Graduation statistical tests serial correlations test

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Unformatted text preview: 1 )2 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 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 50/72 Actuarial Statistics – Module 8: Method of Graduation Methods of graduation Preliminaries 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 Tes...
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This document was uploaded on 04/03/2014.

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