JOINT MODELLING FOR ACTUARIAL GRADUATION AND DUPLICATE POLICIES. BY A. E. RENSHAW, B.Sc., Ph.D. (of The City University, London)

Transcription

1 J.I.A. 119, I, JOINT MODELLING FOR ACTUARIAL GRADUATION AND DUPLICATE POLICIES BY A. E. RENSHAW, B.Sc., Ph.D. (of The City University, London) ABSTRACT In this paper it is demonstrated how recently developed statistical techniques designed to facilitate the joint modelling of the mean and dispersion are well suited to model the presence of duplicate policies in graduation. KEYWORDS Graduation; Duplicate Policies; Models; Life Tables 1. INTRODUCTION The purpose of this paper is to suggest how the possible effects of duplicate policies can be jointly modelled as part of the wider graduation modelling process. It is well known that graduations for the mortality of assured lives are based on the number of policies and not on the number of lives. Consequently, the death of a policyholder carrying more than one policy will appear as more than one death in the raw data, thereby clouding the resulting graduations unless due allowance is made for this eventuality. Various aspects of the effects of such duplicate policies on the distribution of deaths and the implications for graduations have been investigated by a number of researchers, including Seal (1945), Daw(1946), Beard & Perks (1949), Daw (1951), C.M.I. Committee (l957, 1986), and Forfar, McCutcheon & Wilkie (1988). With the exception of Forfar et al. (1988), perhaps one interesting historical side commentary on these developments has been the emphasis placed on adjustments to goodness-of-fit tests for graduations rather than on marginal adjustments to the graduations themselves. The underlying message of these early developments, in particular that of Beard & Perks (1949), is to prescribe an over-dispersed modelling distribution for the number of recorded deaths (more accurately identified as the number of recorded claims). To allow for such over-dispersion during the construction of graduations, Forfar et al. (1988) have advanced a method whereby the data are first transformed before modelling. The method relies on the availability of estimates at each age of the degree of over-dispersion present in the data, viz. a knowledge of the so-called variance ratios. The aims of this paper are to describe and explore the feasibility of modelling for graduations with allowance for over-dispersion attributed to duplicate policies using two-stage generalised linear (or non-linear) models. One clear advantage of such a method accrues from its universality, rendering a detailed 69

2 70 Joint Modelling for Actuarial Graduation and Duplicate Policies collateral knowledge of the specific variance ratios unnecessary. In any event, since these are data based and liable, as a consequence, to contain irregularities, it seems only natural to the author that these, as well as the irregular empirical death rates, should be smoothed by the graduation modelling process. The methodology owes much to recent advances in the joint modelling of mean and dispersion motivated by a certain class of industrial quality improvement experiments, as described in a discussion paper by Nelder & Lee (1991). It has been noted by Renshaw (1990, 1991) that the graduation models currently used by British actuaries are either generalised linear or generalised non-linear models, so that the first stage of the envisaged joint modelling process is already in place. All that is needed is for this to be augmented by a second-stage dispersion modelling process along the lines first proposed by Pregibon (1984) and developed by Nelder & Pregibon (1987); while Chapter 10 of the monograph by McCullagh & Nelder (1989) is also highly relevant here. Further, it is logical that the second stage of the joint modelling process should be formulated in such a way as to reflect the known properties of the empirical variance ratios. Indeed, without such knowledge, the formulation of the predictor-link combination for the second stage generalised linear model would be somewhat speculative. In order to set the issues in context at the outset, it is important to stress that, as Cox (1983) has pointed out, the modelling of excess variation, brought about in this context by the presence of duplicate policies, has only a marginal effect on the estimation of the regression coefficients of primary concern giving rise to graduations, but that statistical tests and confidence intervals may be seriously in error unless its effect is taken into account. Indeed intuitively one would not expect the presence of duplicate policies in the data to have a major effect on any resulting graduation. The nature of the over-dispersion present in graduations due to duplicate policies is discussed in Section 2, and the joint modelling for graduations and duplicate policies described in Section 3. Applications are presented in Section DUPLICATE POLICIES AND OVER-DISPERSION It is necessary to examine the nature of the over-dispersion and, in particular, the empirical evidence for this, if a plausible second stage is to be established for the joint mean and dispersion modelling process. Focus on the graduation target, qx the probability that a policyholder aged x, dies before age (x+ 1). Suppose there are some duplicate policies present, and let A, denote the total number of policies giving rise to claims, or simply, the number of claim, from among the policies held by the Nx policyholders aged x, present at the beginning of a duration period of 1 year. Clearly, if there are no duplicates present, Ax and Nx are synonymous with the actual number of deaths and the initial exposure to risk of death, respectively, but not otherwise. In common with modern actuarial graduation practice, A, is modelled as a

