Spliced Distribution

Study guide: Exam ASTAM

A Spliced Distribution is a severity model built by joining different densities on adjacent intervals, so that, for example, small and medium losses follow one distribution and large losses follow a heavier-tailed one. Loss Models defines a $k$ -component spliced density as a weighted density on each interval, where each piece is a proper density on its own interval and the weights add up to 1. Splicing is one of the operations for creating new distributions.

$f_X(x) = \begin{cases} a_1 f_1(x), & c_0 < x < c_1 \\ a_2 f_2(x), & c_1 < x < c_2 \\ \quad \vdots & \\ a_k f_k(x), & c_{k-1} < x < c_k \end{cases}$

$a_1 + a_2 + \cdots + a_k = 1$

$f_j(x) = \frac{g_j(x)}{G_j(c_j) - G_j(c_{j-1})}, \quad c_{j-1} < x < c_j$

Symbols: $c_0 < c_1 < \cdots < c_k$ are the breakpoints, usually with $c_0 = 0$ and $c_k = \infty$ for a severity. Each weight $a_j > 0$ is the probability that a loss falls in the $j$ -th interval, and each $f_j$ is a density with all of its probability on that interval. The last block makes such a piece out of any density $g_j$ with CDF $G_j$ , by truncating it to the interval and rescaling it.
Continuity: Nothing in the definition makes the density continuous at the breakpoints. When continuity is required, the condition $a_j f_j(c_j) = a_{j+1} f_{j+1}(c_j)$ at each breakpoint, together with the weights summing to 1, fixes the weights.
Working with it: Probabilities, moments and limited expected values are computed piece by piece. For $x$ in the $j$ -th interval, $F_X(x) = a_1 + \cdots + a_{j-1} + a_j F_j(x)$ , and $E[X^m] = \sum_j a_j\,E[X_j^m]$ , where $X_j$ has density $f_j$ . A spliced model is a mixture of its pieces whose components do not overlap, so the mixing rules for moments carry over.
Use: Splicing lets the body be fitted to the plentiful small claims and the tail to the few large ones, for example with a Pareto or generalized Pareto tail above a high threshold.

A Continuous Splice of a Uniform Body and a Pareto Tail Claims below 1,000 are uniform on $(0, 1{,}000)$ . Claims above 1,000 follow a single-parameter Pareto with $\alpha = 2.5$ and $\theta = 1{,}000$ , whose density is $f_2(x) = 2.5(1{,}000)^{2.5}/x^{3.5}$ . Choose the weights so that the spliced density is continuous at 1,000, then find the mean and $\Pr(X > 5{,}000)$ .
Answer At the breakpoint, $f_1(1{,}000) = 1/1{,}000$ and $f_2(1{,}000) = 2.5/1{,}000$ . Continuity requires $a_1/1{,}000 = 2.5\,a_2/1{,}000$ , so $a_1 = 2.5\,a_2$ .
 $\begin{align*} a_2 &= \frac{1}{1 + 2.5} \\ &= 0.285714 \\ a_1 &= 0.714286 \end{align*}$ 
The uniform piece has mean 500, and the Pareto piece has mean $\alpha\theta/(\alpha - 1) = 1{,}666.67$ .
 $\begin{align*} E[X] &= 0.714286(500) + 0.285714(1{,}666.67) \\ &= 357.14 + 476.19 \\ &= 833.33 \\ \Pr(X > 5{,}000) &= 0.285714\left(\frac{1{,}000}{5{,}000}\right)^{2.5} \\ &= 0.005111 \end{align*}$ 
About 29% of claims fall in the Pareto tail, and they supply more than half of the mean.

A Percentile of a Spliced Severity A severity has weight 0.6 on $(0, 2{,}000)$ , where it follows an exponential with mean 1,000 truncated to that interval. It has weight 0.4 on $(2{,}000, \infty)$ , where $X - 2{,}000$ is Pareto with $\alpha = 3$ and $\theta = 4{,}000$ . Find $\Pr(X \le 1{,}000)$ and the 95th percentile.
Answer The body's piece is the exponential density divided by $1 - e^{-2} = 0.864665$ , its probability below 2,000.
 $\begin{align*} \Pr(X \le 1{,}000) &= 0.6 \times \frac{1 - e^{-1}}{1 - e^{-2}} \\ &= 0.6 \times \frac{0.632121}{0.864665} \\ &= 0.438635 \end{align*}$ 
Only 60% of the probability lies below 2,000, so the 95th percentile is in the tail, where $\Pr(X > x) = 0.4\left(\frac{4{,}000}{x + 2{,}000}\right)^3$ .
 $\begin{align*} 0.4\left(\frac{4{,}000}{x + 2{,}000}\right)^3 &= 0.05 \\ \frac{4{,}000}{x + 2{,}000} &= 0.125^{1/3} \\ &= 0.5 \\ x &= 6{,}000 \end{align*}$ 
This density is not continuous: it is 0.000094 just below 2,000 and 0.0003 just above it. The definition allows the jump, and the percentile is unaffected by it.

How it appears on the exam

ASTAM asks candidates to create new distributions by splicing. A question gives the pieces and either the weights or a continuity condition, then asks for the density, the CDF at a point, a percentile or a moment.

Which interval first: A percentile or a Value at Risk is found by first using the cumulative weights to decide which interval it falls in, and then inverting that piece alone.

Moments piece by piece: A mean or a limited expected value is a weighted sum of each piece's own moment on its interval, and most of the work in such a question is the truncated moment of the body.

Exam traps

Unscaled pieces: Each $f_j$ must integrate to 1 on its own interval. Using the untruncated $g_j$ in its place makes the total probability differ from 1, so every answer built on it is off.

Weights are probabilities: The weight $a_j$ is the probability of the $j$ -th interval, not the height of the density. Continuity sets the heights $a_j f_j(c_j)$ equal, which usually makes the weights unequal.

Splice versus mixture: In a mixture, every component covers the whole range of losses. In a splice, each piece covers only its own interval, so the tail probability above a point in the last interval is $a_k$ times that piece's survival function there.