Conditional Empirical Processes Research Articles

Uniform in number of neighbors consistency and weak convergence of $ k $NN empirical conditional processes and $ k $NN conditional $ U $-processes involving functional mixing data

<abstract><p>$ U $-statistics represent a fundamental class of statistics arising from modeling quantities of interest defined by multi-subject responses. $ U $-statistics generalize the empirical mean of a random variable $ X $ to sums over every $ m $-tuple of distinct observations of $ X $. Stute [182] introduced a class of so-called conditional $ U $-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: $ r^{(m)}(\varphi, \mathbf{t}): = \mathbb{E}[\varphi(Y_{1}, \ldots, Y_{m})|(X_{1}, \ldots, X_{m}) = \mathbf{t}], \; \mbox{for}\; \mathbf{ t}\in \mathcal{X}^{m}. $ In this paper, we are mainly interested in the study of the $ k $NN conditional $ U $-processes in a functional mixing data framework. More precisely, we investigate the weak convergence of the conditional empirical process indexed by a suitable class of functions and of the $ k $NN conditional $ U $-processes when the explicative variable is functional. We treat the uniform central limit theorem in both cases when the class of functions is bounded or unbounded satisfying some moment conditions. The second main contribution of this study is the establishment of a sharp almost complete Uniform consistency in the Number of Neighbors of the constructed estimator. Such a result allows the number of neighbors to vary within a complete range for which the estimator is consistent. Consequently, it represents an interesting guideline in practice to select the optimal bandwidth in nonparametric functional data analysis. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The theoretical results established in this paper are (or will be) key tools for further functional data analysis developments. Potential applications include the set indexed conditional <italic>U</italic>-statistics, Kendall rank correlation coefficient, the discrimination problems and the time series prediction from a continuous set of past values.</p></abstract>

Conditional empirical copula processes and generalized measures of association

We study the weak convergence of conditional empirical copula processes indexed by general families of conditioning events that have non zero probabilities. Moreover, we also study the case where the conditioning events are chosen in a data-driven way. The validity of several bootstrap schemes is stated, including the exchangeable bootstrap. We define general multivariate measures of association, possibly given some fixed or random conditioning events. By applying our theoretical results, we prove the asymptotic normality of the estimators of such measures. We illustrate our results with financial data.

Central Limit Theorems for Conditional Empirical and Conditional U-Processes of Stationary Mixing Sequences

In this paper we are concerned with the weak convergence to Gaussian processes of conditional empirical processes and conditional U-processes from stationary β-mixing sequences indexed by classes of functions satisfying some entropy conditions. We obtain uniform central limit theorems for conditional empirical processes and conditional U-processes when the classes of functions are uniformly bounded or unbounded with envelope functions satisfying some moment conditions. We apply our results to introduce statistical tests for conditional independence that are multivariate conditional versions of the Kendall statistics.

Symmetrized Multivariate k-NN Estimators

In this article, we propose a symmetrized multivariate k-NN estimator for the conditional mean and for the conditional distribution function. We establish consistency and asymptotic normality of each estimator. For the estimator of the conditional distribution function, we also establish the weak convergence of the conditional empirical process to a Gaussian process. Compared with the corresponding kernel estimators, the asymptotic distributions of our k-NN estimators do not depend on the existence of the marginal probability density functions of the covariate vector. A small simulation study compares the finite sample performance of our symmetrized multivariate k-NN estimator with the Nadaraya–Watson kernel estimator for the conditional mean.

Conditional large deviations for a sequence of words

Cut an i.i.d. sequence ( X i ) of ‘letters’ into ‘words’ according to an independent renewal process. Then one obtains an i.i.d. sequence of words, and thus the level 3 large deviation behaviour of this sequence of words is governed by the specific relative entropy. We consider the corresponding problem for the conditional empirical process of words, where one conditions on a typical underlying ( X i ) . We find that if the tails of the word lengths decay exponentially, the large deviations under the conditional distribution are almost surely again governed by the specific relative entropy, but the set of attainable limits is restricted. We indicate potential applications of such a conditional LDP to the computation of the quenched free energy for directed polymer models with random disorder.

The invariance principle for conditional empirical processes formed by dependent random variables

We prove the convergence of finite-dimensional distributions and establish density for Nadaraya-Watson conditional empirical processes. The observations are assumed to be described by a strictly stationary sequence of random variables whose mixing coefficients decay polynomially. The proof of density of such processes in the space of continuous functionals uses entropy conditions on the class of indexing functions.

Uniform in bandwidth consistency of kernel-type function estimators

We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya–Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.

Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data

We consider a conditional empirical distribution of the form Fn(C∣x)=∑nt=1ωn(Xt−x)I{Yt∈C} indexed by C∈C, where {(Xt, Yt), t=1, …, n} are observations from a strictly stationary and strong mixing stochastic process, {ωn(Xt−x)} are kernel weights, and C is a class of sets. Under the assumption on the richness of the index class C in terms of metric entropy with bracketing, we have established uniform convergence and asymptotic normality for Fn(·∣x). The key result specifies rates of convergences for the modulus of continuity of the conditional empirical process. The results are then applied to derive Bahadur–Kiefer type approximations for a generalized conditional quantile process which, in the case with independent observations, generalizes and improves earlier results. Potential applications in the areas of estimating level sets and testing for unimodality (or multimodality) of conditional distributions are discussed.

Conditional Empirical Processes Defined by Nonstationary Absolutely Regular Sequences

K. I. Yoshihara (1990,Comput. Math. Appl.19, No. 1, 149–158) proved the weak invariance of the conditional nearest neighbor regression function estimator called the conditional empirical process based onϕ-mixing observations. In this paper, we extend the result for nonstationary and absolutely regular random variables which have applications for Markov processes, for which the initial measure is not necessary, the invariant measure.

Some Large Deviations Limit Theorems in Conditional Nonparametric Statistics

We establish pointwise and uniform large deviations limit theorems of Chernoff-type for the conditional empirical process. On the other hand, we state the pointwise large deviations theorem for the Nadaraya-Watson estimator of the regression function. The estimations are based on sequences of independent and identically distributed random vectors. We derive then some implications of our results in the study of asymptotic efficiency of goodness-of-fit test based on uniform deviation of the conditional empirical distribution function with respect to its theoretical distribution. Moreover, we deduce the inaccuracy rate in conditional distribution functions estimation.

Gaussian approximation of local empirical processes indexed by functions

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.

Koul-DeWet-type estimation in censored regression using the conditional empirical process

Assume that the random variables of interest obey a linear regression model and are randomly censored by independent censoring variables. We consider some minimum distance estimators of the slope parameter. These estimators are asymptotically normal at the assumed regression model and are also qualitatively robust against small Kolmogorov distance deviation from the model. They are adapted from the Koul-DeWet ( Ann. Statist. 11 (1983), 921–932) estimators via a conditioning argument. As a result the conditions on the censoring distributions are very mild.

Conditional empirical, quantile and difference processes for a large class of time series with applications

Abstract The speed of a.s. convergence of conditional empirical distribution functions, quantile functions and difference processes is studied for a wide class of stationary time series. The difference process leads to a Bahadur-Kiefer representation of the conditional quantiles. Applications like autoregression function estimation, estimation of the hazard rate under random censoring, and robust conditional location functionals are also included. When specializing to the i.i.d. case our rates are very close to the best rates known in that special situation. Thus, some recent results on conditional processes with applications are extended to models with dependent data. Rather than modeling this dependence structure by the usual mixing conditions which are hard to precisely understand intuitively since they may fail to hold in very decent cases, we use the notion of mn-decomposability. Time series included are linear (with generalizes ARMA) processes and, more generally, processes with a Volterra expansion ofany finite 1order. Also bilinear processes are in this class.

Nonparametric prediction for random fields

We study prediction for vector valued random fields in a nonparametric setting. The prediction problem is formulated as the problem if estimating certain conditional expectations and a speed of uniform a.s. convergence is obtained, modifying results for conditional empirical processes derived from series with one-dimensional time. As an alternative to the usual mixing conditions we model the dependence by asymptotic decomposability. This includes linear (which generalizes ARMA) fields and random fields with a finite order Volterra expansion. As an example of a linear field we briefly discuss the finite-differences simulation of the heat equation blurred by additive random noise.

A smooth conditional quantile estimator and related applications of conditional empirical processes

Let {( X i , Y i ); i = 1,2,…} be a sequence of i.i.d. r.v.'s and denote by m( y | x 0), −∞ < y < ∞, the conditional distribution function of Y given X = x 0, −∞ < x 0 < ∞. In this paper we propose and discuss certain smooth variants (based both on single as well as double kernel weights) of the standard conditional quantile estimator m n −1( λ | x 0), 0 < λ < 1, of m −1( λ | x 0), where m n ( y | x 0) is a (kernel) estimator of m( y | x 0). The weak convergence of the corresponding conditional quantile process is also established. The same methods are used to study a new estimator of the conditional density and a “robust” estimator of the regression function.

Conditional empirical processes defined by φ - mixing sequences

For a random vector (X, Y) in R 2 let m (y/x)=P(Y≤y|X=x) be the conditional distribution function. It is assumed that (X 1 , Y 1 ), (X 2 , Y 2 ), ... form a strictly stationary Φ-mixing sequence of random vectors with the same distribution as (X, Y). Considering an estimate mn (•|x) of m(•|x) it is shown that (nan) 1/2 {mn(•|x 0 )-m (•|x 0 )} converges to a certain Gaussian process depending on x 0 , where (an) is a sequence of bandwiths

Asymptotics of conditional empirical processes

Using two definitions of the conditional empirical processes we obtain some approximations for these processes. We also prove the functional law of the iterated logarithm for the conditional processes. Our results say that the asymptotic behavior of the conditional and unconditional empirical processes are very similar.

Conditional Empirical Processes

Conditional Empirical Processes

On the Strong Mixing Property for Linear Sequences

On the Strong Mixing Property for Linear Sequences