Simplified estimation of structure parameters in hierarchical ...

Mathematical StatisticsStockholm UniversitySimplified estimation of structure parametersin hierarchical credibilityEsbjörn OhlssonResearch Report 2005:5ISSN 0282-9150

Postal address:Mathematical StatisticsDept. of MathematicsStockholm UniversitySE-106 91 StockholmSwedenInternet:http://www.math.su.se/matstat

Mathematical StatisticsStockholm UniversityResearch Report 2005:5,http://www.math.su.se/matstatSimplified estimation of structure parameters inhierarchical credibilityEsbjörn Ohlsson ∗May 2005AbstractSome of the structure parameters in Jewell’s hierarchical credibilitymodel are commonly estimated by pseudo-estimators. Inthis paper we present alternative, unbiased estimators, similarto those of the Bühlmann-Straub model. The main advantage ofour estimators is that they are given on closed form, while thepseudo-estimators require iterative solution.KeywordsCredibility theory, Jewell’s hierarchical model, structure parameter,pseudo-estimator.∗ Mathematical Statistics, Stockholm University, and Länsförsäkringar insurance group.E-mail: esbjorn.ohlsson@math.su.se

2 1 INTRODUCTION1 IntroductionIn credibility models there are so called structure parameters that mustbe estimated before the calculation of the credibility estimators themselves.In Jewell’s hierarchical credibility model there is one overallmean parameter µ and three variance structure parameters, here calledσ 2 , a and b – see definitions in the next section. This paper is concernedwith the estimation of the variance structure parameters in the hierarchicalcredibility model.For the Bühlmann-Straub model, standard textbooks present unbiasedestimators of all structure parameters. For Jewell’s hierarchical credibilitymodel, there is a simple unbiased estimator of σ 2 , while for aand b only so called pseudo-estimators are provided, see, e.g., Goovaerts& Hoogstad (1987) or Dannenburg, Kaas & Goovaerts (1996).A pseudo-estimator is motivated by a formula a = E[f(X, a)] where fis a function and X is a random vector. The estimator is computed byiteratively solving the equation a = f(x, a), where x is an observationof X. As noted by Sundt (1987), the fact that a = E[f(X, a)] doesnot imply that the resulting estimator of a is unbiased.In the present paper we present alternative, unbiased estimators ofthe structure parameters a and b that are easier to apply than thepseudo-estimators, since they do not require iterative solution. In ourexperience, the new estimators and the pseudo-estimators give rathersimilar results. Hence, the main advantage of the new ones is their simplicityin application. There is also a (minor) pedagogical point in nothaving to introduce the concept of a pseudo-estimator in elementarytexts.

32 Jewell’s hierarchical credibility modelHere we present the classical hierarchical credibility model of Jewell.We call our observations Y jkt , where, in the terminology of Dannenburget al. (1996), j indicates a sector, k is called a cell and t is an exposureunit within the cell (j, k). In practise, (j, k, t) may of course representany hierarchical structure of insurance contracts – for instance, j mightbe a county, k a parish and t an individual insurance taken by a residentof that parish. In motor insurance, j might be a car brand, k a specificcar model and t an individual car.We introduce random effects U j for the sectors j = 1, . . . , J; and U jkfor the cells k = 1, . . . , K j . The basic model is thatE(Y jkt |U j , U jk ) = U jk (2.1)andby which alsoE(U jk |U j ) = U j (2.2)E(Y jkt |U j ) = U j (2.3)The overall expectation is called µ.µ = . E(U j ) = E(U jk ) = E(Y jkt ) (2.4)Note that U jk is the (conditional) mean of all observations for cell (j, k),while U j is the (conditional) mean of all observations in sector j and µis the mean of the entire population.Remark 1.Many texts, like Goovaerts & Hoogstad (1987), introduceabstract risk parameters Θ j and Θ jk and put µ(Θ j , Θ jk ).= E(Y jkt |Θ j , Θ jk ) and ν(Θ j ) . = E(µ(Θ j , Θ jk )|Θ j ) = E(Y jkt |Θ j ). However,since inference is only made on µ and ν, and not on the Θ:sthemselves, there is no loss in generality from using the random effectsnotation instead, where µ(Θ j , Θ jk ) is replaced by U jk and ν(Θ j )

