Abstract: In this paper we investigate an indirect regression model characterized by the Radon transformation. This model is useful for recovery of medical images obtained by computed tomography scans. The indirect regression function is estimated using a series estimator motivated by a spectral cutoff technique. Further, we investigate the empirical process of residuals from this regression, and show that it satisfies a functional central limit theorem. PubDate: 2019-04-01

Abstract: Based on X ∼ Nd(θ, σ X 2 Id), we study the efficiency of predictive densities under α-divergence loss Lα for estimating the density of Y ∼ Nd(θ, σ Y 2 Id). We identify a large number of cases where improvement on a plug-in density are obtainable by expanding the variance, thus extending earlier findings applicable to Kullback-Leibler loss. The results and proofs are unified with respect to the dimension d, the variances σ X 2 and σ Y 2 , the choice of loss Lα; α ∈ (−1, 1). The findings also apply to a large number of plug-in densities, as well as for restricted parameter spaces with θ ∈ Θ ⊂ ℝd. The theoretical findings are accompanied by various observations, illustrations, and implications dealing for instance with robustness with respect to the model variances and simultaneous dominance with respect to the loss. PubDate: 2019-04-01

Abstract: We consider a stationary AR(p) model. The autoregression parameters are unknown as well as the distribution of innovations. Based on the residuals from the parameter estimates, an analog of empirical distribution function is defined and the tests of Kolmogorov’s and ω2 type are constructed for testing hypotheses on the distribution of innovations. We obtain the asymptotic power of these tests under local alternatives. PubDate: 2019-04-01

Abstract: Let X1, X2,... be independent random variables observed sequentially and such that X1,..., Xθ−1 have a common probability density p0, while Xθ, Xθ+1,... are all distributed according to p1 ≠ p0. It is assumed that p0 and p1 are known, but the time change θ ∈ ℤ+ is unknown and the goal is to construct a stopping time τ that detects the change-point θ as soon as possible. The standard approaches to this problem rely essentially on some prior information about θ. For instance, in the Bayes approach, it is assumed that θ is a random variable with a known probability distribution. In the methods related to hypothesis testing, this a priori information is hidden in the so-called average run length. The main goal in this paper is to construct stopping times that are free from a priori information about θ. More formally, we propose an approach to solving approximately the following minimization problem: $$\Delta(\theta;{\tau^\alpha})\rightarrow\min_{\tau^\alpha}\;\;\text{subject}\;\text{to}\;\;\alpha(\theta;{\tau^\alpha})\leq\alpha\;\text{for}\;\text{any}\;\theta\geq1,$$ where α(θ; τ) = Pθ{τ < θ} is the false alarm probability and Δ(θ; τ) = Eθ(τ − θ)+ is the average detection delay computed for a given stopping time τ. In contrast to the standard CUSUM algorithm based on the sequential maximum likelihood test, our approach is related to a multiple hypothesis testing methods and permits, in particular, to construct universal stopping times with nearly Bayes detection delays. PubDate: 2019-04-01

Abstract: In this article, we propose a new method for analyzing longitudinal data which contain responses that are missing at random. This method consists in solving the generalized estimating equation (GEE) of [8] in which the incomplete responses are replaced by values adjusted using the inverse probability weights proposed in [17]. We show that the root estimator is consistent and asymptotically normal, essentially under the some conditions on the marginal distribution and the surrogate correlation matrix as those presented in [15] in the case of complete data, and under minimal assumptions on the missingness probabilities. This method is applied to a real-life data set taken from [13], which examines the incidence of respiratory disease in a sample of 250 pre-school age Indonesian children which were examined every 3 months for 18 months, using as covariates the age, gender, and vitamin A deficiency. PubDate: 2019-04-01

Abstract: We consider a stationary linear AR(p) model with contamination (gross errors in the observations). The autoregression parameters are unknown, as well as the distribution of innovations. Based on the residuals from the parameter estimates, an analog of the empirical distribution function is defined and a test of Pearson’s chi-square type is constructed for testing hypotheses on the distribution of innovations. We obtain the asymptotic power of this test under local alternatives and establish its qualitative robustness under the hypothesis and alternatives. PubDate: 2019-01-01

Abstract: We establish a large deviation approximation for the density of an arbitrary sequence of random vectors, by assuming several assumptions on the normalized cumulant generating function and its derivatives. We give two statistical applications to illustrate the result, the first one dealing with a vector of independent sample variances and the second one with a Gaussian multiple linear regression model. Numerical comparisons are eventually provided for these two examples. PubDate: 2019-01-01

Abstract: Estimation of the predictive probability function of a negative binomial distribution is addressed under the Kullback—Leibler risk. An identity that relates Bayesian predictive probability estimation to Bayesian point estimation is derived. Such identities are known in the cases of normal and Poisson distributions, and the paper extends the result to the negative binomial case. By using the derived identity, a dominance property of a Bayesian predictive probability is studied when the parameter space is restricted. PubDate: 2019-01-01

