To find the 5% quantile for a standardized Fréchet distribution with a shape parameter $\xi = 0.3$, we use the quantile function formula for the generalized extreme value (GEV) distribution. The quantile function $X_p$ for the GEV distribution when $\xi > 0$ is given by:

$$X_p = \mu - \left(\frac{\sigma}{\xi}\right)\left[1 - (-\ln(p))^{-\xi}\right]$$

For the standardized Fréchet distribution, $\mu = 0$ and $\sigma = 1$. Plugging these values into the formula, we get:

$$X_p = 0 - \left(\frac{1}{0.3}\right)\left[1 - (-\ln(p))^{-0.3}\right] = -0.9349$$

Things to Remember

• The Fréchet distribution is a type of extreme value distribution used to model the behavior of extreme values, such as maximum or minimum values in a dataset.
• The shape parameter $\xi$ in the Fréchet distribution determines the tail behavior of the distribution. A higher $\xi$ corresponds to heavier tails.
• The quantile function gives the value below which a certain proportion of the distribution falls. It is the inverse of the cumulative distribution function.
• The standardized Fréchet distribution has a location parameter $\mu = 0$ and a scale parameter $\sigma = 1$.
• The quantile function for the generalized extreme value (GEV) distribution is used to find specific quantiles for the Fréchet distribution.