Estimation of conditional laws given an extreme component

Let (X,Y) be a bivariate random vector. The estimation of a probability of the form P(Y ≤ y |X > t) is challenging when t is large, and a fruitful approach consists in studying, if it exists, the limiting conditional distribution of the random vector (X,Y), suitably normalized, given that X is large. There already exists a wide literature on bivariate models for which this limiting distribution exists. In this paper, a statistical analysis of this problem is done. Estimators of the limiting distribution (which is assumed to exist) and the normalizing functions are provided, as well as an estimator of the conditional quantile function when the conditioning event is extreme. Consistency of the estimators is proved and a functional central limit theorem for the estimator of the limiting distribution is obtained. The small sample behavior of the estimator of the conditional quantile function is illustrated through simulations. Some real data are analysed.     

Tail and dependence behavior of levels that persist for a fixed period of time

This work emerges from a study of the extremal behavior of a daily maximum sea water levels series, {X _i } , presented in Draisma (Duration of extremes at sea. In: Parametric and semi-parametric methods in E. V. T., pp. 137–143. PhD thesis, Erasmus, University, 2001). In its approach, a new series, {Y _i }, is defined, consisting of water levels that persist for a fixed period of time. In this paper, we study the tail behavior of {Y _i } , in case {X _i } is independent and identically distributed (i.i.d.) and in case {X _i } is a max-autoregressive sequence (we will consider two different max-autoregressive processes), whose distribution function is in the Fréchet domain of attraction. We also determine Ledford and Tawn tail dependence index (Ledford and Tawn, Biometrika 83:169–187, 1996, J. R. Stat. Soc. B 59:475–499, 1997) and we analyze the asymptotic tail dependence of the random pair (Y _i , Y _i + m ), in all considered cases. According to Drees (Bernoulli 9:617–657, 2003), we obtain the limit behavior of the tail empirical quantile function associated with a random sample (Y ₁, Y ₂,...Y _n ) and hence the asymptotic normality of a class of estimators of the tail index that includes Hill estimator. Research partially supported by FCT/POCTI and POCI/FEDER.     

The approximation of logarithmic function by q-Bernstein polynomials in the case q > 1

Since in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 0<q<1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0,1] uniformly approximated by their q-Bernstein polynomials (q > 1) remains open. It is known that the approximation occurs for functions admit ting an analytic continuation into a disc {z:|z| < R}, R > 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown that the function f(x) = ln (x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials (q > 1) on the interval [0,1] if and only if a ≥ 1.      

Mixed moment estimator and location invariant alternatives

A new class of estimators of the extreme value index is developed. It has a simple form and is asymptotically very close to the maximum likelihood estimator for a wide class of heavy-tailed models. We also propose an alternative class of estimators, dependent on a tuning parameter p ∈ (0,1) and invariant for changes in both scale and/or location. Such a tuning parameter can help us to choose the number of top order statistics to be used in the estimation of extreme parameters. Research partially supported by FCT / POCTI, POCI, PCDT and PPCDT / FEDER.     

On the robust estimation in poisson processes with periodic intensities

Under some regularity conditions, it is well known that the maximum likelihood estimator (MLE) is asymptotically normal and efficient. However, if the observation is contaminated, the MLE is not always an appropriate estimator. In this paper, we treat M-estimators and study their asymptotic behavior. By choosing estimation equations, robust M-estimators are presented for phase parameters.     

Conditioning exceedances on covariate processes

This article concerns the statistical inference for the upper tail of the conditional distribution of a response variable Y given a covariate X = x based on n random vectors within the parametric extreme value framework. Pioneering work in this field was done by Smith (Stat Sci 4:367–393, 1989) and Smith and Shively (Atmos Environ 29:3489–3499, 1995). We propose to base the inference on a conditional distribution of the point process of exceedances given the point process of covariates. It is of importance that the conditional distribution merely depends on the conditional distribution of the response variable given the covariates. In the special case of Poisson processes such a result may be found in Reiss (1993). Our results are valid within the broader model where the response variables are conditionally independent given the covariates. It is numerically exemplified that the maximum likelihood principle leads to more accurate estimators within the conditional approach than in the previous one.     

