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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Graduation and graduation tests

A graduation is tested at 20 ages. The sum of squared individual standardised deviations is 31.4. The graduation was obtained by fitting a Gompertz-Makeham formula with 3 parameters, and the chi-square test is applied at the 5% level. Which conclusion is correct? (Chi-square 5% critical values: 15 d.f. 25.00; 17 d.f. 27.59; 19 d.f. 30.14; 20 d.f. 31.41.)

Reject the graduation. With 20 ages and 3 fitted parameters there are 17 degrees of freedom, whose 5% critical value is 27.59, and the statistic 31.4 exceeds it, indicating a poor fit.

  1. AAccept the graduation, as 31.4 is below the 20 d.f. critical value of 31.41
  2. BReject the graduation, as 31.4 exceeds the 17 d.f. critical value of 27.59Correct
  3. CReject the graduation, as 31.4 exceeds the 19 d.f. critical value
  4. DAccept the graduation, as 31.4 is below the 17 d.f. critical value
  5. The test cannot be applied because the formula has parameters

Explanation

Degrees of freedom equal the number of age groups minus parameters fitted: 20 − 3 = 17. The critical value is 27.59. Since 31.4 > 27.59, the graduation is rejected. Using 20 d.f. ignores the fitted parameters and wrongly accepts.

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