Tests de structure en régression sur variable fonctionnelle

In this Note we introduce a general approach to construct structural testing procedures in regression on functional variables. In the case of multivariate explanatory variables a well-known method consists in a comparison between a nonparametric estimator and a particular one. We adapt this approach to the case of functional explanatory variables. We give the asymptotic law of the proposed test statistic. The general approach used allows us to cover a large scope of possible applications as tests for no-effect, tests for linearity, …. To cite this article: L. Delsol, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

多元测量误差模型的稳健GM-估计量

由于EV（Errores-in-Variables)模型（也称测量误差模型）的最大似然估计由正交回归给出，而正交回归对污染数据是敏感的，所以，需要采用稳健的统计方法来估计模型参数。本文在多元EV模型中引入稳健GM-估计量，把一元正态EV模型的若干结果推广到多元情形，所得的稳健性结果不仅更具一般性，而且还修正了文献中对一元情形给出的一个错误结果。     

On a mapping approach to investigating the bootstrap accuracy

Summary. A simple mapping approach is proposed to study the bootstrap accuracy in a rather general setting. It is demonstrated that the bootstrap accuracy can be obtained through this method for a broad class of statistics to which the commonly used Edgeworth expansion approach may not be successfully applied. We then consider some examples to illustrate how this approach may be used to find the bootstrap accuracy and show the advantage of the bootstrap approximation over the Gaussian approximation. For the multivariate Kolmogorov–Smirnov statistic, we show the error of bootstrap approximation is as small as that of the Gaussian approximation. For the multivariate kernel type density estimate, we obtain an order of the bootstrap error which is smaller than the order of the error of the Gaussian approximation given in Rio (1994). We also consider an application of the bootstrap accuracy for empirical process to that for the copula process. Received: 23 June 1995 / In revised form: 18 June 1996     

On the Tail Mean-Variance optimal portfolio selection

In the present paper we propose the Tail Mean-Variance (TMV) approach, based on Tail Condition Expectation (TCE) (or Expected Short Fall) and the recently introduced Tail Variance (TV) as a measure for the optimal portfolio selection. We show that, when the underlying distribution is multivariate normal, the TMV model reduces to a more complicated functional than the quadratic and represents a combination of linear, square root of quadratic and quadratic functionals. We show, however, that under general linear constraints, the solution of the optimization problem still exists and in the case where short selling is possible we provide an analytical closed form solution, which looks more “robust” than the classical MV solution. The results are extended to more general multivariate elliptical distributions of risks.     

经济问题的模糊聚类

应用模糊聚类的正交方法，解决经济领域中的一个聚类问题。     

On Location and Approximation of Clusters of Zeros of Analytic Functions

At the beginning of the 1980s, M. Shub and S. Smale developed a quantitative analysis of Newton's method for multivariate analytic maps. In particular, their α-theory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only information concerning the map at the initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem. In this paper we focus on one complex variable function. We study general criteria for detecting clusters and analyze the convergence of Schroder's iteration to a cluster. In the case of a multiple root, it is well known that this convergence is quadratic. In the case of a cluster with positive diameter, the convergence is still quadratic provided the iteration is stopped sufficiently early. We propose a criterion for stopping this iteration at a distance from the cluster which is of the order of its diameter.     

Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with a hermitian functional. We give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case.A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Durán and Grünbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.     

Mixed effects regression trees for clustered data

This paper presents an extension of the standard regression tree method to clustered data. Previous works extending tree methods to accommodate correlated data are mainly based on the multivariate repeated-measures approach. We propose a “mixed effects regression tree” method where the correlated observations are viewed as nested within clusters rather than as vectors of multivariate repeated responses. The proposed method can handle unbalanced clusters, allows observations within clusters to be split, and can incorporate random effects and observation-level covariates. We implemented the proposed method using a standard tree algorithm within the framework of the expectation-maximization (EM) algorithm. The simulation results show that the proposed regression tree method provides substantial improvements over standard trees when the random effects are non negligible. A real data example is used to illustrate the method.     

A generalized Goulden–Jackson cluster method and lattice path enumeration

The Goulden–Jackson cluster method is a powerful tool for obtaining generating functions counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths – both with height bounded and unbounded – by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued fraction formulas.     

Sparse Multivariate Regression With Covariance Estimation

We propose a procedure for constructing a sparse estimator of a multivariate regression coefficient matrix that accounts for correlation of the response variables. This method, which we call multivariate regression with covariance estimation (MRCE), involves penalized likelihood with simultaneous estimation of the regression coefficients and the covariance structure. An efficient optimization algorithm and a fast approximation are developed for computing MRCE. Using simulation studies, we show that the proposed method outperforms relevant competitors when the responses are highly correlated. We also apply the new method to a finance example on predicting asset returns. An R-package containing this dataset and code for computing MRCE and its approximation are available online.