Ridge regression estimators for the extreme value index

Abstract

We consider bias reduced estimators of the extreme value index (EVI) in case of Pareto-type distributions and under all max-domains of attraction. To this purpose we revisit the regression approach started in Feuerverger and Hall (Ann. Stat. 27, 760–781, 1999) and Beirlant et al. (Extremes 2, 177–200, 1999) in the case of a positive EVI, and in Beirlant et al. (2005) for real-valued EVI. We generalize these approaches using ridge regression exploiting the mathematical fact that the bias tends to 0 when the number of top data points used in the estimation is decreased. The penalty parameter is selected by minimizing the asymptotic mean squared error of the proposed estimator. The accuracy and utility of the ridge regression estimators are studied using simulations and are illustrated with case studies on reinsurance claim size data as well as daily wind speed data.

Appendix: Asymptotics

Proof of Theorem 1

Note that

$$\hat{\gamma}_{k}^{\text{+}} = {1 \over k}\sum\limits_{j = 1}^{k} \left\{ 1- \bar{c} \frac{c_{j} -\bar{c}}{S_{cc}+ \hat{\tau}_{k}^{+}} \right\} Z_{j}. $$

Using the consistency of $\hat {\beta }_{k}$ and $\hat {\rho }$ it follows that as M ≠ 0 we have $\hat {\tau }_{k}^{+} \to _{p} \gamma ^{2}/M^{2}$ as $\sqrt {k}b_{n,k} \to M$. So

$$\begin{array}{@{}rcl@{}} && {1 \over \sqrt{k}} \sum\limits_{j = 1}^{k} (c_{j} -\bar{c})\left( {1 \over S_{cc}+ \hat{\tau}_{k}^{+}}- {1 \over S_{cc}+ (\gamma/M)^{2}} \right) Z_{j} \\ && =_{d} \frac{\hat{\tau}_{k}^{+} - (\gamma/M)^{2}} {(S_{cc}+ \hat{\tau}_{k}^{+})(S_{cc}+(\gamma/M)^{2})} {1 \over \sqrt{k}} \sum\limits_{j = 1}^{k} (c_{j} -\bar{c})(\gamma + b_{n,k}c_{j} + \epsilon_{j}) \\ && \to_{p} 0. \end{array} $$

So the limit distribution is obtained from the limit distribution of

$$\sqrt{k} \left( {1 \over k}\sum\limits_{j = 1}^{k} \left\{ 1- \bar{c} \frac{c_{j} -\bar{c}}{S_{cc}+ (\gamma/M)^{2}} \right\} (\gamma + b_{n,k}c_{j}+\epsilon_{j}) -\gamma \right). $$

In case M = 0 we have similarly that $(S_{cc}+ \hat {\tau }_{k}^{+})^{-1} \to _{p} 0$ so that we obtain the limit distribution of

$$\sqrt{k} \left( {1 \over k}\sum\limits_{j = 1}^{k} (\gamma + b_{n,k}c_{j}+\epsilon_{j}) -\gamma \right). \hspace{3cm} $$

□

Proof of Theorem 2

It follows from Beirlant et al. (2005) that the generalised log spacings {Y_j} have the following asymptotic expansions

$$Y_{j} =_{d} \gamma (j + 1)\log\left( 1+\frac{1}{j}\right) + \left\{ b_{n,k} \left( \frac{j}{k + 1}\right)^{|\tilde{\rho}|} + \epsilon_{j}\right\} \left\{1+o_{P}(1)\right\}, $$

where

$$\epsilon_{j} = \left\{ \begin{array}{l l} \frac{j + 1}{\sqrt{k}}\left( \frac{k}{j} W^{(0)}\left( \frac{j}{k}\right)-\frac{k}{j + 1}W^{(0)}\left( \frac{j + 1}{k}\right) \right) & \text{ if } \gamma > 0,\\ \frac{(j + 1)(1-\gamma)}{\sqrt{k}}\left( \left( \frac{k}{j}\right)^{1-\gamma} W^{(\gamma)}\left( \frac{j}{k}\right)- \left( \frac{k}{j + 1}\right)^{1-\gamma}W^{(\gamma)}\left( \frac{j + 1}{k}\right) \right) & \text{ if } \gamma < 0. \end{array} \right. $$

