c x φ 1 2 x 2 σ 2 ln2 πσ 2 α ln αx ln x ln Γ α ln x ln m m x Link g identity

C x φ 1 2 x 2 σ 2 ln2 πσ 2 α ln αx ln x ln γ

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c ( x, φ ) - 1 2 [ x 2 σ 2 + ln(2 πσ 2 )] α ln αx + ln x - ln Γ( α ) - ln( x !) ln ( m m x ) Link g identity reciprocal log logit 7.2 Estimation For an observed random sample x 1 , . . . , x n , the maximum likelihood estimator (MLE) ˆ β is the solution of: n summationdisplay i =1 ( x i - μ i ) φ V ( μ i ) 1 g prime ( μ i ) Y prime i = 0 . (7.4) In the normal model, (7.4) reduces to the usual normal equation : n summationdisplay i =1 Y i ( x i - Y prime i β ) = 0 . In other EDF cases, usually there is no closed form solution to (7.4), but iterative algorithms (Newton–Raphson, Fisher scoring method) provide numerical MLE’s. The MLE for the GLM parameters has some nice properties: 1. ˆ β is an asymptotically unbiased, consistent estimator of β . 2. V ( ˆ β ) Σ = - H - 1 , as n → ∞ . H = - Y prime W o Y is the Hessian matrix, while W o = diag ( w o 1 , . . . , w on ) is a diagonal weight matrix with i-th element w oi = w i φV ( μ i )( g prime ( μ i )) 2 + w i ( x i - μ i ) V ( μ i ) g primeprime ( μ i )+ V prime ( μ i ) g prime ( μ i ) ( V ( μ i )) 2 ( g prime ( μ i )) 3 φ , for known weights w i and matrix Y = ( Y 1 , . . . , Y n ) prime . 3. ˆ β D N ( ˆ β, Σ ) , hence there is convergence in distribution.
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72 CHAPTER 7. CREDIBILITY FOR GLM’S 7.3 Credibility for GLMs If the probability of a small difference between the estimator ˆ μ i and the parameter it estimates, say μ i , is large, then the insurer may find ˆ μ i credible. If this difference is small enough, we say that full credibility is achieved: P braceleftBig | ˆ μ i - μ i | ≤ i bracerightBig π i , (7.5) for a chosen estimation–error tolerance level 0 < r < 1 and confidence prob- ability π i . Equivalently, (7.5) can be expressed graphically as in the figure below. Figure 7.1: Full Credibility a45 x a63 (1 - r ) μ i μ i (1 + r ) μ i a63 ˆ μ i The problem is to translate the tolerance level from the scale of means μ i and ˆ μ i to that of η i = g ( μ i ), as in Figure 7.2. Proposition 7.1. For any GLM, let g be a monotonic increasing link func- tion, then π i = P braceleftBig | ˆ μ i - μ i | ≤ i bracerightBig = P braceleftbig (1 - r ) μ i ˆ μ i (1 + r ) μ i bracerightbig = P braceleftbig g [(1 - r ) μ i ] - g ( μ i ) g μ i ) - g ( μ i ) g [(1 + r ) μ i ] - g ( μ i ) bracerightbig = P braceleftBig g [(1 - r ) μ i ] - Y prime i β Y prime i ˆ β - Y prime i β g [(1 + r ) μ i ] - Y prime i β bracerightBig .
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7.3. CREDIBILITY FOR GLMS 73 Figure 7.2: Full Credibility for the Link Function a45 x a54 g ( x ) g ( x ) μ i (1 - r ) μ i (1 + r ) μ i ˆ μ i g [(1 + r ) μ i ] η i = g ( μ i ) g [(1 - r ) μ i ] ˆ η i = g ( ˆ μ i ) bracerightbig Q 2 bracerightBig Q 1 Similar results can be derived for decreasing link functions (left as an exercise).
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