Estimating the Scale Parameter of a Lévy-stable Distribution via the Extreme Value Approach

The characteristic exponent α of a Lévy-stable law S _α (σ, β, μ) was thoroughly studied as the extreme value index of a heavy tailed distribution. For 1 < α < 2, Peng (Statist. Probab. Lett. 52: 255–264, 2001) has proposed, via the extreme value approach, an asymptotically normal estimator for the location parameter μ. In this paper, we derive by the same approach, an estimator for the scale parameter σ and we discuss its limiting behavior.      

It was 30 years ago today when Laurens de Haan went the multivariate way

Since the publication of his masterpiece on regular variation and its application to the weak convergence of (univariate) sample extremes in 1970, Laurens de Haan (Thesis, Mathematical Centre Tract vol. 32, University of Amsterdam, 1970) is among the leading mathematicians in the world, with a particular focus on extreme value theory (EVT). On the occasion of his 70th birthday it is a great pleasure and a privilege to follow his route through multivariate EVT, which started only seven years later in 1977, when Laurens de Haan published his first paper on multivariate EVT, jointly with Sid Resnick.      

Full likelihood inferences in the Cox model: an empirical likelihood approach

For the regression parameter β ₀ in the Cox model, there have been several estimators constructed based on various types of approximated likelihood, but none of them has demonstrated small-sample advantage over Cox’s partial likelihood estimator. In this article, we derive the full likelihood function for (β ₀, F ₀), where F ₀ is the baseline distribution in the Cox model. Using the empirical likelihood parameterization, we explicitly profile out nuisance parameter F ₀ to obtain the full-profile likelihood function for β ₀ and the maximum likelihood estimator (MLE) for (β ₀, F ₀). The relation between the MLE and Cox’s partial likelihood estimator for β ₀ is made clear by showing that Taylor’s expansion gives Cox’s partial likelihood estimating function as the leading term of the full-profile likelihood estimating function. We show that the log full-likelihood ratio has an asymptotic chi-squared distribution, while the simulation studies indicate that for small or moderate sample sizes, the MLE performs favorably over Cox’s partial likelihood estimator. In a real dataset example, our full likelihood ratio test and Cox’s partial likelihood ratio test lead to statistically different conclusions.     

Estimation of the extreme value index and extreme quantiles under random censoring

In this paper, we consider the estimation of the extreme value index and extreme quantiles in the presence of random right censoring. The generalization of the peaks over threshold method is discussed and an adaptation of the moment estimator is proposed. The corresponding extreme quantile estimators are also introduced. We make a start with the analysis of the asymptotic properties of the moment estimator and the corresponding extreme quantile estimator. The finite sample behaviour is illustrated with a small simulation study and through practical examples from survival data analysis.      

Testing the tail–dependence based on the radial component

Throughout this article we assume that the df H of a random vector (X,Y) is in the max-domain of attraction of an extreme value distribution function (df) G with reverse exponential margins. Therefore, the asymptotic dependence structure of H can be represented by a Pickands dependence function D with D = 1 representing the case of asymptotic independence. One of our aims is to test the null hypothesis of tail-dependence against the alternative of tail-independence. Thus we want to prove the validity of the model where D = 1. The test is based on the radial component X + Y. Under a certain spectral expansion it is verified that the df of X + Y, conditioned on X + Y > c, converges to F(t) = t, as c ↑0, if D ≠ 1 and, respectively, to F(t) = t ^1 + ρ , if D = 1, where ρ > 0 determines the rate at which independence is attained. Based on the limiting dfs we find a uniformly most powerful test procedure for testing tail-dependence against rates of tail-independence. In addition, an estimator of the parameter ρ is proposed. The relationship of ρ to another dependence measure, given in the literature, is indicated.      

