13_AS_8_lec_a

# 3 a lot of values in the tails big absolute

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Unformatted text preview: ion. Purpose: Check for excessively large or small deviations standard deviations too bunched (are concentrated in the central region)-indicates undergraduation ( too close to crude rates) or heterogeneity standard deviations too spread out (errors not centered): Roughly half the standardised deviations should fall in the 2 range (− 3 , 2 ). 3 A lot of values in the tails (big absolute deviations) indicates overgraduation or the existence of duplicates (lives not independent). 25/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Standardised Deviations Test Procedure I Calculate the standard deviation zx for each age group Categorize the standardised deviations into ranges –for example, ∞ to −3,-3 to -2;-2 to -1, -1 to 0, 0 to +1, +1 to +2, +2 to +3, +3 to +∞. (there should be no standardised deviations lying outside range (-3,3)) Note: when the number of age groups is small, use smaller no. of intervals, such that the expected no. of standardised deviations in each interval is not less than ﬁve. Compare the observed no. of the zx that fall in each interval (actual); with the expected no. of the zx that fall in each interval assuming zx ∼ N (0, 1) (expected) 26/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Standardised Deviations Test Procedure II Use the test statistic X= all intervals (actual-expected)2 expected X has a χ2−1 distribution where n is the number of intervals n used Note other tests for normality could be used (such as a Q-Q plot (quantile-quantile)) 27/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Standardised Deviations Test Example For a graduation, we have calculated the standardised deviations as follows x 30 31 32 33 34 35 36 37 zx 2.29 2.71 0.29 -0.55 -2.14 1.18 -0.06 -1.55 x 38 3940 41 42 43 44 45 zx -1.09 -2.05 -1.08 -1.52 -0.04 -0.39 1.48 1.20 x 46 47 48 49 zx 0.21 2.24 0.60 1.99 Analyse the distribution of the standardised deviations for this graduation. 28/72 Actuarial Statistics – Module 8: Method...
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