Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.     

Exceedance probability of the integral of a stochastic process

Let X={X(s)}_s∈S be an almost sure continuous stochastic process (S compact subset of R^d) in the domain of attraction of some max-stable process, with index function constant over S. We study the tail distribution of ∫_SX(s)ds, which turns out to be of Generalized Pareto type with an extra ‘spatial’ parameter (the areal coefficient from Coles and Tawn (1996) [3]). Moreover, we discuss how to estimate the tail probability P(∫_SX(s)ds>x) for some high value x, based on independent and identically distributed copies of X. In the course we also give an estimator for the areal coefficient. We prove consistency of the proposed estimators. Our methods are applied to the total rainfall in the North Holland area; i.e. X represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall.The paper has two main purposes: first to formalize and justify the results of Coles and Tawn (1996) [3]; further we treat the problem in a non-parametric way as opposed to their fully parametric methods.     

Neighborhood radius estimation for variable-neighborhood random fields

We consider random fields defined by finite-region conditional probabilities depending on a neighborhood of the region which changes with the boundary conditions. To predict the symbols within any finite region, it is necessary to inspect a random number of neighborhood symbols which might change according to the value of them. In analogy with the one-dimensional setting we call these neighborhood symbols the context associated to the region at hand. This framework is a natural extension, to d-dimensional fields, of the notion of variable length Markov chains introduced by Rissanen [24] in his classical paper. We define an algorithm to estimate the radius of the smallest ball containing the context based on a realization of the field. We prove the consistency of this estimator. Our proofs are constructive and yield explicit upper bounds for the probability of wrong estimation of the radius of the context.     

Peaks-over-threshold stability of multivariate generalized Pareto distributions

It is well-known that the univariate generalized Pareto distributions (GPD) are characterized by their peaks-over-threshold (POT) stability. We extend this result to multivariate GPDs.It is also shown that this POT stability is asymptotically shared by distributions which are in a certain neighborhood of a multivariate GPD. A multivariate extreme value distribution is a typical example.The usefulness of the results is demonstrated by various applications. We immediately obtain, for example, that the excess distribution of a linear portfolio with positive weights a_i, i≤d, is independent of the weights, if (U₁,…,U_d) follows a multivariate GPD with identical univariate polynomial or Pareto margins, which was established by Macke [On the distribution of linear combinations of multivariate EVD and GPD distributed random vectors with an application to the expected shortfall of portfolios, Diploma Thesis, University of Würzburg, 2004, (in German)] and Falk and Michel [Testing for tail independence in extreme value models. Ann. Inst. Statist. Math. 58 (2006) 261-290]. This implies, for instance, that the expected shortfall as a measure of risk fails in this case.     

On qualitative robustness of support vector machines

Support vector machines (SVMs) have attracted much attention in theoretical and in applied statistics. The main topics of recent interest are consistency, learning rates and robustness. We address the open problem whether SVMs are qualitatively robust. Our results show that SVMs are qualitatively robust for any fixed regularization parameter λ. However, under extremely mild conditions on the SVM, it turns out that SVMs are not qualitatively robust any more for any null sequence λ_n, which are the classical sequences needed to obtain universal consistency. This lack of qualitative robustness is of a rather theoretical nature because we show that, in any case, SVMs fulfill a finite sample qualitative robustness property.For a fixed regularization parameter, SVMs can be represented by a functional on the set of all probability measures. Qualitative robustness is proven by showing that this functional is continuous with respect to the topology generated by weak convergence of probability measures. Combined with the existence and uniqueness of SVMs, our results show that SVMs are the solutions of a well-posed mathematical problem in Hadamard’s sense.     

First exit time probability for multidimensional diffusions: A PDE-based approach

First exit time distributions for multidimensional processes are key quantities in many areas of risk management and option pricing. The aim of this paper is to provide a flexible, fast and accurate algorithm for computing the probability of the first exit time from a bounded domain for multidimensional diffusions. First, we show that the probability distribution of this stopping time is the unique (weak) solution of a parabolic initial and boundary value problem. Then, we describe the algorithm which is based on a combination of the sparse tensor product finite element spaces and an hp-discontinuous Galerkin method. We illustrate our approach with several examples. We also compare the numerical results to classical Monte Carlo methods.     

