CREDIBILITY, HYPOTHESIS TESTING AND REGRESSION SOFTWARE. Greg Taylor

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1 CREDIBILIY, HYPOHESIS ESING AND REGRESSION SOFWARE Greg aylor aylor Fry Consulting Actuaries Level 8, 3 Clarence Street Sydney NSW Australia Professorial Associate Centre for Actuarial Studies Faculty of Economics and Commerce University of Melbourne Parkville VIC 35 Australia Phone: Fax: greg.taylor@taylorfry.com.au February 7

2 Credibility, Hypothesis esting and Regression Software i able of Contents. Introduction.... Basic credibility framework Reducible credibility models Estimation of credibility parameter Numerical evaluation of credibility parameter Hierarchical credibility Conclusion...9 References...

3 Credibility, Hypothesis esting and Regression Software Summary. It has been known since Zehnwirth (977) that a scalar credibility coefficient is closely related to the F-statistic of an analysis of variance between and within risk clauses. he F-statistic may also be viewed as testing a certain regression structure, associated with credibility framework, against the null hypothesis of a simpler structure. his viewpoint is extended to multi-dimensional credibility frameworks in which the credibility coefficient is a matrix (Sections 3 and 4), and to hierarchical regression credibility frameworks (Section 6). In each case the credibility coefficient is expressed in terms of the F-statistic that tests the significance of a defined regression structure against a simpler one. Section 5 prints out how the computation may be implemented by means of regression software. Keywords. Credibility, hierarchical credibility, hypothesis testing, regression.. Introduction A credibility estimate is a linearised Bayes estimate, consisting of a convex combination of a prior quantity and a data-based estimate. he credibility coefficient (which may be a matrix) defining the convex combination also requires estimation from data. Historically, therefore, each new credibility application has tended to be accompanied by an additional analysis indicating how the credibility coefficient may be estimated (see eg Bühlmann and Straub, 97; De Vylder, 978, 985). hese analyses have usually been ad hoc. For complex credibility models, such as the hierarchical regression models considered in Section 6, the determination of the form of credibility coefficient is correspondingly complex, and possibly exceedingly tedious. he purpose of the present paper is to construct a defined procedure by which the estimation of a credibility coefficient may be automated. his is done for a general non-hierarchical credibility framework in Sections 3 and 4, and extended to a hierarchical (regression) framework in Section 6. Section 5 indicates how regression software may be used to carry out the estimation.

4 Credibility, Hypothesis esting and Regression Software. Basic credibility framework Consider the basic framework of regression credibility, as introduced by Hachemeister (975). Let β be a p-vector randomly drawn from a distribution with [ ], [ ] E β = b Var β =Γ (.) where Var [.] is used here to denote the variance-covariance matrix of its argument. Let Y be an observable n-vector satisfying Y = βε (.) with an n x p (n p) design matrix, assumed to be of full rank, and centred stochastic error vector, independent of β and with ε a Var [ ] ε = V (.3) he generalised linear regression estimate of β for model (.) and (.3) is ˆ β= V V Y (.4) Note that ˆ ( Var β = V ) (.5) Let Y % denote the following linear combination of b (prior estimate of Y ) and β ˆ (linear regression fitted value for Y ): ( ) ˆ Y% = Z b Z β (.6) where Z is an as yet unspecified n x n (non-stochastic) matrix. Define L to be the mean square loss function ( L= E Y Y% V Y Y% ) (.7) where the expectation is taken over the oint distribution of β and Y. Choose Z so as to minimise L.

5 Credibility, Hypothesis esting and Regression Software 3 his optimisation of Y % is in fact no less general than if Y % is defined as an affine function of Y (Hachemeister, 975; aylor 977). Substitute (.6) into Y Y % and rearrange to obtain = ( β ) ( ) ( β ) (ˆ β β) Y Y% Y Z b Z (.8) Substitute this into (.7) and make use of the fact that the three members of (.8) are stochastically independent, to obtain L = E Y β V Y β β b ( Z) V ( Z) β b ββ Z V Z ββ ( ˆ ) ( ˆ ) (.9) Note that, for any square (non-stochastic) matrix M and dimensionally compatible centred stochastic vector v, E vmv { [ ] } = rvarvm (.) Application of this result to each of the three members on the right side of (.9), with substitution of (.), (.3) and (.5) for the covariance matrices yields ( ) ( ) L r = Γ Z V Z V Z V Z (.) Note that, for any n x n matrices M, N, ( ) r MZN Z = NM (.) / i i where the subscript denotes the relevant element of the named matrix. Write / Z to denote the matrix of derivatives / Zi, so that (.) becomes r ( MZN )/ Z = ( NM ) (.3) Apply (.3) to (.) and set the derivative to zero so as to minimize L, giving Γ Z V Z = (.4) ransposition of this, followed by post-multiplication by V yields

