Multivariate conditional versions of Spearman's rho and related measures of tail dependence

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.     

Empirical likelihood for semiparametric regression model with missing response data

A bias-corrected technique for constructing the empirical likelihood ratio is used to study a semiparametric regression model with missing response data. We are interested in inference for the regression coefficients, the baseline function and the response mean. A class of empirical likelihood ratio functions for the parameters of interest is defined so that undersmoothing for estimating the baseline function is avoided. The existing data-driven algorithm is also valid for selecting an optimal bandwidth. Our approach is to directly calibrate the empirical log-likelihood ratio so that the resulting ratio is asymptotically chi-squared. Also, a class of estimators for the parameters of interest is constructed, their asymptotic distributions are obtained, and consistent estimators of asymptotic bias and variance are provided. Our results can be used to construct confidence intervals and bands for the parameters of interest. A simulation study is undertaken to compare the empirical likelihood with the normal approximation-based method in terms of coverage accuracies and average lengths of confidence intervals. An example for an AIDS clinical trial data set is used for illustrating our methods.     

Estimating cumulative incidence functions when the life distributions are constrained

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.     

On the block thresholding wavelet estimators with censored data

We consider block thresholding wavelet-based density estimators with randomly right-censored data and investigate their asymptotic convergence rates. Unlike for the complete data case, the empirical wavelet coefficients are constructed through the Kaplan-Meier estimators of the distribution functions in the censored data case. On the basis of a result of Stute [W. Stute, The central limit theorem under random censorship, Ann. Statist. 23 (1995) 422-439] that approximates the Kaplan-Meier integrals as averages of i.i.d. random variables with a certain rate in probability, we can show that these wavelet empirical coefficients can be approximated by averages of i.i.d. random variables with a certain error rate in L². Therefore we can show that these estimators, based on block thresholding of empirical wavelet coefficients, achieve optimal convergence rates over a large range of Besov function classes , p≥2, q≥1 and nearly optimal convergence rates when 1≤p<2. We also show that these estimators achieve optimal convergence rates over a large class of functions that involve many irregularities of a wide variety of types, including chirp and Doppler functions, and jump discontinuities. Therefore, in the presence of random censoring, wavelet estimators still provide extensive adaptivity to many irregularities of large function classes. The performance of the estimators is tested via a modest simulation study.     

Nonparametric estimation of the dependence function for a multivariate extreme value distribution

Understanding and modeling dependence structures for multivariate extreme values are of interest in a number of application areas. One of the well-known approaches is to investigate the Pickands dependence function. In the bivariate setting, there exist several estimators for estimating the Pickands dependence function which assume known marginal distributions [J. Pickands, Multivariate extreme value distributions, Bull. Internat. Statist. Inst., 49 (1981) 859-878; P. Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett. 12 (1991) 429-439; P. Hall, N. Tajvidi, Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli 6 (2000) 835-844; P. Capéraà, A.-L. Fougères, C. Genest, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 (1997) 567-577]. In this paper, we generalize the bivariate results to p-variate multivariate extreme value distributions with p?2. We demonstrate that the proposed estimators are consistent and asymptotically normal as well as have excellent small sample behavior.     

On the covariance of the asymptotic empirical copula process

Conditions are given under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance function than the standard empirical process based on observations from the copula. Illustrations are provided and consequences for inference are outlined.     

Functional-coefficient partially linear regression model

In this paper, the functional-coefficient partially linear regression (FCPLR) model is proposed by combining nonparametric and functional-coefficient regression (FCR) model. It includes the FCR model and the nonparametric regression (NPR) model as its special cases. It is also a generalization of the partially linear regression (PLR) model obtained by replacing the parameters in the PLR model with some functions of the covariates. The local linear technique and the integrated method are employed to give initial estimators of all functions in the FCPLR model. These initial estimators are asymptotically normal. The initial estimator of the constant part function shares the same bias as the local linear estimator of this function in the univariate nonparametric model, but the variance of the former is bigger than that of the latter. Similarly, initial estimators of every coefficient function share the same bias as the local linear estimates in the univariate FCR model, but the variance of the former is bigger than that of the latter. To decrease the variance of the initial estimates, a one-step back-fitting technique is used to obtain the improved estimators of all functions. The improved estimator of the constant part function has the same asymptotic normality property as the local linear nonparametric regression for univariate data. The improved estimators of the coefficient functions have the same asymptotic normality properties as the local linear estimates in FCR model. The bandwidths and the smoothing variables are selected by a data-driven method. Both simulated and real data examples related to nonlinear time series modeling are used to illustrate the applications of the FCPLR model.     

On the transitivity of the comonotonic and countermonotonic comparison of random variables

A recently proposed method for the pairwise comparison of arbitrary independent random variables results in a probabilistic relation. When restricted to discrete random variables uniformly distributed on finite multisets of numbers, this probabilistic relation expresses the winning probabilities between pairs of hypothetical dice that carry these numbers and exhibits a particular type of transitivity called dice-transitivity. In case these multisets have equal cardinality, two alternative methods for statistically comparing the ordered lists of the numbers on the faces of the dice have been studied recently: the comonotonic method based upon the comparison of the numbers of the same rank when the lists are in increasing order, and the countermonotonic method, also based upon the comparison of only numbers of the same rank but with the lists in opposite order. In terms of the discrete random variables associated to these lists, these methods each turn out to be related to a particular copula that joins the marginal cumulative distribution functions into a bivariate cumulative distribution function. The transitivity of the generated probabilistic relation has been completely characterized. In this paper, the list comparison methods are generalized for the purpose of comparing arbitrary random variables. The transitivity properties derived in the case of discrete uniform random variables are shown to be generic. Additionally, it is shown that for a collection of normal random variables, both comparison methods lead to a probabilistic relation that is at least moderately stochastic transitive.     