5By Assumption 1(d), the U jk are identically distributed, and their commonexpected variance is the variance at the cell levela . = E[Var(U jk |U j )] (2.6)Our final structure parameter b is simply the sector varianceb . = Var[U j ] (2.7)which does not depend on j since the U jrandom variables.are identically distributedRemark 3.By the standard laws for computing variances by conditioningwe find the unconditional varianceVar(Y jkt ) = a + b + σ2w jktwhich exhibits the structure parameters σ 2 , a and b as variance components.We further introduce the credibility factorsz jk. =w jk·w jk· + σ 2 /a(2.8)q j. =z j·z j· + a/b(2.9)Here and in the future, the dot notation is used to indicate summation,as in w jk· = ∑ t w jkt . We give weighted means a superindex to indicatewhich weights are used, as inY wjk· =∑t w jktY jkt∑t w jkt(2.10)andY zwj·· =∑k z jkY wjk·∑k z jk(2.11)

6 2 JEWELL’S HIERARCHICAL CREDIBILITY MODELThe well-known (inhomogeneous) credibility estimator at the sectorlevel isÛ j = q j Y zwj·· + (1 − q j )µ (2.12)Further, the credibility estimator at the cell level isÛ jk = z jk Y wjk· + (1 − z jk )Ûj (2.13)See for example Dannenburg et al. (1996, Theorems 3.2.2 and 3.2.3)for a derivation of these estimators. The overall expectation µ may begiven by prior information or estimated by e.g.ˆµ = Y qzw··· =∑j q jY zwj··∑j q j(2.14)In order to apply the credibility estimators in practice, we must firstestimate the variance parameters σ 2 , a and b.2.1 Traditional estimators of variance parametersThe estimators in this section can be found in Goovaerts & Hoogstad(1987, p. 90) or Dannenburg et al. (1996, p. 54), to which we alsorefer for proofs. Firstly, an unbiased estimator of σ 2 is given byˆσ 2 =1∑ ∑j k (T jk − 1)K J∑ ∑ j T∑ jkj=1 k=1 t=1w jkt (Y jkt − Y wjk·) 2 (2.15)where J is the number of sectors, K j is the number of cells in sectorj, and T jk is the number of observations for cell (j, k). For a and b,pseudo-estimators are derived from the fact that⎡⎤Ka = E ⎣1J∑ ∑ j∑j (K z jk (Y wjk· − Y zwj·· ) 2 ⎦ (2.16)j − 1)j=1 k=1[]1J∑b = Eq j (Y zwj·· − Y qzw··· ) 2 (2.17)J − 1j=1Since z jk is a function of a and q j is a function of b, we can not simplydrop the expectation signs to get unbiased estimators. However, the

7equation resulting from omitting the expectation in (2.16) can be solvedfor a by iteration; the solution is called a pseudo-estimator of a. Thisvalue is then inserted into (2.17), the expectation sign is dropped, andthe resulting equation is iterated to give a pseudo-estimator of b.Note that (2.16) and (2.17) do not imply that the pseudo-estimatorsare unbiased.3 The new estimatorsHere we derive the alternative estimators of a and b that is the coreof this paper. The trick is to replace the z- and q-weighted means in(2.16) and (2.17) by means with weights that are known (or at leastalready estimated) at that stage in the calculations. For a, instead ofY zwj··we will use∑kY wwj·· =w jk·Y wjk·∑k w jk·and we suggest the following unbiased estimatorâ =∑j∑k w jk·(Y wjk· − Y wwj·· ) 2 − ˆσ ∑ 2 j (K j − 1)w··· − ∑ j (∑ k w2 jk·)/w j··(3.1)Remark 4. Suppose our hierarchical model had just one sector j,in which case our model was in fact non-hierarchical, i.e. we had theBühlmann-Straub model. We could then omit the index j and (3.1)would reduce to the standard unbiased estimator in the Bühlmann-Straub model, see e.g. Dannenburg et al. (1996), Theorem 2.3.1.Once we know σ 2 and a, we get z jk from (2.8), and thereby we cancompute Y zwj·· in (2.11). In place of Y qzw··· that appears in the pseudoestimatorof b we useY zzw··· =∑j z j·Y zwj··∑j z j·

