Two-part models for assessing misrepresentation on risk status

Abstract

Claims modeling is a classical actuarial task aimed to understand the claim distribution given a set of risk factors. Yet some risk factors may be subject to misrepresentation, giving rise to bias in the estimated risk effects. Motivated by the unique characteristics of real health insurance data, we propose a novel class of two-part aggregate loss models that can (a) account for the semi-continuous feature of aggregate loss data, (b) test and adjust for misrepresentation risk in insurance ratemaking, and (c) incorporate an arbitrary number of correctly measured risk factors. The unobserved status of misrepresentation is captured via a latent factor shared by the two regression models on the occurrence and size of aggregate losses. For the complex two-part model, we derive explicit iterative formulas for the expectation maximization algorithm adopted in parameter estimation. Analytical expressions are obtained for the observed Fisher information matrix, ensuring computational efficiency in large-sample inferences on risk effects. We perform extensive simulation studies to demonstrate the convergence and robustness of the estimators under model misspecification. We illustrate the practical usefulness of the models by two empirical applications based on real medical claims data.

Appendices

Appendix A: Technical proofs

Proof of Proposition 2

The derivation for all the three formulas hinges on the partial derivative of the Q-function (10). We begin with the iterative formula for $\lambda$. Setting

$$\begin{aligned} \frac{\partial }{\partial \lambda } \ Q\left( \varvec{\Psi }\big |\varvec{\Psi }^{(s+1)}\right) =&\sum _{i=1}^{n}(1-v_{i}^{*})\left[ \frac{\eta _{i}^{(s+1)}-1}{1-\lambda } +\frac{\eta _{i}^{(s+1)}}{\lambda }\right] =0 \end{aligned}$$

yields ${{\hat{\lambda }}}^{(s+1)}= \sum _{i=1}^{n}(1-v_{i}^{*})\ \eta _{i}^{(s+1)} \big / \sum _{i=1}^{n}(1-v_{i}^{*})$.

Turning to the estimator for ${\varvec{\alpha }}$, we first recall

$$\begin{aligned} \log (f_v(y))= I_{(y>0)}\ \log \left[ \pi _{v,{\varvec{x}}^{{\mathcal {F}}}}\ g(y;\mu _{v,{\varvec{x}}^{{\mathcal {S}}}},\sigma )\right] + I_{(y=0)}\ \log (1-\pi _{v,{\varvec{x}}^{{\mathcal {F}}}}),\ y\ge 0,\ v=0,1. \end{aligned}$$

For expositional reasons, assume $x_0=1$. It is straightforward that, for $v\in \{0,1\}$ and $j\in \{0,{\mathcal {S}}\}$,

$$\begin{aligned} \frac{\partial \log (f_v(y))}{\partial \alpha _j} = I_{(y_{i}>0)} \left( \frac{\log (y)-\mu _{v,{\varvec{x}}^{{\mathcal {S}}}}}{\sigma ^{2}}\right) \ \frac{\partial }{\partial \alpha _j}\mu _{v,{\varvec{x}}^{{\mathcal {S}}}}= I_{(y_{i}>0)} \left( \frac{\log (y)-\mu _{v,{\varvec{x}}^{{\mathcal {S}}}}}{\sigma ^{2}}\right) x_{j}, \end{aligned}$$

and

$$\begin{aligned} \frac{\partial \log (f_v(y))}{\partial \alpha _{k+1}}= \left\{ \begin{array}{ll} 0, &{} v=0 \\ I_{(y_{i}>0)} \left( \frac{\log (y_{i})-\mu _{v,{\varvec{x}}^{{\mathcal {S}}}}}{\sigma ^{2}}\right) , &{} v=1 \end{array} \right. . \end{aligned}$$

