Pointwise Risk Research Articles

Adaptive minimax estimation of service time distribution in the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from departure data

This article deals with the problem of estimating the service time distribution of the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from observation of the departure epochs. We develop minimax optimal estimators of G and study behavior of the minimax pointwise risk over a suitable family of service time distribution functions. In addition, we address the problem of adaptive estimation and propose a data–driven estimation procedure that adapts to unknown smoothness of the service time distribution function G. Lastly, a numerical study is presented to illustrate practical performance of the developed adaptive procedure.

Adaptive Estimation of a Function from its Exponential Radon Transform in Presence of Noise

In this article we propose a locally adaptive strategy for estimating a function from its Exponential Radon Transform (ERT) data, without prior knowledge of the smoothness of functions that are to be estimated. We build a non-parametric kernel type estimator and show that for a class of functions comprising a wide Sobolev regularity scale, our proposed strategy follows the minimax optimal rate up to a \(\log {n}\) factor. We also show that there does not exist an optimal adaptive estimator on the Sobolev scale when the pointwise risk is used and in fact the rate achieved by the proposed estimator is the adaptive rate of convergence.

Learning the smoothness of noisy curves with application to online curve estimation

Combining information both within and across trajectories, we propose a simple estimator for the local regularity of the trajectories of a stochastic process. Independent trajectories are measured with errors at randomly sampled time points. The proposed approach is model-free and applies to a large class of stochastic processes. Non-asymptotic bounds for the concentration of the estimator are derived. Given the estimate of the local regularity, we build a nearly optimal local polynomial smoother from the curves from a new, possibly very large sample of noisy trajectories. We derive non-asymptotic pointwise risk bounds uniformly over the new set of curves. Our estimates perform well in simulations, in both cases of differentiable or non-differentiable trajectories. Real data sets illustrate the effectiveness of the new approaches.

Pointwise wavelet change-points estimation for dependent biased sample

This paper focuses on the multiple change-points detection for the biased regression function based on the dependent biased sample. A peaks-over-threshold method is utilized to detect the change-points and an equispaced design approach is provided to evaluate the jump sizes. On the basis of the change-points and jump sizes estimations, we propose a (Lp version of the) block thresholding estimator for the biased regression function. This block wavelet change-points estimator is adaptive and the optimal convergence rate of the proposed estimator is established on pointwise risk over Besov space with the presence of dependent bias. Additionally, the effectiveness of the proposed estimator is validated through some examples and numerical simulations.

Nonparametric estimation of low rank matrix valued function

Let $A:[0,1]\rightarrow\mathbb{H}_m$ (the space of Hermitian matrices) be a matrix valued function which is low rank with entries in H\"{o}lder class $\Sigma(\beta,L)$. The goal of this paper is to study statistical estimation of $A$ based on the regression model $\mathbb{E}(Y_j|\tau_j,X_j) = \langle A(\tau_j), X_j \rangle,$ where $\tau_j$ are i.i.d. uniformly distributed in $[0,1]$, $X_j$ are i.i.d. matrix completion sampling matrices, $Y_j$ are independent bounded responses. We propose an innovative nuclear norm penalized local polynomial estimator and establish an upper bound on its point-wise risk measured by Frobenius norm. Then we extend this estimator globally and prove an upper bound on its integrated risk measured by $L_2$-norm. We also propose another new estimator based on bias-reducing kernels to study the case when $A$ is not necessarily low rank and establish an upper bound on its risk measured by $L_{\infty}$-norm. We show that the obtained rates are all optimal up to some logarithmic factor in minimax sense. Finally, we propose an adaptive estimation procedure based on Lepskii's method and model selection with data splitting which is computationally efficient and can be easily implemented and parallelized.

Nonparametric density and survival function estimation in the multiplicative censoring model

In this paper, we consider the multiplicative censoring model, given by Yi = XiUi where (Xi) are i.i.d. with unknown density f on R, (Ui) are i.i.d. with uniform distribution U([0, 1]) and (Ui) and (Xi) are independent sequences. Only the sample (Yi) 1≤i≤n is observed. Nonparametric estimators of both the density f and the corresponding survival function ¯ F are proposed and studied. First, classical kernels are used and the estimators are studied from several points of view: pointwise risk bounds for the quadratic risk are given, upper and lower bounds for the rates in this setting are provided. Then, an adaptive non asymptotic bandwidth selection procedure in a global setting is proved to realize the bias-variance compromise in an automatic way. When the Xi's are nonnegative, using kernels fitted for R +-supported functions, we propose new estimators of the survival function which are proved to be adaptive. Simulation experiments allow us to check the good performances of the estimators and compare the two strategies.

Minimax rate of convergence for an estimator of the functional component in a semiparametric multivariate partially linear model

A multivariate semiparametric partial linear model for both fixed and random design cases is considered. Earlier, in Brown et al. (2014), the model has been analyzed using a difference sequence approach. In particular, the functional component has been estimated using a multivariate Nadaraya–Watson kernel smoother of the residuals of the linear fit. Moreover, this functional component estimator has been shown to be rate optimal if the Lipschitz smoothness index exceeds half the dimensionality of the functional component domain. In the current manuscript, we take this research further and show that, for both fixed and random designs, the rate achieved is the minimax rate under both risk at a point and the L 2 risk. The result is achieved by proving lower bounds on both pointwise risk and the L 2 risk of possible estimators of the functional component.

Adaptive pointwise estimation for pure jump Lévy processes

This paper is concerned with adaptive kernel estimation of the Levy density $$N(x)$$ for bounded-variation pure-jump Levy processes. The sample path is observed at $$n$$ discrete instants in the “high frequency” context ( $$\Delta = \Delta (n) $$ tends to zero while $$n \Delta $$ tends to infinity). We construct a collection of kernel estimators of the function $$g(x)=xN(x)$$ and propose a method of local adaptive selection of the bandwidth. We provide an oracle inequality and a rate of convergence for the quadratic pointwise risk. This rate is proved to be the optimal minimax rate. We give examples and simulation results for processes fitting in our framework. We also consider the case of irregular sampling.