The hyper-Dirichlet process and its discrete approximations: The butterfly model

The aim of this paper is the study of some random probability distributions, called hyper-Dirichlet processes. In the simplest situation considered in the paper these distributions charge the product of three sample spaces, with the property that the first and the last component are independent conditional to the middle one. The law of the marginals on the first two and on the last two components are specified to be Dirichlet processes with the same marginal parameter measure on the common second component. The joint law is then obtained as the hyper-Markov combination, introduced in [A.P. Dawid, S.L. Lauritzen, Hyper-Markov laws in the statistical analysis of decomposable graphical models, Ann. Statist. 21 (3) (1993) 1272-1317], of these two Dirichlet processes. The processes constructed in this way in fact are in fact generalizations of the hyper-Dirichlet laws on contingency tables considered in the above paper. Our main result is the convergence to the hyper-Dirichlet process of the sequence of hyper-Dirichlet laws associated to finer and finer “discretizations” of the two parameter measures, which is proved by means of a suitable coupling construction.     

Grouped Dirichlet distribution: A new tool for incomplete categorical data analysis

Motivated by the likelihood functions of several incomplete categorical data, this article introduces a new family of distributions, grouped Dirichlet distributions (GDD), which includes the classical Dirichlet distribution (DD) as a special case. First, we develop distribution theory for the GDD in its own right. Second, we use this expanded family as a new tool for statistical analysis of incomplete categorical data. Starting with a GDD with two partitions, we derive its stochastic representation that provides a simple procedure for simulation. Other properties such as mixed moments, mode, marginal and conditional distributions are also derived. The general GDD with more than two partitions is considered in a parallel manner. Three data sets from a case-control study, a leprosy survey, and a neurological study are used to illustrate how the GDD can be used as a new tool for analyzing incomplete categorical data. Our approach based on GDD has at least two advantages over the commonly used approach based on the DD in both frequentist and conjugate Bayesian inference: (a) in some cases, both the maximum likelihood and Bayes estimates have closed-form expressions in the new approach, but not so when they are based on the commonly-used approach; and (b) even if a closed-form solution is not available, the EM and data augmentation algorithms in the new approach converge much faster than in the commonly-used approach.     

Some notes on multivariate generalized Pareto distributions

The investigation of multivariate generalized Pareto distributions (GPDs) has begun only recently and there are slightly varying definitions of GPDs available. In this article we investigate the one from Section 5.1 of Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004], which does not differ in the area of interest from those of other authors. We first give an interpretation of the case of independence in terms of the peaks-over-threshold approach. This case is also used in dimension d=3 by Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004] as a counterexample to show that GP functions are not necessarily distribution functions on their entire support. We generalize this counterexample to an arbitrary dimension d≥3 and demonstrate also that other GP functions show this behavior. Finally we show that different GPDs can lead to the same conditional probability measure in the area of interest.     

The contiguity of probability measures and asymptotic inference in continuous time stationary diffusions and Gaussian processes with known covariance

We establish contiguity of families of probability measures indexed by T, as T → ∞, for classes of continuous time stochastic processes which are either stationary diffusions or Gaussian processes with known covariance. In most cases, and in all the examples we consider in Section 4, the covariance is completely determined by observing the process continuously over any finite interval of time. Many important consequences pertaining to properties of tests and estimators, outlined in Section 5, will then apply.     

Nonparametric tests of independence between random vectors

A nonparametric test of the mutual independence between many numerical random vectors is proposed. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of Möbius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. If the number of vectors is p and there are n observations, the test is defined from a collection of processes R_n,A, where A is a subset of {1,…,p} of cardinality |A|>1, which are asymptotically independent and Gaussian. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence cannot be distribution free. The critical values of the proposed test are thus computed with the bootstrap which is shown to be consistent. Another similar test, with the same asymptotic properties, for the serial independence of a multivariate stationary sequence is also proposed. The proposed test works when some or all of the marginal distributions are singular with respect to Lebesgue measure. Moreover, in singular cases described in Section 4, the test inherits useful invariance properties from the general affine invariance property.     

Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.     

Some notes on extremal discriminant analysis

Classical discriminant analysis focusses on Gaussian and nonparametric models where in the second case the unknown densities are replaced by kernel densities based on the training sample. In the present article we assume that it suffices to base the classification on exceedances above higher thresholds, which can be interpreted as observations in a conditional framework. Therefore, the statistical modeling of truncated distributions is merely required. In this context, a nonparametric modeling is not adequate because the kernel method is inaccurate in the upper tail region. Yet one may deal with truncated parametric distributions like the Gaussian ones. Our primary aim is to replace truncated Gaussian distributions by appropriate generalized Pareto distributions and to explore properties and the relationship of discriminant functions in both models.     

