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

• 9

This preview shows page 3 - 6 out of 9 pages.

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.

Subscribe to view the full document.

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 .
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).

Subscribe to view the full document.

• Fall '09
• Dr.D.Dryanov

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes