The Extremal Dependence Measure and Asymptotic Independence

Abstract

Extremal dependence analysis assesses the tendency of large values of components of a random vector to occur simultaneously. This kind of dependence information can be qualitatively different than what is given by correlation which averages over the total body of the joint distribution. Also, correlation may be completely inappropriate for heavy tailed data. We study the extremal dependence measure (EDM), a measure of the tendency of large values of components of a random vector to occur simultaneously and show consistency of an estimator of the EDM. We also show asymptotic normality of an idealized estimator in a restricted case of multivariate regular variation where scaling functions do not have to be estimated.     

Weak Limit of the Sample Extremal Quotient

Necessary and sufficient conditions are obtained for the weak convergence ( W →) of the sample extremal quotient of independent and identically distributed random variables as the number of observations tends to infinity. The paper fully characterizes the class of possible non-degenerate limit distribution functions of the extremal quotient, under general conditions.     

Extremal analysis of processes sampled at different frequencies

The observed extremes of a discrete time process depend on the process itself and the sampling frequency. We develop theoretical results which show how to account for the effect of sampling frequency on extreme values, thus enabling us to analyse systematically extremal data from series with different sampling rates. We present statistical methodology based on these results which we illustrate though simulations and by applications to sea-waves and rainfall data.     

Asymptotic random extremal ratio and product based on generalized order statistics and its dual

The extremal ratio has been used in several fields, most notably in industrial quality control, life testing, small-area variation analysis, and the classical heterogeneity of variance situation. In many biological, agricultural, military activity problems and in some quality control problems, it is almost impossible to have a fixed sample size, because some observations are always lost for various reasons. Therefore, the sample size itself is considered frequently to be an random variable (rv). Generalized order statistics (GOS) have been introduced as a unifying theme for several models of ascendingly ordered rvs. The concept of dual generalized order statistics (DGOS) is introduced to enable a common approach to descendingly ordered rvs. In this article, the limit dfs are obtained for the extremal ratio and product with random indices under non random normalization based on GOS and DGOS. Moreover, this article considers the conditions under which the cases of random and non random indices give the same asymptotic results. Some illustrative examples are obtained, which lend further support to our theoretical results.     

Models for Extremal Dependence Derived from Skew‐symmetric Families

Skew‐symmetric families of distributions such as the skew‐normal and skew‐t represent supersets of the normal and t distributions, and they exhibit richer classes of extremal behaviour. By defining a non‐stationary skew‐normal process, which allows the easy handling of positive definite, non‐stationary covariance functions, we derive a new family of max‐stable processes – the extremal skew‐t process. This process is a superset of non‐stationary processes that include the stationary extremal‐t processes. We provide the spectral representation and the resulting angular densities of the extremal skew‐t process and illustrate its practical implementation.     

Bivariate Tail Dependence and the Generation of Multivariate Extreme Value Distributions

We define, in a probabilistic way, a parametric family of multivariate extreme value distributions. We derive its copula, which is a mixture of several complete dependent copulas and total independent copulas, and the bivariate tail dependence and extremal coefficients. Based on the obtained results for these coefficients, we propose a method to build multivariate extreme value distributions with prescribed tail/extremal coefficients. We illustrate the results with examples.     

Testing upper and lower outlier paris in gamma samples

The problem of discordancy testing for an upper and lower outlier pair in a sample from a gamma distribution with known shape is considered. Three statistics are investigated: the well-known extremal quotient and two likelihood-based procedures. Approximations t o the null distributions of the statistics are obtained where appropriate. The non-null properties are investigated by sensitivity contours and simulation.     

Modelling extremes of time-dependent data by Markov-switching structures

We investigate the extremal clustering behaviour of stationary time series that possess two regimes, where the switch is governed by a hidden two-state Markov chain. We also suppose that the process is conditionally Markovian in each latent regime. We prove under general assumptions that above high thresholds these models behave approximately as a random walk in one (called dominant) regime and as a stationary autoregression in the other (dominated) regime. Based on this observation, we propose an estimation and simulation scheme to analyse the extremal dependence structure of such models, taking into account only observations above high thresholds. The properties of the estimation method are also investigated. Finally, as an application, we fit a model to high-level exceedances of water discharge data, simulate extremal events from the fitted model, and show that the (model-based) flood peak, flood duration and flood volume distributions match their observed counterparts.     

Convergence of random extremal quotient and product

The limit distribution functions are obtained for the extremal quotient, extremal product and the geometric range with random indices under nonrandom centering and normalization. Moreover, this paper considers the conditions under which the cases of random and nonrandom indices give the same asymptotic results. Some illustrative examples are given.     

ESTIMATION OF QUANTILES FOR A FRÉCHET-TYPE DISTRIBUTION

Estimation of high quantiles of a distribution in the domain of attraction of the Fréchet distribution is based on the extremal distribution of the k largest order statistics. The problem is treated by a local maximum likelihood method on a three parameter model. The estimators are shown to be asymptotically consistent for the whole range of the tail index parameter.     

Extremal dependence tests for contagion

A new test for financial market contagion based on changes in extremal dependence defined as co-kurtosis and co-volatility is developed to identify the propagation mechanism of shocks across international financial markets. The proposed approach captures changes in various aspects of the asset return relationships such as cross-market mean and skewness (co-kurtosis) as well as cross-market volatilities (co-volatility). Monte Carlo experiments show that the tests perform well except for when crisis periods are short in duration. Small crisis sample critical values are calculated for use in this case. In an empirical application involving the global financial crisis of 2008–2009, the results show that significant contagion effects are widespread from the US banking sector to global equity markets and banking sectors through either the co-kurtosis or the co-volatility channels, reinforcing that higher order moments matter during crises.     

