Credibility Theory

Credibility Theory provides the framework for weighting observed experience data against prior or external expectations, balancing the statistical reliability of an individual group's data with the stability of broader pooled data.

$\text{Estimate} = Z \cdot \bar{X} + (1 - Z)\cdot \mu_0$

$Z \in [0, 1]$ is the credibility factor, $\bar{X}$ is the observed experience, and $\mu_0$ is the complement of credibility (a class mean, industry data, or manual rate)
Classical (limited fluctuation) credibility assigns full credibility ( $Z = 1$ ) when a dataset meets a minimum size standard; below that threshold, the square-root rule $Z = \sqrt{n / n_0}$ applies
Bühlmann (greatest accuracy) credibility derives $Z$ from the ratio of between-group variance to total variance; it minimizes mean squared error rather than satisfying a probability bound
Bayesian credibility takes the estimate to be the posterior mean $E[\theta \mid \mathbf{X}]$ , and equals Bühlmann exactly for conjugate priors

Classical vs. Bühlmann Credibility A rating class has $n = 300$ claims. The full-credibility standard for frequency is $n_0 = 1{,}082$ claims.

Answer Classical: $Z = \sqrt{300 / 1{,}082} = \sqrt{0.277} = 0.527$

Bühlmann (if between-group variance $v = 0.10$ , process variance $\sigma^2 = 0.20$ ): $Z = \frac{v}{v + \sigma^2/n} = \frac{0.10}{0.10 + 0.20/300} = \frac{0.10}{0.1007} \approx 0.993$

The two methods yield very different credibilities because Bühlmann accounts for variance structure; classical only uses count relative to the full-credibility threshold.