3 Joint Modelling for Actuarial Graduation and Duplicate Policies 71 random variable. Further, adapting the argument presented in Beard & Perks (1949), it is possible to write: where for policyholders k, are independent and identically distributed random variables defined by: (2.1) representing the number of policies giving rise to claim per policyholder, that is, the number of claims per policyholder. Here px = 1 - qx and is the probability that a policyholder, aged x, holds i policies; i = 1, 2, 3,... with: (2.2) Trivially, it follows from equation (2.2) that: so that these expressions, together with equation (2.1), imply that: where: are the initial exposures based on policies; and: (2.3) Note, that when there are no duplicate policies present, so that otherwise; Øx =: 1 and A, has the binomial distribution Ax ~ Bin(Rx, qx). Obviously Rx = Nx also for this case. When there are duplicates present, however, so that n(i) x > 0 for at least one value of i = 2, 3, 4,...; equation (2.3) approximates to: (2.4)

4 72 Joint Modelling for Actuarial Graduation and Duplicate Policies since qx is small. Consequently A, has an over-dispersed binomial distribution. The approximation is a good one because of the relative smallness of qx for all but the very oldest ages x; while the C.M.I. Committee (1957) memorandum contains impressive empirical evidence in support of this. Use of approximation (2.4) is important in-so-far as it renders the over-dispersion parameter X independent of the primary target qx. Defining fx(i) to be the proportion of policyholders, aged x, who have i policies in any empirical study group, the so-called data based variance ratios : estimate the φ xs. Two empirical studies of the number of policies held by policyholders aged x, in which variance ratios have been computed, are available to shed light on a likely structure for the φ xs. Roth studies are based on data culled from the death certificates of policyholders whose deaths occurred during the specific year of investigation. The first of these, relating to duplicate policies held by assured lives in 1954, is reported in the C.M.I. Committee (1957) memorandum; while different aspects of the second study, relating to duplicate Figure 2.1 Variance Ratios Against Age (quinary groups) Various assured lives experiences

5 Variance Ratios Against Age Assured lives experience Figure 2.2 (b) Variance Ratios Against Exposed to Risk Assured lives experience Exposed to Risk

