The risk function of the goodness-of-fit tests for tail models

Abstract

This paper contributes to answering a question that is of crucial importance in risk management and extreme value theory: How to select the threshold above which one assumes that the tail of a distribution follows a generalized Pareto distribution. This question has gained increasing attention, particularly in finance institutions, as the recent regulative norms require the assessment of risk at high quantiles. Recent methods answer this question by multiple uses of the standard goodness-of-fit tests. These tests are based on a particular choice of symmetric weighting of the mean square error between the empirical and the fitted tail distributions. Assuming an asymmetric weighting, which rates high quantiles more than small ones, we propose new goodness-of-fit tests and automated threshold selection procedures. We consider a parameterized family of asymmetric weight functions and calculate the corresponding mean square error as a loss function. Then we explicitly determine the risk function as the expected value of the loss function for finite sample. Finally, the risk function can be used to discuss whether a symmetric or asymmetric weight function should be chosen. With this the goodness-of-fit test which should be used in a new method for determining the threshold value is specified.

Appendix A: Auxiliary calculations

Lemma 1

Let

$$\begin{aligned} m_k(\nu )&= \frac{\nu +1}{n}\sum _{i=1}^n i^k \;\frac{B(i+\nu , n-i+1) }{B(i, n-i+1)} \end{aligned}$$

(32)

then

$$\begin{aligned} m_k(\nu )&= \sum _{l=0}^{k}S_{k+1,l+1}\; \frac{\nu +1}{\nu +1+l} \;(n-1)_{(l)} , \end{aligned}$$

(33)

where $\nu \in {\mathbb {R}}$, $k\in {\mathbb {N}}$, $(n-1)_{(l)}$ is the Pochhammer notation for falling factorials and $S_{k,l}$ are the Stirling numbers of the second kind (Abramowitz and Stegun 2014).

Proof

(Lemma 1) To begin, the beta functions $B(\cdot ,\cdot )$ of Eq. (32) are expressed in terms of the gamma function $\varGamma (\cdot )$ (Abramowitz and Stegun 2014). Simplifying the fraction yields

$$\begin{aligned} m_k(\nu )&= (\nu +1)\; \sum _{i=1}^n i^k \;\frac{\varGamma (n)}{\varGamma (i)} \frac{\varGamma (i-1+\nu +1)}{\varGamma (n +\nu +1)}. \end{aligned}$$

(34)

Depending on $\nu$, the possibly resulting poles must be considered and the gamma function should be considered in its analytic continuation $\varGamma (x+\alpha )= (x)^{(\alpha )}\varGamma (x)$, where $(x)^{(\alpha )}$ is the Pochhammer notation for rising factorials (Abramowitz and Stegun 2014). After simplifying the fraction, we receive

$$\begin{aligned} m_k(\nu ) = (\nu +1)\; \sum _{i=1}^n i^k\;\frac{\varGamma (n)}{\varGamma (i)} \frac{1}{(\nu +1+[i-1])^{(n-[i-1])}}. \end{aligned}$$

(35)

Using the identity $(x)^{(\alpha )} = (x)^{(\beta )} (x+\beta )^{(\alpha - \beta )}$ results in

$$\begin{aligned} m_k(\nu ) = (\nu +1)\; \sum _{i=1}^n i^k\; \frac{\varGamma (n)}{\varGamma (i)} \frac{(\nu +1)^{(i-1)}}{(\nu +1)^{(n)}}. \end{aligned}$$

(36)

Remember that $\frac{\varGamma (n)}{\varGamma (i)} = \left( {\begin{array}{c}n\\ i\end{array}}\right) \frac{i}{n} (n-i)! = \left( {\begin{array}{c}n-1\\ i-1\end{array}}\right) (1)^{(n-i)}$. Then,

$$\begin{aligned} m_k(\nu ) = \frac{(\nu +1)^{}}{(\nu +1)^{(n)}}\; \sum _{i=1}^n i^k\; \left( {\begin{array}{c}n-1\\ i-1\end{array}}\right) (1)^{(n-i)} (\nu +1)^{(i-1)}. \end{aligned}$$

(37)

Now, $i^k$ is decomposed into a sum of the falling factorials, where the coefficients consist of Stirling numbers of the second kind $i^k = \sum _{l=0}^{k} S_{k,l} (i)_{(l)}$. Using the appropriate numbering of the sum with an appropriate extension of the terms gives

$$\begin{aligned} m_k(\nu )&= \frac{(\nu +1)^{}}{(\nu +1)^{(n)}} \nonumber \\&\quad \times \sum _{i=1}^n \sum _{l=1}^{k+1} S_{k+1,l} (n-1)_{(l-1)} \frac{(i-1)_{(l-1)}}{(n-1)_{(l-1)}} \left( {\begin{array}{c}n-1\\ i-1\end{array}}\right) (1)^{(n-i)} (\nu +1)^{(i-1)}. \end{aligned}$$

(38)

By changing the order of the sums and truncating the binomial coefficient, the above equation reduces to

$$\begin{aligned} m_k(\nu )&= \frac{(\nu +1)^{}}{(\nu +1)^{(n)}}\nonumber \\&\quad \times \sum _{l=1}^{k+1} S_{k+1,l} (n-1)_{(l-1)} \sum _{i=1}^n \left( {\begin{array}{c}n-l\\ i-l\end{array}}\right) (1)^{(n-i)} (\nu +1)^{(i-1)}. \end{aligned}$$

(39)

By renumbering the last sum, splitting the term $(\nu +1)^{(l-1)}$ and using Chu-Vandermonde theorem (Oldham et al. 2009, Ch. 18), this equation reduces to

$$\begin{aligned} m_k(\nu )&= \frac{(\nu +1)^{}}{(\nu +1)^{(n)}} \nonumber \\&\quad \times \sum _{l=1}^{k+1} S_{k+1,l} (n-1)_{(l-1)} (\nu +1)^{(l-1)} (\nu +1+l)^{(n-l)}. \end{aligned}$$

(40)

After renumbering the remaining sum and multiplying the rising factorials, Eq. (32) follows immediately. $\square$