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

Under the null hypothesis that a graduation is adequate, with m standardised deviations being independent standard normal, the lag-1 serial correlation statistic r1 is approximately N(0, 1/m). For m = 100 a test statistic of r1 = 0.25 is obtained. At the 5% two-sided level (critical value 1.96), what is the correct conclusion?

Reject the null hypothesis. The standard deviation of r1 is 1/sqrt(100) = 0.1, so the standardised statistic is 0.25/0.1 = 2.5, which exceeds 1.96. This indicates significant positive serial correlation among the deviations and so the graduation is not adequate.

  1. AReject the null; the scaled statistic r1*sqrt(m) = 2.5 exceeds 1.96Correct
  2. BDo not reject; the scaled statistic is 0.25
  3. CDo not reject; the scaled statistic is 1.25
  4. DReject the null; the scaled statistic is 25
  5. Do not reject; the scaled statistic is 0.025

Explanation

Standard deviation of r1 is 1/sqrt(100) = 0.1. Then 0.25/0.1 = 2.5 > 1.96, so reject. Using 0.25 directly without scaling is the main error, and it wrongly suggests non-rejection.

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