Here the confidence interval is transferred from the scale of the GLMestimators ˆμito the scale of the linear components, through the link functiong.If in addition the link function satisfies the conditiong(c μi) =g(μi) +cprime,for anyμi, wherecandcprimeare constants with respect toμi, then we can saymore.Proposition 7.2.PbraceleftBig|ˆμi-μi| ≤rμibracerightBig=PbraceleftBigc1≤Yprimeiˆβ-Yprimeiβ≤c2bracerightBig,(7.6)wherec1andc2are known constants, if and only if the link functiong(x) =cln(x) +τ, wherecis a scale- andτis a shift–parameter.
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74CHAPTER 7.CREDIBILITY FOR GLM’S*Proof:see Garrido and Zhou (2009).squareExample 7.2.Xi∼Poisson number of claims of independent risks fori=1, . . . , n.HereE(Xi) =μi=eyi0β0+···+yi,p-1βi,p-1, with the log–link functiongbracketleftbigE(Xi)bracketrightbig=g(μi) =yi0β0+· · ·+yi,p-1βi,p-1,hence, by Proposition 7.1,Pbraceleftbig|ˆμi-μi| ≤rμibracerightbig=Pbraceleftbigln(1-r)≤Yprimeiˆβ-Yprimeiβ≤ln(1 +r)bracerightbig≤Pbraceleftbigln(1-r)≤Yprimeiˆβ-Yprimeiβ≤ -ln(1-r)bracerightbig≤Pbraceleftbig|Yprimeiˆβ-Yprimeiβ| ≤ |ln(1-r)|bracerightbig.Lets2=V(ˆβ0+· · ·+ˆβp-1) andYi= (1,1, . . . ,1), thenPbraceleftBig|Yprimeiˆβ-Yprimeiβ| ≤ |ln(1-r)|bracerightBig=PbraceleftBigvextendsinglevextendsingle(ˆβ0+· · ·+ˆβp-1)-(β0+· · ·+βp-1)svextendsinglevextendsingle≤|ln(1-r)|sbracerightBig.Approximating by a normal distribution yields|ln(1-r)|s≥Zπ*, whereZπ*istheπ*= 100[1-(1-π2)]-percentile (two–sided) of a standard normal distri-bution. Hence the following asymptoticfull credibility standardis obtained:s2≤bracketleftbiggln(1-r)Zπ*bracketrightbigg2=s2*,which says that the sample sizenmust be sufficiently large to ensure thatthe variance of the sum of the estimatorsˆβ0, . . . ,ˆβp-1be at mosts2*.Forinstance, ifr= 0.1 andπ= 90% thens2*= 0.00410. This result is consistentwith the result given by Schmitter (2004, p.258).triangleAsymptotic results also hold for the linear term.Proposition 7.3.LetΣ= (σij)i,j= (YprimeWY)-1φands2i=V(Yprimeiˆβ), thens2i→YprimeiΣYi,asn→ ∞,(7.7)consistently forYi,WandY.
7.3.CREDIBILITY FOR GLMS75Corollary 7.1.Yprimeiˆβ-Yprimeiβsiconverges toN(0,1) in distribution with the itera-tive algorithm.Approximations can be given for log and general link functions.
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