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Penultimate approximation for the distribution of the excesses

Rym Worms (2010)

ESAIM: Probability and Statistics

Let F be a distribution function (d.f) in the domain of attraction of an extreme value distribution H γ ; it is well-known that Fu(x), where Fu is the d.f of the excesses over u, converges, when u tends to s+(F), the end-point of F, to G γ ( x σ ( u ) ) , where G γ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for γ > - 1 , a function Λ which verifies lim u s + ( F ) Λ ( u ) = γ and is such that Δ ( u ) = sup x [ 0 , s + ( F ) - u [ | F ¯ u ( x ) - G ¯ Λ ( u ) ( x / σ ( u ) ) | converges to 0 faster than d ( u ) = sup x [ 0 , s + ( F ) - u [ | F ¯ u ( x ) - G ¯ γ ( x / σ ( u ) ) | .

Permanents, order statistics, outliers, and robustness.

Narayanaswamy Balakrishnan (2007)

Revista Matemática Complutense

In this paper, we consider order statistics and outlier models, and focus primarily on multiple-outlier models and associated robustness issues. We first synthesise recent developments on order statistics arising from independent and non-identically distributed random variables based primarily on the theory of permanents. We then highlight various applications of these results in evaluating the robustness properties of several linear estimators when multiple outliers are possibly present in the...

Permutation tests for multiple changes

Marie Hušková, Aleš Slabý (2001)

Kybernetika

Approximations to the critical values for tests for multiple changes in location models are obtained through permutation tests principle. Theoretical results say that the approximations based on the limit distribution and the permutation distribution of the test statistics behave in the same way in the limit. However, the results of simulation study show that the permutation tests behave considerably better than the corresponding tests based on the asymptotic critical value.

Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

Elena Di Bernardino, Thomas Laloë, Véronique Maume-Deschamps, Clémentine Prieur (2013)

ESAIM: Probability and Statistics

This paper deals with the problem of estimating the level sets L(c) =  {F(x) ≥ c}, with c ∈ (0,1), of an unknown distribution function F on ℝ+2. A plug-in approach is followed. That is, given a consistent estimator Fn of F, we estimate L(c) by Ln(c) =  {Fn(x) ≥ c}. In our setting, non-compactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by...

Plug-in estimators for higher-order transition densities in autoregression

Anton Schick, Wolfgang Wefelmeyer (2009)

ESAIM: Probability and Statistics

In this paper we obtain root-n consistency and functional central limit theorems in weighted L1-spaces for plug-in estimators of the two-step transition density in the classical stationary linear autoregressive model of order one, assuming essentially only that the innovation density has bounded variation. We also show that plugging in a properly weighted residual-based kernel estimator for the unknown innovation density improves on plugging in an unweighted residual-based kernel estimator....

Poisson convergence for the largest eigenvalues of heavy tailed random matrices

Antonio Auffinger, Gérard Ben Arous, Sandrine Péché (2009)

Annales de l'I.H.P. Probabilités et statistiques

We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in (Electron. Commun. Probab.9 (2004) 82–91), we prove that, in the absence of the fourth moment, the asymptotic behavior of the top eigenvalues is determined by the behavior of the largest entries of the matrix.

Poisson sampling for spectral estimation in periodically correlated processes

Vincent Monsan (1994)

Applicationes Mathematicae

We study estimation problems for periodically correlated, non gaussian processes. We estimate the correlation functions and the spectral densities from continuous-time samples. From a random time sample, we construct three types of estimators for the spectral densities and we prove their consistency.

Power of A Class of Goodness-of-Fit Tests I

Christopher S. Withers, Saralees Nadarajah (2009)

ESAIM: Probability and Statistics

Consider testing whether F = F0 for a continuous cdf on R = (-∞,∞) and for a random sample X1,..., Xn from F. We derive expansions of the associated asymptotic power based on the Cramer-von Mises, Kolmogorov-Smirnov and Kuiper statistics. We provide numerical illustrations using a double-exponential example with a shifted alternative.

Probabilistic methods for semilinear partial differential equations. Applications to finance

Dan Crisan, Konstantinos Manolarakis (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

With the pioneering work of [Pardoux and Peng, Syst. Contr. Lett.14 (1990) 55–61; Pardoux and Peng, Lecture Notes in Control and Information Sciences176 (1992) 200–217]. We have at our disposal stochastic processes which solve the so-called backward stochastic differential equations. These processes provide us with a Feynman-Kac representation for the solutions of a class of nonlinear partial differential equations (PDEs) which appear in many applications in the field of Mathematical Finance....

Probability distribution of transformed random variables with application to nonlinear features extraction

Lubomír Soukup (1998)

Kybernetika

A method for estimation of probability distribution of transformed random variables is presented. The proposed approach admits an approximation of the transformation of the random variables. The approximate probability density function (pdf) is corrected to obtain a resulting pdf which incorporates a prior knowledge of approximation errors. The corrected pdf is not contaminated by any uncontrollable approximation. The method is applied to pattern recognition. It is shown that class conditional pdf...

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