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.     

Consistent tuning parameter selection in high dimensional sparse linear regression

An exhaustive search as required for traditional variable selection methods is impractical in high dimensional statistical modeling. Thus, to conduct variable selection, various forms of penalized estimators with good statistical and computational properties, have been proposed during the past two decades. The attractive properties of these shrinkage and selection estimators, however, depend critically on the size of regularization which controls model complexity. In this paper, we consider the problem of consistent tuning parameter selection in high dimensional sparse linear regression where the dimension of the predictor vector is larger than the size of the sample. First, we propose a family of high dimensional Bayesian Information Criteria (HBIC), and then investigate the selection consistency, extending the results of the extended Bayesian Information Criterion (EBIC), in Chen and Chen (2008) to ultra-high dimensional situations. Second, we develop a two-step procedure, the SIS+AENET, to conduct variable selection in p>n situations. The consistency of tuning parameter selection is established under fairly mild technical conditions. Simulation studies are presented to confirm theoretical findings, and an empirical example is given to illustrate the use in the internet advertising data.     

Asymptotic behavior of central order statistics from stationary processes

In this paper, we show that central order statistics from strictly stationary and ergodic sequences are strongly consistent estimators of population quantiles provided that the quantiles are unique. We generalize this result to strictly stationary but not necessarily ergodic sequences. We also describe three types of possible asymptotic behavior of central order statistics in the case when the corresponding population quantile is not unique. We give applications of the presented results to linear processes with both absolutely continuous and discrete innovations.     

Some inequalities for strong mixing random variables with applications to density estimation

In this paper, we establish an inequality of the characteristic functions for strongly mixing random vectors, by which, an upper bound is provided for the supremum of the absolute value of the difference of two multivariate probability density functions based on strongly mixing random vectors. As its application, we consider the consistency and asymptotic normality of a kernel estimate of a density function under strong mixing. Our results generalize some known results in the literature.     

Bartlett correctable two-sample adjusted empirical likelihood

We propose a two-sample adjusted empirical likelihood (AEL) to construct confidence regions for the difference of two d-dimensional population means. This method eliminates the non-definition of the usual two-sample empirical likelihood (EL) and is shown to be Bartlett correctable. We further show that when the adjustment level is half the Bartlett correction factor for the usual two-sample EL, the two-sample AEL has the same high-order precision as the EL with Bartlett correction. To enhance the performance of the two-sample AEL with adjustment level being half the Bartlett correction factor, we propose a less biased estimate of the Bartlett correction factor. The efficiency of the proposed method is illustrated by simulations and a real data example.     

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.     

Detection of a change point based on local-likelihood

In this paper, we consider the regression function or its νth derivative in generalized linear models which may have a change/discontinuity point at an unknown location. The location and its jump size are estimated with the local polynomial fits based on one-sided kernel weighted local-likelihood functions. Asymptotic distributions of the proposed estimators of location and jump size are established. The finite-sample performances of the proposed estimators with practical aspects are illustrated by simulated and beetle mortality examples.     

Properties of nonparametric estimators of autocovariance for stationary random fields

Summary We introduce nonparametric estimators of the autocovariance of a stationary random field. One of our estimators has the property that it is itself an autocovatiance. This feature enables the estimator to be used as the basis of simulation studies such as those which are necessary when constructing bootstrap confidence intervals for unknown parameters. Unlike estimators proposed recently by other authors, our own do not require assumptions such as isotropy or monotonicity. Indeed, like nonparametric function estimators considered more widely in the context of curve estimation, our approach demands only smoothness and tail conditions on the underlying curve or surface (here, the autocovariance), and moment and mixing conditions on the random field. We show that by imposing the condition that the estimator be a covariance function we actually reduce the numerical value of integrated squared error.     

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.     

Gaussian approximation of local empirical processes indexed by functions

Summary. An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated. Received: 11 January 1995/In revised form: 12 July 1996     

Smooth estimation of survival and density functions for a stationary associated process using Poisson weights

Let {X_n,n≥1} be a sequence of stationary non-negative associated random variables with common marginal density f(x). Here we use the empirical survival function as studied in Bagai and Prakasa Rao (1991) and apply the smoothing technique proposed by Gawronski (1980) (see also Chaubey and Sen, 1996) in proposing a smooth estimator of the density function f and that of the corresponding survival function. Some asymptotic properties of the resulting estimators, similar to those obtained in Chaubey and Sen (1996) for the i.i.d. case, are derived. A simulation study has been carried out to compare the new estimator to the kernel estimator of a density function given in Bagai and Prakasa Rao (1996) and the estimator in Buch-Larsen et al. (2005).     

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^−1/2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.