bigg ] \\ &\quad= \sum _{i=0}^{n} \omega _1^i (1-\omega _1)^{n-i} \binom {n}{i}\omega _1^2\frac {\sigma _1^2 \tau _1^2}{n_1}\bigg [\frac {n_1^2\tau _1^2+n_1\sigma _1^2}{(n_1\tau _1^2+\sigma _1^2)^2} - \frac {i^2 \tau _1^2 +n_1\sigma _1^2}{(i\tau _1^2+\sigma _1^2)^2} \bigg ]\\ &\qquad + \sum _{i=0}^{n} (1-\omega _1)^i (\omega _1)^{n-i} \binom {n}{i}(1-\omega _1)^2\frac {\sigma _2^2 \tau _2^2}{n_2}\bigg [\frac {n_2^2\tau _2^2+n_2\sigma _2^2}{(n_2\tau _2^2+\sigma _2^2)^2} - \frac {i^2 \tau _2^2 +n_2\sigma _2^2}{(i\tau _2^2+\sigma _2^2)^2} \bigg ]\\ &\quad =\sum _{i=0}^{n} \omega _1^i (1-\omega _1)^{n-i} \binom {n}{i} H_{(1,2)}(i), \end{align*}

where

(2) \begin{eqnarray} \nonumber H_{(1,2)}(i)&=&\omega _1^2\frac {\sigma _1^2 \tau _1^2}{n_1}\bigg [\frac {n_1^2\tau _1^2+n_1\sigma _1^2}{(n_1\tau _1^2+\sigma _1^2)^2} - \frac {i^2 \tau _1^2 +n_1\sigma _1^2}{(i\tau _1^2+\sigma _1^2)^2} \bigg ]\\ && +(1-\omega _1)^2\frac {\sigma _2^2 \tau _2^2}{n_2}\bigg [\frac {n_2^2\tau _2^2+n_2\sigma _2^2}{(n_2\tau _2^2+\sigma _2^2)^2} - \frac {(n-i)^2 \tau _2^2 +n_2\sigma _2^2}{((n-i)\tau _2^2+\sigma _2^2)^2} \bigg ] \end{eqnarray}

which arrives by using the fact that $ \binom {n}{i}= \binom {n}{n-i}.$

Now, without loss of generality, assume that the function $H_{(1,2)}(i)$ is a continuous function with respect to $i$ . Therefore, using the first derivative

\begin{eqnarray*} \frac {\partial H_{(1,2)}(i)}{\partial i} = - \omega _1^2 \frac {\sigma _1^2 \tau _1^2}{n_1} \frac {2 \tau _1^2 \sigma _1^2 i}{(i \tau _1^2 + \sigma _1^2)^3} + (1-\omega _1)^2 \frac {\sigma _2^2 \tau _2^2}{n_2} \frac {2 \tau _2^2[(n-i)\sigma _2^2-n_2\sigma _2^2]}{((n-i) \tau _2^2 + \sigma _2^2)^3}, \end{eqnarray*}

one may conclude that $\frac {\partial H_{(1,2)}(i)}{\partial i}|_{i=0} \gt 0$ and $\frac {\partial H_{(1,2)}(i)}{\partial i}|_{i=n} \lt 0$ . On the other hand, since $\frac {\partial ^2 H_{(1,2)}(i)}{\partial ^2 i} \lt 0,$ one may conclude that $H_{(1,2)}(i)$ as a function of $i$ is a concave function which attains its maximum at $i=n_1=n-n_2$ in which $H_{(1,2)}(i=n_1)= 0.$ Therefore, for the case of $k=2,$ we always have $R^{{\rm MCF}}(\!\cdot\!) - R^{{\rm LRC}}(\!\cdot\!)\leq 0.$

Now of the general case of $k\gt 2,$ observe that

\begin{eqnarray*} \omega _k^i(1-\omega _k)^{n-i}&=&(1-\omega _1-\cdots -\omega _{k-1})^i(\omega _1+\cdots +\omega _{k-1})^{n-i}\\ &\leq &\sum _{l=1}^{k-1}\omega _l^i(1-\omega _l)^{n-i}. \end{eqnarray*}

