Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case

Abstract. In general, the risk of joint extreme outcomes in financial markets can be expressed as a function of the tail dependence function of a high‐dimensional vector after standardizing marginals. Hence, it is of importance to model and estimate tail dependence functions. Even for moderate dimension, non‐parametrically estimating a tail dependence function is very inefficient and fitting a parametric model to tail dependence functions is not robust. In this paper, we propose a semi‐parametric model for (asymptotically dependent) tail dependence functions via an elliptical copula. Under this model assumption, we propose a novel estimator for the tail dependence function, which proves favourable compared to the empirical tail dependence function estimator, both theoretically and empirically.     

An Econometric Model of Network Formation With Degree Heterogeneity

I introduce a model of undirected dyadic link formation which allows for assortative matching on observed agent characteristics (homophily) as well as unrestricted agent‐level heterogeneity in link surplus (degree heterogeneity). Like in fixed effects panel data analyses, the joint distribution of observed and unobserved agent‐level characteristics is left unrestricted. Two estimators for the (common) homophily parameter, β₀, are developed and their properties studied under an asymptotic sequence involving a single network growing large. The first, tetrad logit (TL), estimator conditions on a sufficient statistic for the degree heterogeneity. The second, joint maximum likelihood (JML), estimator treats the degree heterogeneity {A_i0}_{i = 1}^N as additional (incidental) parameters to be estimated. The TL estimate is consistent under both sparse and dense graph sequences, whereas consistency of the JML estimate is shown only under dense graph sequences.     

NONPARAMETRIC REGRESSION ON FUNCTIONAL DATA: INFERENCE AND PRACTICAL ASPECTS

We consider the problem of predicting a real random variable from a functional explanatory variable. The problem is tackled using a nonparametric kernel approach, which has been recently adapted to this functional context. We derive theoretical results from a deep asymptotic analysis of the behaviour of the estimate, including mean squared convergence (with rates and precise evaluation of the constant terms) as well as asymptotic distribution. Practical use of these results relies on the ability to estimate these constants. Some perspectives in this direction are discussed. In particular, a functional version of wild bootstrapping ideas is proposed and used both on simulated and real functional datasets.     

Corrected asymptotic distribution of statistics based on the multinomial law

Investigators and epidemiologists often use statistics based on the parameters of a multinomial distribution. Two main approaches have been developed to assess the inferences of these statistics. The first one uses asymptotic formulae which are valid for large sample sizes. The second one computes the exact distribution, which performs quite well for small samples. They present some limitations for sample sizes N neither large enough to satisfy the assumption of asymptotic normality nor small enough to allow us to generate the exact distribution. We analytically computed the 1/N corrections of the asymptotic distribution for any statistics based on a multinomial law. We applied these results to the kappa statistic in 2×2 and 3×3 tables. We also compared the coverage probability obtained with the asymptotic and the corrected distributions under various hypothetical configurations of sample size and theoretical proportions. With this method, the estimate of the mean and the variance were highly improved as well as the 2.5 and the 97.5 percentiles of the distribution, allowing us to go down to sample sizes around 20, for data sets not too asymmetrical. The order of the difference between the exact and the corrected values was 1/N² for the mean and 1/N³ for the variance.     

一类具有时滞的大系统的稳定性

利用常数变易法和李雅普诺夫函数法给出了一类具有时滞的大系统渐近稳定的条件     

Empirical likelihood inference in the presence of measurement error

Suppose that several different imperfect instruments and one perfect instrument are used independently to measure some characteristic of a population. The authors consider the problem of combining this information to make statistical inference on parameters of interest, in particular the population mean and cumulative distribution function. They develop maximum empirical likelihood estimators and study their asymptotic properties. They also present simulation results on the finite sample efficiency of these estimators.     

On the Relationship between Directional and Omnibus Statistical Tests

Abstract.  A common statistical problem involves the testing of a K -dimensional parameter vector. In both parametric and semiparametric settings, two types of directional tests – linear combination and constrained tests – are frequently used instead of omnibus tests in hopes of achieving greater power for specific alternatives. In this paper, we consider the relationship between these directional tests, as well as their relationship to omnibus tests. Every constrained directional test is shown to be asymptotically equivalent to a specific linear combination test under a sequence of contiguous alternatives and vice versa. Even when the direction of the alternative is known, the constrained test in general will not be optimal unless the objective function used to derive it is efficient. For an arbitrary alternative, insight into the power characteristics of directional tests in comparison to omnibus tests can be gained by a chi-square partition of the omnibus test.     

The Asymptotic Distribution of the S–Gini Index

Several generalizations of the classical Gini index, placing smaller or greater weights on various portions of income distribution, have been proposed by a number of authors. For purposes of statistical inference, the large sample distribution theory of the estimators of those measures of economic inequality is required. The present paper was stimulated by the use of bootstrap by Xu (2000) to estimate the variance of the estimator of the S –Gini index. It shows that the theory of L –statistics (Chernoff, Gastwirth & Johns, 1967; Shorack & Wellner, 1986) makes possible the construction of a consistent estimator for the S –Gini index and proof of its asymptotic normality. The paper also presents an explicit formula for the asymptotic variance. The formula should be helpful in planning the size of samples from which the S –Gini index can be estimated with a prescribed margin of error.     

Propriétés dans L2 et estimation des processus purement bilinéaires et strictement superdiagonaux à coefficients périodiques

L₂‐properties and estimation of purely bilinear and strictly superdiagonal time series models with periodic coefficients The authors consider the subclass of purely bilinear and strictly superdiagonal time series models with periodic coefficients. Indeed, thanks to their possible application to a wide variety of fields including economics and finance, bilinear time series models with time‐dependent coefficients have recently been the object of attention in the statistical literature. The authors give conditions ensuring the existence of a causal solution in L², the invertibility and the existence of higher‐order moments. The problem of estimating the parameters is also investigated through an approach based on second and third empirical moments. The authors numerically illustrate their theoretical results via Monte Carlo simulations.     

高阶非线性中立型微分方程解的渐近性与振动性

高阶非线性中立型微分方程解的渐近性与振动性...