Bühlmann model

In credibility theory, a branch of study in actuarial science, the Bühlmann model is a random effects model (or "variance components model" or hierarchical linear model) used in to determine the appropriate premium for a group of insurance contracts. The model is named after Hans Bühlmann who first published a description in 1967.^[1]

Model description

Consider i risks which generate random losses for which historical data of m recent claims are available (indexed by j). A premium for the ith risk is to be determined based on the expected value of claims. A linear estimator which minimizes the mean square error is sought. Write

X_ij for the j-th claim on the i-th risk (we assume that all claims for i-th risk are independent and identically distributed)
$\scriptstyle \bar{X}_i=\frac{1}{m}\sum_{j=1}^{m}X_{ij}$ for the average value.
$\Theta_i$ - the parameter for the distribution of the i-th risk
$m(\vartheta)= \operatorname E\left [ X_{ij} |\Theta_i = \vartheta\right ]$
$\Pi=\operatorname E(m(\vartheta)|X_{i1},X_{i2},...X_{im})$ - premium for the i-th risk
$\mu = \operatorname (m(\vartheta))$
$s^2(\vartheta)=\operatorname{Var}\left [ X_{ij} |\Theta_i = \vartheta\right ]$
$\sigma^2=\operatorname E\left [ s^2(\vartheta) \right ]$
$v^2=\operatorname{Var}\left [ m(\vartheta) \right ]$

Note: $m(\vartheta)$ and $s^2(\vartheta)$ are functions of random parameter $\vartheta$

The Bühlmann model is the solution for the problem:

\underset{a_{i0},a_{i1},...,a_{im}}{\operatorname{arg\,min}} \operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi \right)^2\right ]

where $a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}$ is the estimator of premium $\Pi$ and arg min represents the parameter values which minimize the expression.

Model solution

The solution for the problem is:

$Z\bar{X}_i+(1-Z)\mu$

where:

$Z=\frac{1}{1+\frac{\sigma^2}{v^2m}}$

We can give this result the interpretation, that Z part of the premium is based on the information that we have about the specific risk, and (1-Z) part is based on the information that we have about the whole population.

Proof

The following proof is slightly different from the one in the original paper. It is also more general, because it considers all linear estimators, while original proof considers only estimators based on average claim.^[2]

Lemma: The problem can be stated alternatively as:

$f=\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right )^2\right ]\rightarrow min$

Proof:

$\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right )^2\right ]$ $=\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right )^2\right ]+\operatorname E\left [ \left ( m(\vartheta)-\Pi\right )^2\right ]+2E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right ) \left ( m(\vartheta)-\Pi\right )\right ]$ $=\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right )^2\right ]+\operatorname E\left [ \left ( m(\vartheta)-\Pi\right )^2\right ]$

The last equation follows from the fact that $\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right ) \left ( m(\vartheta)-\Pi\right )\right ]$ $\operatorname E_{\Theta}\left \{ E_X\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right ) \left ( m(\vartheta)-\Pi\right ) | X_{i1},X_{i2},X_{im}\right ]\right \}$ $=\left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right )\operatorname E_{\Theta}\left \{ \operatorname E_X\left [ \left ( m(\vartheta)-\Pi\right ) | X_{i1},X_{i2},X_{im}\right ]\right \}=0$

We are using here the law of total expectation and the fact, that $\Pi=\operatorname E(m(\vartheta)|X_{i1},X_{i2},...X_{im})$

In our previous equation, we decompose minimized function in the sum of two expressions. The second expression does not depend on parameters used in minimization. Therefore, minimizing the function is the same as minimizing the first part of the sum.

Let us find critical points of the function

$\frac{1}{2}\frac{\partial f}{\partial a_{01}}=$

$\operatorname E\left [ a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right ]=a_{i0}+\sum_{j=1}^{m}a_{ij}\operatorname E(X_{ij})-\operatorname E(m(\vartheta))=a_{i0}-\left (\sum_{j=1}^{m}a_{ij}-1 \right )\mu$

$a_{i0}=\left (\sum_{j=1}^{m}a_{ij}-1 \right )\mu$

For $k\neq 0$ we have:

$\frac{1}{2}\frac{\partial f}{\partial a_{ik}}=\operatorname E\left [ X_{ik}\left ( a_{i0} +\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right ) \right ]$ $=\operatorname E\left [ X_{ik} \right ]a_{i0}+\sum_{j=1, j\neq k}^{m}a_{ij}\operatorname E[X_{ik}X_{ij}]+a_{ik}\operatorname E[X^2_{ik}]-\operatorname E[X_{ik}m(\vartheta)]=0$

We can simplify derivative, noting that:

$\operatorname E[X_{ij}X_{ik}]=\operatorname E[\operatorname E[X_{ij}X_{ik}|\vartheta]]=\operatorname E[cov(X_{ij}X_{ik}|\vartheta)+\operatorname E(X_{ij}|\vartheta)\operatorname E(X_{ik}|\vartheta)]=\operatorname E[(m(\vartheta))^2]=v^2+\mu^2$

and

$\operatorname E[X^2_{ik}]=\operatorname E[\operatorname E[X^2_{ik}|\vartheta]]=\operatorname E[s^2(\vartheta)+(m(\vartheta))^2]=\sigma^2+v^2+\mu^2$

and

$\operatorname E[X_{ik}m(\vartheta)]=\operatorname E[\operatorname E[X_{ik}m(\vartheta)|\Theta_i]=\operatorname E[(m(\vartheta))^2]=v^2+\mu^2$

Taking above equations and inserting into derivative, we have:

$\frac{1}{2}\frac{\partial f}{\partial a_{ik}}$ $=\left ( 1-\sum_{j=1}^{m}a_{ij} \right )\mu^2+\sum_{j=1,j\neq k}^{m}a_{ij}(v^2+\mu^2)+a_{ik}(\sigma^2+v^2+\mu^2)-(v^2+\mu^2)=a_{ik}\sigma^2-\left ( 1-\sum_{j=1}^{m}a_{ij} \right )v^2=0$

$\sigma^2a_{ik}=v^2\left (1- \sum_{j=1}^{m} a_{ij}\right)$

Right side doesn't depend on k. Therefore all $a_{ik}$ are constant

$a_{i1}=a_{i2}=...=a_{im}=\frac{v^2}{\sigma^2+mv^2}$

From the solution for $a_{i0}$ we have

$a_{i0}=(1-ma_{ik})\mu=\left ( 1-\frac{mv^2}{\sigma^2+mv^2} \right )\mu$

Finally, the best estimator is

$a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}=\frac{mv^2}{\sigma^2+mv^2}\bar{X_i}+\left ( 1-\frac{mv^2}{\sigma^2+mv^2} \right )\mu=Z\bar{X_i}+(1-Z)\mu$

References

↑ Bühlmann, Hans (1967). "Experience rating and credibility" (PDF) 4 (3). ASTIN Bulletin: 99–207.
↑ Proof can be found on this site: Schmidli, Hanspeter. "Lecture notes on Risk Theory" (PDF). Institute of Mathematics, University of Cologne. Archived from the original (PDF) on August 11, 2013.

Frees, E.W.; Young, V.R.; Luo, Y. (1999). "A longitudinal data analysis interpretation of credibility models". Insurance: Mathematics and Economics 24 (3): 229–247. doi:10.1016/S0167-6687(98)00055-9.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

List of topics Category

This article is issued from Wikipedia - version of the Sunday, April 17, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.