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m−j zi
i =1 z2 = m −j
i =1 (zi +j − z 2 )2 1
m−j m −j z i +j
i =1 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Serial Correlations Test Serial Correlations Test III
For large m, z 1 and z 2 can be approximated by
1
z = m m zi and simplify:
i =1
rj ≈ 1
m −j m−j
i =1 (zi − z ) (zi +j
m
2
1
i =1 (zi − z )
m − z) , which is listed on Page 34 of the Tables
1
rj ∼ Normal 0, m under the hypothesis of no correlation
√
Use test statistic (the T ratio) tj = rj m ∼ Normal (0, 1) Large positive test statistic indicates overgraduation 47/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Serial Correlations Test Serial Correlations Test IV Weakness: In the serial correlation test, it is possible for
correlations in one part of the age range to be canceled out by
opposite correlations in another part. The signs test and
grouping of signs test are usually more powerful in detecting
overgardduation. 48/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Serial Correlations Test Example Consider the deviations on slide 28. We have
1
1
z = 20 49 zx = 20 (2.29 + · · · + 1.99) = 0.19
¯
x =30
49
1
¯2
x =30 (zx − z ) = 2.13
20
49−1
1
¯
¯
x =30 (zx − z )(zx +1 − z ) = 0.94
20−1
The serial correlation coeﬃcient with lag 1: r1 = 0.94/2.13 = 0.44
√
The T ratio with lag 1: t1 = 20 × 0.44 = 1.97
The T ratio is positive and above 1.645, the upper 5% point of the
normal distribution), which suggest the rates are overgraduated. 49/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of 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
50/72 Actuarial Statistics – Module 8: Method of Graduation
Methods of graduation
Preliminaries 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 Tes...
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