Furthermore, write {λ_j(τ)} as

$$\lambda_{j}(\tau) = \alpha(\tau)+\beta(\tau) \left( \frac{j}{k + 1}\right)^{-{\tilde{\rho}}}, $$

where $\alpha (\tau )= 1+ \frac {\bar {c}^{2}}{S_{cc}+\tau }$ and $\beta (\tau ) = -\frac {\bar {c}}{S_{cc}+\tau }$. Also note that $\bar {c} \to \frac {1}{1-{\tilde {\rho }}}$ and $S_{cc} +\bar {c}^{2}\to \frac {1}{1-2\tilde \rho }$ as k →∞ so $\beta (\tau ) = \left (1-\alpha (\tau )\right )(1-{\tilde {\rho }})\left (1+o_{p}(1)\right )$. The proof of Theorem 2 follows then similarly as in the proof of Theorem 1, leading to the asymptotic distribution of

$$S_{k}:= \sqrt{k} \left( {1 \over k}\sum\limits_{j = 1}^{k} \left\{ 1- \bar{c} \frac{c_{j} -\bar{c}}{S_{cc}+ \tau_{M}} \right\} (\gamma + b_{n,k}c_{j}+\epsilon_{j}) -\gamma \right). $$

The expected value of the limit distribution of S_k is given by

$$b_{n,k}\bar{c} \frac{\tau_{M}}{S_{cc}+\tau_{M}}. $$

The asymptotic variance of S_k follows from a more tedious calculation, both in case γ > 0 and γ < 0. □

The asymptotic variance when γ > 0. The covariance of the error terms {𝜖_j} is given by

$$Cov(\epsilon_{i}, \epsilon_{j}) = \left\{ \begin{array}{l l}(1+\gamma^{2})\left( 1+\frac{1}{i}\right)-2\gamma (i + 1) \log\left( 1+\frac{1}{i}\right) & \text{ if } i=j, \\ \gamma \frac{i + 1}{j} \log\left( 1+\frac{1}{i}\right) & \text{ if } i<j. \end{array} \right. $$

It follows that the asymptotic variance of S_k is given by

$$\begin{array}{@{}rcl@{}} && \frac{1}{k^{2}} Var\left( \sum\limits_{j = 1}^{k} \lambda_{j}(\tau)\epsilon_{j}\right) \\ &&= \frac{1}{k^{2}} \left( \sum\limits_{j = 1}^{k}{\lambda_{j}^{2}}(\tau) Var(\epsilon_{j}) + 2\sum\limits_{i = 1}^{k} \sum\limits_{j=i + 1}^{k} \lambda_{i}(\tau) \lambda_{j}(\tau) Cov(\epsilon_{i},\epsilon_{j}) \right) \\ &&= \left\{ \frac{(1-\gamma)^{2}}{k}\left( 1+\frac{\bar{c}^{2} S_{cc}}{(S_{cc}+\tau_{M})^{2}}\right) + \frac{2\gamma}{k}\left( 1 + \bar{c}^{2} S_{cc} \left( \frac{\xi_{1}}{S_{cc}+\tau_{M}}\right.\right.\right.\\ &&\left.\left.\left.{\kern5pt} +\frac{\xi_{2}}{(S_{cc}+\tau_{M})^{2}} \right) \right)\right\}, \end{array} $$

where $\xi _{1}=\frac {-{\tilde {\rho }} \bar {c}}{S_{cc}}$ and $\xi _{2}=\bar {c}$.

