Unformatted text preview: wx (ˆx − µx )2 =
µ S=
x wx [ˆx − (aµs + b )]2
µ
x
x The weights should be proportional toc the reciprocal of the
variance of the crude rates, ie wx = Ex .
µx
ˆ
So the function to be minimised is
S (a, b ) =
x
61/72 c
Ex
[ˆx − (aµs + b )]2
µ
x
µx
ˆ Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Graphical graduation 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
62/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Graphical graduation Graphical Graduation I
Procedure
1 plot the crude rates or plot a function (e.g. log) of the crude
rates
group diﬀerent ages together if data is scarce
choose groups so that the grouped crude rates run more
smoothly
use evenlyspaced groups, eg 5year age bands
group so that at least 5 deaths are in each group
determine average age for the group (weighted versus
unweighted)
2 62/72 plot conﬁdence intervals (eg 95% CI intervals) around the
plotted crude rates. Rough 95% CI intervals are given by
√
√
dx
dx
qx ± 2
or µx ± 2 c
Ex
Ex Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Graphical graduation Graphical Graduation II
3 draw a curve through the crude rates, make sure that
only about 1 point in 20 deviates outside the CI interval curves
the curve is as smooth as possible
the curve captures the overall shape of the crude rates 4 5 Test the graduated rates for goodnessofﬁt and smoothness 6 Redraw if the graduated rates lack smoothness or adherence
to data 7 63/72 determine graduated rates at integer ages from the plotted
curve (avoid over accuracy) Once adequate adherence and a reasonable smooth
progression have been obtained, handpolish the graduated
rates  manually alter the rates to make them smoother steady progression of values Actuarial Statistics – Module 8: M...
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