A coding problem in steganography

To study how to design a steganographic algorithm more efficiently, a new coding problem—steganographic codes (abbreviated stego-codes)—is presented in this paper. The stego-codes are defined over the field with q(q ≥ 2) elements. A method of constructing linear stego-codes is proposed by using the direct sum of vector subspaces. And the problem of linear stego-codes is converted to an algebraic problem by introducing the concept of the tth dimension of a vector space. Some bounds on the length of stego-codes are obtained, from which the maximum length embeddable (MLE) code arises. It is shown that there is a corresponding relation between MLE codes and perfect error-correcting codes. Furthermore the classification of all MLE codes and a lower bound on the number of binary MLE codes are obtained based on the corresponding results on perfect codes. Finally hiding redundancy is defined to value the performance of stego-codes.      

Stable Subconstructs: A Correction and New Results

In Lowen and Wuyts (Appl Categ Struct 8:235–245, 2000) the authors studied the simultaneously concretely reflective and concretely coreflective subconstructs of the category Ap of approach spaces. For the sake of shortness we call such subconstructs stable. Using a technique introduced in Herrlich and Lowen (1999) it was possible to explicitly describe such stable subconstructs by a condition on the objects which used certain subsets of [0, ∞ ]. Thus each stable subconstruct Ap _m described in [9] corresponds to the subset {0} ∪ [m, ∞ ] ⊂ [0, ∞ ] for m ∈ [0, ∞ ]. Although this characterization is correct, Theorem 4.7 in [9] stating that the subconstructs Ap _m were the only stable subconstructs of Ap is not. The main results, which together prove that the only stable subconstructs are those where a restriction is put on the range of the distances of the objects, are upheld, but it turns out that not only the sets {0} ∪ [m, ∞ ], but actually each closed subsemigroup of [0, ∞ ] determines a stable subconstruct (albeit again in exactly the same way as characterized in [9]). In the first part of our paper, Sections 1 and 2, we develop the general technique, which is totally different to the one from [3], and in Theorem 2.13 we prove the main result for the case of approach spaces. The technique which we develop is also applicable to other cases. Thus, in Section 3, more precisely in Theorems 3.9 and 3.11, we give the complete solution to the corresponding characterization problem for the constructs pq Met ^∞ of pseudo-quasi-metric spaces and p Met ^∞ of pseudometric spaces and in Section 4 we briefly sketch how the technique can be adapted and used to also completely solve the problem in the case of more general types of approach spaces and metric spaces. At the same time, in all cases, we are able to give necessary and sufficient conditions under which two stable subconstructs of one of these topological constructs are concretely isomorphic. It turns out that in all cases there are 2^à₀2^{\aleph_0} non-concretely isomorphic stable subconstructs.     

Orders on Multisets and Discrete Cones

We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of independence of equal submultisets (IES) introduced by Sertel and Slinko (Ranking committees, words or multisets. Nota di Laboro 50.2002. Center of Operation Research and Economics. The Fundazione Eni Enrico Mattei, Milan, 2002, Econ. Theory 30(2):265–287, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (J. Math. Psychol., 40:64–77, 1996) and Conder and Slinko (J. Math. Psychol., 48(6):425–431, 2004), we define functions f(n,k) and g(n,k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that g(n,k) = n − 1 whenever n ≥ 3 and (n, k) ≠ (5, 2). In the exceptional case, g(5,2) = 3. We also prove that g(n,k) ≤ f(n,k) ≤ n and establish that for small n and k the functions g(n,k) and f(n,k) coincide.      

The efficiency of the second-order nonlinear least squares estimator and its extension

We revisit the second-order nonlinear least square estimator proposed in Wang and Leblanc (Anne Inst Stat Math 60:883–900, 2008) and show that the estimator reaches the asymptotic optimality concerning the estimation variability. Using a fully semiparametric approach, we further modify and extend the method to the heteroscedastic error models and propose a semiparametric efficient estimator in this more general setting. Numerical results are provided to support the results and illustrate the finite sample performance of the proposed estimator.     

