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Advantages of using baseball data
 constant set of risk, vs insurance where risks leave or enter the database
 loss data is readily available, accurate and final
 each team plays roughly the same number of games (i.e. no consideration for size of risk)

Testing for differences between teams
 calculate average and standard deviation of losing % by team
 compare to binomial distribution with p = 50%, σ = √[np(1  p)]
 since many teams losing percentage is outside of 2σ, conclude teams are different

Testing for shifting risk parameters  standard 𝝌^{2} test
 test if the risk process could have the same mean over period
 compare 5 years actual losses vs expected losses
 𝝌^{2} test = ∑(Actual  Expected)^{2}/Expected
 𝝌^{2} has n  1 d.f. → compare to tabular value (higher = confirms risks are different)

Testing for shifting risk parameters  correlations test
 compute correlation between the results for all risks for pairs of years
 take average correlation with a given difference in time
 examine how the average correlation depends of this time difference
 correlation decreases over time, which wouldn't be the case if parameters were stable
 however it's high for small time difference, suggesting there’s value to using recent experience to predict the future.

Credibility weighting methods
 every risk is average: only use μ = 50%
 most recent years repeat: use 100% credibility on the latest year
 most recent year and μ: Z * most recent year, (1  Z) * μ_{}
 equal weight to most recent years: Z/N for the N most recent years, Z to μ
 exponential smoothing: apply Z to prior year actual, and (1  Z) to prior year estimate
 generalized method: apply given factors Zi to prior years, and what’s left to μ

Criteria to decide between solutions
 least square error: smaller the MSE, the better the solution
 limited fluctionation: Pr(act > k% different from est) = Pr(X_{est}  X_{act}/X_{est}> k%) < P
 Meyers/Dorweiler: calculate correlation between actual/predicted, and predicted/overall actual mean; want correlation as close to zero as possible using Kendall 𝝉; this would confirm that there is no evidence that large predictions lead to large errors and small predictions lead to small errors

Meyers/Dorweiler vs Other Criteria
 least square & limited fluctuations both attempt to eliminate large errors
 Meyers/Dorweiler is concerned with the pattern of the errors. Large errors are not a problem, as long as there is no pattern relating errors to experience rating modifications
 Most actuaries would lean towards least square & limited fluctuations

Conclusion
 when there are shifting risk parameters, older years are less relevant in predicting the future
 not having the most recent year of historical data significantly increase the SSE of estimate

