JIA 119 (1992) 69-85
J.I.A. 119, I, 69-85
JOINT
MODELLING
FOR ACTUARIAL
GRADUATION
DUPLICATE
POLICIES
AND
BY A. E. RENSHAW, B.Sc., Ph.D.
(of The City University, London)
ABSTRACT
In this paper itis demonstrated how recently developed statistical techniques designedto 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
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 4.
2. 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
Joint Modelling for Actuarial
random variable. Further, adapting
(1949), it is possible to write:
Graduation
the argument
and Duplicate
presented
Policies
71
in Beard & Perks
(2.1)
where
random
for policyholders
variables defined by:
k, are independent
and identically
distributed
(2.2)
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:
Trivially,
it follows from equation
so that these expressions,
together
(2.2) that:
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)
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 φxs. 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 φxs. 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
Variance Ratios Against Age
Assured lives 1970-82 experience
Figure 2.2 (b)
Variance Ratios Against Exposed to Risk
Assured lives 1979-82 experience
Exposed to Risk
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 φxs. 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,
or as over-dispersed
Poisson
µx-graduations,
so that:
Var(Ax)
variables
= φ xRxqx (1-qx)
with central
Current graduation
practice, as comprehensively
(1988), is to use the Gompertz-Makeham
predictor:
exposures
described
Rx in the case of
by Forfar
et al.
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
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
φx = 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
x)
for the φ
xs basedon:
dx = dx(Ax;
either the Pearson
squared
residual,
where:
dx = dx(Ax; mx) =
or the deviance
squared
residual,
(Ax – mx)2
V(mx)
(3.2)
where:
(3.3)
Here, in each expression,
V(e) is the variance
graduation
modelling process, namely:
V(mx) = mx(l –
function
of the first stage
mx/Rx) for qx-graduations,
and:
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
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
E(Ax) = mx,
Var(Ax)
variables
with:
= φ xV(mx)
and predictor-link
(2) Model the unknown
dispersion
dispersion statistic dx with:
parameters
φ x using
an appropriate
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
with the log-likelihood
based
Poisson residuals. The properties
discussed in papers by Carroll
1988) and Breslow (1990).
When d, is based on deviance
(3.2). The reader will immediately associate this
on assumed independent
normally distributed
and applications of the pseudo-likelihood
Pare
& Ruppert (1982), Davidian & Carroll (1987,
squared
residuals,
the estimation
of both sets of
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:
(3.6)
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:
(3.7)
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
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 1967-70 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 A1967-70 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.
Joint Modelling for Actuarial
X
X
Graduation
and Duplicate
X
Policies
79
10.5
qx
0.0017047
4x
1.000
37.5
9x
0.0011041
4,
1.392
64.5
e(
0.0232648
4x
1.779
11.5
0.0016203
1.000
38.5
0.0012298
1.426
65.5
0.0256827
1.774
66.5
0.0283249
1.768
1.761
12.5
0.0015377
1.000
39.5
0.0013772
1.460
13.5
0.0014572
1.000
40.5
0.0015484
1.492
67.5
0.0312099
14.5
0.0013790
1.000
41.5
0.0017456
1.523
68.5
0.0343571
1.753
15.5
0.0013032
1.001
42.5
0.0019713
1.552
69.5
0.0377878
1.743
16.5
0.0012302
1.001
43.5
0.0022280
1.580
70.5
0.0415240
1.731
17.5
0.0011601
1.002
44.5
0.0025187
1.605
71.5
0.0455893
1.718
18.5
0.0010933
1.003
45.5
0.0028466
1.629
72.5
0.0500085
1.704
73.5
0.0548076
1.688
74.5
0.0600139
1.669
19.5
0.0010301
1.006
46.5
0.0032150
1.651
20.5
0.0009708
1.009
47.5
0.0036278
1.670
21.5
0.0009157
1.013
48.5
0.0040891
1.688
75.5
0.0656556
1.650
22.5
0.0008653
1.019
49.5
0.0046033
1.705
76.5
0.0717621
1.628
23.5
0.0008201
1.027
50.5
0.0051753
1.719
77.5
0.0783634
1.604
24.5
0.0007805
I .037
51.5
0.0058104
1.732
78.5
0.0854902
1.678
25.5
0.0007470
1.050
52.5
0.0065145
1.743
79.5
0.0931737
1.551
26.5
0.0007203
1.065
53.5
0.0072937
1.753
80.5
0.1014447
1.522
27.5
0.0007009
1.084
54.5
0.0081549
1.762
81.5
0.1103343
1.491
28.5
0.0006896
1.105
55.5
0.0091056
1.769
82.5
0.1198724
1.458
29.5
0.0006872
1.130
56.5
0.0101537
1.775
83.5
0.1300880
1.424
30.5
0.0006944
1.157
67.5
0.0113081
1.779
84.5
0.1410081
1.390
31.5
0.0007122
1.186
58.5
0.0125782
1.783
85.5
0.1526577
1.355
32.5
0.0007415
1.218
59.5
0.0139743
1.785
86.5
0.1650585
1.319
33.5
0.0007836
1.251
60.5
0.0155074
1.786
87.5
0.1782287
1.284
34.5
0.0008395
1.286
61.5
0.0171896
1.786
88.5
0.1921822
1.250
35.5
0.0009105
1.321
62.5
0.0190337
1.785
89.5
0.2069278
1.216
36.5
0.0009982
1.356
63.5
0.0210538
1.783
x- age, qx- graduated, ax- estimated over-dispersion
Table 4.1. Assured lives 1967- 70: Duration
5 and over
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 setequal 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 l979-82 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
Joint Modelling for Actuarial
Graduation
and Duplicate
Policies
81
Residuals
-
-I
15
30
45
60
Figure 4.1 (a)
Residuals Against Age. GM(2,2) predictor
Assured lives 1979-82 experience
canonical log-link and with offsets log (Rx). To set this in
actuarial graduation
practice, it is equivalent to the use
combination
with the exponentiated
predictor GMx(0,s)
(3.1) in which the polynomial
expression β jxj has been
spline expression defined immediately
above. To accord
usage, see for example, Chapter 16 of Benjamin & Pollard
use the following alternative form for the predictor:
75
90
Age (years)
the context of existing
of the identity link in
defined by equation
replaced by the cubic
with current actuarial
(1980), it is possible to
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.
