On Estimation and Testing for Pareto Tails

2010 Mathematics Subject Classification: 62F10, 62F12.The t-Hill estimator for independent data was introduced by Fabian and Stehlik (2009). It estimates the extreme value index of distribution function with regularly varying tail. This paper considers sampling of an infinite moving average model. We prove that in the discussed case the t-Hill estimator is weak consistent. However, in contrast to independent identically distributed case here it is shown that the t-Hill and the Hill estimator applied to the moving average model are not robust with respect to large observations

Extreme Value Index Estimators and Smoothing Alternatives: A Critical Review

Extreme-value theory and corresponding analysis is an issue extensively applied in many different fields. The central point of this theory is the estimation of a parameter γ, known as the extreme-value index. In this paper we review several extreme-value index estimators, ranging from the oldest ones to the most recent developments. Moreover, some smoothing and robustifying procedures of these estimators are presented.Extreme value index, Semi-parametric estimation, Smoothing modification

Robust estimator of distortion risk premiums for heavy-tailed losses

We use the so-called t-Hill tail index estimator proposed by Fabi\'an(2001), rather than Hill's one, to derive a robust estimator for the distortion risk premium of loss. Under the second-order condition of regular variation, we establish its asymptotic normality. By simulation study, we show that this new estimator is more robust than of Necir and Meraghni 2009 both for small and large samples.Comment: submitte

Testing for Changes in Kendall's Tau

For a bivariate time series $((X_i,Y_i))_{i=1,...,n}$ we want to detect whether the correlation between $X_i$ and $Y_i$ stays constant for all $i = 1,...,n$. We propose a nonparametric change-point test statistic based on Kendall's tau and derive its asymptotic distribution under the null hypothesis of no change by means a new U-statistic invariance principle for dependent processes. The asymptotic distribution depends on the long run variance of Kendall's tau, for which we propose an estimator and show its consistency. Furthermore, assuming a single change-point, we show that the location of the change-point is consistently estimated. Kendall's tau possesses a high efficiency at the normal distribution, as compared to the normal maximum likelihood estimator, Pearson's moment correlation coefficient. Contrary to Pearson's correlation coefficient, it has excellent robustness properties and shows no loss in efficiency at heavy-tailed distributions. We assume the data $((X_i,Y_i))_{i=1,...,n}$ to be stationary and P-near epoch dependent on an absolutely regular process. The P-near epoch dependence condition constitutes a generalization of the usually considered $L_p$-near epoch dependence, $p \ge 1$, that does not require the existence of any moments. It is therefore very well suited for our objective to efficiently detect changes in correlation for arbitrarily heavy-tailed data

Comovements of Different Asset Classes During Market Stress

This paper assesses the linkages between the most important U.S.financial asset classes (stocks, bonds, T-bills and gold) during periods of financial turmoil. Our results have potentially important implications for strategic asset allocation and pension fund management. We use multivariate extreme value theory to estimate the exposure of one asset class to extreme movements in the other asset classes. By applying structural break tests to those measures we study to what extent linkages in extreme asset returns and volatilities are changing over time. Univariate results andch bivariate comovement results exhib significant breaks in the 1970s and 1980s corresponding to the turbulent times of e.g. the oil shocks, Volcker's presidency of the Fed or the stock market crash of 1987.Flight to quality, financial market distress, extreme value theory