A.02.Mahler 1

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.

• 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

• all weight in mean (Z = 0)
• all weight in latest data (Z = 1)
• X = ZY₁ + (1 - Z)Y₂
• X = (Z/M)ΣY_i + (1-Z)M
• Note: A simplification of a general method is equal or inferior

• goal is to reflect the fact that most recent yrs have more value
• exponential smoothing: X_j+1 = ZY_j + (1-Z)Y_j-1 + Z(1-Z)²Y_j-2 + ... + (1-Z)ⁿX₀
• fully general: X_i+1 = ΣZ_iX_i + (1-Z_i)M
• (+) subsumes all other methods
• (-) difficult to select optimal weights

• 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

• between variance: var between risks
• within variace = process var excl shift param + due to s.p.

• constant set of risks over the period
• data is accurate & final
• same amount of exposure
• grand mean is fixed

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.