Simple algebraic operation yields the following partial derivative formula:

$$\begin{aligned} \frac{\partial }{\partial \alpha _j} Q\left( \varvec{\Psi }\big |\varvec{\Psi }^{(s)}\right)&\propto \sum _{i=1}^{n} I_{(y_{i}>0)} v_{i}^{*} (\log (y_{i})-\mu _{1,{\varvec{x}}_1^{\mathcal {S}}}) x_{ij}\\&\quad +\sum _{i=1}^{n} I_{(y_{i}>0)} (1-v_{i}^{*}) \bigg [\left( 1-\eta ^{(s+1)}_{i}\right) (\log (y_{i})-\mu _{0,{\varvec{x}}_1^{\mathcal {S}}}) x_{ij}+ \eta ^{(s+1)}_{i}(\log (y_{i})-\mu _{1,{\varvec{x}}_1^{\mathcal {S}}}) x_{ij} \bigg ]\\&= \sum _{i=1}^{n} I_{(y_{i}>0)} \log (y_{i}) x_{ij} -\sum _{h\in {\mathcal {S}}}\alpha _h\sum _{i=1}^{n} I_{(y_{i}>0)} x_{ij} x_{ih} \\&\quad - \alpha _{k+1}\bigg [\sum _{i=1}^{n} I_{(y_{i}>0)} x_{ij}\left( v_{i}^{*} +(1-v^{*}_{i})\eta ^{(s+1)}_{i} \right) \bigg ], \end{aligned}$$

for $j\in \{0,{\mathcal {S}}\}$, and

$$\begin{aligned}\frac{\partial }{\partial \alpha _{k+1}} Q\left( \varvec{\Psi }\big |\varvec{\Psi }^{(s)}\right)&\propto \sum _{i=1}^{n} I_{(y_{i}>0)} v_{i}^{*} (\log (y_{i})-\mu _{1,{\varvec{x}}_{i}^{\mathcal {S}}})\\&\qquad +\sum _{i=1}^{n} I_{(y_{i}>0)} (1-v_{i}^{*}) \left[ \eta ^{(s+1)}_{i}\left( \log (y_{i})-\mu _{1,{\varvec{x}}_{i}^{\mathcal {S}}}\right) \right] \\& =\sum _{i=1}^{n} I_{(y_{i}>0)} \log (y_{i}) \left( v_{i}^{*} +(1-v^{*}_{i})\eta ^{(s+1)}_{i} \right) \\&\qquad -\sum _{j\in \{0,{\mathcal {S}},k+1\}} \alpha _{j} \sum _{i=1}^{n} I_{(y_{i}>0)} x_{ij} \left( v_{i}^{*} +(1-v^{*}_{i})\eta ^{(s+1)}_{i} \right) . \end{aligned}$$

Setting $\frac{\partial }{\partial \alpha _{j}} Q\left( \varvec{\Psi }\big |\varvec{\Psi }^{(s)}\right) =0$ for $j\in \{0,{\mathcal {S}},k+1\}$ forms a system of $(|{\mathcal {S}}|+2)$ linear equations with the same number of unknowns. The linear equation system can be solved via $({\varvec{B}}^T {\varvec{B}}+\varvec{E})^{-1}\ {\varvec{B}}^T \varvec{t}$. This yields the estimator for ${\varvec{\alpha }}$.

Finally, set $\frac{\partial }{\partial \sigma ^{2}} Q\left( \varvec{\Psi }\big |\varvec{\Psi }^{(s)}\right) =0$. With some simple algebraic calculations, we obtain

$$\begin{aligned}&({\hat{\phi }}^{2})^{(s+1)}\\&\quad =\frac{\sum _{i=1}^{n} \bigg \{v_{i}^{*}\big (\log (y_{i})-{\hat{\mu }}_{1,{\varvec{x}}_{i}^{{\mathcal {S}}}}^{(s+1)}\big )^{2}+(1-v_{i}^{*})\bigg [ (1-\eta ^{(s+1)}_{i})\big (\log (y_{i})-{\hat{\mu }} _{0,{\varvec{x}}_{i}^{{\mathcal {S}}}}^{(s+1)}\big )^{2}+ \eta ^{(s+1)}_{i}\big (\log (y_{i})-{\hat{\mu }}_{1,{\varvec{x}}_{i}^{{\mathcal {S}}}}^{(s+1)}\big )^{2} \bigg ]\bigg \}I_{(y_{i}>0)} }{\sum _{i=1}^{n} I_{(y_{i}>0)}}. \end{aligned}$$