Using the above inequality, the difference between the two risk functions can be bounded above by

\begin{align*}& R^{{\rm MCF}}(P^{{\rm MCF}}) - R^{{\rm LRC}}(P^{{\rm LRC}})\\&\quad =\sum _{l=1}^{k} \omega _l^2 \bigg [ \alpha _l^2 \frac {\sigma _l^2}{n_l} + (1-\alpha _l)^2 \tau _l^2 \bigg ] - \sum _{l=1}^{k}\omega _l^2\bigg [ \xi _l^2 \frac {\sigma _l^2}{n_l} + (1-\xi _l)^2 \tau _l^2 \bigg ] \\ &\quad =\sum _{l=1}^{k} \omega _l^2\frac {\sigma _l^2}{n_l}\bigg [ \alpha _l^2 -\xi _l^2\bigg ] +\sum _{l=1}^{2} \omega _l^2 \tau _l^2 \bigg [(1-\alpha _l)^2 - (1-\xi _l)^2\bigg ]\\ &\quad =\sum _{l=1}^{k} \omega _l^2\frac {\sigma _l^2}{n_l} \sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i}\bigg [\frac {n_l\tau _l^2}{n_l\tau _l^2 + \sigma _l^2} +\frac {i\tau _l^2}{i\tau _l^2 + \sigma _l^2}\bigg ]\bigg [\frac {n_l\tau _l^2}{n_l\tau _l^2 + \sigma _l^2} -\frac {i\tau _l^2}{i\tau _l^2 + \sigma _l^2}\bigg ]\\ &\qquad+\sum _{l=1}^{k} \omega _l^2 \tau _l^2 \sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i}\bigg [\frac {\sigma _l^2 }{n_l\tau _l^2 + \sigma _l^2} + \frac {\sigma _l^2}{i\tau _l^2+\sigma _l^2}\bigg ]\bigg [\frac {\sigma _l^2}{n_l\tau _l^2 + \sigma _l^2} -\frac {\sigma _l^2}{i\tau _l^2+\sigma _l^2}\bigg ] \\ &\quad=\sum _{l=1}^{k} \omega _l^2\frac {\sigma _l^2 \tau _l^2}{n_l} \sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i}\tau _l^2\bigg [\frac {n_l}{n_l\tau _l^2 + \sigma _l^2} +\frac {i}{i\tau _l^2 + \sigma _l^2}\bigg ]\bigg [\frac {n_l}{n_l\tau _l^2 + \sigma _l^2} -\frac {i}{i\tau _l^2 + \sigma _l^2}\bigg ] \\ &\qquad\!\!+\sum _{l=1}^{k} \omega _l^2\frac {\sigma _l^2 \tau _l^2}{n_l} \sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i} n_l\sigma _l^2\bigg [\frac {1}{n_l\tau _l^2 + \sigma _l^2} + \frac {1}{i\tau _l^2+\sigma _l^2}\bigg ]\bigg [\frac {1}{n_l\tau _l^2 + \sigma _l^2} -\frac {1}{i\tau _l^2+\sigma _l^2}\bigg ] \end{align*}

\begin{align*}&=\sum _{l=1}^{k}\sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i}\omega _l^2\frac {\sigma _l^2 \tau _l^2}{n_l} \bigg [\frac {n_l^2\tau _l^2+n_l\sigma _l^2}{(n_l\tau _l^2+\sigma _l^2)^2} - \frac {i^2 \tau _l^2 +n_l\sigma _l^2}{(i\tau _l^2+\sigma _l^2)^2} \bigg ]\qquad \qquad \\ &\qquad\leq \sum _{l=1}^{k-1}\sum _{i=0}^{n} \omega _l^i (1-\omega _l)^{n-i} \binom {n}{i} H_{(l,k)}(i), \end{align*}

where $H_{(l,k)}(i)$ is a general version of the $H_{(1,2)}(i)$ given by Equation (2). The same argument, as we did above for the case $k=2,$ leads to desired result for the general case $k\gt 2$ , that is, $ R^{{\rm MCF}}(\!\cdot\!) - R^{{\rm LRC}}(\!\cdot\!) \leq 0$ .