82
x
Joint Modelling for Actuarial
Graduation
and Duplicate
Policies
øx
µx
0.0008091
Øx
1.870
x
64
µx
0.0168150
38
0.0008851
1.870
65
0.0186114
1.870
39
0.0009744
1.870
66
0.0206604
1.870
0.0010790
1.870
67
0.0229924
1.869
x
10
µx
0.0030744
øx
1.000
11
0.0027562
1.000
12
0.0024709
1.000
13
0.0022151
1.000
40
37
1.870
14
0.0019858
1.000
41
0.0012009
1.870
68
0.0256331
1.868
15
0.0017802
1.002
42
0.0013429
1.870
69
0.0286074
1.867
16
0.0015960
1.020
43
0.0015077
1.870
70
0.0319377
1.865
17
0.0014308
1.055
44
0.0016988
1.870
71
0.0356422
1.863
18
0.0012826
1.093
45
0.0019197
1.870
72
0.0397327
1.861
19
0.0011502
1.136
46
0.0021746
1.870
73
0.0442123
1.858
20
0.0010332
1.264
47
0.0024679
1.870
74
0.0490723
1.853
21
0.0009313
1.719
48
0.0028044
1.870
75
0.0542995
1.846
22
0.0008437
1.863
49
0.0031887
1.870
76
0.0590098
1.835
23
0.0007696
1.869
50
0.0036253
1.870
77
0.0659327
1.822
24
0.0007081
1.870
51
0.0041184
1.870
78
0.0724039
1.804
25
0.0006582
1.870
52
0.0046717
1.870
79
0.0793666
1.783
26
0.0006192
1.870
53
0.0052878
1.870
80
0.0868730
1.746
27
0.0005905
1.870
54
0.0059680
1.870
81
0.0949865
1.703
28
0.0005719
1.870
55
0.0067117
1.870
82
0.1037824
1.657
29
0.0005631
1.870
56
0.0075159
1.870
83
0.1133520
1.604
30
0.0005634
1.870
57
0.0083771
1.870
84
0.1237965
1.535
31
0.0005723
1.870
58
0.0092992
1.870
85
0.1352032
1.470
32
0.0005893
1.870
59
0.0102904
1.870
86
0.1476611
1.401
33
0.0006147
1.870
60
0.0113618
1.870
87
0.1612670
1.339
34
0.0006487
1.870
61
0.0125282
1.870
88
0.1761264
1.271
35
0.0006919
1.870
62
0.0138085
1.870
89
0.1923550
1.212
36
0.0007450
1.870
63
0.0152271
1.870
90
0.2100789
1.164
x- age, µx- graduated, x,- estimated over-dispersion
Table 4.2. Assured lives 1979-82:
Duration
5 and over
Joint Modelling for Actuarial
Graduation
and Duplicate
Policies
83
4 Residuals
3
2
1
0
-1
-2
-3
-4
16
30
46
30
76
90
Age (years)
Figure 4.1 (b)
Residuals Against Age. Cubic Spline Predictor
Assured lives 1979-82experience
The Assured Lives 1979-82 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 1967 –70 Experience, for the Assured Lives 1979-82,
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.
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 Effectof
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 QuasiLikelihood Models. J. Amer. Statist. Ass., 85, 565.
CARROLL,
R. J. & RUPPERT,
D. (1982). Robust Estimation of Heteroscedastic Linear Models. Ann.
Stat., 10, 1079.
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,
1079.
DAVIDIAN,
M.& CARROLL,
R. J. (1988).A Note on Extended Quasi-Likelihood. J.R. Statist. Soc. B,
50, 74.
DAW,R. H. (l946). Onthe Validity of Statistical Tests of the Graduation ofMortality
a
Table.J.I.A.,
72, 174.
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). GeneralizedLinear 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, 1589.
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.