6 74 Joint Modelling for Actuarial Graduation and Duplicate Policies policies held by assured lives in 1981 and 1982, are reported by the C.M.I. Committee (1986) and in Section 17 of Forfar et al. (1988). The variance ratios from each study, based on quinquennial age groups, are plotted against age in Figure 2.1. For the former study, two sets of variance ratios are displayed; one based on data from all offices participating in the study and the other based on non-industrial offices reporting not less than 300 policy claims; the latter being interpreted as a realisation of an upper estimate for the φ xs. For the more recent study and in the interests of greater clarity, only variance ratios based on 1981 and 1982 combined are displayed. In any event, values for the 2 years taken separately lie fairly closely on either side of these plotted values. Also available from the more recent study are variance ratios for individual ages. These are plotted both against age and against exposure to risk in Figures 2.2(a) and (b). It is remarked that Figure 2.2(b) is substantially the same as Figure 2.2(a) folded in half, since the exposed to risk rises to a peak at age 36 and then falls away again. The patterns observed in all three figures form the basis for the choice of structure imparted to the dispersion modelling process introduced in the next section. 3. JOINT MODELLING FOR GRADUATIONS AND DUPLICATE POLICIES The first stage graduation modelling process is a generalised linear or nonlinear model in which the responses, Ax, are modelled either as over-dispersed binomial variables with initial exposures Rx in the case of qx-graduations, so that: E(Ax) = Rxqx, Var(Ax) = φ xrxqx (1-qx) or as over-dispersed Poisson variables with central exposures Rx in the case of µx-graduations, so that: Current graduation practice, as comprehensively described by Forfar et al. (1988), is to use the Gompertz-Makeham predictor: subject to the convention that r = 0 implies the exponentiated polynomial term only, and s = 0 implies the polynomial term only; in conjunction with either the odds-link: with inverse in the case of qx-graduations, or the identity link nx=µx for µx-graduations. Renshaw (1990, 1991) has described how this practice falls within the generalised linear and non-linear modelling framework and, in particular, how the scope of current actuarial graduation practice can be appreciably enhanced by the use of

7 Joint Modelling for Actuarial Graduation and Duplicate Policies 75 alternative and somewhat more conventional specific linkages of the general type x = g(qx) or x = g(µx), as the case may be. Here g is a monotonic differentiable function. Graduated values are computed using the inverse linkage and predictor combination once the parameters in the predictor (equation (3.1)) have been estimated. If duplicate policies are either non-existent or simply ignored, so that φ x = 1 for all ages X, the AxS are modelled as conventional independent binomial or Poisson variables, and the parameter estimates follow by maximum likelihood using the likelihood for the binomial or Poisson response variables as the case may be. The question now to be addressed is what to do when the effects of duplicate policies are to be incorporated into the graduation modelling process, so that the over-dispersion paramters, φ x 1, are not all equal to unity and are unknown. It is proposed to model the unknown dispersion paramters φ x a secondary interrelating generalised linear model in much the same way as the unknown parameters qx or µx in the mean, mx = E(Ax) = Rxqx or Rxµx+½, are modelled as the primary target. There are two possible candidates to act as dispersion statistics dx = dx(ax; x) for the φ xs based on: either the Pearson squared residual, where: dx = dx (Ax; mx) = or the deviance squared residual, where: (Ax mx)2 V(mx) (3.2) (3.3) Here, in each expression, V(e) is the variance function of the first stage graduation modelling process, namely: and: V(mx) = mx(l mx/rx) for qx-graduations, V (mx) = mx for µx-graduations. Expressions (3.2) and (3.3) are identical if the variance function is a constant, corresponding to a normal modelling distribution, but not otherwise. Since Var(Ax) = fxv(mx), it follows trivially from expression (3.2), that when evaluated at the true value of for the Pearson dispersion statistic; while for the deviance dispersion statistic. It is further necessary to specify the variance function VD( ) for the second stage generalised linear model so that with scale factor t. Note that when the Axs are normally distributed, dx has the distribution so that Var(dx)=2& and the gamma model would be chosen for the dxs with Vn(+,) = 4:. The choice of the predictorlink structure for the dispersion modelling generalised linear model is guided by

