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Statistical tests
Standardised Deviations Test Interval (−∞, −3) (−3, −2) (−2, −1) (−1, 0) (0, 1)
Actual
0
2
4
4
3
Expected
0.0
0.4
2.8
6.8
6.8
Interval
(1, 2)
(2, 3)
(3, ∞)
Actual
4
3
0
2.8
0.4
0
Expected
If we combine the small groups by pooling the values in the ranges
(−∞, −1) and (1, ∞), applying a chi square test to the resulting 4
groups gives:
χ2 = (6 − 3.2)2 (4 − 6.82)2 (3 − 6.8)2 (7 − 3.2)2
+
+
+
= 10.24
3.2
6.8
6.8
3.2 10.24 > 7.815 (the upper 95% point of the χ2 ), so the deviations
3
do not conform to a standard normal distribution.
(Strictly speaking, we should not use χ2 test here
since we have E = 3.2 < 5)
29/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Standardised Deviations Test Furthermore:
2
Only 7 values among 20 fall in the range (− 3 , 2 ).
3 Too few values in the center and too many in the tails ie
variance too higher.
This might indicate overgraduation or the presence of duplicates. 30/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Signs 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
31/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Signs test Signs test
Tests balance between positive and negative deviations
Rationale: if graduated rates adhere to the crude rates:
Roughly half the graduated rates should be above and half
below the crude rates
Null hypothesis: the number above ( denoted by P ) should
have a Binomial(m, 1 ) distribution (assuming that there are m
2
age groups) If the test shows that the no. of positive values is very high or
very low, this indicates that the rates are on average too low
or too high, respectively. The pattern of the signs will indicate...
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This document was uploaded on 04/03/2014.
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