Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk

8.4.4.1 Weighted Anderson–Darling Goodness-of-Fit Tests for Heavy Tails

In the case of the AD form of the CvM test statistic Q, the test is performed according to the following stages in Algorithm 8.3. As it turns out two of the more important results one can derive for the asymptotic expansions of the right tail of the AD test statistic, under the nominal claim, are based on a result derived by Zolotarev (1961); we provide this result in Theorem 8.8.

Theorem 8.8 (Distribution of positively weighted quadratic Gaussian infinite sums)

Consider Z₁, Z₂,… as i.i.d. random variables with Z_i ~ Normal (0, 1). Then the limiting distribution of the positively weighted quadratic sequence given by

(8.79)

with λ_i ≥ 0 for all i ∈ {1, 2,…, n} and decreasing λ₁ > λ₂ > … > λ_n for all n, satisfies F_n → F as n → ∞ with the distribution right tail given explicitly by the product

(8.80)

See details in Zolotarev (1961).

In Figure 8.7, the distribution for the random sum

(8.81)

is displayed as a function of sample size n with two different weight functions for given by the following:

• Case 1: weight ...