13_AS_8_lec_a

Test chi square 2 test i purpose to test for goodness

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Unformatted text preview: Graduation by Reference to a Standard Table Graphical graduation Statistical Tests The Effect of Duplicate policies 18/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Chi-square (χ2 ) test Chi-square (χ2 ) test I Purpose: to test for goodness-of-fit of the graduated rates o o q x or µx + 1 2 The test statistic: 2 zx X= all ages x Rationale: a high value of the statistic indicates that the discrepancies between the observed numbers and those predicted by the graduated rates are large, i.e., the fit is not very good (overgraduation). the test statistic X is a χ2 random variable. degrees of freedom are equal to the no. of groups MINUS the no. of parameters estimated when carrying out the graduation. 18/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Chi-square (χ2 ) test Chi-square (χ2 ) test II If the test statistic X exceeds the 95% upper point of the chi-square distribution, this indicates a poor fit or overgraduation For mortality rates - test can be for individual ages or age groups 19/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Chi-square (χ2 ) test Markov/Poisson models o c dx − Ex µx + 1 2 co Ex µx + 1 2 ∼ Normal (0, 1) and o c dx − Ex µx + 1 2 x co Ex µ x + 1 2 2 ∼ χ2 , n where n is the number of ages minus no. of parameters estimated when calculating a graduation, if any. 20/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Chi-square (χ2 ) test Binomial model o o o Dx ∼ Normal Ex q x , Ex q x 1 − q x so that o d x − Ex q x o ∼ Normal (0, 1) o Ex q x 1 − q x and o dx − E x q x o x o Ex q x 1 − q x 2 ∼ χ2 , n where n is the number of age groups minus no. of parameters estimated when calculating a graduation, if any. 21/72 Actuarial Statistics – Module 8: Method of Graduation Statistical tests Chi-square (χ2 ) test Example A graduation was carried out by assuming that log( qx ) satisfies a px linear...
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