Estimation and uncertainty quantification for extreme quantile regions

Abstract

Estimation of extreme quantile regions, spaces in which future extreme events can occur with a given low probability, even beyond the range of the observed data, is an important task in the analysis of extremes. Existing methods to estimate such regions are available, but do not provide any measures of estimation uncertainty. We develop univariate and bivariate schemes for estimating extreme quantile regions under the Bayesian paradigm that outperforms existing approaches and provides natural measures of quantile region estimate uncertainty. We examine the method’s performance in controlled simulation studies. We illustrate the applicability of the proposed method by analysing high bivariate quantiles for pairs of pollutants, conditionally on different temperature gradations, recorded in Milan, Italy.

Appendix: The Extremal-t model with restriction to the positive reals

Consider a student-t distribution restricted to $(0,\infty )$. Using Beirlant et al. (2004, p.59) the norming constants required in Eq. 21 are

$$ \begin{array}{ccc} a_{n} = n^{1/\nu} \left( \frac{ 2 {\Gamma}\left( \frac{\nu + 1}{2}\right) \nu^{\frac{\nu}{2}-1} }{\sqrt{\pi} {\Gamma}\left( \frac{\nu}{2}\right) } \right)^{1/\nu} & \text{and} & b_{n} =0. \end{array} $$

Applying the conditional tail dependence function framework of Nikoloulopoulos et al. (2009), the exponent function can be written as:

$$ \begin{array}{@{}rcl@{}} V(x,y) =& \lim_{n \to \infty} x^{-\nu} \mathbb{P} \left( Z_{2} \leq a_{n} y | Z_{1} \leq a_{n} x \right) + y^{-\nu} \mathbb{P} \left( Z_{1} \leq a_{n} x | Z_{2} \leq a_{n} y \right), \end{array} $$

where (Z ₁,Z ₂)^⊤ follows a centred bivariate-t distribution on $(0, \infty )^{2}$ with unit variance, correlation ρ and degree of freedom ν. The conditional distribution of Z ₂|Z ₁ = z ₁ is a truncated t distribution on $(0, \infty )$ with mean ρ z ₁, variance $(\nu + {z_{1}^{2}}) (1-\rho ^{2}) / (\nu +1)$ and ν + 1 degrees of freedom. As a consequence, we obtain

$$ \mathbb{P} \left( Z_{2} \leq a_{n} y | Z_{1} \leq a_{n} x \right) = \frac{T_{1,\nu+1} \left( \sqrt{\frac{\nu +1}{1-\rho^{2}}} \frac{a_{n} (y-\rho x)}{\sqrt{\nu + {a_{n}^{2}} x^{2}}} \right) - T_{1,\nu+1} \left( - \rho \sqrt{\frac{\nu +1}{1-\rho^{2}}} \frac{a_{n} x}{\sqrt{\nu + {a_{n}^{2}} x^{2}}} \right)} {1 - T_{1,\nu+1} \left( - \rho \sqrt{\frac{\nu +1}{1-\rho^{2}}} \frac{a_{n} x}{\sqrt{\nu + {a_{n}^{2}} x^{2}}} \right) }, $$

and

$$ \lim_{n \to \infty} \mathbb{P} \left( Z_{2} \leq a_{n} y | Z_{1} \leq a_{n} x \right) = \frac{T_{1,\nu+1} \left( \sqrt{\frac{\nu +1}{1-\rho^{2}}} \left( \frac{y}{x}-\rho \right) \right) - T_{1,\nu+1} \left( - \rho \sqrt{\frac{\nu +1}{1-\rho^{2}}} \right)} {1- T_{1,\nu+1} \left( - \rho \sqrt{\frac{\nu +1}{1-\rho^{2}}} \right) }. $$

Identical calculations can be applied to the second term in the exponent function. Transforming the margins to unit-Fréchet margins allows expression of the exponent function as

$$ \begin{array}{@{}rcl@{}} V(x,y) &=& \frac{1}{1- T_{1,\nu+1} \left( - \rho \sqrt{\frac{\nu +1}{1-\rho^{2}}} \right) }\\ && \times \left\{ \frac{1}{x} \left[ T_{1,\nu+1} \left( \sqrt{\frac{\nu +1}{1-\rho^{2}}} \left( \frac{y}{x}-\rho \right) \right) - T_{1,\nu+1} \left( - \rho \sqrt{\frac{\nu +1}{1-\rho^{2}}} \right) \right] \right.\\ && \left. + \frac{1}{y} \left[ T_{1,\nu+1} \left( \sqrt{\frac{\nu +1}{1-\rho^{2}}} \left( \frac{x}{y}-\rho \right) \right) - T_{1,\nu+1} \left( - \rho \sqrt{\frac{\nu +1}{1-\rho^{2}}} \right) \right] \right\}. \end{array} $$

It is easy to verify that $\lim _{y \to 0} \partial / \partial x V(x,y) = 0$ and $\lim _{x \to 0} \partial / \partial y V(x,y) = 0$ which implies H({0}) = H({1}) = 0. Finally, note that due to the form of V (x,y), taking the double partial derivative with respect to x and y is equivalent to the double partial derivative of the exponent function of the Extremal-t model multiplied by a scaling term. Hence the angular density on the interior of the 2-dimensional unit simplex is as given.