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Credibility vs time shifting parameters
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 = ZY1 + (1 - Z)Y2X = (Z/M)ΣYi + (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
- exponential smoothing: Xj+1 = ZYj + (1-Z)Yj-1 + Z(1-Z)2Yj-2 + ... + (1-Z)nX0
- fully general: Xi+1 = ΣZiXi + (1-Zi)M
- (+) subsumes all other methods
- (-) difficult to select optimal weights
Criterias to assess quality of Z
- 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.
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