### Article

## Density deconvolution under general assumptions

In this paper we study the problem

of density deconvolution under general assumptions on the measurement error distribution. Typically

deconvolution estimators are constructed using Fourier transform techniques, and

it is assumed that

the characteristic function of

the measurement errors does not have zeros

on the real line. This assumption is rather strong and is not fulfilled

in many cases of interest. In this paper we develop a

methodology for constructing optimal density deconvolution estimators in the general setting that covers

vanishing and non--vanishing characteristic functions of the measurement errors.

We derive upper bounds on the risk of the proposed estimators and

provide sufficient conditions under which zeros of the corresponding characteristic function have no effect on estimation accuracy.

Moreover, we show that the derived conditions are also necessary in some

specific problem instances.

In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post– Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models. We construct a nonparametric estimator for the mixing density based on the generalized Post–Widder formula, derive bounds for its root mean square error and give a brief numerical example.

This paper generalizes recent proposals of density forecasting models and it develops theory for this class of models. In density forecasting, the density of observations is estimated in regions where the density is not observed. Identification of the density in such regions is guaranteed by structural assumptions on the density that allows exact extrapolation. In this paper, the structural assumption is made that the density is a product of one-dimensional functions. The theory is quite general in assuming the shape of the region where the density is observed. Such models naturally arise when the time point of an observation can be written as the sum of two terms (e.g., onset and incubation period of a disease). The developed theory also allows for a multiplicative factor of seasonal effects. Seasonal effects are present in many actuarial, biostatistical, econometric and statistical studies. Smoothing estimators are proposed that are based on backfitting. Full asymptotic theory is derived for them. A practical example from the insurance business is given producing a within year budget of reported insurance claims. A small sample study supports the theoretical results

We study a problem of designing an optimal two-dimensional circularly symmetric convolution kernel (or point spread function (PSF)) with a circular support of a chosen radius *R*. Such function will be optimal for estimating an unknown signal (image) from an observation obtained through a convolution-type distortion with the additive random noise. This technique is then generalized to the case of an imprecisely known or random PSF of the measurement distortion. It is shown that the construction of the optimal convolution kernel reduces to a one-dimensional Fredholm equation of the first or a second kind on the interval [0,*R*]. If the reconstruction PSF is sought in a finite-dimensional class of functions, the problem naturally reduces to a finite-dimensional optimization problem or even a system of linear equations. We also analyze how reconstruction quality depends on the radius of the convolution kernel. It allows finding a good balance between computational complexity and quality of the image reconstruction.

Despite the increasingly broad use of perfusion applications, we still have no generally accessible means for their verification: The common sense of perfusion maps and "bona fides" of perfusion software vendors remain the only grounds for acceptance. Thus, perfusion applications are one of a very few clinical tools considerably lacking practical objective hands-on validation. MATERIALS AND METHODS. To solve this problem, we introduce digital perfusion phantoms (DPPs) - numerically simulated DICOM image sequences specifically designed to have known perfusion maps with simple visual patterns. Processing DPP perfusion sequences with any perfusion algorithm or software of choice and comparing the results with the expected DPP patterns provide a robust and straightforward way to control the quality of perfusion analysis, software, and protocols. RESULTS. The deviations from the expected DPP maps, observed in each perfusion software, provided clear visualization of processing differences and possible perfusion implementation errors. CONCLUSION. Perfusion implementation errors, often hidden behind real-data anatomy and noise, become very visible with DPPs. We strongly recommend using DPPs to verify the quality of perfusion applications.

Positive-Unlabeled (PU) learning is an analog to supervised binary classification for the case when only the positive sample is clean, while the negative sample is contaminated with latent instances of positive class and hence can be considered as an unlabeled mixture. The objectives are to classify the unlabeled sample and train an unbiased positive-negative classifier, which generally requires to identify the mixing proportions of positives and negatives first. Recently, unbiased risk estimation framework has achieved state-of-the-art performance in PU learning. This approach, however, exhibits two major bottlenecks. First, the mixing proportions are assumed to be identified, i.e. known in the domain or estimated with additional methods. Second, the approach relies on the classifier being a neural network. In this paper, we propose DEDPUL, a method that solves PU Learning without the aforementioned issues. The mechanism behind DEDPUL is to apply a computationally cheap post-processing procedure to the predictions of any classifier trained to distinguish positive and unlabeled data. Instead of assuming the proportions to be identified, DEDPUL estimates them alongside with classifying unlabeled sample. Experiments show that DEDPUL outperforms the current state-of-the-art in both proportion estimation and PU Classification and is flexible in the choice of the classifier.

Positive-Unlabeled (PU) learning is an analog to supervised binary classification for the case when only the positive sample is clean, while the negative sample is contaminated with latent instances of positive class and hence can be considered as an unlabeled mixture. The objectives are to classify the unlabeled sample and train an unbiased positive-negative classifier, which generally requires to identify the mixing proportions of positives and negatives first. Recently, unbiased risk estimation framework has achieved state-of-the-art performance in PU learning. This approach, however, exhibits two major bottlenecks. First, the mixing proportions are assumed to be identified, i.e. known in the domain or estimated with additional methods. Second, the approach relies on the classifier being a neural network. In this paper, we propose DEDPUL, a method that solves PU Learning without the aforementioned issues. The mechanism behind DEDPUL is to apply a computationally cheap postprocessing procedure to the predictions of any classifier trained to distinguish positive and unlabeled data. Instead of assuming the proportions to be identified, DEDPUL estimates them alongside with classifying unlabeled sample. Experiments show that DEDPUL outperforms the current state-of-the-art in both proportion estimation and PU Classification and is flexible in the choice of the classifier.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.