Location estimation in nonparametric regression with censored data

Consider the heteroscedastic model Y=m(X)+σ(X)?, where ? and X are independent, Y is subject to right censoring, m(·) is an unknown but smooth location function (like e.g. conditional mean, median, trimmed mean…) and σ(·) an unknown but smooth scale function. In this paper we consider the estimation of m(·) under this model. The estimator we propose is a Nadaraya-Watson type estimator, for which the censored observations are replaced by ‘synthetic’ data points estimated under the above model. The estimator offers an alternative for the completely nonparametric estimator of m(·), which cannot be estimated consistently in a completely nonparametric way, whenever high quantiles of the conditional distribution of Y given X=x are involved.We obtain the asymptotic properties of the proposed estimator of m(x) and study its finite sample behaviour in a simulation study. The method is also applied to a study of quasars in astronomy.     

Approximating conditional density functions using dimension reduction

We propose to approximate the conditional density function of a random variable Y given a dependent random d-vector X by that of Y given θ^τX, where the unit vector θ is selected such that the average Kullback-Leibler discrepancy distance between the two conditional density functions obtains the minimum. Our approach is nonparametric as far as the estimation of the conditional density functions is concerned. We have shown that this nonparametric estimator is asymptotically adaptive to the unknown index θ in the sense that the first order asymptotic mean squared error of the estimator is the same as that when θ was known. The proposed method is illustrated using both simulated and real-data examples.     

Least squares cross-validation for the kernel deconvolution density estimatorValidation croisée pour l'estimateur à noyau de la déconvolution d'une densité

Assume we have i.i.d. replications from the corrupted random variable Y=X+ε, where X and ε are independent. We propose a data-driven bandwidth based on cross-validation ideas, for the kernel deconvolution estimator of the density of X. The proposed method is shown to be asymptotically optimal. To cite this article: É. Youndjé, M.T. Wells, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 509–513.     

A note on the conditional density estimate in the single functional index model

In this paper, we consider the estimation of the conditional density of a scalar response variable Y, given a Hilbertian random variable X when the observations are linked with a single-index structure. We establish the pointwise and the uniform almost complete convergence (with the rate) of the kernel estimate of this model. As an application, we show how our result can be applied in the prediction problem via the conditional mode estimate. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is also discussed but not tackled.     

Bivariate distributions characterized by one family of conditionals and conditional percentile or mode functions

It is well known that full knowledge of all conditional distributions will typically serve to completely characterize a bivariate distribution. Partial knowledge will often suffice. For example, knowledge of the conditional distribution of X given Y and the conditional mean of Y given X is often adequate to determine the joint distribution of X and Y. In this paper, we investigate the extent to which a conditional percentile function or a conditional mode function (of Y given X), together with knowledge of the conditional distribution of X given Y will determine the joint distribution. Finally, using this methodology a new characterization of the classical bivariate normal distribution is given.     

Estimation in nonparametric location-scale regression models with censored data

Consider the random vector (X, Y), where X is completely observed and Y is subject to random right censoring. It is well known that the completely nonparametric kernel estimator of the conditional distribution ${F(\cdot|x)}$ of Y given X = x suffers from inconsistency problems in the right tail (Beran 1981, Technical Report, University of California, Berkeley), and hence any location function m(x) that involves the right tail of ${F(\cdot|x)}$ (like the conditional mean) cannot be estimated consistently in a completely nonparametric way. In this paper, we propose an alternative estimator of m(x), that, under certain conditions, does not share the above inconsistency problems. The estimator is constructed under the model Y = m(X) + σ(X)ε, where ${\sigma(\cdot)}$ is an unknown scale function and ε (with location zero and scale one) is independent of X. We obtain the asymptotic properties of the proposed estimator of m(x), we compare it with the completely nonparametric estimator via simulations and apply it to a study of quasars in astronomy.     