8 3 THE NEW ESTIMATORSOur suggested unbiased estimator of b is now∑j ˆb z j·(Y zwj·· − Y zzw··· ) 2 − â(J − 1)=z·· − ∑ j z2 j·/z··(3.2)The estimators â and ˆb are simpler than the pseudo-estimators in thatthey do not require iteration. The unbiasedness is proved in Theorem3.1 below.Note that even though ˆb as it stands is unbiased, in practice we plugin â and ˆσ 2 into z jk and then ˆb is no longer strictly unbiased. A similarremark goes for the equation (2.17) that defines the pseudo-estimator– this equation is not strictly true when we plug in â and ˆσ 2 into z jk .This kind of problem is of course common in statistics, and will not bediscussed further here.Remark 5. Negative values. As with the corresponding estimatorin the Bühlmann-Straub model, the suggested estimators may take onnegative values. This problem is discussed for the Bühlmann-Straubmodel in Example 2.3.2 of Dannenburg et al. (1996). The conclusionis that we can not reject the hypothesis that the parameter is equalto zero, and so the corresponding level of random effects should beremoved from the model.In our experience, in situations when our â or ˆb is negative, the correspondingpseudo-estimator iterations converge (slowly) to zero. This isin accordance with the result for the Bühlmann-Straub model in Dubey& Gisler (1981, Theorem 2) which states that the pseudo-estimatorequation has one and only one positive solution if and only if theunbiased estimator gives a strictly positive value; if not, the pseudoestimatorconverges to zero.Remark 6.Multi-level models. We believe that similar estimators

9could be found for hierarchical models with three or more levels. Itremains to work out the details for these multi-level models, though.Theorem 3.1 (a) With â as in (3.1), we have E[â] = a.(b) With ˆb as in (3.2), we have E[ˆb] = b.Proof. For part (a), we first note thatVar(Y wjk·|U j ) = E[Var(Y wjk·|U j , U jk )|U j ] + Var(E[Y wjk·|U j , U jk ]|U j )By (2.1), (2.5) and (2.6)E[Var(Y wjk·|U j )] = E[Var(Y wjk·|U j , U jk )] + E[Var(U jk |U j )] = σ2w jk·+ aFurther, by (2.3),E[Yjk·|U wj ] = U j and E[Y wwj·· |U j ] = U j(3.3)This justifies the second equality in the following derivation, in whichwe further use the fact from Assumption 1(b) that {Y wjk·; k = 1, . . . , K j }are conditionally independent given U j .E [ (Y wjk· − Y wwj·· ) 2] = E= E [ Var ( Y wjk· − Y ww )]j·· |U j= E=[ [ (Y wE jk· − Y ww ) 2]]j·· |Uj[ (1 − w ) 2jk·Var(Ywjk·|U wj ) + ∑j··l≠k)(1 − w jk·w j··) 2 ( σ2w jk·+ a(= σ21 − w )jk·+ aw jk· w j··(+ ∑ l≠k1 − 2 w jk·w j··+ ∑ l(wjl·w j··) 2Var(Y wjl·|U j )(wjl·w j··) 2 ( σ2w jl·+ a(wjl·w j··) 2))]

