13_AS_6_lec_a

# n d actuarial statistics module 6 parametric models

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , 1, 2, . . . N . d 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 d N −d ∂ ln L (q ) = − . ∂q q 1−q First order conditions then yield q= d N Noting ∂2 d (N − d ) ln L (q ) |q= d = − 2 − <0 N ∂ q2 q (1 − q )2 4/30 conﬁrms that q is an MLE. Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Without censoring Properties of maximum likelihood estimator: unbiased E [q ] = q Var [q ] = eﬃcient) q (1−q ) N (minimum variance of all estimators - − asymptotically q ∼ Normal q , q(1N q) 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 q = ˆ 45 27000 5/30 ± 1.96 45 27000 so q (1−q ) ˆ ˆ N that the conﬁdence interval is 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 Actuarial Statistics – Mod...
View Full Document

## This document was uploaded on 04/03/2014.

Ask a homework question - tutors are online