8 76 Joint Modelling for Actuarial Graduation and Duplicate Policies the known properties of the variance ratios presented graphically in Figures 2.1 and 2.2. As a consequence, it is proposed to use either a quadratic or possibly higher order polynomial age predictor or a straight line exposure-to-risk predictor in combination with a monotonic increasing differentiable link function h mapping the interval [1, 1+ k], K > 0 to the whole of the real line. Further, Figures 2.1 and 2.2 would seem to support that K = 1, which is assumed throughout with one exception. More precise detail is presented in Section 4. It is appropriate to summarise the foregoing model structure in outline as follows, before proceeeding to discuss how to fit both stages of the model: (1) Model the Axs as independent response variables with: and predictor-link E(Ax) = mx, Var(Ax) = φ xv(mx) (2) Model the unknown dispersion parameters φ x using an appropriate dispersion statistic dx with: and predictor-link Here it is assumed that the first stage predictor x as well as the second stage predictor lx is linear, although it is possible to relax this condition, as discussed previously. The u,,5 and v,,5 are known covariate components, while it is necessary to estimate the β js and γ js in order to fit the model. To do this an optimisation or estimating function is needed to play the role of the likelihood or log-likelihood function in the estimation of parameters for simpler, probably more familiar, model structures. There are two cases to consider depending on the specific form of d,. When d, is based on Poisson squared residuals, the estimation of both sets of regression parameters is based on the optimisation of the so-called pseudolikelihood (strictly the pseudo-log-likelihood) P defined by: (3.4) where d, is defined by equation (3.2). The reader will immediately associate this with the log-likelihood based on assumed independent normally distributed Poisson residuals. The properties and applications of the pseudo-likelihood Pare discussed in papers by Carroll & Ruppert (1982), Davidian & Carroll (1987, 1988) and Breslow (1990). When d, is based on deviance squared residuals, the estimation of both sets of

9 Joint Modeling for Actuarial Graduation and Duplicate Policies 77 regression parameters is based on the optimisation of the Nelder & Pregibon (1987) extended quasi-likelihood (strictly the extended quasi-log-likelihood) Q defined by: (3.5) where this time dx is defined by equation (3.3). The properties of the extended quasi-likelihood function are also discussed in McCullagh & Nelder (1989). The β is enter into expressions (3.4) and (3.5) via the mean mx, in the corresponding expressions for dx, viz. equations (3.2) and (3.3) respectively. Using the chain rule for first order partial derivatives, it follows that the optimum values for the β is satisfy the following system of equations: irrespective of which version of dx is selected. These are the standard Wedderburn quasi-likelihood estimation equations with weights 1/ φ x see Wedderburn (1974) or McCullagh & Nelder (1989), which may be solved using the GLIM software package when the weights 1/ φ x are known. It is instructive to tie up loose ends at this stage by noting that if the weights 1/ φ x are replaced by their estimates 1/rx based on the variance ratios of Section 2; then it is easy to see that the resulting parameter estimating equations (for a linear predictor) based on the optimisation of the likelihood (and also chi-square) expressions (6.2.23) and (6.3.17) of Forfar et al. (1988) are special cases of the quasi-likelihood estimation equations (3.6) and would, therefore, lead to the same graduations. Forfar et al. (1988) do not, however, appear to have adopted this method of allowing for duplicate policies but have opted instead to transform the data before using the conventional Poisson likelihood to obtain predictor estimates for µx-graduations in Section 17 of their paper. The γ is enter into expressions (3.4) and (3.5) via the over-dispersion parameters φ x. Again using the chain-rule for first-order partial derivatives, it follows this time that the optimum values for the γ is satisfy the following system of equations: where dx is defined by the appropriate expression (3.2) or (3.3). This time these are the Wedderburn quasi-likelihood estimation equations based on independent responses dx, with variance function VD( φ x) = φ &corresponding 2x to a second stage generalised linear model based on gamma response variates. The parameter estimating equations (3.6) and (3.7) point the way forward. The two stages of the modelling process are clearly inter-dependent in-so-far as fitted values are needed from the second-stage generalised linear model to provide estimates for the weights in order to fit the first-stage generalised linear (3.6) (3.7)

