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Many authors have exploited the fact that the distribution of the multivariate probability integral transformation (PIT) of a continuous random vector X ∈ R d with cumulative distribution function F X is free of the marginal distributions. While most of these methods are based on the cdf of W = F X (X) , this paper introduces the weighted characteristic function (WCf) of W. A sample version of the WCf of W based on pseudo-observations is proposed and its weak limit in a space of complex functions is formally established. This result can be used to define test statistics for multivariate independence and goodness-of-fit in copula models, whose asymptotic behaviour comes from the weak convergence of the empirical WCf process. Simulations show the good sampling properties of these new tests, and an illustration is given on the multivariate Cook and Johnson dataset. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with modeling the dependence structure of two (or more) time series in the presence of a (possibly multivariate) covariate which may include past values of the time series. We assume that the covariate influences only the conditional mean and the conditional variance of each of the time series but the distribution of the standardized innovations is not influenced by the covariate and is stable in time. The joint distribution of the time series is then determined by the conditional means, the conditional variances and the marginal distributions of the innovations, which we estimate nonparametrically, and the copula of the innovations, which represents the dependency structure. We consider a nonparametric and a semiparametric estimator based on the estimated residuals. We show that under suitable assumptions, these copula estimators are asymptotically equivalent to estimators that would be based on the unobserved innovations. The theoretical results are illustrated by simulations and a real data example. [ABSTRACT FROM AUTHOR]

A common assumption in pair-copula constructions is that the copula of the conditional distribution of two random variables given a covariate does not depend on the value of that covariate. Two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators of that copula. On the one hand, the best possible rates for estimating the marginal conditional distribution functions are slower than the parametric one. On the other hand, the invariance of the conditional copula given the value of the covariate suggests the possibility of parametric convergence rates. The more optimistic intuition is shown to be correct, confirming a conjecture supported by extensive Monte Carlo simulations by Hobæk Haff and Segers (2015) and improving upon the nonparametric rate obtained theoretically by Gijbels et al. (2015). The novelty of the proposed approach lies in a double smoothing procedure for the estimator of the marginal conditional distribution functions. The copula estimator itself is asymptotically equivalent to an oracle empirical copula, as if the marginal conditional distribution functions were known. [ABSTRACT FROM AUTHOR]

For a strictly stationary sequence of R + d –valued random vectors we derive functional convergence of partial maxima stochastic processes under joint regular variation and weak dependence conditions. The limit process is an extremal process and the convergence takes place in the space of R + d –valued càdlàg functions on [ 0 , 1 ] , with the Skorohod weak M 1 topology. We also show that this topology in general cannot be replaced by the stronger (standard) M 1 topology. The theory is illustrated on three examples, including the multivariate squared GARCH process with constant conditional correlations. [ABSTRACT FROM AUTHOR]

A general result on weak convergence of the empirical measure of discretely observed functional data is shown. It is applied to the problem of estimation of functional mean value, and the problem of consistency of various types of depth for functional data. Counterexamples illustrating the fact that the assumptions as stated cannot be dropped easily are given. [ABSTRACT FROM AUTHOR]

This article studies the weak convergence and associated Central Limit Theorem for blurring and nonblurring processes. Then, they are applied to the estimation of location parameter. Simulation studies show that the location estimation based on the convergence point of blurring process is more robust and often more efficient than that of nonblurring process. [ABSTRACT FROM AUTHOR]

Abstract: We propose a new sequential procedure to detect change in the parameters of a process belonging to a large class of causal models (such as AR( ), ARCH( ), TARCH( ), or ARMA–GARCH processes). The procedure is based on a difference between the historical parameter estimator and the updated parameter estimator, where both these estimators are quasi-likelihood estimators. Unlike classical recursive fluctuation test, the updated estimator is computed without the historical observations. The asymptotic behavior of the test is studied and the consistency in power as well as an upper bound of the detection delay is obtained. Some simulation results are reported with comparisons to some other existing procedures exhibiting the accuracy of our new procedure. This procedure coupled with retrospective tests is applied to solve off-line multiple breaks detection in the daily closing values of the FTSE 100 stock index. [Copyright &y& Elsevier]