6 Credibility, Hypothesis esting and Regression Software 4 ( ) Z Γ V = Γ V which is solved by Z = Γ V (.5) his is the classic result obtained by Hachemeister (975). he matrix Z will be referred to henceforth as the credibility matrix. 3. Decomposition of credibility models Consider the model described by (.) (.3), but now with and as and. Express Γ in the following block form: Γ Γ written Γ Γ = * Γ * (3.) where Γ is of dimension p x p. Partition in a dimensionally consistent manner: [ ] = (3.) where is of dimension nx p. No assumption is made about V other than the standard one of positive definiteness. he notation here is that, are the designs representing a regression null hypothesis and alternative hypothesis, and is the augmenting matrix connecting the two designs. Example 3. (Bühlmann-Straub). Let K u n = [ un ] = M O M K un r (3.3) where u n denotes the n-dimensional column vector with all entries unity and

7 Credibility, Hypothesis esting and Regression Software 5 r ni = n (3.4) i= where n is the dimension of the top left zero vector in. he interpretation of this model is as follows. If the r columns of are associated with r risk classes, then the model represents n i observations on risk class i, i,..., r, which is assumed characterised by parameter β for i =, = and by β β i otherwise. his is the data set-up of the model of Bühlmann and Straub (97). he model represented by the design matrix treats all risk classes as subect to the same parameter. Example 3. (Hachemeister regression). Consider the case in which K um tm um t m = M M = M M O um t m K um tm where tm is the time trend covariate vector [,,,m], n = rm, and dimension n x r. (3.5) is of Here the r parameters consist of r pairs and each pair may be regarded as defining a time trend for one of r risk classes. his corresponds to the model of Hachemeister (975), in which the risk classes were states. he model represented by the design matrix treats all states as subect to the same pair of risk parameters. Example 3.3. A further example may be constructed by merging a model of the Hachemeister type with the sort of econometric model found in the workers compensation literature. For example, Butler (994) found that statistical significant explanatory variables for real indemnity costs per employee included: wage replacement ratio; risky employment measure (proportion of workforce employed in manufacturing and construction); waiting period. One might therefore define a model in which decomposes further:

8 Credibility, Hypothesis esting and Regression Software 6 () () 3 ( H ) ( H ) = x x x = (3.6), where () () 3 x, x, x are the n-vectors of log (replacement ratio), log (risky employment measure) and waiting period respectively, and are the Hachemeister versions of and defined by (3.5). Moreover, one might choose ( H ) and ( H ) Γ =, Γ =Γ Γ ( H ) H (3.7) where the top left block of Γ as for the Hachemeister model. corresponds to () () 3 x, x, x and ( H ) ( H ) Γ Γ are, he credibility model (.6) then reduces to: 3 ( ) % () i () i ( H) ( H) ( H) ( H) (3.8) Y = β x Z b Z i= ˆ β where the terms with superscript (H) all relate to the Hachemeister portion of the model, ie the model with the x (i) components deleted. In this example, the regression coefficients associated with the first three regressors, which are not state-specific (the regressors themselves may be, but their coefficients are not), are given full credibility in (3.8). However, the state-specific trends comprising the full extension model are credibilityweighted. he distinction between the x (i) H and the is the distinction between fixed and random effects in the regression model (see eg Ohlsson and Johansson, 6). 4. Estimation of credibility parameter Application of the credibility formula (3.5) and (3.6) requires a knowledge, or estimate, of some properties of Γ e and V. Full estimation of these matrices is a substantial task. Hachemeister (975) shows how to estimate the former. he present paper will be concerned with the restricted case in which the structure of each of the two matrices is known, ie each is known up to a multiplier, V W, c, e G =σ Γ = Γ =τ e (4.)

