More on extremal ranks of the matrix expressions A − BX ± X*B* with statistical applications

Through a Hermitian‐type (skew‐Hermitian‐type) singular value decomposition for pair of matrices (A, B) introduced by Zha (Linear Algebra Appl. 1996; 240 :199–205), where A is Hermitian (skew‐Hermitian), we show how to find a Hermitian (skew‐Hermitian) matrix X such that the matrix expressions A ? BX ± X^*B^* achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations BX ± X^*B^* = A, we give general solutions through the two kinds of generalized singular value decompositions. As applications to the general linear model {y, Xβ, σ²V}, we discuss the existence of a symmetric matrix G such that Gy is the weighted least‐squares estimator and the best linear unbiased estimator of Xβ, respectively. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive density estimation of stationary β-mixing and τ-mixing processes

We propose an algorithm to estimate the common density s of a stationary process X ₁, ..., _{X n} . We suppose that the process is either β or τ-mixing. We provide a model selection procedure based on a generalization of Mallows’ C _p and we prove oracle inequalities for the selected estimator under a few prior assumptions on the collection of models and on the mixing coefficients. We prove that our estimator is adaptive over a class of Besov spaces, namely, we prove that it achieves the same rates of convergence as in the i.i.d. framework.      

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.     

Generalized time-dependent conditional linear models under left truncation and right censoring

Consider the model f(S(z|X)){\phi(S(z|X))} = \pmbb(z) [(X)\vec]{\pmb{\beta}(z) {\vec{X}}}, where f{\phi} is a known link function, S(·|X) is the survival function of a response Y given a covariate X, [(X)\vec]{\vec{X}} = (1, X, X ² , . . . , X ^p ) and \pmbb(z){\pmb{\beta}(z)} is an unknown vector of time-dependent regression coefficients. The response is subject to left truncation and right censoring. Under this model, which reduces for special choices of f{\phi} to e.g. Cox proportional hazards model or the additive hazards model with time dependent coefficients, we study the estimation of the vector \pmbb(z){\pmb{\beta}(z)} . A least squares approach is proposed and the asymptotic properties of the proposed estimator are established. The estimator is also compared with a competing maximum likelihood based estimator by means of simulations. Finally, the method is applied to a larynx cancer data set.     

Estimating the Diffusion Coefficient for Diffusions Driven by fBm

When the Hurst coefficient of a fBm B _t ^H is greater than 1/2, it is possible to define a stochastic integral with respect to B _t ^H as the pathwise limit of Riemann sums. In this article we consider diffusion equations of the type X_t = x₀ + ₀ ^T (X_s) dB_s ^H. We then construct a simple-to-use estimator of the diffusion coefficient (x), based on the number of crossings of level x of the process X _t. We then study consistency in probability of this estimator and calculate convergence rates in probability.     

Parameter estimation for a misspecified arma model with infinite variance innovations

We consider a general linear model , where the innovations Z_t belong to the domain of attraction of an α-stable law for α<2, so that neither Z_t nor X_t have a finite variance. We do not assume that (X_t) is a standardARMA process of the form φ(B)X_t=ϕ(B)Z_t, but we fit anARMA process of a given order to the data X₁,...,X_n by estimating the coefficients of φ and ϕ. Given that (X_t) is anARMA process, it has been proved that the Whittle estimator is a consistent estimator of the true coefficients of ϕ and φ. Moreover, it then has a heavytailed limit distribution and the rate of convergence is (n/logn)^1/α, which compares favorably with the L² situation with rate . In this note we study the limit properties of the Whittle estimator when the underlying model is not necessarily anARMA process. Under general conditions we show that the Whittle estimate converges in probability. It converges weakly to a distribution which does not have a finite moment of order a and the rate of convergence is again (n/logn)^1/α. We also give an analytic expression for the limit distribution. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

Estimation in a model with infinite-dimensional nuisance parameter

Let X₁ be a random variable with density function f(t), Ψ(t) be an increasing absolutely continuous function, Φ(t) be the inverse function to Ψ(t), and X₂ be the random variable X₂ = Φ(X₁). We consider the maximum likelihood estimator for the density ψ of the function Ψ in the case when we observe two independent samples from the distributions of X₁ and X₂. Under appropriate conditions on the involved distributions, we prove the consistency of the maximum likelihood estimator. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 160–165. An erratum to this article is available at .     

Non parametric estimation of smooth stationary covariance functions by interpolation methods

In this paper we introduce a nonparametric approach for the estimation of the covariance function of a stationary stochastic process X _t indexed by The data consist of a finite number of observations of the process at irregularly spaced time points and the aim is to estimate the covariance at any lag point without parametric assumptions and in such a way that it is a positive definite function. After interpolating the process, we use the estimator designed by Parzen (Technometrics 3:167–190,1961) for continuous-time data. Our estimator is shown to be consistent under smoothness assumptions on the covariance. Its performance is evaluated by simulations.     

