000523 3421 030 49 61110 184 0003011 0002600 15889

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Unformatted text preview: function α + β x , which was fitted using the method of least squares. 2 o o qx ˆ qx Ex q x = E (A−E ) x Ex dx = A E 30 70,000 39 0.000557 0.000460 27.09 5.24 0.000645 0.000428 28.54 7.33 31 66,672 43 32 68,375 34 0.000497 0.000473 32.34 0.09 33 65,420 31 0.000474 0.000523 34.21 0.30 ··· ··· ··· ··· ··· 49 61,110 184 0.003011 0.002600 158.89 3.97 22/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Chi-square (χ2 ) test χ2 test: The test statistic is X= (A − E )2 = 5.24 + 7.33 + · · · + 3.97 = 43.30 E The graduated rates are calculated by estimating two parameters α and β and there are 20 age groups. So the degrees of freedom are 20 − 2 = 18. The upper 95% point for the χ2 distribution is 28.88 18 The observed value of the test statistic (43.30) exceeds 28, 87. Hence, we reject the null hypothesis, which indicates that the mortality experience does not conform to a formula of the type assumed in the graduation. 23/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Chi-square (χ2 ) test Discussion The χ2 test is a good test for overall goodness of fit but it tells us nothing about the direction of any bias there could be a few large deviations offset by a lot of very small deviations there could be groups of all positive or all negative deviations could be consistently biased up or down with small deviations 24/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Standardised Deviations 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 25/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Standardised Deviations Test Standardised deviations test This test essentially checks for normality of the zx ’s: Null hypothesis: the observed pattern of the individual standardised deviations (ie the numbers falling in each interval) is consistent with a standard normal distribut...
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