Lecture 4- Credibility 3

Lecture 4- Credibility 3 - Credibility III ActSCi 432/832 1...

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Credibility III ActSCi 432/832 1 / 20

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Linear Exponential Family - A quick review Deﬁnition (LEF) X LEF if the density is given by f ( x ; θ ) = p ( x ) e r ( θ ) x q ( θ ) r ( θ ) is called the canonical parameter. It turns out that the LEF class contains well-known distns and we can compute its mean and variance in a uniﬁed framework: Theorem (Mean and variance of LEF) If X LEF, 1 E ( X | θ ) = μ ( θ ) = q 0 ( θ ) r 0 ( θ ) q ( θ ) 2 Var ( X | θ ) = v ( θ ) = μ 0 ( θ ) r 0 ( θ ) 2 / 20
Deﬁnition (Conjugate prior) A prior distn is a conjugate prior for a given model if the posterior is of the same type as the prior (but perhaps diﬀerent parameters) Theorem (Conjugate prior and the LEF) Suppose that given Θ = θ the rv X 1 , . . . , X n are iid with pf f ( x j | θ ) = p ( x j ) e r ( θ ) x j q ( θ ) , and Θ has pdf π ( θ ) = [ q ( θ )] - k e r ( θ ) μ k r 0 ( θ ) c ( μ, k ) , where k and μ are parameters and c ( μ, k ) is the normalizing const. Then the posterior π ( θ | X ) is of the same type as π ( θ ) . 3 / 20

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Exact credibility for LEF For some LEF models the B-S credibility premium and the Bayesian premium are identical For the linear exponential family (LEF) class, let us assume: E ( X j ) , Var ( X j ) , and Cov ( X i , X j ) are all ﬁnite The support of π ( θ ) is explicitly deﬁned as θ 0 < θ < θ 1 , where -∞ ≤ θ 0 < θ 1 ≤ ∞ Following two equality holds (typically so as numerators are zeros) π ( θ 0 ) r 0 ( θ 0 ) = π ( θ 1 ) r 0 ( θ 1 ) and π ( θ 0 | x ) r 0 ( θ 0 ) = π ( θ 1 | x ) r 0 ( θ 1 ) 4 / 20
1. E [ E ( X j | θ )] = E [ μ ( θ )] = μ , demonstrating that the symbol μ was chosen deliberately inside the prior. 2. π ( θ | x ) = [ q ( θ )] - k * e r ( θ ) μ * k * r 0 ( θ ) c ( μ * , k * ) where k * = k + n and μ * = μ k + n ¯ x k + n 3. The Bayesian premium is E ( X n +1 | x ) = μ * = μ k + n ¯ x k + n Since this Bayesian premium is a linear function of the data we see that the credibility is exact. 5 / 20

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This note was uploaded on 02/08/2012 for the course ACTSC 432 taught by Professor Davidlandriault during the Fall '09 term at Waterloo.

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Lecture 4- Credibility 3 - Credibility III ActSCi 432/832 1...

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