Estimation of the density of regression errors by pointwise model selection

This paper presents two results: a density estimator and an estimator of regression error density. We first propose a density estimator constructed by model selection, which is adaptive for the quadratic risk at a given point. Then we apply this result to estimate the error density in a homoscedastic regression framework Y _i = b(X _i ) + ε _i from which we observe a sample (X _i , Y _i ). Given an adaptive estimator $ \hat b $ \hat b of the regression function, we apply the density estimation procedure to the residuals $ \hat \varepsilon _i = Y_i - \hat b(X_i ) $ \hat \varepsilon _i = Y_i - \hat b(X_i ) . We get an estimator of the density of ε _i whose rate of convergence for the quadratic pointwise risk is the maximum of two rates: the minimax rate we would get if the errors were directly observed and the minimax rate of convergence of $ \hat b $ \hat b for the quadratic integrated risk.     

Integrated squared error of kernel-type estimator of distribution function

Let X ₁ ,...,X _n be a random sample drawn from distribution function F(x) with density function f(x) and suppose we want to estimate X(x). It is already shown that kernel estimator of F(x) is better than usual empirical distribution function in the sense of mean integrated squared error. In this paper we derive integrated squared error of kernel estimator and compare the error with that of the empirical distribution function. It is shown that the superiority of kernel estimators is not necessarily true in the sense of integrated squared error.     

On the Kaplan–Meier Estimator of Long-Range Dependent Sequences

The limiting distributions are obtained for the Kaplan–Meier estimator of unknown distribution function of stationary time series of the form G(X _j), where X _j is stationary Gaussian process with long-range dependence and G(·) is non-random function.     

Uniform Hölder exponent of a stationary increments Gaussian process: Estimation starting from average values

Let {X(t)}_t∈R be a stationary increments Gaussian process satisfying some assumptions. By using the notion of generalized quadratic variation we build a strongly consistent and asymptotically normal estimator of the uniform Hölder exponent of X, over a compact interval. Our estimator is obtained starting from average values of the process over a regular grid.     

Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure

We consider a linear multivariate errors-in-variables model AX ≈ B, where the matrices A and B are observed with errors and the matrix parameter X is to be estimated. In the case of lack of information about the error covariance structure, we propose an estimator that converges in probability to X as the number of rows in A tends to infinity. Sufficient conditions for this convergence and for the asymptotic normality of the estimator are found. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1026–1033, August, 2007.     

On Bayes and unbiased estimators of loss

We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In 3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988, On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly, that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss. Research supported by NSF Grant DMS-97-04524.     

Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation

Let X _t be a one-dimensional Harris recurrent diffusion, with a drift depending on an unknown parameter θ belonging to some metric compact Θ. We firstly show that all integrable additive functionals of X _t are asymptotically equivalent in probability to some deterministic process v _t . Then we use this result to study the behavior of the maximum likelihood estimator for the parameter θ. Under mild regularity assumptions, we find an upper rate of its convergence as a function of v _t , extending some recent results for ergodic diffusions.      

Nonparametric bayesian estimation of a survival curve with dependent censoring mechanism

Summary Let (X ₁,Y ₁),..., (X _N ,Y _N ) be a random sample from a bivariate distribution functionF. Based on observing only (δ _i ,Z _i ) whereδ _i =1 ifX _i ≦X _i and =0 otherwise andZ _i =min{X _i ,Y _i } fori=1,...,n, we obtain the Bayes estimator ofF whenF is a Dirichlet process under the usual integrated squared error loss function. It should be pointed out here thatX _i andY _i neednot be independent which is the usual assumption in survival analysis models. The effect of this dependence can be seen clearly in the estimators obtained and also in the given example which illustrates the estimator when Freunds' bivariate exponential distribution is taken as the parameter of the Dirichlet process. The research of this author is supported in part by an NSF Grant MCS80-03244. The research of this author is supported by the NIH Grant NO: 1 R01 GM 28405.     

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.     

Adaptive estimation in circular functional linear models

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y ₁, ..., Y _n are modeled in dependence of 1-periodic, second order stationary random functions X ₁, ...,X _n . We consider an orthogonal series estimator of the slope function β, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. We propose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill-posedness to be known. Then we generalize the procedure to a random set of admissible m’s and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in terms of a general weighted L ²-risk. This means that we provide adaptive estimators of both β and its derivatives.     

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.