4. Application to the real-world data

To illustrate the practical application of the above findings, we now consider a real-world dataset from an Iranian insurance company. This dataset contains demographic information (such as gender, age, etc.) as well as the size of claims for 1,180,255 individuals. Table 1 represents the list of available information in the dataset.

Table 1.

Output of command str(Data)

Figure 1(a) illustrates the box plot of claim size. However, Figure 1(b) illustrates the box plot of claim size after outlier data has been removed from the dataset. We defined an outlier observation as any observation that exceeds the value of 181,727,944 rials (the Iranian currency), and after removing such outlier data, we obtained 1,179,470 individuals.

Figure 1.

The box plot of damages’ size before removing outliers (a) and after removing outliers (b).

Figure 2 shows our attempt to fit a statistical distribution to the claim size.

Figure 2.

Histogram and density plots of the claim size.

Figure 3.

The box plot of damages’ size for different categories.

As one may observe, there is considerable non-homogeneity in the data. Therefore, we decided to derive some subpopulations to homogenize the data.

To get started, we categorized the covariates “Age” into 5 classes and “Province” into 8 classes. The Spearman test validated a significant relationship between age, gender, province, type of insured, and claim size. Moreover, the nonparametric Kruskal–Wallis test revealed a significant difference between groups associated with gender, type of insured, age, and province. Figure 3 shows the box plot of claim size regarding the aforementioned covariates.

Using Figure 3, one may conclude that these covariates can be employed to define some more homogenous subpopulations.

In the next step, using the $k$ -means clustering method, we derived two subpopulations, which classify insured individuals into high-risk and low-risk categories. A bivariate visualization of the two subpopulations is presented in Figure 4. Moreover, Figure 5 illustrates a histogram of claim size for those subpopulations.

Figure 4.

A bivariate visualization of the two subpopulations.

Figure 5.

Histogram of claim size for the high-risk and the low-risk classes.

To determine the probability that a given individual belongs to a given subpopulation, we implement the following logistic regression:

(3) \begin{equation} \omega _1 = P(Y_i\in Pop_1|\textbf{Z}_{\textbf{i}}=\textbf{z}_{\textbf{i}}) =1/\{1+e^{-\beta _{0} -\beta _{1}{z_1}-\beta _{2}{z_2}-\beta _{3}{z_3}-\beta _{4}{z_4}}\}, \end{equation}

where $z_1, z_2, z_3, z_4,$ respectively, stand for the re-categorized province, age, gender, and type of insured. Using the least square error method against observed data, we estimated the logistic regression parameters. Table 2 reports such estimated parameters.

Now, as an example, consider a $50$ -year-old single man who lives in a location labeled 6. Moreover, suppose that the logarithms of his $11$ years claim experiences are $16.91,$ $17.68,$ $15.97,$ $19.23,$ $15.63,$ $16.03,$ $14.85,$ $18.41,$ $14.20,$ $16.20$ , and $14.75258.$

Using the logistic regression model (3) against his information, with probability $0.2378$ ( $\omega =0.2378$ ), he would fall into Class 1. Therefore, the logarithm of the MCF for his next year is

\begin{eqnarray*} \log(P^{{\rm MCF}})&=&\omega \left [\left (\frac {n_1}{n_1+\frac {2}{{\hat \tau }_1^2}}\right )\bar {x}_1+\left (1-\frac {n_1}{n_1+\frac {{\hat \sigma }_1^2}{{\hat \tau }_1^2}}\right ){\hat \mu _1}\right ]\\ &=&+ (1-\omega ) \left [\left (\frac {n_2}{n_2+\frac {{\hat \sigma }_2^2}{{\hat \tau }_2^2}}\right )\bar {x}_2+\left (1-\frac {n_2}{n_2+\frac {{\hat \sigma }_2^2}{{\hat \tau }_2^2}}\right ){\hat \mu _2}\right ]\\ &=&0.2378\left [\left (\frac {597253}{597253+\frac {2}{1.48}}\right )17.42+\left (1-\frac {597253}{597253+\frac {2}{1.48}}\right )15\right ]\\ &&+0.7622\left [\left (\frac {582217}{582217+\frac {3}{1.51}}\right )17.83+\left (1-\frac {582217}{582217+\frac {3}{1.51}}\right )20\right ]\\ &=&17.73251. \end{eqnarray*}