When there are shifting param over time, older years of data should be given substantially less cred than more recent years. There may be only a minimal gain in efficiency from using add'l yrs of data.
Testing for shifting parameters over time
Chi-square: |actual - exp|^{2}/exp, df = n - 1
Interpretation: x% chance diff obs from same distribution
Avg Correlation btwn pair of yrs
Interpretation: look if evolves or not over time
Simple methods to derive new estimate
all weight in mean (Z = 0)
all weight in latest data (Z = 1)
X = ZY_{1} + (1 - Z)Y_{2}
X = (Z/M)ΣY_{i} + (1-Z)M
Note: A simplification of a general method is equal or inferior
General methods to derive new estimate
goal is to reflect the fact that most recent yrs have more value
least square error: smaller the better; max = 75% of prev
small chance of large error: minimize prob
Meyers/Dorweiler: correlation btwn (actual/exp) and (pred/avg) → look for evidence that there's no pattern suggesting that larger predictions lead to larger errors
First 2 methods minimize large errors; 3rd concerned w pattern of error
Variance of Data
between variance: var between risks
within variace = process var excl shift param + due to s.p.
Mahler's approach: baseball vs insurance
constant set of risks over the period
data is accurate & final
same amount of exposure
grand mean is fixed
Credibility of delayed data
Mahler shows that using delayed data substantially increases the squared arror btwn pred & obs. In particular, incr btwn 1 & 2 yrs of separation is dramatic.