Abstract: We consider the problem of nonparametric density estimation of a random environment from the observation of a single trajectory of a random walk in this environment. We build several density estimators using the beta-moments of this distribution. Then we apply the Goldenschluger-Lepski method to select an estimator satisfying an oracle type inequality. We obtain non-asymptotic bounds for the supremum norm of these estimators that hold when the RWRE is recurrent or transient to the right. A simulation study supports our theoretical findings. PubDate: 2019-01-01

Abstract: In this work we suppose that the random vector (X, Y) satisfies the regression model Y = m(X) + ϵ, where m(·) belongs to some parametric class { \({m_\beta}(\cdot):\beta \in \mathbb{K}\) } and the error ϵ is independent of the covariate X. The response Y is subject to random right censoring. Using a nonlinear mode regression, a new estimation procedure for the true unknown parameter vector β0is proposed that extends the classical least squares procedure for nonlinear regression. We also establish asymptotic properties for the proposed estimator under assumptions of the error density. We investigate the performance through a simulation study. PubDate: 2019-01-01

Abstract: The aim of the paper is to show that the presence of one possible type of outliers is not connected to that of heavy tails of the distribution. In contrary, typical situation for outliers appearance is the case of compactly supported distributions. PubDate: 2019-01-01

Authors:B. Levit Pages: 245 - 267 Abstract: For the Hardy classes of functions analytic in the strip around real axis of a size 2β, an optimal method of cardinal interpolation has been proposed within the framework of Optimal Recovery [12]. Below this method, based on the Jacobi elliptic functions, is shown to be optimal according to the criteria of Nonparametric Regression and Optimal Design. In a stochastic non-asymptotic setting, the maximal mean squared error of the optimal interpolant is evaluated explicitly, for all noise levels away from 0. A pivotal role is played by the interference effect, in which the oscillations exhibited by the interpolant’s bias and variance mutually cancel each other. In the limiting case β → ∞, the optimal interpolant converges to the well-knownNyquist–Shannon cardinal series. PubDate: 2018-10-01 DOI: 10.3103/s1066530718040014 Issue No:Vol. 27, No. 4 (2018)

Authors:E. Caron; S. Dede Pages: 268 - 293 Abstract: We consider the usual linear regression model in the case where the error process is assumed strictly stationary.We use a result of Hannan, who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and the error process.We show that for a large class of designs, the asymptotic covariance matrix is as simple as in the independent and identically distributed (i.i.d.) case.We then estimate the covariance matrix using an estimator of the spectral density whose consistency is proved under very mild conditions. PubDate: 2018-10-01 DOI: 10.3103/s1066530718040026 Issue No:Vol. 27, No. 4 (2018)

Authors:M. V. Boldin; M. N. Petriev Pages: 294 - 311 Abstract: We consider a stationary linear AR(p) model with observations subject to gross errors (outliers). The distribution of outliers is unknown and arbitrary, their intensity is γn−1/2 with an unknown γ, n is the sample size. The autoregression parameters are unknown, they are estimated by any estimator which is n1/2-consistent uniformly in γ ≤ Γ < ∞. Using the residuals from the estimated autoregression, we construct a kind of empirical distribution function (e.d.f.), which is a counterpart of the (inaccessible) e.d.f. of the autoregression innovations. We obtain a stochastic expansion of this e.d.f., which enables us to construct a test of Pearson’s chi-square type for testing hypotheses about the distribution of innovations. We establish qualitative robustness of this test in terms of uniform equicontinuity of the limiting level with respect to γ in a neighborhood of γ = 0. PubDate: 2018-10-01 DOI: 10.3103/s1066530718040038 Issue No:Vol. 27, No. 4 (2018)

Authors:B. Berckmoes; G. Molenberghs Pages: 312 - 323 Abstract: An observation of a cumulative distribution function F with finite variance is said to be contaminated according to the inflated variance model if it has a large probability of coming from the original target distribution F, but a small probability of coming from a contaminating distribution that has the same mean and shape as F, though a larger variance. It is well known that in the presence of data contamination, the ordinary sample mean looses many of its good properties, making it preferable to use more robust estimators. It is insightful to see to what extent an intuitive estimator such as the sample mean becomes less favorable in a contaminated setting. In this paper, we investigate under which conditions the sample mean, based on a finite number of independent observations of F which are contaminated according to the inflated variance model, is a valid estimator for the mean of F. In particular, we examine to what extent this estimator is weakly consistent for the mean of F and asymptotically normal. As classical central limit theory is generally inaccurate to copewith the asymptotic normality in this setting, we invokemore general approximate central limit theory as developed in [3]. Our theoretical results are illustrated by a specific example and a simulation study. PubDate: 2018-10-01 DOI: 10.3103/s106653071804004x Issue No:Vol. 27, No. 4 (2018)

