Scale and Shape Parameters

Study guide: Exam FAM

Scale and Shape Parameters are the two roles a parameter can play in a continuous severity model. A scale parameter $\theta$ sets the units: multiplying the loss by a positive constant $c$ multiplies $\theta$ by $c$ and leaves every other parameter unchanged. A shape parameter, such as $\alpha$ or $\tau$ , sets the form of the distribution, including its skewness, its hazard rate and the weight of its tail, and a change of units leaves it alone.

$F_X(x) = F_1\!\left(\frac{x}{\theta}\right)$

$E[X^k] = \theta^k\,E[X_1^k]$

$E[(cX) \wedge u] = c\,E\!\left[X \wedge \frac{u}{c}\right]$

Definitions: Loss Models calls a family a scale distribution if a member multiplied by a positive constant is still in the family, and calls a parameter a scale parameter if that multiplication multiplies it by the same constant and changes no other parameter. In the first two blocks, $X_1$ is the member with $\theta = 1$ and the same shape. In the exam tables, $\theta$ is the scale parameter of the exponential, gamma, Weibull and Pareto families and of their transformed and inverse versions. The lognormal is a scale distribution, but $cX$ has $\mu + \ln c$ in place of $\mu$ , so $\mu$ is not a scale parameter, and $e^{\mu}$ plays that role instead.
What the scale carries: The mean, the standard deviation, every percentile and the limited expected value at a proportionally scaled limit all move in proportion to $\theta$ . Unit-free quantities, such as the coefficient of variation, the skewness and the probability of exceeding a multiple of $\theta$ , depend on the shape alone. This is why uniform inflation or a change of currency is handled by changing $\theta$ and nothing else.
What the shape carries: The Pareto's $\alpha$ decides which moments exist, since $E[X^k]$ is finite only for $k < \alpha$ . The gamma's $\alpha$ sets the coefficient of variation $1/\sqrt{\alpha}$ and the skewness $2/\sqrt{\alpha}$ . The Weibull's $\tau$ makes the hazard rate decrease when $\tau < 1$ , stay constant when $\tau = 1$ and increase when $\tau > 1$ . The shape therefore decides the tail weight, while the scale only stretches the axis.

Inflation Acts on the Scale Parameter Losses are gamma with $\alpha = 2$ and $\theta = 500$ . Next year, every loss is 8% larger. Show that next year's losses are gamma with $\theta = 540$ , and compare the mean, the coefficient of variation and the probability that a loss exceeds 3,000 in the two years. For $\alpha = 2$ , $S(x) = e^{-x/\theta}(1 + x/\theta)$ .
Answer If $Y = 1.08X$ , then $F_Y(y) = F_X(y/1.08)$ . The gamma CDF is $\Gamma(\alpha;\, x/\theta)$ , so its argument becomes $y/(1.08 \times 500) = y/540$ . That is the gamma CDF with $\alpha = 2$ and $\theta = 540$ , so $\theta$ absorbs the inflation and $\alpha$ is unchanged.
 $\begin{align*} \Pr(X > 3{,}000) &= e^{-6}(1 + 6) \\ &= 0.017351 \\ \Pr(Y > 3{,}000) &= e^{-5.5556}(1 + 5.5556) \\ &= 0.025343 \end{align*}$ 
The mean rises from $2(500) = 1{,}000$ to $2(540) = 1{,}080$ , by exactly 8%. The coefficient of variation stays at $1/\sqrt{2} = 0.7071$ , because it depends on the shape alone. The probability of exceeding the fixed threshold of 3,000 rises by 46%, because the threshold has fallen from 6 to 5.56 in units of $\theta$ .

Same Mean, Different Shape Two Weibull severities both have mean 1,000, one with $\tau = 0.5$ and one with $\tau = 2$ . Find each scale parameter, coefficient of variation and probability of a loss above 5,000, and describe each hazard rate. The Weibull mean is $\theta\,\Gamma(1 + 1/\tau)$ .
Answer The scale is the mean divided by $\Gamma(1 + 1/\tau)$ , and $S(x) = e^{-(x/\theta)^{\tau}}$ .
 $\begin{align*} \theta_{0.5} &= \frac{1{,}000}{\Gamma(3)} \\ &= 500 \\ \theta_{2} &= \frac{1{,}000}{\Gamma(1.5)} \\ &= 1{,}128.38 \\ S_{0.5}(5{,}000) &= e^{-\sqrt{10}} \\ &= 0.042329 \\ S_{2}(5{,}000) &= e^{-(4.4311)^2} \\ &= 2.97 \times 10^{-9} \end{align*}$ 
The coefficients of variation are $\sqrt{\Gamma(5)/\Gamma(3)^2 - 1} = 2.236$ and $\sqrt{\Gamma(2)/\Gamma(1.5)^2 - 1} = 0.523$ . The hazard rate $\tau x^{\tau-1}/\theta^{\tau}$ falls from 0.001 at 500 to 0.0005 at 2,000 when $\tau = 0.5$ , and it rises from 0.00079 to 0.00314 when $\tau = 2$ . With the same mean, the shape alone makes a loss above 5,000 about fourteen million times as likely.

How it appears on the exam

FAM asks candidates to identify the role of scale and shape parameters in continuous models. A typical question applies inflation or a currency change to a tabled distribution and asks for a probability, a moment or a limited expected value afterwards, often with a deductible or limit that stays fixed.

ASTAM asks candidates to describe how changes in the parameters affect the distributions. A question may ask which parameter moves the mean without changing the shape, or which one changes the tail.

Spotting the scale: In a scale family, $x$ enters the CDF only through $x/\theta$ . That test identifies the scale parameter of a distribution the question defines from scratch.

Exam traps

The lognormal: Inflation of $r$ adds $\ln(1+r)$ to $\mu$ and leaves $\sigma$ unchanged. Multiplying $\mu$ by $1 + r$ is a common error.

Counting inflation twice: With a fixed deductible $d$ , either inflate $\theta$ to $(1+r)\theta$ or keep $\theta$ and deflate the deductible to $d/(1+r)$ , then multiply by $1 + r$ . Doing both counts the inflation twice.

Shape parameters do not absorb inflation: Inflation never changes $\alpha$ or $\tau$ . Lowering a Pareto's $\alpha$ to match a higher mean also makes the tail heavier, which uniform inflation does not do.

Unit-free does not mean unchanged: The probability of exceeding a fixed amount changes with $\theta$ , even though the probability of exceeding a fixed multiple of $\theta$ does not.