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Unformatted text preview: of Graduation 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 Chi-square (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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