9 Credibility, Hypothesis esting and Regression Software 7 with W,G e known and σ, τ unknown. Substitution of (4.) in (3.6) yields Z ( = νegeew ) (4.) with ν =τ / σ (4.3) and all other terms in (4.) known. An estimate of only the ratio (4.3), rather than of its separate components, is required for computation of Z. he ratio ν will be referred to as the credibility parameter in view of its central role. he following paragraphs address its estimation. hey require the following elementary results. Result 4.. For any conformable non-stochastic matrix A and centred stochastic vector x, E x Ax { [ ]} = r AVar x (4.4) Result 4. For any matrices A,B of dimensions m x n and n x m respectively ( r AB = r BA) (4.5) It follows that the trace of a matrix product (of any number of factors) is invariant under cyclic permutation of the matrix factors. Consider the following two regression models: Model : Y = β ε Model : Y = β ε subect to (.3) and (3.7). Generally in the following a subscript or will be used to indicate which of the models is under discussion. he fitted value of Y, denoted Y, i =,, is ˆi Y ˆi = PY (4.6) i with P = W W i i i i i (4.7) Define the residual sum of squares for Model as

10 Credibility, Hypothesis esting and Regression Software 8 ( ˆ ) ( ˆ) { ( ) ( σ r W P W P) } RSS = Y Y W Y Y = [using Results 4. and 4.] which, after some minor manipulation, yields [ ] σ ( E RSS = r P) (4.8) where the matrices on the right are of dimension n x n. By Result 4., (4.8) gives r P reduces to the trace of the p x p identity matrix, and so [ ] E RSS = n p σ (4.9) Now define the regression sum of squares reg ( ˆ ˆ ) ( ˆ ˆ ) SS = Y Y W Y Y = Y P P W P P Y (4.) Consider the matrix ( P P W P P ). It is elementary to check that PW P= W P, i=, (4.) i i i ( Now consider the cross terms in P P W P P. By (4.7), ) PW P W W = W W W (4.) Expand according to (3.7) within the square bracket: W W W W W W W W = = (4.3) Substitution of (4.3) in (4.) yields [ ] PW P = W W W W P = (4.4) It may also be noted that

11 Credibility, Hypothesis esting and Regression Software 9 [ by (4.4)] PW P= PW P = W P = W P [ by (4.7)] (4.5) Substitution of (4.), (4.4) and (4.5) in (4.) yields SSreg Y W P P Y ( = ) (4.6) and then, by Result 4., { ( ) [ ]} E SS = r W P P Var Y reg (4.7) o evaluate this quantity, write Y = b ( β b) ( Y β) and note that covariances between all three terms on the right are zero. herefore, by (4.), (.) and (.3), [ ] e Var Y = Γ V τ G = e σ W (4.8) where (4.) has been used again. By substitution of (4.8) into (4.7), { ( τ )} e σ E SS = r W P P G W reg (4.9) Now ( i ) r W PW = r (4.) p p i x i by (4.7) and Result 4., where i is of dimension n x p i. hus { σ } = ( ) r W P P W p p σ (4.) Note that, by Result 4., r W P P Ge = r W PP Ge (4.) o evaluate this, consider W P P = W W P (4.3)

12 Credibility, Hypothesis esting and Regression Software by (4.7). Now W P = W W W W = ( W ) W W W W W = W ( W )( W ) ( W ) (4.4) Substitute (4.4) into (4.3) and expand obtain W in a similar block form to W ( P P ) = Substitution of (4.5) in (4.) gives W ( P ) (4.5) r W P P Ge = r W P G (4.6) Combining (4.9), (4.) and (4.6) reg = σ ( ) τ (4.7) E SS p p t D where D is the regression design that recognises,, in addition to W and G, and = ( ) t D tr W P G (4.8) Combine (4.7) with (4.9) to obtain ( reg )/ ( ) E SS p p [ ]/ ( ) E RSS n p where ν was defined in (4.3). hus, ν is estimated by = ν ( p p ) t( D) (4.9) ( F ) ˆ ν = max, p p / t D (4.3) where F is the conventional regression F-statistic for testing Model against Model, ie