Indeed, as k →∞

$$\begin{array}{@{}rcl@{}} && \frac{1}{k} \sum\limits_{j = 1}^{k}{\lambda_{j}^{2}}(\tau_{M}) Var(\epsilon_{j}) \\ && = \frac{1}{k} \sum\limits_{j = 1}^{k} \left( \alpha(\tau_{M})+\beta(\tau_{M}) \left( \frac{j}{k + 1}\right)^{-{\tilde{\rho}}} \right)^{2} \left( (1+\gamma^{2})\left( 1+\frac{1}{j}\right)\right.\\ &&{\kern5pt} \left. -2\gamma (j + 1) \log\left( 1+\frac{1}{j}\right) \right)\\ && = \left\{ {\int}_{\frac{1}{k}}^{1} \left( \alpha(\tau_{M})+\beta(\tau_{M}) u^{-{\tilde{\rho}}}\right)^{2} \left( (1+\gamma^{2})\left( 1+\frac{1}{ku}\right) \right.\right.\\ &&{\kern5pt} \left.\left.- 2\gamma \left( ku+ 1\right) \log\left( 1+\frac{1}{ku}\right) \right) du \right\} \left\{1+o(1)\right\} \\ && = \left\{ (1-\gamma)^{2} {{\int}_{0}^{1}} \left( \alpha(\tau_{M})+\beta(\tau_{M}) u^{-{\tilde{\rho}}}\right)^{2} du \right\} \left\{1+o(1)\right\} \\ && = (1-\gamma)^{2}\left\{ \alpha^{2}(\tau_{M})+\frac{2\alpha(\tau_{M})\beta(\tau_{M})}{1-{\tilde{\rho}}} + \frac{\beta^{2}(\tau_{M})}{1-2{\tilde{\rho}}}\right\} \left\{1+o(1)\right\} \\ && = (1-\gamma)^{2} \left\{ 1+\frac{\bar{c}^{2} S_{cc}}{(S_{cc}+\tau_{M})^{2}}\right\} \left\{1+o(1)\right\}, \end{array} $$

and

$$\begin{array}{@{}rcl@{}} && \frac{1}{k} \sum\limits_{i = 1}^{k} \sum\limits_{j=i + 1}^{k} \lambda_{i}(\tau_{M}) \lambda_{j}(\tau_{M}) Cov(\epsilon_{i},\epsilon_{j}) \\ && = \frac{\gamma}{k^{2}} \sum\limits_{i = 1}^{k} \left( \alpha(\tau_{M})+\beta(\tau_{M}) \left( \frac{i}{k + 1}\right)^{-{\tilde{\rho}}} \right) (i + 1) \log \left( 1+\frac{1}{i}\right) \sum\limits_{j=i + 1}^{k} \left( \alpha(\tau_{M})\right.\\ &&{\kern5pt} \left.+\beta(\tau_{M}) \left( \frac{j}{k + 1}\right)^{-{\tilde{\rho}}} \right) \frac{k}{j} \\ && = \left\{ \gamma {\int}_{\frac{1}{k}}^{1} \left( \alpha(\tau_{M})+\beta(\tau_{M}) u^{-{\tilde{\rho}}} \right) \left( ku+ 1\right) \log \left( 1+\frac{1}{ku}\right) {{\int}_{u}^{1}} \left( \alpha(\tau_{M})\right.\right.\\ &&{\kern5pt} \left.\left.+\beta(\tau_{M}) v^{-{\tilde{\rho}}} \right) \frac{dv}{v} du \right\} \left\{ 1+o(1)\right\} \\ && = \left\{ \gamma {{\int}_{0}^{1}} \left( \alpha(\tau_{M})+\beta(\tau_{M}) u^{-{\tilde{\rho}}} \right) \left( -\alpha(\tau_{M}) \log u -\frac{\beta(\tau_{M})}{{\tilde{\rho}}}\left( 1-u^{-{\tilde{\rho}}}\right)\right) du \right\} \left\{ 1+o(1)\right\} \\ && = \gamma \left\{ \alpha^{2}(\tau_{M})+\frac{\alpha(\tau_{M})\beta(\tau_{M})}{1-{\tilde{\rho}}} \left( 1+\frac{1}{1-{\tilde{\rho}}}\right) + \frac{\beta^{2}(\tau_{M})}{(1-{\tilde{\rho}})(1-2{\tilde{\rho}})}\right\} \left\{ 1+o(1)\right\} \\ && = \gamma \left\{ 1 -\frac{{\tilde{\rho}}}{1-{\tilde{\rho}}} \frac{\bar{c}^{2}}{S_{cc}+\tau_{M}}+ \frac{S_{cc}}{1-{\tilde{\rho}}}\frac{\bar{c}^{2}}{(S_{cc}+\tau_{M})^{2}} \right\} \left\{ 1+o(1)\right\} \end{array} $$

The asymptotic variance when γ < 0. Here

$$Cov(\epsilon_{i}, \epsilon_{j}) = \left\{ \begin{array}{l l} \frac{1-\gamma}{1-2\gamma}\left\{ (2-(1+\gamma)(1-2\gamma))\left( 1+\frac{1}{i}\right)+ 4(j + 1)\left( 1-\left( 1+\frac{1}{i}\right)^{\gamma}\right) \right\} & \text{ if } i=j, \\ \frac{2(1-\gamma)}{1-2\gamma}(i + 1)(j + 1) \left( i^{-\gamma}-(i + 1)^{-\gamma}\right) \left( j^{\gamma-1}-(j + 1)^{\gamma-1}\right) & \text{ if } i<j. \end{array} \right. $$