10 78 Joint Modelling for Actuarial Graduation and Duplicate Policies model; while fitted values are needed from the first-stage generalised linear model in order to compute realisations of the responses dx, for the second-stage generalised linear model. Thus, the full model structures are fitted by alternating between the two separate stages until convergence is achieved. An effective starting model to assume for this iterative process is that based on no duplicate policies, so that initially for all x. 4. APPLICATIONS To investigate the potential of the method, the joint modelling process outlined in Section 3 has been applied to a number of data sets including the Assured Lives Experience with Duration 5 and over, reported by the C.M.I. Committee in J.I.A. (1974). The model used by the C.M.I. Committee to produce the A qx-graduation reported as Table 4 on pages 160-l63 of C.M.I. Committee J.I.A. (1974) using the data for ages 10.5 years to 89.5 years comprises a generalised non-linear model with binomial responses, odds link and non-linear predictor GMx(2,2) defined by equation (3.1). Both the formal statistical tests applied by the C.M.I. Committee together with an examination of various residual plots, which are not reproduced here, indicate a highly satisfactory fit. Since, however, the raw data contain duplicate policies, it is desirable to make fine adjustments to this graduation through the introduction of an over-dispersed binomial modelling distribution and the estimation of its parameters through the introduction of an associated dispersion modelling stage. To proceed, it is necessary to declare the precise form for the dispersion link function h where Since this is designed to map the interval [1,2] to the whole of the real line, three choices of links immediately come to mind based on the appropriate translation of either the complementary log-log link, the logit link or the probit link; all of which map the interval [0,1] to the whole of the real line, while it is not too difficult to suggest other possibilities. The detail needed to implement these links using the GLIM computer software package is as follows: (1) The translated complementary log-log link with inverse: and derivative (2) The translated logit link with inverse: and derivative (3) The translated probit link with inverse: and derivative where Q) is the distribution function of the standard normal deviate.

11 Joint Modelling for Actuarial Graduation and Duplicate Policies 79 X qx x X 9x 4, X e( 4x I x- age, qx- graduated, ax- estimated over-dispersion Table 4.1. Assured lives : Duration 5 and over

12 80 Joint Modelling for Actuarial Graduation and Duplicate Policies Implementation of the full joint modelling of the mean and dispersion is possible using the GLIM computer software package and the interactive facility to program user defined models as GLIM macros. Notably, empirical studies based on this, and other data sets, indicate that both choices of dispersion statistics defined by equations (3.2) and (3.3) in combination with all three of the above link functions produce near identical results when applied to the-same data set. The resulting qxgraduation and dispersion parameter estimates φ x, are reproduced in Table 4.1. These involve the use of a two-stage modelling process comprising a primary graduation stage based on over-dispersed binomial responses, with odds link and non-linear predictor GMx(2,2); and a secondary dispersion stage based on the Pearson statistic defined by equation (3.2), the translated probit link and quadratic predictor It is instructive to compare these results with those contained in Table 4 of the CMI Committee (1974) presentation. There the variance ratios for quinquennial age groups are not based on the actual data, but quoted rather from the earlier CMI Committee (1957) study. One possible small reservation concerning the smooth nature of the over-dispersion parameter estimates at ages 60 and 65 and over the years between these ages is that, in reality, it is quite likely that multiple policies drop rather sharply as multiple endowment policies mature and policyholders are left with, perhaps, a single whole of life policy. Figure 2.2(a) gives evidence for this discontinuity. It is suggested that it may be possible to cater for this effect by experimenting with different second-stage predictor types, perhaps involving the use of splines. Convergence of the alternating meandispersion modelling stages was achieved to appreciable accuracy after 10 iterations; while the qx-graduation of Table 4 of the C.M.I. Committee (1974) has been verified using the GLIM macros with φ x set equal to 1 for all ages x, and the dispersion modelling stage suppressed. One further, perhaps not surprising feature revealed by these empirical studies, is the apparent failure of the joint modelling process to converge when the structure of the primary graduation modelling stage is known to be inadequate. One such case in point concerns the µx-graduation of the Assured Lives l Experience with Duration 5 and over, based on the generalised non-linear model with Poisson responses, identity link and non-linear predictor GMx(2,2); which is discussed at length in Section 17 of Forfar et al. (1988). The lack of fit, identified by Forfar et al. (1988) through the application of their standard battery of statistical tests, is graphically illustrated in Figure 4.1(a) by the developing cyclical pattern in the residuals with increasing age. To combat this, Forfar et al. (1988) resort to the higher order nonlinear predictor GMx(3,6) which would appear to be GLIM unfriendly. As a viable alternative, it is possible to resort to the natural cubic spline predictor: with n fixed knots {xj}, such that xj<xj+1 cases, in combination with the

