Related papers: Nonstandard Analysis and Generalized Functions
We give a general approach to infinite dimensional non-Gaussian Analysis for measures which need not have a logarithmic derivative. This framework also includes the possibility to handle measures of Poisson type.
We consider high-dimensional estimation problems where the number of parameters diverges with the sample size. General conditions are established for consistency, uniqueness, and asymptotic normality in both unpenalized and penalized…
In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
In distributional or average-case analysis, the goal is to design an algorithm with good-on-average performance with respect to a specific probability distribution. Distributional analysis can be useful for the study of general-purpose…
We establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.
Paper is devoted to extremal problems in geometric function theory of complex variables associated with estimates of functionals defined on the systems of non-overlapping domains. In particular, we strengthen some known result in this…
Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are…
The main phases of applied statistical work are discussed in general terms. The account starts with the clarification of objectives and proceeds through study design, measurement and analysis to interpretation. An attempt is made to extract…
Seemingly unrelated linear regression models are introduced in which the distribution of the errors is a finite mixture of Gaussian components. Identifiability conditions are provided. The score vector and the Hessian matrix are derived.…
These notes are concerned with the existence and the basic properties of the set-theoretic universes for nonstandard analysis, compiled by a beginner in the subject. It assumes a basic background in first-order logic, though the necessary…
We apply methods of nonstandard mathematics in order to regard analytic geometry in a very different way. For example, complex spaces are seen to be the "standard part" of certain algebraic nonstandard schemes. We construct a category of…
Multivariate functions emerge naturally in a wide variety of data-driven models. Popular choices are expressions in the form of basis expansions or neural networks. While highly effective, the resulting functions tend to be hard to…
One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and non-stationary regimes is the lack of a suitable limit object. Instead, suitable distributional approximations can be used, where…
We generalize Taylor's theorem by introducing a stochastic formulation based on an underlying Poisson point process model. We utilize this approach to propose a novel non-linear regression framework and perform statistical inference of the…
The non isotropic noncentral elliptical shape distributions via pseudo-Wishart distribution are founded. This way, the classical shape theory is extended to non isotropic case and the normality assumption is replaced by assuming a…
This paper presents a general framework for unifying functional interpretations. It is based on families of parameters allowing for different degrees of freedom on the design of the interpretation. In this way we are able to generalise…
We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.
Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued…
We extend for the second time the Nonstandard Analysis by adding the left monad closed to the right, and right monad closed to the left, while besides the pierced binad (we introduced in 1998) we add now the unpierced binad - all these in…
A gauge invariant partition function is defined for gauge theories which leads to the standard quantization. It is shown that the descent equations and consequently the consistent anomalies and Schwinger terms can be extracted from this…