Abstract: Case-cohort designs provide a cost effective way to conduct epidemiological follow-up studies in which event times are the outcome variables. This paper develops a quantile regression approach to the analysis of case-cohort data. Quantile regression is a highly useful tool to delineate relationships between the outcome variable and covariates. Unbiased functional estimating equations are constructed, resulting in asymptotically unbiased estimators. Efficient algorithms based on minimizing -type convex functions are given. Uniform consistency and weak convergence of the resulting estimators are established. Error estimation and confidence intervals are obtained by applying a specially designed resampling procedure for case-cohort data. Simulation studies are conducted to assess the performance of the proposed method. An example is also provided for illustration. [Copyright &y& Elsevier]

Abstract: Empirical and sequential empirical copula processes play a central role for statistical inference on copulas. However, as pointed out by Johan Segers [J. Segers, Asymptotics of empirical copula processes under non-restrictive smoothness assumptions, Bernoulli 18 (3) (2012) 764–782] the usual assumptions under which these processes have been studied so far are too restrictive. In this paper, we provide a unified approach to the analysis of empirical and sequential empirical copula processes that circumvents those restrictive assumptions in a very general setting. In particular, our methods allow for an easy analysis of copula processes and appropriate bootstrap approximations in the setting of sequentially dependent data. One particularly useful finding is that certain sequential empirical copula processes converge without any smoothness assumptions on the copula. [Copyright &y& Elsevier]

Abstract: We observe two sequences of curves which are connected via an integral operator. Our model includes linear models as well as autoregressive models in Hilbert spaces. We wish to test the null hypothesis that the operator did not change during the observation period. Our method is based on projecting the observations onto a suitably chosen finite dimensional space. The testing procedure is based on functionals of the weighted residuals of the projections. Since the quadratic form is based on estimating the long-term covariance matrix of the residuals, we also provide some results on Bartlett-type estimators. [Copyright &y& Elsevier]

Abstract: The peakedness of a random variable (RV) about a point is defined by . A RV is said to be less peaked about than a RV about , denoted by , if for all , i.e., is stochastically larger than . These generalize the original definitions of Birnbaum (1948) who considered the cases where and were symmetric about and , respectively. Statistical inferences about the distribution functions of continuous and under peakedness ordering in the symmetric case have been treated in the literature. Rojo et al. (2007) provided estimators of the distributions in the general case and analyzed their properties. We show that these estimators could have poor asymptotic properties relative to those of the empiricals. We provide improved estimators of the DFs, show that they are consistent, derive the weak convergence of the estimators, compare them with the empirical estimators, and provide formulas for statistical inferences. An example is also used to illustrate our theoretical results. [Copyright &y& Elsevier]

Abstract: Truncation occurs when the variable of interest can be observed only if its value satisfies certain selection criteria. Most existing methods for analyzing such data critically rely on the assumption that the truncation variable is quasi-independent of the variable of interest. In this article, the authors propose a likelihood-based inference approach under the assumption that the dependence structure of the two variables follows a general form of copula model. They develop a model selection method for choosing the best-fitted copula among a broad class of model alternatives, and they derive large-sample properties of the proposed estimators, including the inverse Fisher information matrix. The treatment of ties is also discussed. They apply their methods to the analysis of a transfusion-related AIDS data set and compare the results with existing methods. Simulation results are also provided to evaluate the finite-sample performances of all the competing methods. [Copyright &y& Elsevier]

Abstract: We prove general theorems that characterize situations in which we could have asymptotic closeness between the original statistics and its bootstrap version , without stipulating the existence of weak limits. As one possible application we introduce a novel goodness of fit test based on the modification of Total Variation metric. This new statistic is more sensitive than the Kolmogorov–Smirnov statistic, it applies to higher dimensions, and it does not converge weakly; but we show that it can be bootstrapped. [Copyright &y& Elsevier]

Abstract: A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding . 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. [ABSTRACT FROM AUTHOR]

Abstract: In competing risks studies, the Kaplan–Meier estimators of the distribution functions (DFs) of lifetimes and the corresponding estimators of cumulative incidence functions (CIFs) are used widely when no prior information is available for these distributions. In some cases better estimators of the DFs of lifetimes are available when they obey some inequality constraints, e.g., if two lifetimes are stochastically or uniformly stochastically ordered, or some functional of a DF obeys an inequality in an empirical likelihood estimation procedure. If the restricted estimator of a lifetime differs from the unrestricted one, then the usual estimators of the CIFs will not add up to the lifetime estimator. In this paper we show how to estimate the CIFs in this case. These estimators are shown to be strongly uniformly consistent. In all cases we consider, when the inequality constraints are strict the asymptotic properties of the restricted and the unrestricted estimators are the same, thus providing the asymptotic properties of the restricted estimators essentially “free of charge”. We give an example to illustrate our procedure. [Copyright &y& Elsevier]