Decomposition for multivariate extremal processes

The probability distribution of an extremal process in R^d with independent max-increments is completely determined by its distribution function. The df of an extremal process is similar to the cdf of a random vector. It is a monotone function on (0, ∞) × R^d with values in the interval [0,1]. On the other hand the probability distribution of an extremal process is a probability measure on the space of sample functions. That is the space of all increasing right continuous functions y: (0, ∞) → R^d with the topology of weak convergence. A sequence of extremal processes converges in law if the probability distributions converge weakly. This is shown to be equivalent to weak convergence of the df's.

An extremal process Y: [0, ∞) → R^d is generated by a point process on the space [0, ∞) × [-∞, ∞)^d and has a decomposition Y = X v Z as the maximum of two independent extremal processes with the same lower curve as the original process. The process X is the continuous part and Z contains the fixed discontinuities of the process Y. For a real valued extremal process the decomposition is unique: for a multivariate extremal process uniqueness breaks down due to blotting.     

Local power and size properties of the LR,Wald, score and gradient tests in dispersion models

We derive asymptotic expansions for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of dispersion models, under a sequence of Pitman alternatives. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing the precision parameter. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

A Smooth Nonparametric,Multivariate, Mixed-Data Location-Scale Test

Abstract

A number of tests have been proposed for assessing the location-scale assumption that is often invoked by practitioners. Existing approaches include Kolmogorov–Smirnov and Cramer–von Mises statistics that each involve measures of divergence between unknown joint distribution functions and products of marginal distributions. In practice, the unknown distribution functions embedded in these statistics are typically approximated using nonsmooth empirical distribution functions (EDFs). In a recent article, Li, Li, and Racine establish the benefits of smoothing the EDF for inference, though their theoretical results are limited to the case where the covariates are observed and the distributions unobserved, while in the current setting some covariates and their distributions are unobserved (i.e., the test relies on population error terms from a location-scale model) which necessarily involves a separate theoretical approach. We demonstrate how replacing the nonsmooth distributions of unobservables with their kernel-smoothed sample counterparts can lead to substantial power improvements, and extend existing approaches to the smooth multivariate and mixed continuous and discrete data setting in the presence of unobservables. Theoretical underpinnings are provided, Monte Carlo simulations are undertaken to assess finite-sample performance, and illustrative applications are provided.     

Extremal limit theorems for observations separated by random power law waiting times

This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.     

Characterizations of geometric distribution through progressively Type-II right-censored order statistics

In this paper, we consider characterizations of geometric distribution based on some properties of progressively Type-II right-censored order statistics. Specifically, we establish characterizations through conditional expectation, identical distribution, and independence of functions of progressively Type-II right-censored order statistics. Moreover, extensions of these results to generalized order statistics are also sketched. These generalize the corresponding results known for the case of ordinary order statistics.     

The distribution of the ratio of independent central wishart determinants

Ratios of independent central Wishart determinants are useful statistics in multivariate analyses, particularly in the study of multivariate linear models. A method based on the inversion of characteristic functions is outlined for deriving new experessions for the probability distribution functions of the logarithms of these statistics. Accurate tables of the percentiles of these distributions have been obtained covering many bivariate and trivariate cases which have been computed by approximating these expression.     

Extremes of Markov chains with tail switching potential

Summary.  A recent advance in the utility of extreme value techniques has been the characteri- zation of the extremal behaviour of Markov chains. This has enabled the application of extreme value models to series whose temporal dependence is Markovian, subject to a limitation that prevents switching between extremely high and extremely low levels. For many applications this is sufficient, but for others, most notably in the field of finance, it is common to find series in which successive values switch between high and low levels. We term such series Markov chains with tail switching potential, and the scope of this paper is to generalize the previous theory to enable the characterization of the extremal properties of series displaying this type of behaviour. In addition to theoretical developments, a modelling procedure is proposed. A simulation study is made to assess the utility of the model in inferring the extremal dependence structure of autoregressive conditional heteroscedastic processes, which fall within the tail switching Markov family, and generalized autoregressive conditional heteroscedastic processes which do not, being non-Markov in general. Finally, the procedure is applied to model extremal aspects of a financial index extracted from the New York Stock Exchange compendium.     

Tables of critical values of the normal probability paper test

The present paper relates to Holm's normal probability paper test for simple hypotheses. It is shown how to obtain the statistics for the one-sided and two-sided tests using normal probability paper. Tables of critical values of these test statistics are given for practical use of the test; the equations for calculating the distribution functions of the test statistics are listed. Two different methods lead to these equations; but details are beyond the scope of the present paper and the interested reader is referred to another paper.     

Inference for clusters of extreme values

Summary. Inference for clusters of extreme values of a time series typically requires the identification of independent clusters of exceedances over a high threshold. The choice of declustering scheme often has a significant effect on estimates of cluster characteristics. We propose an automatic declustering scheme that is justified by an asymptotic result for the times between threshold exceedances. The scheme relies on the extremal index, which we show may be estimated before declustering, and supports a bootstrap procedure for assessing the variability of estimates.