Unformatted text preview: function α + β x , which was ﬁtted using the method of least
squares.
2
o
o
qx
ˆ
qx
Ex q x = E (A−E )
x
Ex
dx = A
E
30 70,000
39
0.000557 0.000460
27.09
5.24
0.000645 0.000428
28.54
7.33
31 66,672
43
32 68,375
34
0.000497 0.000473
32.34
0.09
33 65,420
31
0.000474 0.000523
34.21
0.30
···
···
···
···
···
49 61,110
184
0.003011 0.002600
158.89
3.97 22/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Chisquare (χ2 ) test χ2 test:
The test statistic is
X= (A − E )2
= 5.24 + 7.33 + · · · + 3.97 = 43.30
E The graduated rates are calculated by estimating two
parameters α and β and there are 20 age groups. So the
degrees of freedom are 20 − 2 = 18.
The upper 95% point for the χ2 distribution is 28.88
18
The observed value of the test statistic (43.30) exceeds 28, 87.
Hence, we reject the null hypothesis, which indicates that the
mortality experience does not conform to a formula of the type
assumed in the graduation.
23/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Chisquare (χ2 ) test Discussion The χ2 test
is a good test for overall goodness of ﬁt
but it tells us nothing about the direction of any bias
there could be a few large deviations oﬀset by a lot of very
small deviations
there could be groups of all positive or all negative deviations
could be consistently biased up or down with small deviations 24/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Standardised Deviations Test 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
25/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Standardised Deviations Test Standardised deviations test
This test essentially checks for normality of the zx ’s:
Null hypothesis: the observed pattern of the individual
standardised deviations (ie the numbers falling in each
interval) is consistent with a standard normal distribut...
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 Three '14
 Statistics, Normal Distribution, Statistical tests, actuarial statistics, Parametric Formula

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