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Vol 25, No 1 (2016)
- Year: 2016
- Articles: 4
- URL: https://journal-vniispk.ru/1066-5307/issue/view/13887
Article
Tikhonov–Phillips regularizations in linear models with blurred design
Abstract
The paper deals with recovering an unknown vector β ∈ ℝp based on the observations Y = Xβ + єξ and Z = X + σζ, where X is an unknown n × p matrix with n ≥ p, ξ ∈ ℝp is a standard white Gaussian noise, ζ is an n × p matrix with i.i.d. standard Gaussian entries, and є, σ ∈ ℝ+ are known noise levels. It is assumed that X has a large condition number and p is large. Therefore, in order to estimate β, the simple Tikhonov–Phillips regularization (ridge regression) with a data-driven regularization parameter is used. For this estimation method, we study the effect of noise σζ on the quality of recovering Xβ using concentration inequalities for the prediction error.
1-25
Structural adaptive deconvolution under \({\mathbb{L}_p}\)-losses
Abstract
In this paper, we address the problem of estimating a multidimensional density f by using indirect observations from the statistical model Y = X + ε. Here, ε is a measurement error independent of the random vector X of interest and having a known density with respect to Lebesgue measure. Our aim is to obtain optimal accuracy of estimation under \({\mathbb{L}_p}\)-losses when the error ε has a characteristic function with a polynomial decay. To achieve this goal, we first construct a kernel estimator of f which is fully data driven. Then, we derive for it an oracle inequality under very mild assumptions on the characteristic function of the error ε. As a consequence, we getminimax adaptive upper bounds over a large scale of anisotropic Nikolskii classes and we prove that our estimator is asymptotically rate optimal when p ∈ [2,+∞]. Furthermore, our estimation procedure adapts automatically to the possible independence structure of f and this allows us to improve significantly the accuracy of estimation.
26-53
Efficiency of exponentiality tests based on a special property of exponential distribution
Abstract
New goodness-of-fit tests for exponentiality based on a particular property of exponential law are constructed. Test statistics are functionals of U-empirical processes. The first of these statistics is of integral type, the second one is a Kolmogorov type statistic.We show that the kernels corresponding to our statistics are nondegenerate. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various parametric alternatives is calculated and is comparedwith simulated powers of new tests. Conditions of local optimality of new statistics in Bahadur sense are discussed and examples of “most favorable” alternatives are given. New tests are applied to reject the hypothesis of exponentiality for the length of reigns of Roman emperors which was intensively discussed in recent years.
54-66
Extremal problems for hypotheses testing with set-valued decisions
Abstract
We consider a class of extremal problems for multiple hypothesis testing with set-valued decisions and given total variation distances between hypotheses. The quality of a test is measured by an arbitrary piecewise linear continuous function of the error probabilities. We show that the extremal value of the test quality may be found as a solution of some linear programming problem, so the original infinite-dimensional problem is reduced to a certain finite-dimensional one.
67-77