This completes the proof of the proposition. $\square$

Proof of Corollary 3

Under the simplified assumption $\pi _{v,{\varvec{x}}}=\pi _v=\beta _0+\beta _1v$ for $v=0,1$, we first aim to find ${{\hat{\pi }}}_0$ and $\hat{\pi }_1$ that maximize Q-function (10). Note that

$$\begin{aligned} \frac{\partial }{\partial \pi _v }\log (f_v(y))=I_{(y>0)} \frac{1}{\pi _v} - (1-I_{(y>0)}) \frac{1}{1-\pi _v},\ v=0,1. \end{aligned}$$

Set

$$\begin{aligned} \frac{\partial }{\partial \pi _0} Q\left( \varvec{\Psi }\big |\varvec{\Psi }^{(s)}\right) = \sum _{i=1}^{n}(1-v_{i}^{*}) \left( 1-\eta _{i}^{(s+1)}\right) \left( I_{(y>0)} \frac{1}{\pi _0} - (1-I_{(y>0)}) \frac{1}{1-\pi _0} \right) =0, \end{aligned}$$

and

$$\begin{aligned} \frac{\partial }{\partial \pi _1} Q\left( \varvec{\Psi }\big |\varvec{\Psi }^{(s)}\right) = \sum _{i=1}^{n} \left( v_{i}^{*}+ (1-v_{i}^{*}) \eta _{i}^{(s+1)}\right) \left( I_{(y>0)} \frac{1}{\pi _1} - (1-I_{(y>0)}) \frac{1}{1-\pi _1} \right) =0. \end{aligned}$$

We obtain

$$\begin{aligned} {{\hat{\pi }}}_0& = \frac{\sum _{i=1}^{n} I_{(y_{i}>0)}(1-v^{*}_{i}) \left( 1-\eta _{i}^{(s+1)}\right) }{\sum _{i=1}^{n} (1-v^{*}_{i})\left( 1-\eta _{i}^{(s+1)}\right) } \text { and } \\{{\hat{\pi }}}_1 & = \frac{\sum _{i=1}^{n} I_{(y_{i}>0)}\left( v_{i}^{*}+(1-v^{*}_{i})\eta _{i}^{(s+1)} \right) }{\sum _{i=1}^{n} \left( v_{i}^{*}+(1-v^{*}_{i})\eta _{i}^{(s+1)} \right) }. \end{aligned}$$

The iterative formula for $\varvec{\beta }$ can be obtained via the one-to-one correspondence between $(\beta _0,\beta _1)$ and $(\pi _0,\pi _1)$. The proof is completed. $\square$

Proof of Proposition 5

For ease of presentation, we only report the Fisher information matrix for the $i$-th sample, $i=1,\ldots ,n$. The associated complete-data log likelihood function is denoted by $l_{c,i}$. With an observed sample of size $n$, we can compute the observed Fisher information matrix by summing the individual information matrices over the $n$ observations.

First, we study the expected Fisher information matrix associated with the complete-data log likelihood function used in the EM algorithm. To this end, we need the expected second derivatives of the complete-data log likelihood function. Tedious yet manageable calculations yield