Authors:E. Belitser; N. Nurushev Pages: 163 - 183 Abstract: We study the problem of inference (estimation and uncertainty quantification problems) on the unknown parameter in the biclustering model by using the penalization method. The underlying biclustering structure is that the high-dimensional parameter consists of a few blocks of equal coordinates. The quality of the inference procedures is measured by the local quantity, the oracle rate, which is the best trade-off between the approximation error by a biclustering structure and the complexity of that approximating biclustering structure. The approach is also robust in that the additive errors are assumed to satisfy only certain mild condition (allowing non-iid errors with unknown joint distribution). By using the penalization method, we construct a confidence set and establish its local (oracle) optimality. Interestingly, as we demonstrate, there is (almost) no deceptiveness issue for the uncertainty quantification problem in the biclustering model. Adaptive minimax results for the biclustering, stochastic block model (with implications for network modeling) and graphon scales follow from our local results. PubDate: 2018-07-01 DOI: 10.3103/s1066530718030018 Issue No:Vol. 27, No. 3 (2018)

Authors:A. Bibi Pages: 184 - 204 Abstract: In this paper, we explore some probabilistic and statistical properties of constant conditional correlation (CCC) multivariate periodic GARCH models (CCC − PGARCH for short). These models which encompass some interesting classes having (locally) long memory property, play an outstanding role in modelling multivariate financial time series exhibiting certain heteroskedasticity. So, we give in the first part some basic structural properties of such models as conditions ensuring the existence of the strict stationary and geometric ergodic solution (in periodic sense). As a result, it is shown that the moments of some positive order for strictly stationary solution of CCC − PGARCH models are finite.Upon this finding, we focus in the second part on the quasi-maximum likelihood (QML) estimator for estimating the unknown parameters involved in the models. So we establish strong consistency and asymptotic normality (CAN) of CCC − PGARCH models. PubDate: 2018-07-01 DOI: 10.3103/s106653071803002x Issue No:Vol. 27, No. 3 (2018)

Authors:Yu. Golubev Pages: 205 - 225 Abstract: The paper considers a simple Errors-in-Variables (EiV) model Yi = a + bXi + εξi; Zi= Xi + σζi, where ξi, ζi are i.i.d. standard Gaussian random variables, Xi ∈ ℝ are unknown non-random regressors, and ε, σ are known noise levels. The goal is to estimates unknown parameters a, b ∈ ℝ based on the observations {Yi, Zi, i = 1, …, n}. It is well known [3] that the maximum likelihood estimates of these parameters have unbounded moments. In order to construct estimates with good statistical properties, we study EiV model in the large noise regime assuming that n → ∞, but \({\epsilon ^2} = \sqrt n \epsilon _ \circ ^2,{\sigma ^2} = \sqrt n \sigma _ \circ ^2\) with some \(\epsilon_\circ^2, \sigma_\circ^2>0\) . Under these assumptions, a minimax approach to estimating a, b is developed. It is shown that minimax estimates are solutions to a convex optimization problem and a fast algorithm for solving it is proposed. PubDate: 2018-07-01 DOI: 10.3103/s1066530718030031 Issue No:Vol. 27, No. 3 (2018)

Authors:D. N. Politis; V. A. Vasiliev; S. E. Vorobeychikov Pages: 226 - 243 Abstract: The problem of estimation of the heavy tail index is revisited from the point of view of truncated estimation. A class of novel estimators is introduced having guaranteed accuracy based on a sample of fixed size. The performance of these estimators is quantified both theoretically and in simulations over a host of relevant examples. It is also shown that in several cases the proposed estimators attain — within a logarithmic factor — the optimal parametric rate of convergence. The property of uniform asymptotic normality of the proposed estimators is established. PubDate: 2018-07-01 DOI: 10.3103/s1066530718030043 Issue No:Vol. 27, No. 3 (2018)

Authors:Yu. S. Kharin; V. A. Voloshko; E. A. Medved Pages: 103 - 118 Abstract: The problem of statistical parameter estimation is considered for binary GLM-based autoregression with the link function of general form and the base functions (regressors) nonlinear w.r.t. the lagged variables. A new consistent asymptotically normal frequencies-based estimator (FBE) is constructed and compared with the classical MLE. It is shown that the FBE has less restrictive sufficient conditions of uniqueness than the MLE (does not need log-concavity of the inverse link) and can be computed recursively under the model extension. The sparse version of the FBE is proposed and the optimal model-dependent weight matrices (parameterizing the FBE) are found for the FBE and for the sparse FBE. The proposed empirical choice of the subset of s-tuples for the sparse FBE is examined by numerical and analytical examples. Computer experiments for comparison of the FBE versus the MLE are performed on simulated and real (genetic) data. PubDate: 2018-04-01 DOI: 10.3103/s1066530718020023 Issue No:Vol. 27, No. 2 (2018)