One focus of data analysis in formal concept analysis is attribute-significance measure, and another is attribute reduction. From the perspective of information granules, we propose information entropy in formal contexts and conditional information entropy in formal decision contexts, and they are further used to measure attribute significance. Moreover, an approach is presented to measure the consistency of a formal decision context in preparation for calculating reducts. Finally, heuristic ideas are integrated with reduction technique to achieve the task of calculating reducts of an inconsistent data set.
In the framework of a stochastic optimization problem, it is assumed that the stochastic characteristics of optimized system are estimated from randomly right-censored data. Such a case is frequently encountered in time-to-event or lifetime studies. The analysis of precision of such a solution is based on corresponding theoretical properties of estimated stochastic characteristics. The main concern is to show consistency of optimal solution even in the random censoring case. Behavior of solutions for finite data sizes is studied with the aid of randomly generated example.
The Perona-Malik nonlinear parabolic problem, which is widely used in image processing, is investigated in this paper from the numerical point of view. An explicit finite volume numerical scheme for this problem is presented and consistency property is proved.
The paper studies polynomial approximation models with a new type of constraints that enable to get estimates with significant properties. Recently we enhanced a representation of polynomials based on three reference points. Here we propose a two-part cubic smoothing scheme that leverages this representation. The presence of these points in the model has several consequences. The most important one is the fact that by appropriate location of the reference points the resulting approximant of two successively assessed neighboring approximants will be smooth. We also show that the considered models provide estimates with appropriate statistical properties such as consistency and asymptotic normality.
Interpolating and approximating polynomials have been living separately more than two centuries. Our aim is to propose a general parametric regression model that incorporates both interpolation and approximation. The paper introduces first a new r-point transformation that yields a function with a simpler geometrical structure than the original function. It uses r≥2 reference points and decreases the polynomial degree by r−1. Then a general representation of polynomials is proposed based on r≥1 reference points. The two-part model, which is suited to piecewise approximation, consist of an ordinary least squares polynomial regression and a reparameterized one. The later is the central component where the key role is played by the reference points. It is constructed based on the proposed representation of polynomials that is derived using the r-point transformation Tr(x). The resulting polynomial passes through r reference points and the other points approximates. Appropriately chosen reference points ensure quasi smooth transition between the two components and decrease the dimension of the LS normal matrix. We show that the model provides estimates with such statistical properties as consistency and asymptotic normality.
The paper investigates generalized linear models (GLM's) with binary responses such as the logistic, probit, log-log, complementary log-log, scobit and power logit models. It introduces a median estimator of the underlying structural parameters of these models based on statistically smoothed binary responses. Consistency and asymptotic normality of this estimator are proved. Examples of derivation of the asymptotic covariance matrix under the above mentioned models are presented. Finally some comments concerning a method called enhancement and robustness of median estimator are given and results of simulation experiment comparing behavior of median estimator with other robust estimators for GLM's known from the literature are reported.
New statistical procedures for a change in means problem within a very general panel data structure are proposed. Unlike classical inference tools used for the changepoint problem in the panel data framework, we allow for mutually dependent panels, unequal variances across the panels, and possibly an extremely short follow up period. Two competitive ratio type test statistics are introduced and their asymptotic properties are derived for a large number of available panels. The proposed tests are proved to be consistent and their empirical properties are investigated in an extensive simulation study. The suggested testing approaches are also applied to a real data problem.
The consistency of the least trimmed squares estimator (see Rousseeuw \cite{Rous} or Hampel et al. \cite{HamRonRouSta}) is proved under general conditions. The assumptions employed in paper are discussed in details to clarify the consequences for the applications.
Recently, the parameter estimations for normal fuzzy variables in the Nahmias' sense was studied by Cai [4]. These estimates were also studied for general T-related, but not necessarily normal fuzzy variables by Hong [10] In this paper, we report on some properties of estimators that would appear to be desirable, including unbiasedness. We also consider asymptotic or "large-sample" properties of a particular type of estimator.