Best Linear Unbiased Predictor

Study guide: Exam MAS-II

The best linear unbiased predictor (BLUP) is the estimate a Linear Mixed Model produces for a random effect. It is the group's raw deviation shrunk toward zero by a factor that is, in the simplest case, exactly a credibility weight.

\hat{\mathbf{b}} = \mathbf{D}\mathbf{Z}^{\top}\mathbf{V}^{-1}\left(\mathbf{y} - \mathbf{X}\hat{\boldsymbol{\beta}}\right)

\text{random intercept: } \hat{b}_i = \frac{n_i\sigma_b^{2}}{n_i\sigma_b^{2} + \sigma^{2}}\left(\bar{y}_i - \bar{y}\right)

• Fixed effects are estimated, random effects are predicted — a random effect is a realized value of a random variable, not a fixed constant, so "prediction" is the correct word
• The shrinkage factor \dfrac{n_i}{n_i + \sigma^2/\sigma_b^2} is Bühlmann's Z = n/(n+k) with k = \sigma^2/\sigma_b^2 = \mathrm{EPV}/\mathrm{VHM} — the mixed model and the credibility model are the same estimator
• A group with plenty of data keeps nearly all of its own deviation; a thin group is pulled almost all the way back to the population mean
• BLUPs are shrunk on purpose: they minimize mean squared prediction error, so their spread is deliberately narrower than the true effects'
• Adding \mathbf{X}\hat{\boldsymbol\beta} + \mathbf{Z}\hat{\mathbf{b}} gives the group-specific (conditional) fitted values, against the population-average fit \mathbf{X}\hat{\boldsymbol\beta}

BLUP as a Credibility Estimate A random-intercept model of pure premium by class has \hat\sigma_b^2 = 900 and \hat\sigma^2 = 3{,}600. Class C has 6 policy-years averaging 260 against a grand mean of 200. What is its BLUP and its fitted pure premium?

Answer k = \sigma^2/\sigma_b^2 = 3{,}600/900 = 4, so Z = \frac{6}{6 + 4} = 0.60 \qquad \hat b_C = 0.60(260 - 200) = 36 Fitted pure premium = 200 + 36 = 236, not the raw 260. Six policy-years earn 60% credibility; the remaining 40% stays with the class mean.