Adaptive estimation in the functional nonparametric regression model

In this paper, we consider nonparametric regression estimation when the predictor is a functional random variable (typically a curve) and the response is scalar. Starting from a classical collection of kernel estimates, the bias–variance decomposition of a pointwise risk is investigated to understand what can be expected at best from adaptive estimation. We propose a fully data-driven local bandwidth selection rule in the spirit of the Goldenshluger and Lepski method. The main result is a nonasymptotic risk bound which shows the optimality of our tuned estimator from the oracle point of view. Convergence rates are also derived for regression functions belonging to Hölder spaces and under various assumptions on the rate of decay of the small ball probability of the explanatory variable. A simulation study also illustrates the good practical performances of our estimator.

Adaptive function estimation in nonparametric regression with one-sided errors

We consider the model of nonregular nonparametric regression where smoothness constraints are imposed on the regression function $f$ and the regression errors are assumed to decay with some sharpness level at their endpoints. The aim of this paper is to construct an adaptive estimator for the regression function $f$. In contrast to the standard model where local averaging is fruitful, the nonregular conditions require a substantial different treatment based on local extreme values. We study this model under the realistic setting in which both the smoothness degree $\beta>0$ and the sharpness degree $\mathfrak{a}\in(0,\infty)$ are unknown in advance. We construct adaptation procedures applying a nested version of Lepski’s method and the negative Hill estimator which show no loss in the convergence rates with respect to the general $L_{q}$-risk and a logarithmic loss with respect to the pointwise risk. Optimality of these rates is proved for $\mathfrak{a}\in(0,\infty)$. Some numerical simulations and an application to real data are provided.

Uniform Convergence and Rate Adaptive Estimation of Convex Functions via Constrained Optimization

This paper discusses asymptotic analysis and adaptive design of convex estimators over the Holder class under the sup-norm risk and the pointwise risk using constrained optimization and asymptotic statistical techniques. Specifically, convex B-spline estimators are proposed to achieve uniform optimal convergence rates and adaptive procedures. The presence of the convex shape constraint complicates asymptotic performance analysis, particularly uniform convergence analysis. This in turn requires deep understanding of a family of size varying constrained optimization problems on spline coefficients. To address these issues, we establish the uniform Lipschitz property of optimal spline coefficients in the $\ell_\infty$-norm by exploiting the structure of underlying constrained optimization problems. By using this property, polyhedral theory, and statistical techniques, we show that the convex B-spline estimator attains uniform consistency and optimal rates of convergence on the entire interval of interest ove...

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 \( \hat b \) of the regression function, we apply the density estimation procedure to the residuals \( \hat \varepsilon _i = Y_i - \hat b(X_i ) \). 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 \( \hat b \) for the quadratic integrated risk.

Universal pointwise selection rule in multivariate function estimation

In this paper, we study the problem of pointwise estimation of a multivariate function. We develop a general pointwise estimation procedure that is based on selection of estimators from a large parameterized collection. An upper bound on the pointwise risk is established and it is shown that the proposed selection procedure specialized for different collections of estimators leads to minimax and adaptive minimax estimators in various settings.

Spatial aggregation of local likelihood estimates with applications to classification

This paper presents a new method for spatially adaptive local (constant) likelihood estimation which applies to a broad class of nonparametric models, including the Gaussian, Poisson and binary response models. The main idea of the method is, given a sequence of local likelihood estimates (weak estimates), to construct a new aggregated estimate whose pointwise risk is of order of the smallest risk among all weak estimates. We also propose a new approach toward selecting the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation. We establish a number of important theoretical results concerning the optimality of the aggregated estimate. In particular, our oracle result claims that its risk is, up to some logarithmic multiplier, equal to the smallest risk for the given family of estimates. The performance of the procedure is illustrated by application to the classification problem. A numerical study demonstrates its reasonable performance in simulated and real-life examples.

Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data

We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared quantum systems. The state is represented through the Wigner function, a generalized probability density on $\mathbb{R}^2$ which may take negative values and must respect intrinsic positivity constraints imposed by quantum physics. The effect of the losses due to detection inefficiencies, which are always present in a real experiment, is the addition to the tomographic data of independent Gaussian noise. We construct a kernel estimator for the Wigner function, prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions, and implement it for numerical results. We construct adaptive estimators, that is, which do not depend on the smoothness parameters, and prove that in some setups they attain the minimax rates for the corresponding smoothness class.

Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors

We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared pulses. The state is represented through the Wigner function, a “quasi-probability density” on ℝ2 which may take negative values and must satisfy intrinsic positivity constraints imposed by quantum physics. The data consists of n i.i.d. observations from a probability density equal to the Radon transform of the Wigner function. We construct an estimator for the Wigner function and prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions. A similar result was previously derived by Cavalier in the context of positron emission tomography. Our work extends this result to the space of smooth Wigner functions, which is the relevant parameter space for quantum homodyne tomography.

Lower bounds for the asymptotic minimax risk with spherical data

Lower bounds for the asymptotic minimax risk are given when estimating the density function with spherical data. Both the pointwise risk and the integrated risk are studied. Results are derived by constructing a sequence of experiments that converge weakly to a Gaussian shift experiment. Lower bounds derived in this article complement some previously derived upper bounds. We give one upper bound for the pointwise risk which is only a factor of 1.18 larger than the corresponding lower bound. The lower bounds are formulated in a general setting so that also the Euclidean results follow as a special case.