13 Credibility, Hypothesis esting and Regression Software ( F = SS reg / p p / RSS / n p ) (4.3) he credibility matrix Z in (4.) is thus estimated by replacing ν with νˆ. he dependence of Z on the F-statistic was demonstrated by Zehnwirth (977) in the simple case of -dimensional credibility. It may also be remarked that the estimator of Z derived here is different from Hachemeister s (975) estimate because he did not make the reducibility assumption (3.3) and assumed no prior knowledge of. He therefore estimated Γ in its entirety Γ rather than ust the scaling parameter in (4.). τ he regression whose statistics appear on the left side of (4.9) may have a number of equivalent designs. Example 3., for example, might have been formulated with taking any of the block diagonal forms (,...,,,..., n ns ns n ) r = diag u u u u (4.3) However, changing from one design D to another would not change the left side of (4.9), and so t(d) is invariant over D D, were D denotes the set of all regression designs equivalent to (and including) the one of interest. Example 3. (continued). Suppose that [ β] τ Var = W = diag w w with w = w.,,..., n Note that, since the parameter vector for module H is represented in the form ( β, β β,..., β β r ) instead of ust ( β,..., β r ), it is necessary for the n i= former vector to have covariance matrix τ. hat is [ ] = τ Var M β with i M = u r It may be checked that, in this representation, u r Var [ β] = τ M ( M ) = τ ur ur ur and so G = u u (4.33) r r

14 Credibility, Hypothesis esting and Regression Software Now W = u W u = w n n (4.34) P = w ununw (4.35) W unw u n = O (4.36) unw r ( r) u nr where is written in block diagonal form W = diag W W with W (),..., ( r) blocks corresponding to those in. hus (,..., ) W = diag w w r where w = r W, and so ( r ) ( () ) r W G = w... w = w w (4.37) Also ( n)( n) W P G = w W u W u G (4.38) and W un w = M (4.39) w ( r) herefore, by substitution of (4.39) in (4.38) and use of (4.33) and Result 4., ( ) = ( n) ( r r )( n) r W P G w r W u u u W u r r = w w w = = r = w w w ww = () () (4.4)

15 Credibility, Hypothesis esting and Regression Software 3 It then follows from (4.7) and (4.8) that r ( ) ( () ) () E SS reg = ( r ) σ τ w w ww() w w w w = r = ( r ) σ τ ww w (4.4) = Comparison with (4.7) indicates that r t D = ww w (4.4) = Note that t(d) is indeed independent of D, as predicted earlier. he estimator (4.3) now takes the explicit form r ˆ ν = max, ( F )( r )/ ww w (4.43) = with F = SSreg / r / RSS / n r (4.44) Here r = ( ) reg = SS w Y Y (4.45) where Y is the weighted mean of Y over the -th risk class with weight vector w and Y is the grand weighted mean. he estimator (4.43) is the same as that obtained by Bühlmann and Straub (97). It is emphasised, however, that, despite its algebraic development for illustrative purposes here, it could have been derived numerically (with no algebra) as described in Section 5 below. 5. Numerical evaluation of credibility parameter Suppose one is faced with the credibility regression design represented by (3.), (3.3) and (4.). One wishes to apply the credibility formulas (3.5) and (4.), and needs an estimate of ν in order to do so. One may proceed by means of the following steps.

