Simplifications in baseball data compared to insurance data

1. A constant set of risks (teams).

2. The baseball loss data is readily available, accurate, and not subject to development.

3. Each risk (team) is of equal size; they each play roughly the same # of games each year.

Tests to determine if risk parameters are shifting over time

1. chi^{2} test. For each risk, calculate the chi^{2} test statistic as follows: chi^{2} test statistic = Σ (i=1 to n) (Actual Losses_{i} - Expected Losses_{i})^{2} / Expected Losses_{i}. If there are n time period groups, the chi^{2} table value to compare against will have n - 1 degrees of freedom. If the test statistic exceeds the tabular value with the acceptable percentage level, then conclude that the distributions are different, and thus risk parameters for that risk have shifted over time.

2. Compare correlations between years.

6 methods to estimate X

1. X_{est} = mu

2. X_{est} = Y_{1}

3. X_{est} = ZY_{1} + (1 - Z)mu

4. X_{est} = Z/n Σ (i = 1 to n) Y_{i} + (1 - Z) mu

5. X_{est,i+1} = ZY_{i} + (1 - Z) X_{est,i}

6. X_{est} = Σ (i = 1 to n) Z_{i}Y_{i} + (1 - Σ (i = 1 to n) Z_{i}) * mu

3 criteria to evaluate possible estimates

1. Least Squared Error - Calculate the Mean Squared Error of the prediction compared with the actual observed result. The Bühlmann/Bayesian credibility methods attempt to minimize this criteria.

2. Limited Fluctuation - Also known as Small Chance of Large Errors. Measure the probability that the observed result differs by more than a certain percent from the predicted result. The classical credibility method targets this criteria.

3. Meyers/Dorweiler - Calculate the correlation between the ratio of actual to expected losses and the ratio of predicted losses to overall losses. This criteria confirms that there is no evidence that large predictions lead to large errors and small predictions lead to small errors.