13 Joint Modelling for Actuarial Graduation and Duplicate Policies Residuals I Figure 4.1 (a) Age (years) Residuals Against Age. GM(2,2) predictor Assured lives experience canonical log-link and with offsets log (Rx). To set this in the context of existing actuarial graduation practice, it is equivalent to the use of the identity link in combination with the exponentiated predictor GMx(0,s) defined by equation (3.1) in which the polynomial expression β jxj has been replaced by the cubic spline expression defined immediately above. To accord with current actuarial usage, see for example, Chapter 16 of Benjamin & Pollard (1980), it is possible to use the following alternative form for the predictor: where: Further, implementation of this predictor in the current actuarial graduation context of generalised liner models does not require the determination of specific weights, constructed either by reference to a similar standard mortality table as described in Benjamin & Pollard (1980) or by resorting to the McCutcheon (1981) iterative process.

14 82 Joint Modelling for Actuarial Graduation and Duplicate Policies x µx ø x x µx Øx x µx øx x- age, µx- graduated, x,- estimated over-dispersion Table 4.2. Assured lives : Duration 5 and over

15 4 Residuals Joint Modelling for Actuarial Graduation and Duplicate Policies Figure 4.1 (b) Residuals Against Age. Cubic Spline Predictor Assured lives experience 90 Age (years) The Assured Lives Experience with Duration 5 and over has been regraduated using a two-stage generalised linear model comprising a primary graduation stage based on over-dispersed Poisson responses A,, with canonical long-link, natural cubic spline predictor with offsets log(&); and a secondary dispersion stage based on the Pearson dispersion statistic dx, defined by equation (3.2), the scaled and translated probit link with inverse and straight-line predictor The resulting µx-graduation and dispersion parameter estimates, x, for eight approximately equally spaced knots l8, 28, 38, 47, 56, 65, 74, 83 with K = 0.87 corresponding to the maximum observed variance ratios for these data--are reproduced in Table 4.2. Convergence is achieved after six iterations. The goodness-of-fit of the primary graduation stage improves significantly as the number of knots is increased from seven to eight. The standard battery of statistical tests, including the run test and serial correlation tests at lags 1 to 3 inclusive all indicate a highly satisfactory fit, as may be judged from Figure 4.1 (b) in which residuals are once again plotted against age. Unlike the Assured Lives Experience, for the Assured Lives , the dispersion parameter estimates attain the upper bound set by the dispersion link; a feature which is non-attributable to the difference in predictors. It is conjectured that one possible explanation as to why this should occur may well lie in the quality of the fit achieved in the primary graduation stage.

