13_AS_6_lec_a

Censoring assume now that all lives are not observed

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Unformatted text preview: ule 6: Parametric models: Binomial and Poisson models Binomial model With censoring Assume now that All lives are not observed over the complete year x to x + 1, that is, there may be other decrements other than death (right censoring); and also the possibility of Increments (left censoring) Hence, we observe a life from age x + ai to x + bi (0 ≤ ai &lt; bi ≤ 1) For each life we know di , ai and bi Consider the random variable Di = 0 if the i th life survives from age x + ai to x + bi , and 1 if the i th life dies so that Pr [Di = 1] = bi −ai qx +ai Pr [Di = 0] = 1 −bi −ai qx +ai 6/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model With censoring Maximum Likelihood Estimation Assuming independence, the likelihood of the total sample will be N (bi −ai qx +ai )di (1 −bi −ai qx +ai )1−di L q; d = i =1 where q = (b1 −a1 qx +a1 ,b2 −a2 qx +a2 , · · · ,bN −aN qx +aN ) d = (d1 , d2 , · · · , dN ). Note: unless we are using a (continuous) parametric model for the probabilites of death, there are potentially as many probabilities of death to estimate than data points another possibility is to reformulate all as a function of qx using assumptions (0 ≤ t ≤ 1): 7/30 uniform distribution of deaths: t qx = t · qx , or Balducci assumption: 1−t qx +t = (1 − t ) · qx , or constant force of mortality: t qx = 1 − e −µt Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson model...
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This document was uploaded on 04/03/2014.

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