Abstract: In this paper we consider elliptical random vectors in with stochastic representation , where is a positive random radius independent of the random vector which is uniformly distributed on the unit sphere of and is a given matrix. Denote by the Euclidean norm in , and let be the distribution function of . The main result of this paper is an asymptotic expansion of the probability for in the Gumbel or the Weibull max-domain of attraction. In the special case that is a mean zero Gaussian random vector our result coincides with the one derived in Hüsler et al. (2002) . [Copyright &y& Elsevier]

Abstract: Consider the model , where is unknown but smooth. It is often assumed that and are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between and by means of a copula model, i.e. , where is a copula function depending on an unknown parameter , and and are the marginals of and . Since many parametric copula families contain the independent copula as a special case, the so-obtained regression model is more flexible than the ‘classical’ regression model. We estimate the parameter via a pseudo-likelihood method and prove the asymptotic normality of the estimator, based on delicate empirical process theory. We also study the estimation of the conditional distribution of given . The procedure is illustrated by means of a simulation study, and the method is applied to data on food expenditures in households. [Copyright &y& Elsevier]

Abstract: There are many ways to measure the dispersion of a random variable. One such method uses the concept of peakedness. If the random variable is symmetric about a point , then Birnbaum [Z.W. Birnbaum, On random variables with comparable peakedness, The Annals of Mathematical Statistics 19 (1948) 76–81] defined the function , as the peakedness of . If two random variables, and , are symmetric about the points and , respectively, then is said to be less peaked than , denoted by , if for all , i.e.,  is stochastically larger than . For normal distributions this is equivalent to variance ordering. Peakedness ordering can be generalized to the case where and are arbitrary points. However, in this paper we study the comparison of dispersions in two continuous random variables, symmetric about their respective medians, using the peakedness concept where normality, and even moment assumptions are not necessary. We provide estimators of the distribution functions under the restriction of symmetry and peakedness ordering, show that they are consistent, derive the weak convergence of the estimators, compare them with the empirical estimators, and provide formulas for statistical inferences. An example is given to illustrate the theoretical results. [Copyright &y& Elsevier]

Abstract: For a {0, 1}-pattern of finite length, an empirical process is introduced in order to describe the number of overlapping occurrences of the pattern at each level in a sequence of the corresponding indicators of i.i.d. [0, 1]-valued observations of length . A method for obtaining the exact finite-dimensional distributions of the empirical process is given. The weak convergence of the process to a Gaussian process in as tends to infinity is also established. The limiting process depends on the given pattern. The exact covariance function is compared with the asymptotic covariance function in a numerical example. [Copyright &y& Elsevier]

Abstract: Let be iid elliptical random vectors in and let be two non-empty disjoint index sets. Denote by the subvectors of with indices in , respectively. For any such that is in the support of the conditional random sample consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of has distribution function in the max-domain of attraction of a univariate extreme value distribution. [Copyright &y& Elsevier]

Abstract: This paper establishes the weak convergence of a class of marked empirical processes of possibly non-stationary and/or non-ergodic multivariate time series sequences under martingale conditions. The assumptions involved are similar to those in Brown''s martingale central limit theorem. In particular, no mixing conditions are imposed. As an application, we propose a test statistic for the martingale hypothesis and we derive its asymptotic null distribution. Finally, a Monte Carlo study shows that the asymptotic results provide good approximations for small and moderate sample sizes. An application to the S&P 500 is also considered. [Copyright &y& Elsevier]

Abstract: The functional autoregressive model is a Markov model taylored for data of functional nature. It revealed fruitful when attempting to model samples of dependent random curves and has been widely studied along the past few years. This article aims at completing the theoretical study of the model by addressing the issue of weak convergence for estimates from the model. The main difficulties stem from an underlying inverse problem as well as from dependence between the data. Traditional facts about weak convergence in non-parametric models appear: the normalizing sequence is not an , a bias term appears. Several original features of the functional framework are pointed out. [Copyright &y& Elsevier]

