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STAT3801_2011Unit4a[1]

STAT3801_2011Unit4a[1] - THE UNIVERSITY OF HONG KONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3801 ADVANCED LIFE CONTINGENCIES Unit 4a 2010 - 11 2 nd semester 4.9 Associated single decrement model For each decrement in multiple-decrement table a single decrement model can be defined operation of that decrement independent of others Could be presented in a single decrement table Represent lives affected by only one decrement Rare to have a group subject to one decrement The concept may appear unrealistic But is useful both in theory and practice Single decrement model associated with decrement j t p 0 ( j ) x = exp - Z t 0 μ 0 ( j ) x ( s ) ds p 0 ( j ) x = 0 ( j ) x +1 0 ( j ) x q 0 ( j ) x = 1 - p 0 ( j ) x t q 0 ( j ) x = 1 - t p 0 ( j ) x
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S&AS: STAT3801 ADVANCED LIFE CONTINGENCIES 2 q 0 ( j ) x is often referred to as an independent rate of decrement q ( j ) x is often referred to as a dependent probability of decrement Investigate relationship between given multiple decrement table and associated single tables Not possible to develop exact relationships Need to make certain assumptions Force of decrement is not affected by operation of other decrements present μ 0 ( j ) x = μ ( j ) x for each j t p ( τ ) x = exp " - Z t 0 r X j =1 μ ( j ) x ( t ) dt # = r Y j =1 exp - Z t 0 μ ( j ) x ( t ) dt = r Y j =1 t p 0 ( j ) x If some other type of decrement other than j is operating t p ( τ ) x < t p 0 ( j ) x t p ( τ ) x μ ( j ) x ( t ) < t p 0 ( j ) x μ ( j ) x ( t ) q ( j ) x = Z 1 0 t p ( τ ) x μ ( j ) x ( t ) dt < Z 1 0 t p 0 ( j ) x μ ( j ) x ( t ) dt = q 0 ( j ) x
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S&AS: STAT3801 ADVANCED LIFE CONTINGENCIES 3 Estimation of decrement rates Equality of central rates Constant force assumption Uniform distribution for multiple decrements Uniform distribution in single decrement models 4.10 Uniform distribution of decrements in multiple decrement model t q ( j ) x = t · q ( j ) x 0 t 1 t p ( τ ) x μ ( j ) x ( t ) = q ( j ) x 0 t 1 by differentiation. Summing over all values of j t q ( τ ) x = t · q ( τ ) x t p ( τ ) x = 1 - t · q ( τ ) x Under the given assumption μ ( j ) x ( t ) = q ( j ) x t p ( τ ) x = q ( j ) x 1 - t · q ( τ ) x
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S&AS: STAT3801 ADVANCED LIFE CONTINGENCIES 4 Hence s q 0 ( j ) x = 1 - exp - Z s 0 μ ( j ) x ( t ) dt = 1 - exp " - Z s 0 q ( j ) x 1 - t · q ( τ ) x dt # = 1 - exp ( q ( j ) x q ( τ ) x log h 1 - t q ( τ ) x i )fl fl fl fl fl t = s t =0 = 1 - exp ( q ( j ) x q ( τ ) x log h 1 - s q ( τ ) x i ) Result: q ( j ) x UDD Multiple Decr.
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