It follows that the asymptotic variance of S_k is given by

$$\begin{array}{@{}rcl@{}} && \frac{1}{k^{2}} Var\left( \sum\limits_{j = 1}^{k} \lambda_{j}(\tau_{M})\epsilon_{j}\right) \\ &&= \frac{1}{k^{2}} \left( \sum\limits_{j = 1}^{k}{\lambda_{j}^{2}}(\tau_{M}) Var(\epsilon_{j}) + 2\sum\limits_{i = 1}^{k} \sum\limits_{j=i + 1}^{k} \lambda_{i}(\tau_{M}) \lambda_{j}(\tau_{M}) Cov(\epsilon_{i},\epsilon_{j}) \right) \\ &&= \left\{ \frac{(1-\gamma)^{2}}{k}\left( 1+\frac{\bar{c}^{2} S_{cc}}{(S_{cc}+\tau_{M})^{2}}\right) \right. \\ && \hspace{10mm} \left. + \frac{2\gamma}{k} \left( \frac{2(1-\gamma)}{1-2\gamma} +\bar{c}^{2} S_{cc}\left( \frac{\xi_{1}(\gamma)} {S_{cc}+\tau_{M}}+\frac{\xi_{2}(\gamma)}{(S_{cc}+\tau_{M})^{2}}\right) \right) \right\}, \end{array} $$

where $\xi _{1}(\gamma )=\frac {-2{\tilde {\rho }}(1-\gamma )}{S_{cc}(1-\gamma -{\tilde {\rho }})(1-2\gamma )}$ and $\xi _{2}(\gamma )=\frac {2(1-\gamma )^{2}}{(1-\gamma -{\tilde {\rho }})(1-2\gamma )}$. Indeed, as k →∞

$$\begin{array}{@{}rcl@{}} &&\frac{1}{k} \sum\limits_{j = 1}^{k}{\lambda_{j}^{2}}(\tau_{M}) Var(\epsilon_{j}) \\ & &= \frac{1-\gamma}{1-2\gamma} \frac{1}{k} \left\{ (2-(1+\gamma)(1-2\gamma)) \sum\limits_{j = 1}^{k} \left( \alpha(\tau_{M})+\beta(\tau_{M}) \left( \frac{j}{k + 1}\right)^{-{\tilde{\rho}}} \right)^{2}\left( 1+\frac{1}{j}\right) \right.\\ & &\hspace{30mm} \left. + 4 \sum\limits_{j = 1}^{k} \left( \alpha(\tau_{M})+\beta(\tau_{M}) \left( \frac{j}{k + 1}\right)^{-{\tilde{\rho}}} \right)^{2} (j + 1) \left( 1-\left( 1+\frac{1}{j}\right)^{\gamma}\right) \right\} \\ &&= \frac{1-\gamma}{1-2\gamma}\left\{ (2-(1+\gamma)(1-2\gamma)) {\int}_{\frac{1}{k}}^{1} \left( \alpha(\tau_{M})+\beta(\tau_{M}) u^{-{\tilde{\rho}}} \right)^{2}\left( 1+\frac{1}{ku}\right) du \right.\\ && \hspace{30mm} \left. + 4 {\int}_{\frac{1}{k}}^{1} \left( \alpha(\tau_{M})+\beta(\tau_{M}) u^{-{\tilde{\rho}}} \right)^{2} (ku+ 1) \left( 1-\left( 1+\frac{1}{ku}\right)^{\gamma}\right) du \right\} \left\{ 1+o(1)\right\} \\ && = \frac{(1-\gamma)\left( 2-(1+\gamma)(1-2\gamma)-4\gamma\right)}{1-2\gamma} \left\{ {\int}_{\frac{1}{k}}^{1} \left( \alpha(\tau_{M})+\beta(\tau_{M}) u^{-{\tilde{\rho}}} \right)^{2} \left( 1+\frac{1}{ku}\right) du \right\} \left\{ 1+o(1)\right\} \\ && = (1-\gamma)^{2}\left( 1+\frac{\bar{c}^{2} S_{cc}}{(S_{cc}+\tau_{M})^{2}}\right) \left\{ 1+o(1)\right\}, \end{array} $$

