025 if k p m k accept the null hypothesis at the

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Unformatted text preview: the range of ages where is the bias is worst. Weaknesses of the signs test: the test provides no indication of the extent of the discrepancy. The test is qualitative rather than quantitative. 31/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Signs test Procedure calculate the statistic P : the number of zx that are positive find the value k ∗ -the smallest k such that k P (P ≤ k ) = j =0 m j 1 2 m ≥ 0.025, . If k ∗ ≤ P ≤ m − k ∗ , accept the Null hypothesis (at the 5% level). Another way is to use the p -value. (can be found on Pages 186-188 of the Tables if the sample size is 2, 3, · · · , 12 or 20) Alternatively, use the Normal approximation for large m: 1 1 P ∼ N 2 m, 4 m 32/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Signs test Example A graduation covers 20 age groups and has resulted in 6 positive and 14 negative deviations. We have then: 1 under the null hypothesis, P ∼ Bin(20, 2 ). the p -value of the test is p = 2P (P ≤ 6) = 0.1154 > 5% Hence, we cannot reject the null hypothesis at a 5% significance level. 33/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Cumulative 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 34/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Cumulative Deviations Test Cumulative Deviations Test Tests whether the overall no. of deaths conforms to the model with the mortality rates assumed in the graduation. We can apply the test to the whole age range or parts of the age range Conclusion: high value of test statistic indicates either the graduated rates are biased (too low if the test statistic is positive, too high if the test statistic is negative), or actual variance is higher that predicted by the assumed model for the range of ages considered 34/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Cumulative Deviations Test Multiple-state Markov model The deviation has the approximate distribution o o c c dx − Ex µx + 1 ∼ Normal 0, Ex µx + 1 2 2 The accumulated deviation has...
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This document was uploaded on 04/03/2014.

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