10 3 THE NEW ESTIMATORSThis yields⎡K∑ jE ⎣k=1w jk·(Y w⎤jk· − Y wwj·· ) 2 ⎦= σ 2 (K j − 1) + a(w j·· − 2 ∑ kw 2 jk·w j··+ ∑ l)wjl·2w j··and thus⎡K J∑ ∑ jE ⎣j=1 k=1= σ 2 ∑ jw jk·(Y w⎤jk· − Y wwj·· ) 2 ⎦(K j − 1) + a(w··· − ∑ j∑k)wjk·2w j··which we solve for a. Since ˆσ 2 is unbiased for σ 2 we conclude that (3.1)gives an unbiased estimator of a.We turn to part (b) of the theorem and first note that by (3.3)E[Var(Y wjk·|U j )] = az jkand hence E[Var(Y zwj·· |U j )] = az j·,by the conditional independence assumption in Assumption 1(b). Now,since E(Y zwj·· |U j ) = U j ,Var(Y zwj·· ) = E[Var(Y zwj·· |U j )] + Var(E[Y zwj·· |U j ])= E[Var(Y zwj·· |U j )] + Var[U j ]= az j·+ bNote that since E[Y jkt ] = µ we have E[Y zwj·· ] = E[Y zzw··· ] = µ, whichjustifies the first equality below, where we further use the independence

11of Assumption 1(a),E [ (Y zw==j··− Y zzw··· ) 2] = Var [ Y zwj··(1 − z ) 2j·Var(Y zwj·· ) + ∑z··l≠j)(1 − z j·z··) 2 ( az j·+ b= a (1 − z )j·+ bz j· z··(+ ∑ l≠j1 − 2 z j·− Y zzw ]···(zl·) 2Var(Y zwl·· )z··( ) 2 (zl· a+ bz·· z l·z··+ ∑ l(zl·z··) 2))Finally we get the result[ J∑] (E z j·(Y zwj·· − Y zzw··· ) 2 = a(J − 1) + bj=1z·· −∑ )j z2 j·z··and since â is unbiased, this completes the proof of part (b) of thetheorem.4 Numerical exampleIn our applications of hierarchical credibility at Länsförsäkringar wehave found that the differences between the pseudo and the unbiasedestimators are usually rather small. For confidentiality reasons, we donot present the results here. Instead we use the artificial populationin Section 3.3 of Dannenburg et al. (1996) for illustration. In theirTable 3.1, the values of Y wjk· are given, but not the original data Y jktthemselves. This is, however, enough for our purposes, if we take thestated value ˆσ 2 = 15.89 as given and only estimate a and b. (Rememberthat our approach uses the same unbiased ˆσ 2 as Dannenburg et al.)The results are given in Table 4.1. Supposedly due to round-off errorsin the data, our pseudo-estimates differ slightly from those provided

12 4 NUMERICAL EXAMPLEby Dannenburg et al. In parenthesis we give the values when all theweights w jkt are set equal to one.True value Pseudo-est. Unbiased est.a 1.000 1.152 (1.093) 1.209 (1.093)b 25.000 25.309 (25.259) 25.300 (25.259)Table 4.1: Comparison of estimators for the data in Table 3.1 of Dannenburget al. (1996). (Equal weights example in parenthesis.)The difference between the estimators is negligible for b and less than5% for a. In this case, we know the true values since the populationis artificially generated. The pseudo-estimator of a is somewhat closerto the true value here, but this might well be due to chance. In thecase with equal weights and balanced design (all K j equal and all T jkequal) it is not hard to show that the pseudo-estimator equations haveexplicit solutions that are equal to our unbiased estimators – as verifiedhere by the numbers given in parenthesis.In their comparison of the unbiased and pseudo-estimator of the parametera in the Bühlmann-Straub model, Dubey & Gisler (1981) foundthat “neither of the two estimators is universally better than the other”(in terms of variance). Since the estimators considered in the presentpaper are closely related to the quoted estimators, one might conjecturethat the conclusion of Dubey & Gisler are valid in our hierarchicalcase, too.Conditional on this conjecture being true, our choice in the hierarchicalcase falls on the unbiased estimators for the sake of simplicity inapplication.

135 ReferencesDannenburg, D.R., Kaas, R. & Goovaerts, M.J. (1996): Practical actuarialcredibility models. IAE, Amsterdam.Dubey, A. & Gisler, A. (1981): On Parameter Estimators in Credibility.MVSVM, 81:2, 187-212.Goovaerts, M.J. & Hoogstad, W.J. (1987): Credibility theory. Surveyof Actuarial Studies, Nationale-Nederlanden N.V., Rotterdam.Sundt, B. (1987): Book review of Goovaerts & Hoogstad (1987). ASTINBulletin 17, No. 2.