On the estimation of parameters for linear stochastic differential equations

The estimation of arbitrary number of parameters in linear stochastic differential equation (SDE) is investigated. The local asymptotic normality (LAN) of families of distributions corresponding to this SDE is established and the asymptotic efficiency of the maximum likelihood estimator (MLE) is obtained for the wide class of loss functions with polynomial majorants. As an example a single-degree of freedom mechanical system is considered. The results generalize [8], where all elements of the drift matrix are estimated and the asymptotic efficiency is proved only for the bounded loss functions. Received: 12 March 1997 / Revised version: 22 June 1998     

Kullback-Leibler Information Consistent Estimation for Censored Data

This paper is intended as an investigation of parametric estimation for the randomly right censored data. In parametric estimation, the Kullback-Leibler information is used as a measure of the divergence of a true distribution generating a data relative to a distribution in an assumed parametric model M. When the data is uncensored, maximum likelihood estimator (MLE) is a consistent estimator of minimizing the Kullback-Leibler information, even if the assumed model M does not contain the true distribution. We call this property minimum Kullback-Leibler information consistency (MKLI-consistency). However, the MLE obtained by maximizing the likelihood function based on the censored data is not MKLI-consistent. As an alternative to the MLE, Oakes (1986, Biometrics, 42, 177–182) proposed an estimator termed approximate maximum likelihood estimator (AMLE) due to its computational advantage and potential for robustness. We show MKLI-consistency and asymptotic normality of the AMLE under the misspecification of the parametric model. In a simulation study, we investigate mean square errors of these two estimators and an estimator which is obtained by treating a jackknife corrected Kaplan-Meier integral as the log-likelihood. On the basis of the simulation results and the asymptotic results, we discuss comparison among these estimators. We also derive information criteria for the MLE and the AMLE under censorship, and which can be used not only for selecting models but also for selecting estimation procedures.     

On q *-bisimple IC semigroups of type-E

A semigroup is called type-E if the band of its idempotents can be expressed as a direct product of a rectangular band and an ω-chain. For brevity, we call an IC ^*-bisimple quasi-adequate semigroup of type-E a q ^*-bisimple IC semigroup of type-E. In this paper, we characterize q ^*-bisimple semigroups by using some kind of generalized Bruck-Reilly extensions. As a consequence, some results concerning *-bisimple type-A ω-semigroups given by Asibong-Ibe (Semigroup Forum 31:99–117, 1985) are generalized.     

Complete classification of the positive solutions of −Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We study the equation −Δu + u ^q = 0, q > 1, in a bounded C ² domain Ω ⊂ ℝ ^N . A positive solution of the equation is moderate if it is dominated by a harmonic function and σ-moderate if it is the limit of an increasing sequence of moderate solutions. It is known that in the subcritical case, 1 < q <, q _c = (N + 1)/(N − 1), every positive solution is σ-moderate [32]. More recently, Dynkin proved, by probabilistic methods, that this remains valid in the supercritical case for q ≤ 2, [15]. The question remained open for q > 2. In this paper, we prove that for all q ≥ q _c , every positive solution is σ-moderate. We use purely analytic techniques, which apply to the full supercritical range. The main tools come from linear and non-linear potential theory. Combined with previous results, our result establishes a one-to-one correspondence between positive solutions and their boundary traces in the sense of [36].     

Variance estimation for sample quantiles using them out ofn bootstrap

We consider the problem of estimating the variance of a sample quantile calculated from a random sample of sizen. Ther-th-order kernel-smoothed bootstrap estimator is known to yield an impressively small relative error of orderO(n ^−r/(2r+1) ). It nevertheless requires strong smoothness conditions on the underlying density function, and has a performance very sensitive to the precise choice of the bandwidth. The unsmoothed bootstrap has a poorer relative error of orderO(n ^−1/4), but works for less smooth density functions. We investigate a modified form of the bootstrap, known as them out ofn bootstrap, and show that it yields a relative error of order smaller thanO(n ^−1/4) under the same smoothness conditions required by the conventional unsmoothed bootstrap on the density function, provided that the bootstrap sample sizem is of an appropriate order. The estimator permits exact, simulation-free, computation and has accuracy fairly insensitive to the precise choice ofm. A simulation study is reported to provide empirical comparison of the various methods. Supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7131/00P).