$$\begin{aligned} -{\mathbb E}[{\partial ^{2} l_{c,i}}/{\partial \alpha _j \partial \alpha _t}]& = I_{(y_{i}>0)} \sigma ^{-2} x_{ij}x_{ih} ,\ j,h\in \{0,{\mathcal {S}}\};\\ -{\mathbb E}[{\partial ^{2} l_{c,i}}/{\partial \alpha _j \partial \alpha _{k+1}}]& = I_{(y_{i}>0)}\sigma ^{-2} x_{ij} \left[ v_{i}^{*}+(1-v_{i}^{*}) \eta _{i} \right] ,\ j\in \{0,{\mathcal {S}},k+1 \};\\ -{\mathbb E}[{\partial ^{2} l_{c,i}}/{\partial \beta _j \partial \beta _t}]& = x_{ij}x_{ih}\left[ (1-\pi _{1,{\varvec{x}}_{i}^{\mathcal {F}}})\pi _{1,{\varvec{x}}_{i}^{\mathcal {F}}} \left[ v_{i}^{*}+(1-v_{i}^{*}) \eta _{i} \right] \right. \\&\quad \left. + \left[ (1-\pi _{0,{\varvec{x}}_{i}^{\mathcal {F}}})\pi _{0,{\varvec{x}}_{i}^{\mathcal {F}}} \right] (1-v^{*}_{i})(1-\eta _{i})\right] ,\ j,h\in \{0,{\mathcal {F}}\};\\ -{\mathbb E}[{\partial ^{2} l_{c,i}}/{\partial \beta _j \partial \beta _{k+1}}]& = x_{ij}\left[ (1-\pi _{1,{\varvec{x}}_{i}^{\mathcal {F}}})\pi _{1,{\varvec{x}}_{i}^{\mathcal {F}}} \right] \left[ v_{i}^{*}+ (1-v_{i}^{*}) \eta _{i} \right] ,\\&\quad \quad \ j\in \{0,{\mathcal {F}},k+1 \};\\ -{\mathbb E}[{\partial ^{2} l_{c,i}}/{\partial \alpha _{j}\partial \sigma ^{2} }]& = I_{(y_{i}>0)} \sigma ^{-4} x_{ij} \left[ (\log (y_{i})-\mu _{1,{\varvec{x}}_{i}^{{\mathcal {S}}}})+\alpha _{k+1}(1-v_{i}^{*}) (1-\eta _{i}) \right] ,\\&\quad \quad \ j\in \{0,{\mathcal {S}}\};\\ -{\mathbb E}[{\partial ^{2} l_{c,i}}/{\partial \alpha _{k+1}\partial \sigma ^{2} }]& = I_{(y_{i}>0)} \sigma ^{-4} \left[ (\log (y_{i})-\mu _{1,{\varvec{x}}_{i}^{{\mathcal {S}}}})(v^{*}_{i}+(1-v_{i}^{*}) z_{i})\right] ;\\ -{\mathbb E}[{\partial ^{2} l_{c,i}}/{\partial \sigma ^{2}\partial \sigma ^{2} }]& = I_{(y_{i}>0)} \sigma ^{-6} \left[ (\log (y_{i})-\mu _{1,{\varvec{x}}_{i}^{{\mathcal {S}}}})^{2} (v_{i}^{*}+(1-v_{i}^{*})\eta _{i})\right. \\&\qquad \left. + \left( \log (y_{i})-\mu _{0,{\varvec{x}}_{i}^{{\mathcal {S}}}}\right) ^{2}(1-v_{i}^{*}) (1-\eta _{i}) -0.5 \sigma ^{2} \right] ;\\ -{\mathbb E}[{\partial ^{2} l_{c,i}}/{\partial \lambda^2}]& = (1-v^{*}_{i}) \left[ \lambda ^{-2}-(1-2\lambda )[\lambda (1-\lambda )]^{-2} (1-\eta _{i}) \right] . \end{aligned}$$

The expected second derivatives are equal to zero otherwise. By using the aforementioned derivative formulas, the Fisher information matrix associated with the complete-data log likelihood function can now be constructed according to Eq. (13).

Next, we study the covariance matrix of the gradient vector of the complete data log likelihood function. We again set $x_{i0}=1$, $i=1,\ldots ,n$, for notational convenience. It holds that

