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Mahler alternative method of estimating R(L)
- use direct calculation at low limits
- use a fitted curve at higher limits
- mean residual life (MRL): $ expected XS of L given claim > L
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Steps to combine R(L) data
- truncate at d
- Fy(x) = 0 if x ≤ d
- Fy(x) = Fx(x) - Fx(d) / (1 - Fx(d)) otherwise
- shift to zero: Fw(x) = Fy(x+d)
- normalize to achieve mean unity
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Mahler's mixture
- mixture of Pareto A(x;s,b) & exponential B(x;c)
- Pareto: cdf = 1 - (1 + x/b)-s
- Exponential: cdf = 1 - e-x/c; R(r) = e-r
- F(x;s,b,c) = pA(x) + (1-p)B(x)
- RF(L) = [pmARA(L) + (1-p)mBRB(L)\ / [pmA - (1-p)mB]
- (+) R(L) closer fit (influenced by 4 parameters)
- (+) for mid-lvl values, steady drop off similar to exponential, and pareto keeps R(L) from deteriorating too quickly at high values
- R(L), L>d = (base using data) * R(hat)(L - d)
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Truncation point
- low enough to have enough data to fit reasonable curve
- high enough that it provides a reasonable ballast to the base factor
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