13_AS_8_lec_a

Deviations test example a graduation of the mortality

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Unformatted text preview: distribution o c dx − Ex µx + 1 ∼ Normal 2 ages o c Ex µx + 1 0, ages 2 The test statistic is o ages c d x − E x µx + 1 2 co 1 ages Ex µx + 2 35/72 ∼ Normal (0, 1) Actuarial Statistics – Module 8: Method of Graduation Statistical tests Cumulative Deviations Test Binomial Model Given selected age groups, the test statistic is o ages d x − Ex q x o o ages Ex q x (1 − q x ) 36/72 ∼ Normal (0, 1) Actuarial Statistics – Module 8: Method of Graduation Statistical tests Cumulative Deviations Test Example A graduation of the mortality experience of the female population of a region has been carried out. We have: Age 35 36 37 38 39 40 41 42 43 44 37/72 Obs deaths 241 204 212 231 242 220 213 218 185 193 o Ex q x 270 202 245 217 225 230 200 195 187 166 Dev -29 2 -33 14 17 -10 13 23 -2 27 22 Variance 270 202 245 217 225 230 200 195 187 166 2137 St dev -1.76 0.14 -2.11 0.95 1.13 -0.66 0.92 1.65 -0.16 2.10 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Cumulative Deviations Test The total deviation = 22, Variance = 2137. 22 The observed value of the statistic = √2137 = 0.48 The statistic follows Normal(0,1). So the critical value is 1.96 at level 5%. The observed value of the statistic= 0.48 < 1.96 (the critical value), so we cannot reject the null hypothesis. 38/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Cumulative Deviations Test Discussion Care with graduation methods that result in a zero cumulative deviation, in which case this test cannot be applied. Strength: detects overall bias or long runs of deviations of the same sign. It addresses the problem of the inability of the χ2 test to detect a large positive or negative cumulative deviation over part of the whole of the age range. Weakness: only detects features that are present over the whole age range considered. Positive deviations over one age range may cancel out negatives over another range. 39/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Grouping of 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 Corr...
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