On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

Characterizations of degree one bivariate measures of concordance

We call a measure of concordance κ of an ordered pair (X,Y) of two continuous random variables a bivariate measure of concordance. This κ may be considered to be a function κ(C) of the copula C associated with (X,Y). κ is considered to be of degree n if, given any two copulas A and B, the value of their convex sum, κ(tA+(1−t)B), is a polynomial in t of degree n. Examples of bivariate measures of concordance are Spearman’s rho, Blomqvist’s beta, Gini’s measure of association, and Kendall’s tau. The first three of these are of degree one, but Kendall’s tau is of degree two. We exhibit three characterizations of bivariate measures of concordance of degree one.     

Concordance measures for multivariate non-continuous random vectors

A notion of multivariate concordance suitable for non-continuous random variables is defined and many of its properties are established. This allows the definition of multivariate, non-continuous versions of Kendall’s tau, Spearman’s rho and Spearman’s footrule, which are concordance measures. Since the maximum values of these association measures are not +1 in general, a special attention is given to the computation of upper bounds. The latter turn out to be multivariate generalizations of earlier findings made by Nešlehová (2007) [9] and Denuit and Lambert (2005) [2]. They are easy to compute and can be estimated from a data set of (possibly) discontinuous random vectors. Corrected versions are considered as well.     

多参数情形下 MLE 的大偏差

在大样本点估计理论中,我们一般只考虑相合估计,对于任何一个相合估计 T_n 以及任何给定的ε>0,其尾概率     

Copula parameter estimation by maximum-likelihood and minimum-distance estimators: a simulation study

The purpose of this paper is to present a comprehensive Monte Carlo simulation study on the performance of minimum-distance (MD) and maximum-likelihood (ML) estimators for bivariate parametric copulas. In particular, I consider Cramér-von-Mises-, Kolmogorov-Smirnov- and L ₁-variants of the CvM-statistic based on the empirical copula process, Kendall’s dependence function and Rosenblatt’s probability integral transform. The results presented in this paper show that regardless of the parametric form of the copula, the sample size or the location of the parameter, maximum-likelihood yields smaller estimation biases at less computational effort than any of the MD-estimators. The MD-estimators based on copula goodness-of-fit metrics, on the other hand, suffer from large biases especially when used for estimating the parameters of archimedean copulas. Moreover, the results show that the bias and efficiency of the minimum-distance estimators are strongly influenced by the location of the parameter. Conversely, the results for the maximum-likelihood estimator are relatively stable over the parameter interval of the respective parametric copula.     

From Archimedean to Liouville copulas

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall’s tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.     

A multivariate version of Hoeffding’s Phi-Square

A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.     

Local efficiency of a Cramér-von Mises test of independence

Deheuvels proposed a rank test of independence based on a Cramér-von Mises functional of the empirical copula process. Using a general result on the asymptotic distribution of this process under sequences of contiguous alternatives, the local power curve of Deheuvels’ test is computed in the bivariate case and compared to that of competing procedures based on linear rank statistics. The Gil-Pelaez inversion formula is used to make additional comparisons in terms of a natural extension of Pitman's measure of asymptotic relative efficiency.     

Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models

Parameters of Gaussian multivariate models are often estimated using the maximum likelihood approach. In spite of its merits, this methodology is not practical when the sample size is very large, as, for example, in the case of massive georeferenced data sets. In this paper, we study the asymptotic properties of the estimators that minimize three alternatives to the likelihood function, designed to increase the computational efficiency. This is achieved by applying the information sandwich technique to expansions of the pseudo-likelihood functions as quadratic forms of independent normal random variables. Theoretical calculations are given for a first-order autoregressive time series and then extended to a two-dimensional autoregressive process on a lattice. We compare the efficiency of the three estimators to that of the maximum likelihood estimator as well as among themselves, using numerical calculations of the theoretical results and simulations.     

二元Friday-Patil型指数分布的特征及其应用

导出了二元Friday-Patil型指数分布的一个特征,利用该特征获得了二元Friday-Patil型指数分布参数的最大似然估计及矩估计,给出了强度服从二元Friday-Patil型指数分布时系统可靠度的估计.     

二元Block～Basu型指数分布的特征及其应用

导出了二元Block~Basu型指数分布的一个特征,利用该特征,获得了二元Block~Basu型指数分布参数的最大似然估计及矩估计,给出了强度服从二元Block~Basu型分布时并联结构系统可靠度的估计,并给出了二元Block~Basu型指数分布的一个随机模拟.     