16 Credibility, Hypothesis esting and Regression Software 4 5. Choose a regression design D and evaluate t(d) according to (4.8). his may be done algebraically or numerically. 5. Evaluate the F-statistic for testing the regression Model against the Model null hypothesis. his may be done by applying regression software to the data, or by direct calculation. he latter would amount to a re-creation of the regression software. 5.3 Assemble the result into the estimator ˆ ν given by (4.3), and hence an estimator of Z. Step 5. provides a quick and systematic way of evaluating the credibility matrix in cases of complicated design. An example is given in the next section. 6. Hierarchical credibility 6. Definition of hierarchical model A general hierarchical credibility model (aylor 979, Sundt 979, 98) can be extensive, and the procedure set out in Section 5 may be helpful in the evaluation of the various credibility matrices. he standard hierarchical credibility framework, as defined by aylor, is one in which risk classes consist of sub-classes, and sub-sub-classes, and so on. If a risk class is labelled, it will be composed of sub-classes,,, and generally. his will consist of sub-sub-classes 3, and ultimately q. he nodes ( k ), k =,,, q form a tree. A regression structure may be placed at each node. his sort of structure is studied in full generality by Sundt (979, 98) who sets up a framework involving an observable vector,..., Y = Y Y q, in which the q components represent the q levels of the hierarchy. Latent parameters θ,..., θq are associated with the different levels of the hierarchy. A slightly simplified version of Sundt s assumptions, sufficient for present purposes, is as follows: (i) Y and θk are conditionally independent given θ,..., θ if k >. (ii) ( Y Y ) and ( Y Yq ) are conditionally independent given θ,..., θ, =,, q. (iii) θ θ = β ( θ θ ) E Y,...,,...,, =,,... q (6.)

17 Credibility, Hypothesis esting and Regression Software 5 for non-stochastic matrix and vector function., (iv) E β θ,..., θ θ,..., θ = β θ,..., θ, =,..., q (6.) and E β ( θ) = β (6.3) Condition (i) is actually stronger than Sundt, who assumed only that Y and were unconditionally independent. Let β () θ k ( θ ) G = EVar β θ,..., θ θ,..., θ, =,..., q = Var β for = (6.4) where the expectation operator is tacitly assumed taken over all conditioning variables of its operand. Similarly, let W = EVar Y θ,..., θ, =,..., q (6.5) Define Y = Y,..., Y q, =,..., q (6.6) =,..., q,,..., = q (6.7) W = EVar Y θ,..., θ, =,..., q (6.8) N = EVar Y θ,..., θ,,..., q = = Var Y () for = (6.9) hen and N may be calculated recursively as follows (Sundt, 98): W ( ) W = W for = q W =, q, q,..., N = ( ) (6.) N = W G,,,..., = q q (6.)

18 Credibility, Hypothesis esting and Regression Software 6 Finally the credibility estimator of,...,,..., E β θ θ θ θ (ie least squares estimator, linear in Y), denoted as follows (Sundt, 98). Define based on data Y % β, is calculated recursively M = G W (6.) Z = M ( M ) (6.3) ˆ = N N Y β (6.4) which may be recognized as a generalised least squares regression estimator of E β ( θ,..., θ ) θ,..., θ. hen % β = Z % β Z ˆ β, =,,..., q (6.5) with % β = β. It is of interest to consider the special case in which the covariance matrices G and W are independent of θ,..., θ and are known up to scaling constants, and the relation between the different W also known: G = τ G, W = σ W (6.6) where Define G and W = W N = N W are known and and σ are to be estimated from data. τ / σ, / σ (6.7) It may be checked from (6.) and (6.) that W = W, = q W =, = q, q,..., N ( ) (6.8) N = W ν, G = q, q,..., (6.9) where

19 Credibility, Hypothesis esting and Regression Software 7 ν τ σ = / (6.) By (6.), (6.7) and (6.), M = ν G W (6.) and (6.3) then yields Z in a form parallel to (4.) with the ν now referred to as the credibility parameters in parallel with ν in (4.3). 6. Estimation of the credibility parameters he credibility parameters of the hierarchical regression model may be estimated by the approach established in Sections 4 and 5. the details of the application will depend on the parametric details of the model. An example follows. Begin with the regression (,..., ) Y = β θ θ ε (6.) where, by (6.), (6.8) and (6.6), E ε = (6.3) Example 6.. Consider the model defined in Section 6., and note that, for = fixed k, the required credibility parameter ν k arises from τ k and σ. If the former is set to zero, then (6.4) and (6.6) imply that Var βk θ,..., θk θ,..., θ K = (6.4) and so, by (6.), ( β θ,..., θ = β θ,..., θ (6.5) k k k k) By (6.), (6.), (6.6) and (6.7), E Y θ,..., θ = β θ,..., θ, =,,..., q (6.6) By (6.) with = k and (6.6) with = k, Yk θ,..., θk kβk θ,..., θk E Y θ,..., θ = k k βk ( θ,..., θ k k) k k βk ( θ,..., θk) = k ηk (6.7)

