SOA Exam ASTAM Fall 2025, Question 3

Study guide: Exam ASTAM

The loss severity random variable $X$ has a Uniform distribution on the interval $[a, b]$ . We denote this by $X \sim \text{Unif}(a, b)$ .

You are given that the survival function of $X$ is $S_X(x) = \dfrac{b - x}{b - a}, \quad a < x < b$ .

Let $Y = X - d \mid X > d$ denote the excess loss random variable where $0 \le a \le d < b$ .

Part a (2 points)

Show that $Y \sim \text{Unif}(0, b - d)$ .

Part b (3 points)

Now let $a = 0$ , $b = 10$ , and $d = 2$ .

Let $H_\xi$ denote the Generalized Extreme Value distribution with parameter $\xi$ . You are given that $Y$ is in the MDA of $H_\xi$ with $\xi = -1$ . Derive the normalizing sequences, $c_n$ and $d_n$ , for the limiting extreme value distribution of $Y$ .

Part c (2 points)

Let $Y_1, Y_2, \ldots, Y_n$ be independent and identically distributed random variables such that $Y_j \sim \text{Unif}(0, 8)$ for $j = 1, 2, \cdots, n$ , and let $M_n = \max\{Y_1, Y_2, \cdots, Y_n\}$ .

(i) Calculate $\Pr[M_{10} \le 7]$ directly.

(ii) Estimate $\Pr[M_{10} \le 7]$ using the limiting extreme value distribution.

Part d (2 points)

(i) State with reasons whether $Y$ is in the MDA of the Fréchet, Gumbel, or Weibull Extreme Value (EV) distribution.

(ii) Describe one key difference between the Weibull EV distribution and the Gumbel distribution.

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SOA Exam ASTAM Fall 2025, Question 3

Part a (2 points)

Part b (3 points)

Part c (2 points)

Part d (2 points)

Concepts tested

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