Here the confidence interval is transferred from the scale of the GLM

Here the confidence interval is transferred from the

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Here the confidence interval is transferred from the scale of the GLM estimators ˆ μ i to the scale of the linear components, through the link function g . If in addition the link function satisfies the condition g ( c μ i ) = g ( μ i ) + c prime , for any μ i , where c and c prime are constants with respect to μ i , then we can say more. Proposition 7.2. P braceleftBig | ˆ μ i - μ i | ≤ i bracerightBig = P braceleftBig c 1 Y prime i ˆ β - Y prime i β c 2 bracerightBig , (7.6) where c 1 and c 2 are known constants, if and only if the link function g ( x ) = c ln( x ) + τ , where c is a scale- and τ is a shift–parameter.
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74 CHAPTER 7. CREDIBILITY FOR GLM’S *Proof: see Garrido and Zhou (2009). square Example 7.2. X i Poisson number of claims of independent risks for i = 1 , . . . , n . Here E ( X i ) = μ i = e y i 0 β 0 + ··· + y i,p - 1 β i,p - 1 , with the log–link function g bracketleftbig E ( X i ) bracketrightbig = g ( μ i ) = y i 0 β 0 + · · · + y i,p - 1 β i,p - 1 , hence, by Proposition 7.1, P braceleftbig | ˆ μ i - μ i | ≤ i bracerightbig = P braceleftbig ln(1 - r ) Y prime i ˆ β - Y prime i β ln(1 + r ) bracerightbig P braceleftbig ln(1 - r ) Y prime i ˆ β - Y prime i β ≤ - ln(1 - r ) bracerightbig P braceleftbig | Y prime i ˆ β - Y prime i β | ≤ | ln(1 - r ) | bracerightbig . Let s 2 = V ( ˆ β 0 + · · · + ˆ β p - 1 ) and Y i = (1 , 1 , . . . , 1), then P braceleftBig | Y prime i ˆ β - Y prime i β | ≤ | ln(1 - r ) | bracerightBig = P braceleftBig vextendsingle vextendsingle ( ˆ β 0 + · · · + ˆ β p - 1 ) - ( β 0 + · · · + β p - 1 ) s vextendsingle vextendsingle | ln(1 - r ) | s bracerightBig . Approximating by a normal distribution yields | ln(1 - r ) | s Z π * , where Z π * is the π * = 100[1 - ( 1 - π 2 )]-percentile (two–sided) of a standard normal distri- bution. Hence the following asymptotic full credibility standard is obtained: s 2 bracketleftbigg ln(1 - r ) Z π * bracketrightbigg 2 = s 2 * , which says that the sample size n must be sufficiently large to ensure that the variance of the sum of the estimators ˆ β 0 , . . . , ˆ β p - 1 be at most s 2 * . For instance, if r = 0 . 1 and π = 90% then s 2 * = 0 . 00410. This result is consistent with the result given by Schmitter (2004, p.258). triangle Asymptotic results also hold for the linear term. Proposition 7.3. Let Σ = ( σ ij ) i,j = ( Y prime WY ) - 1 φ and s 2 i = V ( Y prime i ˆ β ), then s 2 i Y prime i Σ Y i , as n → ∞ , (7.7) consistently for Y i , W and Y .
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7.3. CREDIBILITY FOR GLMS 75 Corollary 7.1. Y prime i ˆ β - Y prime i β s i converges to N (0 , 1) in distribution with the itera- tive algorithm. Approximations can be given for log and general link functions.
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