A.02.Mahler 1

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A.02.Mahler 1
2012-08-13 21:13:44
credibility least square meyers dorweiler shifting parameters

An example of credibility and shifting risk parameters
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  1. 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.
  2. 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
  3. Simple methods to derive new estimate
    • all weight in mean (Z = 0)
    • all weight in latest data (Z = 1)
    • X = ZY1 + (1 - Z)Y2
    • X = (Z/M)ΣYi + (1-Z)M
    • Note: A simplification of a general method is equal or inferior
  4. 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
  5. 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
  6. Variance of Data
    • between variance: var between risks
    • within variace = process var excl shift param + due to s.p.
  7. 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
  8. 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.