16 84 Joint Modelling for Actuarial Graduation and Duplicate Policies 5. SUMMARY The probability based argument developed in Section 2, in which the effects of duplicate policies on the graduation process are modelled as over-dispersion parameters Øx, differs conceptually from the modelling of duplicate policies used hitherto by actuaries through their use of variance ratios rx, in the sense that variance ratios are here perceived as data-based estimates of the over-dispersion model parameters. One way to allow for the over-dispersion attributable to duplicate policies, that suggested by Forfar et al. (1988), is to transform the data by dividing both the actual numbers of deaths and the exposures to risk of death by the variance ratios before modelling; using either familiar binomial or Poisson models as the case may be. The approach described here differs inasmuch as the untransformed data are modelled directly, while the effects of overdispersion are built into the model structure. The joint modelling process then generates smoothed estimates for the over-dispersion parameters, different from the alternative irregular variance ratio estimates which are not always available, but nevertheless constructed in such a way as to preserve the essential patterns observed in variance ratios. These are then used as weights in the primary graduation modelling stage. The method has clear advantages when variance ratio estimates are unavailable for the data in question, while, as a viable alternative to the data transformation method suggested by Forfar et al. (1988) when variance ratio estimates are available, the variance ratios can be used as alternative weights and the untransformed data modelled directly. REFERENCES BEARD, R. E. & PERKS, W. (1949). The Relation between the Distribution of Sickness and the Effect of Duplicates on the Distribution of Deaths. J.I.A., 75, 75. BENJAMIN, B. & POLLARD, J. H. (1980). The Analysis of Mortality and other Actuarial Statistics. Heinemann, London. BRESLOW, N. (1990). Tests of Hypotheses in Overdispersed Poisson Regression and Other Quasi- Likelihood Models. J. Amer. Statist. Ass., 85, 565. CARROLL, R. J. & RUPPERT, D. (1982). Robust Estimation of Heteroscedastic Linear Models. Ann. Stat., 10, CMI COMMITTEE (1957). Continuous Investigation into the Mortality of Assured Lives, Memorandum on a Special Inquiry into the Distribution of Duplicate Policies. J.I.A., 83,34 and T.F.A., 24,94. CMI COMMITTEE: (1974). Considerations Affecting the Preparation of Standard Tables of Mortality. J.I.A., 101, 133. CMI COMMITTEE (1986). An Investigation into the Distribution of Policies per Life Assured in the Cause of Death Investigation Data. CMIR, 8,49. COX, D. R. (1983). Some Remarks on Over-Dispersion. Biometrika. 70, 269. DAVIDIAN, M. & CARROLL., R. J. (1987). Variance Function Estimation. J. Amer. Statist. Ass., 82, DAVIDIAN, M.& CARROLL, R. J. (1988). A Note on Extended Quasi-Likelihood. J.R. Statist. Soc. B, 50, 74. DAW, R. H. (l946). On the Validity of Statistical Tests of the Graduation of Mortality a Table. J.I.A., 72, 174.

17 Joint Modelling for Actuarial Graduation and Duplicate Policies 85 DAW, R. H. (1951). Duplicate Policies in Mortality Data. J.I.A., 77, 261. FORFAR, D. O., McCUTCHEON, J. J. & WILKIE, A. D. (1988). On Graduation by Mathematical Formula. J.I.A., 115, 1 and T.F.A., 41, 97. McCULLAGH, P. & NELDER, J. A. (1989). Generalized Linear Models. Chapman & Hall, London. McCUTCHEON, J. J. (1981). Some Remarks on Splines. T.F.A., 37, 421. NELDER, J. A. & LEE, Y. (1991). Generalized Linear Models for the Analysis of Taguchi-type Experiments. Applied Stoch. Mod. & Data Anal. 7, 107. NELDER, J. A. & PREGIBON, D. (1987). An Extended Quasi-Likelihood Function. Biometrika, 74,221. PREGIBON, D. (1984). Review of Generalized Linear Models. Ann. Statist. 12, RENSHAW, A. E. (1990). Graduation by Generalized Linear Modelling Techniques. XXII ASTIN Colloquium, Montreux, Switzerland. RENSHAW, A. E. (1991). Actuarial Graduation Practice and Generalised Linear and Non-Linear Models. J.I.A., 118, 295. SEAL, H. (1945). Tests of a Mortality Table Graduation. J.I.A., 71, 3. WEDDERBURN, R. W. M. (1974). Quasi-Likelihood Functions, Generalized Linear Models and the Gauss - Newton Method. Biometrika, 61, 439.