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Unformatted text preview: 8: Method of Graduation
Statistical tests
Grouping of Signs Test Grouping of Signs test (Stevens’ test) IV For large m (say ≥ 20) , using a normal approximation
G ∼ Normal n1 (n2 + 1) (n1 n2 )2
,
n1 + n2 (n1 + n2 )3 The test can lead to diﬀerent conclusions depending on
whether positive or negative groups are considered. 43/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Grouping of Signs Test Example We refer to the data on slide 37 an perform the grouping of signs
test:
The observed value of the statistic G = 4 (no. of groups of
‘+’ve sign observed)
The critical value (with n1 = 6 and n2 = 4) from Table is 1.
Finally, 4 > 1, so we cannot reject the null hypothesis. 44/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Serial Correlations 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
45/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Serial Correlations Test Serial Correlations Test I
Tests for overgraduation
Detects excessive agglomeration of deviations of the same sign
Rationale
For a good graduation, deviations should be more or less
independent at consecutive ages
Overgraduation: the graduated curve will tend to stay the
same side of the crude rates for relatively long periods and we
would expect the values of consecutive deviations to have
similar values. This will result in positive correlation of
deviations
Undergraduation: the graduated curve will cross the crude
rates quite frequently and the values of consecutive deviations
tend to oscillate, ie they will be negatively correlated.
45/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Serial Correlations Test Serial Correlations Test II
Null hypothesis: the individual standardised deviations at
consecutive ages are independent.
Under the null hypothesis, for any j ,
the sequence z1 , z2 , · · · , zm−j and the j th lagged sequence
zj +1 , zj +2 , · · · , zm are independent
Deﬁne the serial correlation coeﬃcients by
rj = m −j
i =1
m−j
i =1 where
z1 =
46/72 1
m−j (zi − z 1 ) (zi +j − z 2 ) (zi − z...
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