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Том 26, № 4 (2017)
- Год: 2017
- Статей: 5
- URL: https://journal-vniispk.ru/1066-5307/issue/view/13895
Article
Laguerre deconvolution with unknown matrix operator
Аннотация
In this paper we consider the convolutionmodel Z = X + Y withX of unknown density f, independent of Y, when both random variables are nonnegative. Our goal is to estimate the unknown density f of X from n independent identically distributed observations of Z, when the law of the additive process Y is unknown. When the density of Y is known, a solution to the problem has been proposed in [17]. To make the problem identifiable for unknown density of Y, we assume that we have access to a preliminary sample of the nuisance process Y. The question is to propose a solution to an inverse problem with unknown operator. To that aim, we build a family of projection estimators of f on the Laguerre basis, well-suited for nonnegative random variables. The dimension of the projection space is chosen thanks to a model selection procedure by penalization. At last we prove that the final estimator satisfies an oracle inequality. It can be noted that the study of the mean integrated square risk is based on Bernstein’s type concentration inequalities developed for random matrices in [23].
237-266
Statistical foundations for assessing the difference between the classical and weighted-Gini betas
Аннотация
The ‘beta’ is one of the key quantities in the capital asset pricing model (CAPM). In statistical language, the beta can be viewed as the slope of the regression line fitted to financial returns on the market against the returns on the asset under consideration. The insurance counterpart of CAPM, called the weighted insurance pricing model (WIPM), gives rise to the so-called weighted-Gini beta. The aforementioned two betas may or may not coincide, depending on the form of the underlying regression function, and this has profound implications when designing portfolios and allocating risk capital. To facilitate these tasks, in this paper we develop large-sample statistical inference results that, in a straightforward fashion, imply confidence intervals for, and hypothesis tests about, the equality of the two betas.
267-281
Minimax signal detection under weak noise assumptions
Аннотация
We consider minimax signal detection in the sequence model. Working with certain ellipsoids in the space of square-summable sequences of real numbers, with a ball of positive radius removed, we obtain upper and lower bounds for the minimax separation radius in the non-asymptotic framework, i.e., for a fixed value of the involved noise level. We use very weak assumptions on the noise (i.e., fourth moments are assumed to be uniformly bounded). In particular, we do not use any kind of Gaussian distribution or independence assumption on the noise. It is shown that the established minimax separation rates are not faster than the ones obtained in the classical sequence model (i.e., independent standard Gaussian noise) but, surprisingly, are of the same order as the minimax estimation rates in the classical setting. Under an additional condition on the noise, the classical minimax separation rates are also retrieved in benchmark well-posed and ill-posed inverse problems.
282-298
On estimation in some reduced rank extended growth curve models
Аннотация
The general multivariate analysis of variance model has been extensively studied in the statistical literature and successfully applied in many different fields for analyzing longitudinal data. In this article, we consider the extension of this model having two sets of regressors constituting a growth curve portion and a multivariate analysis of variance portion, respectively. Nowadays, the data collected in empirical studies have relatively complex structures though often demanding a parsimonious modeling. This can be achieved for example through imposing rank constraints on the regression coefficient matrices. The reduced rank regression structure also provides a theoretical interpretation in terms of latent variables. We derive likelihood based estimators for the mean parameters and covariance matrix in this type of models. A numerical example is provided to illustrate the obtained results.
299-310
On joint weak reversed hazard rate order under symmetric copulas
Аннотация
In this paper, a weak version of the joint reversed hazard rate order, useful for stochastic comparison of non-independent random variables, has been defined and discussed. In particular, some relationships between the joint weak reversed hazard rate order and the usual reversed hazard rate order are established when the underlying copulas are symmetric.
311-318