13_AS_6_lec_a - Actuarial Statistics Module 6 Parametric...

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Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Actuarial Statistics Benjamin Avanzi c University of New South Wales (2013) School of Risk and Actuarial Studies [email protected] Module 6: Parametric models: Binomial and Poisson models 1/30
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Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models 1 Binomial model Without censoring With censoring The actuarial estimate Central exposed to risk Discussion 2 Poisson model Introduction Maximum Likelihood Estimation Link to 2-state Markov models 3 Discussion: Binomial, Poisson and Markov multi-state models 2/30
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Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Without censoring 1 Binomial model Without censoring With censoring The actuarial estimate Central exposed to risk Discussion 2 Poisson model Introduction Maximum Likelihood Estimation Link to 2-state Markov models 3 Discussion: Binomial, Poisson and Markov multi-state models 3/30
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Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Without censoring Binomial model Assume we observe N independent lives at exactly aged x at the beginning of the year, for one whole year we observe the number of lives who die d each life has probability of death of q x over that year(initial rate of mortality) Then the random variable D , number of deaths, is D Binomial ( N , q x ) that is, Pr [ D = d ] = N d q d (1 - q ) N - d d = 0 , 1 , 2 , . . . N . 3/30
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Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Without censoring Maximum Likelihood estimation Under the Binomial model, the loglikelihood is ln L ( q ) = ln N d + d ln q + N - d ln (1 - q ) so that q ln L ( q ) = d q - N - d 1 - q . First order conditions then yield e q = d N Noting 2 q 2 ln L ( q ) | q = d N = - d q 2 - ( N - d ) (1 - q ) 2 < 0 confirms that e q is an MLE. 4/30
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Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Without censoring Properties of maximum likelihood estimator: unbiased E [ e q ] = q Var [ e q ] = q (1 - q ) N (minimum variance of all estimators - efficient) asymptotically e q Normal q , q (1 - q ) N Numerical example: Consider a mortality investigation over 1 year. The population at the beginning of the year numbered 27000 and there were 45 deaths. Estimate q , and also provide a 95% confidence interval. We have ˆ q = 45 27000 so that the confidence interval is 45 27000 ± 1 . 96 q ˆ q (1 - ˆ q ) N 5/30
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Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model With censoring 1 Binomial model Without censoring With censoring The actuarial estimate Central exposed to risk Discussion 2 Poisson model Introduction Maximum Likelihood Estimation Link to 2-state Markov models 3 Discussion: Binomial, Poisson and Markov multi-state models 6/30
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Actuarial Statistics – Module 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
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