and

$$\begin{array}{@{}rcl@{}} && \frac{1}{k} \sum\limits_{i = 1}^{k} \sum\limits_{j=i + 1}^{k} \lambda_{i}(\tau_{M}) \lambda_{j}(\tau_{M}) Cov(\epsilon_{i},\epsilon_{j}) \\ && = \frac{2(1-\gamma)}{1-2\gamma} \left\{\frac{1}{k} \sum\limits_{i = 1}^{k} \left( \alpha(\tau_{M}) + \beta(\tau_{M}) \left( \frac{i}{k + 1}\right)^{-{\tilde{\rho}}} \right) (i + 1) \left( i^{-\gamma}-(i + 1)^{-\gamma}\right) \right. \\ && \hspace{30mm} \left. \sum\limits_{j=i + 1}^{k} \left( \alpha(\tau_{M}) + \beta(\tau_{M}) \left( \frac{j}{k + 1}\right)^{-{\tilde{\rho}}} \right) (j + 1) \left( j^{\gamma-1}-(j + 1)^{\gamma-1}\right) \right\} \\ && = \frac{2(1-\gamma)^{2} \gamma}{1-2\gamma} \left\{\frac{1}{k^{2}} \sum\limits_{i = 1}^{k} \left( \alpha(\tau_{M}) + \beta(\tau_{M}) \left( \frac{i}{k + 1}\right)^{-{\tilde{\rho}}} \right) \frac{i + 1}{k} \left( \frac{i}{k}\right)^{-\gamma-1} \right. \\ && \hspace{35mm} \left. \sum\limits_{j=i + 1}^{k} \left( \alpha(\tau_{M}) + \beta(\tau_{M}) \left( \frac{j}{k + 1}\right)^{-{\tilde{\rho}}} \right) \frac{j + 1}{k} \left( \frac{j}{k}\right)^{\gamma-2} \right\} \\ && = \frac{2(1-\gamma)^{2} \gamma}{1-2\gamma} \left\{ {\int}_{\frac{1}{k}}^{1} \left( \alpha(\tau_{M}) + \beta(\tau_{M}) u^{-{\tilde{\rho}}} \right) u^{-\gamma} {{\int}_{u}^{1}} \left( \alpha(\tau_{M}) + \beta(\tau_{M}) v^{-{\tilde{\rho}}} \right) v^{\gamma-1} dv du \right\} \left\{ 1+o(1)\right\} \\ && = \frac{2(1-\gamma)^{2} \gamma}{1-2\gamma} \left\{ {\int}_{\frac{1}{k}}^{1} \left( \alpha(\tau_{M})u^{-\gamma} + \beta(\tau_{M}) u^{-{\tilde{\rho}}-\gamma} \right) \left( \frac{\alpha(\tau_{M})}{\gamma} \left( 1-u^{\gamma}\right)+ \frac{\beta(\tau_{M})}{\gamma-{\tilde{\rho}}}\left( 1-u^{\gamma-{\tilde{\rho}}}\right) \right) du \right\} \left\{ 1+o(1)\right\} \\ && = \frac{2(1-\gamma) \gamma}{1-2\gamma} \left\{\alpha^{2}(\tau_{M})+ \frac{\alpha(\tau_{M}) \beta(\tau_{M})}{1-{\tilde{\rho}}} \left( 1+\frac{1-\gamma}{1-\gamma-{\tilde{\rho}}} \right)+\frac{\beta^{2}(\tau_{M})}{1-2{\tilde{\rho}}}\frac{1-\gamma}{1-\gamma-{\tilde{\rho}}} \right\}\left\{ 1+o(1)\right\} \\ && = \frac{2(1-\gamma) \gamma}{1-2\gamma} \left\{ 1 -\frac{{\tilde{\rho}}}{1-\gamma-{\tilde{\rho}}} \frac{\bar{c}^{2}}{S_{cc}+\tau_{M}} + \frac{(1-\gamma)S_{cc}}{1-\gamma-{\tilde{\rho}}} \frac{\bar{c}^{2}}{(S_{cc}+\tau_{M})^{2}} \right\} \left\{ 1+o(1)\right\}. \hspace{1cm} \Box \end{array} $$