20 Credibility, Hypothesis esting and Regression Software 8 where = (,..., ) (,..., ) ηk βk θ θk βk θ θk (6.8) Let Model denote (6.7), and Model denote the same subect to (6.5), ie with ηk =. he situation is now parallel to Example 3., with k ( ), k = k = = ( k ) (6.9) hen ν = τ / σ is estimated by (4.3). he detail is as follows. First note k k that, by condition (i) of the hierarchical regression model, Y k- and conditionally independent given θ,..., θk, and it then follows that Yk θ,..., θk and Yk θ,..., θk are equal in distribution. hen (6.7) becomes θ k are (,..., ) Y k,..., k k β θ θ E θ θ k Y k = k k ηk (6.3) In the notation associated with (4.3), [ ] G = Var ηk θ,..., θk = Var βk θ,..., θk θ,..., θk (6.3) Yk W Var θ,..., Y θ = k ( k ) (6.3) Now consider the special case in which the covariance matrices in (6.3) and (6.3) do not depend on the conditioning parameters shown. hen they may be written as G EVar G = βk θ,..., θk θ,..., θk = k (6.33) Yk Wk W = EVar θ,..., θ Y k = W k ( k) (6.34) where the entries in the matrix are ustified as follows. he bottom right entry follows directly from (6.8). he top left follows from (6.5) and the fact that Y θ,..., θ and Y θ,..., θ are equal in distribution. he off-diagonal k k k zeros are ustified by observing that Y θ,..., θ and Y θ,..., θk are independent, by condition (ii) of the model. Also k k k θk θ,..., θk are independent, from condition (i). It follows that Yk θ,..., θk and Y θ,..., θk are independent. his is ust an application of the general ( k ) k Y θ,..., θ k k and

21 Credibility, Hypothesis esting and Regression Software 9 result that independence of U A and V A, and of U A and B A, implies independence of U AB and V AB, whose proof is left to the reader. he credibility parameter ν k is now estimated by ˆk ν given by (4.3). 7. Conclusion raditionally, credibility coefficients have been estimated by manipulation of squared error terms of some sort. he manipulations have been ad hoc, and each new credibility model has required a new exercise in estimation. his paper has developed an estimate of a credibility coefficient on the basis of an analysis of variance for the testing of one regression model against another. he credibility coefficient is expressed in terms of the F test statistic. Sections 4 and 6 give examples of the procedure s application. Some of these, especially Example 6., are complex, and the ad hoc algebra involved in manipulating squared error terms in order to arrive at estimators for the credibility coefficients would be laborious and possibly error-prone. he suggested procedure reduces this to an algorithm once null and alternative hypotheses have been formulated. Further, as pointed out in Section 5, the reduction of the estimation to hypothesis testing of regression models means that the required F-statistic may be computed by standard regression software.

22 Credibility, Hypothesis esting and Regression Software References Bühlmann H and Straub E (97). Glaubwürdigkeit für Schadensätze. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, 7. Butler RJ (994). Economic determinants of workers compensation trends. Journal of Risk and Insurance, 6, De Vylder F (978). Parameter estimation in credibility theory. Astin Bulletin,, 99. De Vylder F (985). Non-linear regression in credibility theory. Insurance: mathematics and economics, 4, Hachemeister CA (975). Credibility for regression with application to trend. Appears in Kahn (975). Kahn PM (975) (ed). Credibility: theory and applications. Academic Press, New York, NY. Ohlsson E and Johansson B (6). Exact credibility and weedie models. Astin Bulletin, 36, 33. Sundt B (979). A hierarchical credibility regression model. Scandinavian Actuarial Journal, 7 4. Sundt B (98). A multi-level hierarchical credibility regression model. Scandinavian Actuarial Journal, 5 3. aylor GC (977). Abstract credibility. Scandinavian Actuarial Journal, aylor GC (979). Credibility analysis of a general hierarchical model. Scandinavian Actuarial Journal,. Zehnwirth B (977). he credible distribution function is an admissible Bayes rule. Scandinavian Actuarial Journal, 7.