Bühlmann model ( $m = E [m (Θ)], v = E [v (Θ)], a = Var (m (Θ))$ )

Given a latent risk parameter

Θ

, observations are independent and identically distributed random variables with conditional expectation

m (Θ)

and conditional variance

v (Θ)

. The Bühlmann credibility premium estimates

m (Θ)

by an affine function of past observations under mean squared error. Its population parameters are

m = E m (Θ)

, expected process variance

v = E v (Θ)

, and variance of hypothetical means

a = Var (m (Θ))

.

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Bühlmann–Straub model ( $Var (X_{j} ∣ Θ) = σ^{2} (Θ) / m_{j}$ )

Conditional on a common latent variable

Θ

, annual averages

X_{j}

are independent, have common conditional expectation

μ (Θ)

, and have conditional variance

σ^{2} (Θ) / m_{j}

for known positive exposures

m_{j}

. Its structural parameters are

m_{0} = E μ (Θ)

, expected process variance

v = E σ^{2} (Θ)

and variance of hypothetical means

a = Var (μ (Θ))

. It generalizes the equal-exposure Bühlmann model.

Bühlmann–Straub credibility estimate ( $μ = Z \overline{X}_{w} + (1 - Z) m_{0}$ )

For total exposure

W = \sum_{j} m_{j}

and weighted average

\overline{X}_{w} = \sum_{j} m_{j} X_{j} / W

, the best affine mean squared error estimate of the conditional mean is

Z \overline{X}_{w} + (1 - Z) m_{0}

, where

Z = Wa / (Wa + v)

. If the coefficients sum to

B

, their error is

a (1 - B)^{2} + v \sum_{j} b_{j}^{2} / m_{j}

. Cauchy-Schwarz inequality minimizes the latter term at

b_{j} = B m_{j} / W

, and a one-dimensional quadratic minimization then gives

B = Z

.

Exact Bühlmann–Straub credibility for Poisson-gamma counts ( $μ_{cred} = μ_{Bayes}$ )

With a Poisson-gamma conjugacy with unequal exposures model,

m_{0} = v = α / β

and

a = α / β^{2}

, so the Bühlmann–Straub credibility factor is

W / (W + β)

. The credibility estimate simplifies to

(α + \sum_{j} Y_{j}) / (β + W)

, exactly the posterior mean and hence the Bayes estimator under squared error loss. The agreement is exact because that posterior mean is already affine in the exposure-weighted data.

Bühlmann–Straub credibility factor ( $Z = \frac{Wa}{Wa + v}$ )

The experience weight is

Z = Wa / (Wa + v)

, where

W

is total observed exposure,

a

the variance of hypothetical means, and

v

the expected process variance. With positive

a

, it is

W / (W + v / a)

. More exposure raises the weight; larger process noise lowers it. If

a = 0

, the unknown conditional mean is constant and one can set

Z = 0

.

Bühlmann credibility premium ( $m = Z \overline{X} + (1 - Z) m$ )

The Bühlmann credibility premium is the best affine mean squared error estimate of a risk’s conditional claim expected value. In the Bühlmann model it is

Z \overline{X} + (1 - Z) m

, where

Z = na / (na + v)

. The linear least-squares projection equations use

Var (X_{j}) = a + v

and

Cov (X_{i}, X_{j}) = a

for distinct years.

Credibility factor ( $Z = na / (na + v)$ )

The credibility factor in the Bühlmann model is the weight

Z = na / (na + v)

placed on the sample mean after

n

observations. The remaining weight is placed on the population expected value. More experience or larger variance of hypothetical means increases

Z

; larger expected process variance decreases it.

Variance of hypothetical means ( $a = Var (E [X ∣ Θ])$ )

The variance of hypothetical means in the Bühlmann model is the between-risk variance of the conditional claim expected value. It creates the shared covariance

a

between different years of the same risk and determines how much individual experience should influence the premium.

Expected process variance ( $v = E [Var (X ∣ Θ)]$ )

The expected process variance in the Bühlmann model averages within-risk conditional variance over the population of risks. It measures variation that repeated observations can average out and contributes

v / n

to the variance of the sample mean.

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The Bühlmann model, introduced by Hans Bühlmann in the context of actuarial science, is a method for estimating risk or making predictions, particularly in the field of insurance. It is designed to improve the estimation of claims or losses by considering both historical data and additional information, which may help refine predictions.

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