Of groups of ve sign observed the critical value with

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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 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 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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This document was uploaded on 04/03/2014.

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