Limit distribution of a roundoff error

Let [X] and {X} be the integer and the fractional parts of a random variable X. The conditional distribution function F_n(x)=P({X}≤x|[X]=n) for an integer n is investigated. F_n for a large n is regarded as the distribution of a roundoff error in an extremal event. For most well-known continuous distributions, it is shown that F_n converges as n→∞ and three types of limit distributions appear as the limit distribution according to the tail behavior of F.     

Empirical distributions in marked point processes

We study the asymptotic behaviour of the empirical distribution function derived from a stationary marked point process when a convex sampling window is expanding without bounds in all directions. We consider a random field model which assumes that the marks and the points are independent and admits dependencies between the marks. The main result is the weak convergence of the empirical process under strong mixing conditions on both independent components of the model. Applying an approximation principle weak convergence can be also shown for appropriately weighted empirical process defined from a stationary d-dimensional germ-grain process with dependent grains.     

Change point estimators by local polynomial fits under a dependence assumption

We study a random design regression model generated by dependent observations, when the regression function itself (or its ν-th derivative) may have a change or discontinuity point. A method based on the local polynomial fits with one-sided kernels to estimate the location and the jump size of the change point is applied in this paper. When the jump location is known, a central limit theorem for the estimator of the jump size is established; when the jump location is unknown, we first obtain a functional limit theorem for a local dilated-rescaled version estimator of the jump size and then give the asymptotic distributions for the estimators of the location and the jump size of the change point. The asymptotic results obtained in this paper can be viewed as extensions of corresponding results for independent observations. Furthermore, a simulated example is given to show that our theory and method perform well in practice.     

On Pickands coordinates in arbitrary dimensions

Pickands coordinates were introduced as a crucial tool for the investigation of bivariate extreme value models. We extend their definition to arbitrary dimensions and, thus, we can generalize many known results for bivariate extreme value and generalized Pareto models to higher dimensions and arbitrary extreme value margins.In particular we characterize multivariate generalized Pareto distributions (GPs) and spectral δ-neighborhoods of GPs in terms of best attainable rates of convergence of extremes, which are well-known results in the univariate case. A sufficient univariate condition for a multivariate distribution function (df) to belong to the domain of attraction of an extreme value df is derived. Bounds for the variational distance in peaks-over-threshold models are established, which are based on Pickands coordinates.     

Nonparametric comparison of regression functions

In this work, we provide a new methodology for comparing regression functions m₁ and m₂ from two samples. Since apart from smoothness no other (parametric) assumptions are required, our approach is based on a comparison of nonparametric estimators and of m₁ and m₂, respectively. The test statistics incorporate weighted differences of and computed at selected points. Since the design variables may come from different distributions, a crucial question is where to compare the two estimators. As our main results we obtain the limit distribution of (properly standardized) under the null hypothesis H₀:m₁=m₂ and under local and global alternatives. We are also able to choose the weight function so as to maximize the power. Furthermore, the tests are asymptotically distribution free under H₀ and both shift and scale invariant. Several such ’s may then be combined to get Maximin tests when the dimension of the local alternative is finite. In a simulation study we found out that our tests achieve the nominal level and already have excellent power for small to moderate sample sizes.     

Moment properties of multivariate infinitely divisible laws and criteria for multivariate self-decomposability

Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples.     

A unified approach to testing for and against a set of linear inequality constraints in the product multinomial setting

A problem that is frequently encountered in statistics concerns testing for equality of multiple probability vectors corresponding to independent multinomials against an alternative they are not equal. In applications where an assumption of some type of stochastic ordering is reasonable, it is desirable to test for equality against this more restrictive alternative. Similar problems have been considered heretofore using the likelihood ratio approach. This paper aims to generalize the existing results and provide a unified technique for testing for and against a set of linear inequality constraints placed upon on any probability vectors corresponding to r independent multinomials. The paper shows how to compute the maximum likelihood estimates under all hypotheses of interest and obtains the limiting distributions of the likelihood ratio test statistics. These limiting distributions are of chi bar square type and the expression of the weighting values is given. To illustrate our theoretical results, we use a real life data set to test against second-order stochastic ordering.