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Unformatted text preview: the range of ages where is the bias is worst.
Weaknesses of the signs test: the test provides no indication
of the extent of the discrepancy. The test is qualitative
rather than quantitative.
31/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Signs test Procedure
calculate the statistic P : the number of zx that are positive
ﬁnd the value k ∗ the smallest k such that
k P (P ≤ k ) =
j =0 m
j 1
2 m ≥ 0.025, . If k ∗ ≤ P ≤ m − k ∗ , accept the Null hypothesis (at the 5%
level). Another way is to use the p value. (can be found on
Pages 186188 of the Tables if the sample size is 2, 3, · · · , 12
or 20)
Alternatively, use the Normal approximation for large m:
1
1
P ∼ N 2 m, 4 m
32/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Signs test Example A graduation covers 20 age groups and has resulted in 6 positive
and 14 negative deviations. We have then:
1
under the null hypothesis, P ∼ Bin(20, 2 ). the p value of the test is p = 2P (P ≤ 6) = 0.1154 > 5%
Hence, we cannot reject the null hypothesis at a 5% signiﬁcance
level. 33/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Cumulative 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
34/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Cumulative Deviations Test Cumulative Deviations Test Tests whether the overall no. of deaths conforms to the model
with the mortality rates assumed in the graduation.
We can apply the test to the whole age range or parts of the
age range
Conclusion: high value of test statistic indicates either
the graduated rates are biased (too low if the test statistic is
positive, too high if the test statistic is negative), or
actual variance is higher that predicted by the assumed model
for the range of ages considered 34/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Cumulative Deviations Test Multiplestate Markov model
The deviation has the approximate distribution
o o c
c
dx − Ex µx + 1 ∼ Normal 0, Ex µx + 1
2 2 The accumulated deviation has...
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