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Graduation by Reference to a Standard Table 1 Introduction
2 Testing smoothness
3 Statistical tests
Preliminaries
Chisquare (χ2 ) test
Standardised Deviations Test
Signs test
Cumulative Deviations Test
Grouping of Signs Test
Serial Correlations Test
4 Methods of graduation
Preliminaries
Graduation by Parametric Formula
Graduation by Reference to a Standard Table
Graphical graduation
Statistical Tests
The Eﬀect of Duplicate policies
58/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Graduation by Reference to a Standard Table Graduation by Reference to a Standard Table Rationale: if the class of lives involved in the graduation is
suﬃciently similar to the class of lives whose experience formed the
basis of a particular standard table, the underlying mortality of
both class should be similar
(Overall levels of mortality may diﬀer between the two, the overall
progression of rates from age to age should be similar).
When we do not have much data, a standard table is useful 58/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Graduation by Reference to a Standard Table Graduation by Reference to a Standard Table I
Procedure
1
2 Select an appropriate standard table
Decide a relationship between standard table and graduated
o
o
s
rates q x = f (qx ) or µx = f (µs )
x
For example,
o s
q x = a + bqx o s
q x = (a + bx ) qx o qx
= (a + bx )
s
qx o µx = µs + k
x To decide a suitable function f (), we can use some simple
plots. For example,
s
a plot of qx against qx
s
a plot of ln(− ln (1 − qx )) against ln(− ln (1 − qx ))
59/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Graduation by Reference to a Standard Table Graduation by Reference to a Standard Table II 3 Determine parameter values
maximum likelihood
least squares:
minimize o o x wx (ˆx − µx )2 or
µ x wx (ˆx − q x )2
q with the weight wx being the inverse of the estimated variance
for µx or qx .
˜
˜
4
5 60/72 Determine the graduated rates from the ﬁtted relationship
Test the graduation (goodnessofﬁt) Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Graduation by Reference to a Standard Table Example
Suppose you wish to graduate a set of mortality rates µ by
reference to a standard table µs using the relationship
o
µx = aµs + b . The parameters are to be determined using
x
weighted least squares. In general, we need to minimise:
o...
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