Asymptotic optimality of the generalized bayes estimator in multiparameter cases

The higher order asymptotic efficiency of the generalized Bayes estimator is discussed in multiparameter cases. For all symmetric loss functions, the generalized Bayes estimator is second order asymptotically efficient in the classA ₂ of the all second order asymptotically median unbiased (AMU) estimators and third order asymptotically efficient in the restricted classD of estimators.     

Minimum f-divergence estimators and quasi-likelihood functions

Maximum quasi-likelihood estimators have several nice asymptotic properties. We show that, in many situations, a family of estimators, called the minimum f-divergence estimators, can be defined such that each estimator has the same asymptotic properties as the maximum quasi-likelihood estimator. The family of minimum f-divergence estimators include the maximum quasi-likelihood estimators as a special case. When a quasi-likelihood is the log likelihood from some exponential family, Amari's dual geometries can be used to study the maximum likelihood estimator. A dual geometric structure can also be defined for more general quasi-likelihood functions as well as for the larger family of minimum f-divergence estimators. The relationship between the f-divergence and the quasi-likelihood function and the relationship between the f-divergence and the power divergence is discussed.This work was supported by National Science Foundation grant DMS 88-03584.     

Gaining effciency via weighted estimators for mul-tivariate failure time data

Multivariate failure time data arise frequently in survival analysis.A commonly used tech-nique is the working independence estimator for marginal hazard models.Two natural questions are how to improve the effciency of the working independence estimator and how to identify the situations under which such an estimator has high statistical effciency.In this paper,three weighted estimators are proposed based on three different optimal criteria in terms of the asymptotic covariance of weighted estimators.Simpli...     

Testing for small bias of tail index estimators

We determine the joint asymptotic normality of kernel and weighted least-squares estimators of the upper tail index of a regularly varying distribution when each estimator is a bivariate function of two parameters: the tuning parameter is motivated by possible underlying second-order behavior in regular variation, while no such behavior is assumed, and the fraction parameter determines that upper portion of the sample on which the estimator is based. Under the hypothesis that the scaled asymptotic biases of the estimators vanish uniformly in the parameter points considered, these results imply joint asymptotic normality for deviations of ratios of the estimators from 1, which in turn yield asymptotic chi-square tests for checking the small-bias hypothesis, equivalent to the constructibility of asymptotic confidence intervals. The test procedure suggests adaptive choices of the tuning and fraction parameters: data-driven (t)estimators.     

Accurate distribution and its asymptotic expansion for the tetrachoric correlation coefficient

Accurate distributions of the estimator of the tetrachoric correlation coefficient and, more generally, functions of sample proportions for the 2 by 2 contingency table are derived. The results are obtained given the definitions of the estimators even when some marginal cell(s) are empty. Then, asymptotic expansions of the distributions of the parameter estimators standardized by the population asymptotic standard errors up to order O(1/n) and those of the studentized ones up to the order next beyond the conventional normal approximation are derived. The asymptotic results can be obtained in a much shorter computation time than the accurate ones. Numerical examples were used to illustrate advantages of the studentized estimator of Fisher’s z transformation of the tetrachoric correlation coefficient.     

Extreme behavior of bivariate elliptical distributions

This paper exploits a stochastic representation of bivariate elliptical distributions in order to obtain asymptotic results which are determined by the tail behavior of the generator. Under certain specified assumptions, we present the limiting distribution of componentwise maxima, the limiting upper copula, and a bivariate version of the classical peaks over threshold result.     

Superefficient estimation of the marginals by exploiting knowledge on the copula

We consider the problem of estimating the marginals in the case where there is knowledge on the copula. If the copula is smooth, it is known that it is possible to improve on the empirical distribution functions: optimal estimators still have a rate of convergence n^−1/2, but a smaller asymptotic variance. In this paper we show that for non-smooth copulas it is sometimes possible to construct superefficient estimators of the marginals: we construct both a copula and, exploiting the information our copula provides, estimators of the marginals with the rate of convergence logn/n.     

Wavelets-based estimation of nonlinear canonical analysis

Using a wavelets-based estimator of the bivariate density, we introduce an estimation method for nonlinear canonical analysis. Consistency of the resulting estimators of the canonical coefficients and the canonical functions is established. Under some conditions, asymptotic normality results for these estimators are obtained. Then it is shown how to compute in practice these estimators by usingmatrix computations, and the finite-sample performance of the proposed method is evaluated through simulations.     

Representation of Downton’s bivariate exponential random vector and its applications

Downton’s bivariate exponential distribution is one of the most important bivariate distributions in reliability theory. In this paper a simple representation for Downton’s bivariate exponential random vector is given. As an application of this representation, we consider a reliability model where an item is subject to shocks and obtain an explicit expression for the long-run cost rate.