Vitesse de convergence presque sûre de l'estimateur à noyau du maximum de vraisemblance local

We consider the local maximum likelihood estimation of $\theta (x)$, unknown parameter of the conditional distribution of Y given $X=x$. The aim of this Note is the study of strong uniform consistency rates of the local maximum likelihood kernel estimator. Under suitable regularity conditions, we establish a uniform law of the logarithm for the maximal deviation of this estimator. The method of proof is based upon functional limit laws derived by modern empirical process theory. To cite this article: D. Blondin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Some inverse problems involving conditional expectations

Let (Ω, F, P) be a probability space, let H be a sub-σ-algebra of F, and let Y be positive and H-measurable with E[Y] = 1. We discuss the structure of the convex set CE(Y; H) = {X ∈ pF: Y = E[X|H]} of random variables whose conditional expectation given H is the prescribed Y. Several characterizations of extreme points of CE(Y; H) are obtained. A necessary and sufficient condition is given in order that CE(Y; H) be the closed, convex hull of its extreme points. For the case of finite F we explicitly calculate the extreme points of CE(Y; H), identify pairs of adjacent extreme points, and characterize extreme points of CE(Y; H) ? CE(Z; G), where G is a second sub-σ-algebra of F and Z ∈ pG. When H = σ(Y) and appropriate topological hypotheses hold, extreme points of CE(Y; H) are shown to be in explicit one-to-one correspondence with certain left inverses of Y. Finally, it is shown how the same approach can be applied to the problem of extremal random measures on $\text{R}$₊ with a prescribed compensator, to deduce that the number of extreme points is zero or one.     

Estimation of conditional laws given an extreme component

Let (X,Y) be a bivariate random vector. The estimation of a probability of the form P(Y ≤ y |X > t) is challenging when t is large, and a fruitful approach consists in studying, if it exists, the limiting conditional distribution of the random vector (X,Y), suitably normalized, given that X is large. There already exists a wide literature on bivariate models for which this limiting distribution exists. In this paper, a statistical analysis of this problem is done. Estimators of the limiting distribution (which is assumed to exist) and the normalizing functions are provided, as well as an estimator of the conditional quantile function when the conditioning event is extreme. Consistency of the estimators is proved and a functional central limit theorem for the estimator of the limiting distribution is obtained. The small sample behavior of the estimator of the conditional quantile function is illustrated through simulations. Some real data are analysed.     

Asymptotic confidence intervals for Poisson regression

Let (X,Y) be a R^d×N₀-valued random vector where the conditional distribution of Y given X=x is a Poisson distribution with mean m(x). We estimate m by a local polynomial kernel estimate defined by maximizing a localized log-likelihood function. We use this estimate of m(x) to estimate the conditional distribution of Y given X=x by a corresponding Poisson distribution and to construct confidence intervals of level α of Y given X=x. Under mild regularity conditions on m(x) and on the distribution of X we show strong convergence of the integrated L₁ distance between Poisson distribution and its estimate. We also demonstrate that the corresponding confidence interval has asymptotically (i.e., for sample size tending to infinity) level α, and that the probability that the length of this confidence interval deviates from the optimal length by more than one converges to zero with the number of samples tending to infinity.     

Likelihood-Based Local Polynomial Fitting for Single-Index Models

The parametric generalized linear model assumes that the conditional distribution of a response Y given a d-dimensional covariate X belongs to an exponential family and that a known transformation of the regression function is linear in X. In this paper we relax the latter assumption by considering a nonparametric function of the linear combination β^TX, say η₀(β^TX). To estimate the coefficient vector β and the nonparametric component η₀ we consider local polynomial fits based on kernel weighted conditional likelihoods. We then obtain an estimator of the regression function by simply replacing β and η₀ in η₀(β^TX) by these estimators. We derive the asymptotic distributions of these estimators and give the results of some numerical experiments.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

More on Stochastic Comparisons and Dependence among Concomitants of Order Statistics

For a sample of iid observations {(X_i, Y_i)} from an absolutely continuous distribution, the multivariate dependence of concomitants Y_[]=(Y_[1], Y_[2], …, Y_[n]) and the stochastic order of subsets of Y_[] are studied. If (X, Y) is totally positive dependent of order 2, Y_[] is multivariate totally positive dependent of order 2. If the conditional hazard rate function of Y given X, h_YX(yx), is decreasing in x for every y, Y_[] is multivariate right corner set increasing. And if Y is stochastically increasing in X, the concomitants are increasing in multivariate stochastic order.     