Abstract: A new family of conditional-dependence measures based on Spearman''s rho is introduced. The corresponding multidimensional versions are established. Asymptotic distributional results are derived for related estimators which are based on the empirical copula. Particular emphasis is placed on a new type of multidimensional tail-dependence measure and its relationship to other measures of tail dependence is shown. Multivariate tail dependence describes the limiting amount of dependence in the vertices of the copula''s domain. [Copyright &y& Elsevier]

Abstract: In this paper we consider elliptical random vectors in with stochastic representation where R is a positive random radius independent of the random vector which is uniformly distributed on the unit sphere of and is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman''s result to Type III elliptical random vectors in . Further, we derive an asymptotic approximation for the conditional distribution of such random vectors. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Semiparametric models to describe the functional relationship between k groups of observations are broadly applied in statistical analysis, ranging from nonparametric ANOVA to proportional hazard (ph) rate models in survival analysis. In this paper we deal with the empirical assessment of the validity of such a model, which will be denoted as a “structural relationship model”. To this end Hadamard differentiability of a suitable goodness-of-fit measure in the k-sample case is proved. This yields asymptotic limit laws which are applied to construct tests for various semiparametric models, including the Cox ph model. Two types of asymptotics are obtained, first when the hypothesis of the semiparametric model under investigation holds true, and second for the case when a fixed alternative is present. The latter result can be used to validate the presence of a semiparametric model instead of simply checking the null hypothesis “the model holds true”. Finally, various bootstrap approximations are numerically investigated and a data example is analyzed. [Copyright &y& Elsevier]

In this paper, we consider the problem of testing for a parameter change in stochastic processes. In performing a test, we employ the cusum test considered in Lee et al. (Scand. J. Statist. 30 (2003) 651). The cusum test is based on the conditional least-squares estimator introduced by Klimko and Nelson (Ann. Statist. 6 (1978) 629). Special attention is paid to the nonlinear autoregressive processes including TAR and ARCH processes. It is shown that under regularity conditions, the test statistic behaves asymptotically the same as the sup of the squares of independent standard Brownian bridges. Simulation results as to ARCH(1) processes and an example of real data analysis are provided for illustration. [Copyright &y& Elsevier]

Statistical depth functions are being used increasingly in nonparametric multivariate data analysis. In a broad treatment of depth-based methods, Liu, Parelius, and Singh (“Multivariate analysis by date depth: Descriptive statistics, graphics and inference (with discussion),” 1999) include several devices for visualizing selected multivariate distributional characteristics by one-dimensional curves constructed in terms of given depth functions. Here we show how these tools may be represented as special depth-based cases of generalized quantile functions introduced by J. H. J. Einmahl and D. M. Mason (1992, Ann. Statist.20, 1062–1078). By specializing results of the latter authors to the depth-based case, we develop an easily applied general result on convergence of sample depth-based generalized quantile processes to a Brownian bridge. As applications, we obtain the asymptotic behavior of sample versions of depth-based curves for “scale” and “kurtosis” introduced by Liu, Parelius and Singh. The kurtosis curve is actually a Lorenz curve designed to measure heaviness of tails of a multivariate distribution. We also obtain the asymptotic distribution of the quantile process of the sample depth values. [Copyright &y& Elsevier]

Each α-stable distribution can be approximated either by an α-stable distribution with a discrete Poisson spectrum or by a sum of i.i.d. random vectors. Here we give results on the accuracy that can be achieved under both these ways of approximation. They are purely theoretical and aim to outline possible direction of further investigations. [Copyright &y& Elsevier]

We consider testing the validity of the generalized logit model with I+1 categories based on case-control data. After reparametrization, the assumed logit model is equivalent to an (I+1)-sample semiparametric model in which the I log ratios of two unspecified density functions are linear in data. By identifying this (I+1)-sample semiparametric model, which is of intrinsic interest in general (I+1)-sample problems, with a biased sampling model, we propose a weighted Kolmogorov–Smirnov-type statistic to test the validity of the generalized logit model. We establish some asymptotic results associated with the proposed test statistic. We also propose a bootstrap procedure along with some results on simulation and on analysis of three real data sets. [Copyright &y& Elsevier]