STAT3801_2011Unit2a[1]

# STAT3801_2011Unit2a[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 2a 2010 - 11 2 nd semester Multiple life functions Dependent lifetime models Insurance and annuity beneﬁts Simple contingent functions 2.10 Common shock Z, T * ( x ) and T * ( y ) are independent s T * ( x ) T * ( y ) ( t 1 , t 2 ) = Pr [ T * ( x ) > t 1 T * ( y ) > t 2 ] = s T * ( x ) ( t 1 ) s T * ( y ) ( t 2 ) s Z ( z ) = e - λz , z > 0 , λ 0 T ( x ) = min [ T * ( x ) , Z ] and T ( y ) = min [ T * ( y ) , Z ] Joint survival function of [ T ( x ) , T ( y )] s T ( x ) T ( y ) ( t 1 , t 2 ) = Pr { min [ T * ( x ) , Z ] > t 1 min [ T * ( y ) , Z ] > t 2 } = Pr { [ T * ( x ) > t 1 Z > t 1 ] [ T * ( y ) > t 2 Z > t 2 ] } = Pr [ T * ( x ) > t 1 T * ( y ) > t 2 Z > max ( t 1 , t 2 )] = s T * ( x ) ( t 1 ) s T * ( y ) ( t 2 ) e - λ [ max ( t 1 ,t 2 )]

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2 Domain of common shock probability density function 6 - ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 0 t 1 t 2 t 1 < t 2 t 1 = t 2 t 1 > t 2 Joint probability density function of [ T ( x ) , T ( y )] s T ( x ) T ( y ) ( t 1 , t 2 ) = Z t 2 Z t 1 f T ( x ) T ( y ) ( u, v ) du dv f T ( x ) T ( y ) ( t 1 , t 2 ) = 2 ∂t 1 ∂t 2 s T * ( x ) ( t 1 ) s T * ( y ) ( t 2 ) e - λ [ max ( t 1 ,t 2 )] For 0 < t 2 < t 1 f T ( x ) T ( y ) ( t 1 , t 2 ) = 2 ∂t 1 ∂t 2 s T * ( x ) ( t 1 ) s T * ( y ) ( t 2 ) e - λt 1 = d dt 1 s T * ( x ) ( t 1 ) s 0 T * ( y ) ( t 2 ) e - λt 1 = [ s 0 T * ( x ) ( t 1 ) - λ s T * ( x ) ( t 1 )] s 0 T * ( y ) ( t 2 ) e - λt 1
3 Similarly for 0 < t 1 < t 2 f T ( x ) T ( y ) ( t 1 , t 2 ) = [ s 0 T * ( y ) ( t 2 ) - λ s T * ( y ) ( t 2 )] s 0 T * ( x ) ( t 1 ) e - λt 2 Common shock contribution to probability density function when t 1 = t 2 = t f T ( x ) T ( y ) ( t, t ) = λ e - λt s T * ( x ) ( t ) s T * ( y ) ( t ) t 0 Proof Let s x ( t 1 ) = s T * ( x ) ( t 1 ) , s y ( t 2 ) = s T * ( y ) ( t 2 ) By diﬀerentiation of products and integration - s x ( u ) s y ( u ) e - λu = Z u [ s 0 x ( t 1 ) s y ( t 1 ) + s x ( t 1 ) s 0 y ( t 1 ) - λs x ( t 1 ) s y ( t 1 )] e - λt 1 dt 1 - s x ( u ) e - λu = Z u [ s 0 x ( t 1 ) - λ s x ( t 1 )] e - λt 1 dt 1 - s y ( u ) e - λu = Z u [ s 0 y ( t 2 ) - λ s y ( t 2 )] e - λt 2 dt 2 Consider survival function Pr [ T * ( x ) > u T * ( y ) > u Z > u T ( x ) = T ( y )] = s x ( u ) s y ( u ) e - λu - Z u Z t 1 u [ s 0 x ( t 1 ) - λ s x ( t 1 )] s 0 y ( t 2 ) e - λt 1 dt 2 dt 1 - Z u Z t 2 u [ s 0 y ( t 2 ) - λ s y ( t 2 )] s 0 x ( t 1 ) e - λt 2 dt 1 dt 2 = s x ( u ) s y ( u ) e - λu - Z u [ s 0 x ( t 1 ) - λ s x ( t 1 )][ s y ( t 1 ) - s y ( u )] e - λt 1 dt 1 - Z u [ s 0 y ( t 2 ) - λ s y ( t 2 )][ s x ( t 2 ) - s x ( u )] e - λt 2 dt 2 By expanding terms = s x ( u ) s y ( u ) e - λu - Z u [ s 0 x ( t 1 ) s y ( t 1 ) - λ s x ( t 1 ) s y ( t 1 )] e - λt 1 dt 1 + s y ( u ) Z u [ s

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STAT3801_2011Unit2a[1] - THE UNIVERSITY OF HONG KONG...

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