Régression non-paramétrique pour des variables aléatoires fonctionnelles mélangeantesNonparametric regression for mixing functional random variables

This paper concerns a nonparametric regression model for dependent functional variables. The aim is to explain a r.r.v. Y by a functional regressor namely a variable X which is valued in some semi-normed vector space. A kernel type estimator is proposed and asymptotics with rates are proved under strong mixing assumption. To cite this article: F. Ferraty, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 217–220.     

On characterization of power series distributions by a marginal distribution and a regression function

The conditional distribution of Y given X=x, where X and Y are non-negative integer-valued random variables, is characterized in terms of the regression function of X on Y and the marginal distribution of X which is assumed to be of a power series form. Characterizations are given for a binomial conditional distribution when X follows a Poisson, binomial or negative binomial, for a hypergeometric conditional distribution when X is binomial and for a negative hypergeometric conditional distribution when X follows a negative binomial.     

Flexible modeling based on copulas in nonparametric median regression

Consider the model Y=m(X)+ε, where m(⋅)=med(Y|⋅) is unknown but smooth. It is often assumed that ε and X are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between ε and X by means of a copula model, i.e. (ε,X)∼C_θ(F_ε(⋅),F_X(⋅)), where C_θ is a copula function depending on an unknown parameter θ, and F_ε and F_X are the marginals of ε and X. 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 θ 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 Y given X. The procedure is illustrated by means of a simulation study, and the method is applied to data on food expenditures in households.     

Weak consistency of the Support Vector Machine Quantile Regression approach when covariates are functions

This paper deals with a nonparametric estimation of conditional quantile regression when the explanatory variable X takes its values in a bounded subspace of a functional space X and the response Y takes its values in a compact of the space Y?R. The functional observations, X₁,…,X_n, are projected onto a finite dimensional subspace having a suitable orthonormal system. The X_i’s will be characterized by their coordinates in this basis. We perform the Support Vector Machine Quantile Regression approach in finite dimension with the selected coefficients. Then we establish weak consistency of this estimator. The various parameters needed for the construction of this estimator are automatically selected by data-splitting and by penalized empirical risk minimization.     

Inference with a single fuzzy conditional proposition

A topic in the area of approximate reasoning that has been extensively researched is the following: Given a single fuzzy conditional proposition ifXthenY is B with A and B being fuzzy sets defining particular values of the linguistic variables X and Y respectively and given some A1 ≠ A, then what can be inferred about the value of Y when A us replaced by A1In the works of L. Zadeh, J. Baldwin, E. Hisdal, R. Yager, M. Mizumoto et al. different solutions to this problem were proposed. Though different, they all share one common feature: they ignore the fact that a single fuzzy conditional proposition represents only one specific manifestation of the causal influence exerted by X on Y which is being described by binding the fuzzy set B to the fuzzy set A. Thus, the general pattern of the causal relatinship between the two linguistic variables if forgotten. As a result the inference made when the original value of X is replaced by a new one might, in some cases, contradict the general pattern of the causal relationship.The present paper makes an attempt to specify these types of inference with a single fuzzy conditional proposition which do not depend on the general pattern of the causal relationship between X and Y. It also discusses some associated problems.     

Nonparametric Empirical Bayes Estimation of the Matrix Parameter of the Wishart Distribution

We consider independent pairs (X₁, Σ₁), (X₂, Σ₂), …, (X_n, Σ_n), where eachΣ_iis distributed according to some unknown density functiong(Σ) and, givenΣ_i=Σ,X_ihas conditional density functionq(xΣ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observationX_n+1is obtained, the objective is to estimateΣ_n+1corresponding toX_n+1. This estimator is called the empirical Bayes (EB) estimator ofΣ. An EB estimator ofΣis constructed without any parametric assumptions ong(Σ). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal.