Multivariate Student Distributions

We introduce another class of elliptical distributions, namely certain multivariate Student distributions, and deal with statistical questions. At the end of this section, we also introduce non-elliptical Student distributions. Multivariate Student Distribution with Common Denominators Recall from Section 6.3 that Z X 2Y a 1 2 is a standard Student random variable with shape parameter a gt 0, where X is standard normal and Y is a gamma random variable with parameter r a 2. Next, X will be...

Multivariate Peaks Over Threshold

We already realized in the univariate case that, from the conceptual viewpoint, the peaks-over-threshold method is a bit more complicated than the annual maxima method. One cannot expect that the questions are getting simpler in the multivariate setting. Subsequently, our attention is primarily restricted to the bivariate case, this topic is fully worked out in 16 , 2nd ed., for any dimension. A new result about the testing of tail dependence is added in Section 13.3. 13.1 Nonparametric and...

PLRy 1 x1Tlr y

The p-value, modified with a Bartlett correction, is for b 4.0. Recall that u is replaced by xn-k n if the largest ordered values are taken instead of exceedances. 5.3 Testing Extreme Value Conditions with Applications co authored by J. Hiisler and D. Li24 We introduce two methods of testing one dimensional extreme value conditions and apply them to two financial data sets and a simulated sample. Introduction and the Test Statistics Extreme value theory EVT can be applied in many fields of...

Px 1 x1Tlrx1 28n

There is a general device to replace an LR-statistic TLR by TLR 1 b n to achieve a higher accuracy in the x2-approximation12. A Test-Statistic Suggested by LAN-Theory Marohn13 applies the LAN-approach to deduce a test statistic for the testing problem 4.5 . Let x - x - y a x - y a 2 1 - exp - x - y a 2, x -1 a x - y a 1- exp - x - y a a, x 1 exp - x - y a a . The test-statistic can be approximately represented by x N.6449 xj - a x 0.5066 V2iMjCT xj p x 2 Tn,A x ,i x x 0.69 - 1, where t x and a...

Generalized Jackknife Estimators of the Tail Index

Whenever we are dealing with semi-parametric estimators of the tail index, or even other parameters of extreme events, we have usually information about the asymptotic bias of these estimators. We may thus choose estimators with similar asymptotic properties, and build the associated Generalized Jackknife statistic. 36Martins, M.J., Gomes, M.I. and Neves, M. 1999 . Some results on the behavior of Hill estimator. J. Statist. Comput. and Simulation 63, 283-297. 37Beirlant, J., Dierckx, G.,...

Adding a Skewness Parameter a Continuous Parameterization

To represent the family of all non-degenerate, sum-stable distributions one must include a skewness parameter 1 lt ft lt 1 in addition to the index of stability 0 lt a lt 2. For a 2 one gets the normal df S 2 for each ft. We choose a parameterization so that the densities and dfs vary continuously in the parameters. Such a property is indispensable for statistical inference and visualization techniques. The continuous location-scale parameterization introduced by Nolan9 is a variant of the M...

The Minimum Distance Method

By visually fitting a normal df or density to the sample df or, respectively, to a kernel density or histogram, one is essentially applying a minimum distance MD method. Let d be a distance on the family of dfs. Then, tn, an is an MDE2, if d Fn, A ,f inf d Fn, , where Fn again denotes the sample df. The distance may also be based on a distance between normal densities and sample densities fn. One must apply the Hellinger distance H, fn y f x - f1 2 x 3.4 to obtain asymptotically efficient...

Mean Excess Functions

The mean excess function eF of a df F respectively, of a random variable X is given by the conditional expectation of X u given X gt u. We have eF u E X u X gt u xdF u x , u lt w F . 2.20 It is evident that eF u is the mean of the excess df at u. The mean excess function eF is also called the mean residual life function, see, e.g., 28 . Note that u aeF u M a , 2.21 where and a are the location and scale parameters. In conjunction with the visualization of data, we are expressly interested in...

Reciprocal Hazard Functions a Von Mises Condition

The reciprocal 1 hW of the hazard function of a GP df W is a straight line which is clearly of interest for visual investigations. Moreover, this observation leads to a condition due to von Mises that guarantees that a df belongs to the max and the pot-domain of an EV and GP distribution. Thus, the reciprocal hazard function is also of theoretical interest. Reciprocal hazard and mean excess functions are related to each other. We ew, a 1 a 1 ahwi gt a , 2.29 if y lt 1. Note that eWo hWo 1. The...

Serial Analysis of Stationary Data the Autocovariance Function For random variables X Y with EX2 EY2 to the covariance

Loosely speaking, there is a tendency that the random variables X and Y simultaneously exceed or fall below their expectations EX and EY, if there is a positive covariance. The random variables X and Y are uncorrelated if Cov X, Y 0. Recall that independent random variables are uncorrelated, yet the converse conclusion is not valid. where X and y are the sample means of the data x , ,xn and y , , yn. Subsequently, we study the serial dependence structure of a detrended and deseasonalized time...

Grrr Xin28

Where x1 n lt lt xn n are the data arranged in increasing order. In statistical publications, it is also suggested to employ plotting positions i 0.5 n in place of i n 1 in 2.8 . In addition, let F,-1 be constant between consecutive points i n 1 . Likewise, one may employ a linear interpolation. Note that the sample qf remains constant in case of multiple points. The sample qf F 1 is the qf in the sense of 1.64 of the sample df Fn if the sample qf is taken left-continuous. Also, Fn-1 q is the...

The Seasonal Component

If the moving average exhibits a variation that is annual in period in other words, seasonal then a refined decomposition of the measurements yj is suggested- For simplicity, let tj i for i l, ,n, where n Ip and l, p are the number and length of periods. Now we also single out a periodic component sn, i l, ,p, satisfying Sn i jp Sn i , j 0, ,l l i l, ,p, 2.49 and J2i lt p sn i 0. Thus, we have a decomposition in mind yj m i s i Xj, 2.50 where the mn i represent the smooth trend components and...

GY Giaaa126

With i 1 if 7 gt 0 and i 2 if 7 lt 0. The pertaining densities are go x G0 x e-x, for all x, x Gy x 1 7x - 1 1 y , 1 7x gt 0, 7 0. Also, check that x go x as 7 0. Some EV densities around the Gumbel density are displayed in Fig. 1.4. 1 Yx gt 0, y 0. 1.25 Yx 1 7 exp x as y 0, one Fig. 1.4. left. Gumbel density dashed and two Frechet solid densities for parameters Y .28, .56. right. Gumbel density dashed and two Weibull solid densities for parameters y -.28, -.56. On the right-hand side, we see...

Statistical Analysis of Extreme Values

With Applications to Insurance, Finance, Hydrology and Other Fields Rolf-Dieter Reiss Michael Thomas FB Mathematik Universit t Siegen Walter-Flex-Str. 3 57068 Siegen Germany e-mail reiss stat.math.uni-siegen.de 2000 Mathematics Subject Classification 60G70, 62P05, 62P99, 90A09, 62N05, 68N15, 62-07, 62-09, 62F10, 62G07, 62G09, 62E25, 62G30, 62M10 Library of Congress Control Number 2007924804 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this...

Parametric Modeling

Chapter 1 is basic for the understanding of the main subjects treated in this book. It is assumed that the given data are generated according to a random mechanism that can be linked to some parametric statistical model. In this chapter, the parametric models will be justified by means of mathematical arguments, namely by limit theorems. In this manner, extreme value EV and generalized Pareto GP models are introduced that are central for the statistical analysis of maxima or minima and of...