Asymptotics for in-sample density forecasting
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
Nowadays insurance market is one of the most rapidly developing sectors of economy, the purpose of which is to protect the property interests of individuals and legal entities under ensuing of specific events (insured accidents) at the expense of monetary funds formed from insurance dues (insurance premiums) paid by them. Probabilistic nature of insured accidents as well as the uncertainty of the moment of their occurrence and the severity of losses leads to the necessity of forming loss reserves. Reserves for incurred but not reported claims (hereinafter referred to as IBNR reserves) seem to be the most challenging in terms of actuarial calculations. The following article provides the descriptions of various actuarial techniques of loss reserving and examples of their application to a real insurance portfolio. In this paper the point estimating methods such as Chain Ladder, Bornhuetter-Fergusson, multiplicative techniques are compared with the stochastic method of Bootstrap and the most accurate estimate is determined using run-off analysis.
Abstract We study two types of testing problems in a nonparametric additive model setting: We develop methods to test (i) whether an additive component function has a given parametric form and (ii) whether an additive component has a structural break. We apply the theory to a nonparametric extension of the linear heterogeneous autoregressive model which is widely employed to describe realized variance data. We find that the linearity assumption is often rejected, but actual deviations from linearity are mild. © 2015 Elsevier B.V.
Varying coefficient regression models are known to be very useful tools for analysing the relation between a response and a group of covariates. Their structure and interpretability are similar to those for the traditional linear regression model, but they are more flexible because of the infinite dimensionality of the corresponding parameter spaces. The aims of this paper are to give an overview on the existingmethodological and theoretical developments for varying coefficientmodels and to discuss their extensions with some new developments. The new developments enable us to use different amount of smoothing for estimating different component functions in the models. They are for a flexible form of varying coefficient models that requires smoothing across different covariates’ spaces and are based on the smooth backfitting technique that is admitted as a powerful technique for fitting structural regression models and is also known to free us from the curse of dimensionality.
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.
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.