STAT3801_2011Unit4a[1]

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

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Unformatted text preview: 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 ( j ) x = exp •- Z t μ ( j ) x ( s ) ds ‚ p ( j ) x = ‘ ( j ) x +1 ‘ ( j ) x q ( j ) x = 1- p ( j ) x t q ( j ) x = 1- t p ( j ) x S&AS: STAT3801 ADVANCED LIFE CONTINGENCIES 2 q ( 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 ( j ) x = μ ( j ) x for each j t p ( τ ) x = exp "- Z t r X j =1 μ ( j ) x ( t ) dt # = r Y j =1 exp •- Z t μ ( j ) x ( t ) dt ‚ = r Y j =1 t p ( j ) x If some other type of decrement other than j is operating t p ( τ ) x < t p ( j ) x t p ( τ ) x μ ( j ) x ( t ) < t p ( j ) x μ ( j ) x ( t ) q ( j ) x = Z 1 t p ( τ ) x μ ( j ) x ( t ) dt < Z 1 t p ( j ) x μ ( j ) x ( t ) dt = q ( j ) x 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 ≤ t ≤ 1 t p ( τ ) x μ ( j ) x ( t ) = q ( j ) x ≤ 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 S&AS: STAT3801 ADVANCED LIFE CONTINGENCIES 4 Hence s q ( j ) x = 1- exp •- Z s μ ( j ) x ( t ) dt ‚ = 1- exp "- Z s q ( j ) x 1- t · q ( τ ) x dt # = 1- exp ( q...
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This note was uploaded on 07/30/2011 for the course STAT 3801 taught by Professor Kc during the Fall '11 term at HKU.

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

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