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

A mortality investigation graduates by a parametric formula with 3 fitted parameters over ages 30 to 39 inclusive (10 age groups). The chi-squared goodness-of-fit statistic is 17.0. Using the usual convention for degrees of freedom, and given the 5% upper chi-squared critical values of 12.59 (6 df), 14.07 (7 df), 15.51 (8 df) and 18.31 (10 df), what is the correct conclusion?

Degrees of freedom are 10 groups minus 3 fitted parameters, which is 7. The 5% critical value is 14.07, and the statistic of 17.0 exceeds it, so the graduation is rejected as a poor fit. Using 10 degrees of freedom would wrongly accept it.

  1. AAccept the graduation, as 17.0 is below 18.31 with 10 degrees of freedom
  2. BReject the graduation, as 17.0 exceeds 14.07 with 7 degrees of freedomCorrect
  3. CReject the graduation, as 17.0 exceeds 12.59 with 6 degrees of freedom
  4. DAccept the graduation, as 17.0 is above 15.51 with 8 degrees of freedom
  5. The test cannot be applied because the parameters were fitted from the data

Explanation

Degrees of freedom = number of age groups minus parameters fitted = 10 − 3 = 7. The critical value is 14.07, and 17.0 exceeds it, so the graduation is rejected. Using 10 df ignores the fitted parameters.

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