$$\begin{aligned} {\partial l_{c,i}}/{\partial \alpha _j}& = I_{(y_{i}>0)} \sigma ^{-2} x_{ij} \left[ (\log (y_{i})-\mu _{1,{\varvec{x}}_{i}^{{\mathcal {S}}}})+\alpha _{k+1}(1-v_{i}^{*}) (1-z_{i}) \right] ,\ j\in \{0,{\mathcal {S}}\};\\ {\partial l_{c,i} }/{\partial \alpha _{k+1}}& = I_{(y_{i}>0)} \sigma ^{-2} \left[ (\log (y_{i})-\mu _{1,{\varvec{x}}_{i}^{{\mathcal {S}}}})(v^{*}_{i}+(1-v_{i}^{*}) z_{i})\right] ;\\ {\partial l_{c,i}}/{\partial \beta _j}& = x_{ij}\bigg [\left( I_{(y_{i}>0)}(1-\pi _{1,{\varvec{x}}_{i}^{{\mathcal {F}}}})-I_{(y_{i}=0)}\pi _{1,{\varvec{x}}_{i}^{{\mathcal {F}}}} \right) \\&\quad + \left( \pi _{1,{\varvec{x}}_{i}^{{\mathcal {F}}}}-\pi _{0,{\varvec{x}}_{i}^{{\mathcal {F}}}} \right) (1-v_{i}^{*}) (1-z_{i})\bigg ],\ j\in \{0,{\mathcal {F}}\};\\ {\partial l_{c,i}}/{\partial \beta _{k+1}}& = \left[ I_{(y_{i}>0)}(1-\pi _{1,{\varvec{x}}_{i}^{{\mathcal {F}}}})-I_{(y_{i}=0)} \pi _{1,{\varvec{x}}_{i}^{{\mathcal {F}}}} \right] \left( v^{*}_{i}+(1-v_{i}^{*}) z_{i}\right) ;\\ {\partial l_{c,i}}/{\partial \sigma ^{2}}& = \frac{1}{2} I_{(y_{i}>0)} \sigma ^{-4} \left[ (\log (y_{i})-\mu _{1,{\varvec{x}}_{i}^{{\mathcal {S}}}})^{2}-\sigma ^{2}\right. \\&\quad \left. +\left[ (\log (y_{i})-\mu _{0,{\varvec{x}}_{i}^{{\mathcal {S}}}})^{2}-(\log (y_{i})-\mu _{1,{\varvec{x}}_{i}^{{\mathcal {S}}}})^{2}\right] (1-v_{i}^{*}) (1-z_{i})\right] ;\\ {\partial l_{c,i}}/{\partial \lambda }& = (1-v^{*}_{i}) \left[ \lambda ^{-1}-(\lambda (1-\lambda ))^{-1} (1-z_{i}) \right] . \end{aligned}$$

Note that all the partial derivatives reported above are linear in $z_{i}$. Thereby, we have readily got

$$\begin{aligned} {\mathrm {Cov}}(\varvec{S}_{c,i}(z_{i};\varvec{\Psi })|y_{i},{\varvec{x}}_{i},v_{i}^{*})={\mathrm {Var}}(z_{i})\ {\varvec{b}}^{T}_{i}{\varvec{b}}_{i}=\eta _{i}(1-\eta _{i})\ {\varvec{b}}_{i}^{T}{\varvec{b}}_{i}. \end{aligned}$$

The application of the observed Fisher information formula in Lemma 4 completes the proof for the proposition. $\square$

Appendix B: Bayesian implementation

For the Bayesian implementation, we may use the MCMC methods based on the complete-data log likelihood function (8). Either Gibbs sampling or the Metropolis-Hastings algorithm can be used for the posterior simulations. References such as Hurn et al. [14], McLachlan and Peel [20] give comprehensive reviews of such algorithms for mixture models. In the current paper, we focus on the implementation of the two-part misrepresentation models using the BUGS language [18]. Owing to the excellent introductions by Scollnik [23, 24], the BUGS language has been widely used in the actuarial literature for implementing Bayesian models.

For the sake of illustration, let us consider the setting in Sect. 4 where there are four rating factors $(X_1,X_2,X_3,V)$ with ${\mathcal {S}}=\{1,2,3\}$ and ${\mathcal {F}}=\{1,2\}$. The rating factor $V$ is subject to misrepresentation. For the parameters in $\varvec{\Psi }$, we assume normal $Normal(0,10)$ priors for the regression coefficients in ${\varvec{\alpha }}$ and $\varvec{\beta }$, an inverse gamma prior $IG(0.001,0.001)$ for the shape parameter $\sigma$, and a $Uniform(0,1)$ prior for the misrepresentation prevalence parameters $\lambda$ and $\theta =\mathbb {P}[V=1]$. Such a non-informative prior specification represents a situation where we have no prior knowledge on the parameters. The following BUGS implementation of the two-part misrepresentation model utilizes the ones trick in specifying the complete-data log likelihood. In an attempt to maximize the likelihood function, in the ones trick a Bernoulli trial is assumed for a vector of ones, with the probability of each observation being the likelihood divided by a large number.

For the application studies, the Bayesian implementation gives similar results on the estimation of the parameters and their standard errors. Hence, we will not present the results here. For the simulation study, Bayesian MCMC simulations are much slower than the frequentist counterparts. Thus, running repeated simulations with large sample sizes is computationally impossible. When there is no prior knowledge on the misrepresentation behaviors, the frequentist methods discussed in Sect. 3 are more convenient for the implementation of the proposed two-part misrepresentation models.