Tail distribution estimates of the mixed-fractional CEV model

The aim of this paper is to study the tail distribution of the CEV model driven by Brownian motion and fractional Brownian motion. Based on the techniques of Malliavin calculus and a result established recently in [<a href="#1">1</a>], we obtain an explicit estimate for tail distributions.

Asymptotic methods of testing statistical hypotheses

<p>For a long time, the goodness of fit (GOF) tests have been one of the main objects of the theory of testing of statistical hypotheses. These tests possess two essential properties. Firstly, the asymptotic distribution of GOF test statistics under the null hypothesis is free from the underlying distribution within the hypothetical family. Secondly, they are of omnibus nature, which means that they are sensitive to every alternative to the null hypothesis. GOF tests are typically based on non-linear functionals from the empirical process. The first idea to change the focus from particular functionals to the transformation of the empirical process itself into another process, which will be asymptotically distribution free, was first formulated and accomplished by {\bf Khmaladze} \cite{Estate1}. Recently, the same author in consecutive papers \cite{Estate} and \cite{Estate2} introduced another method, called here the {\bf Khmaladze-2} transformation, which is distinct from the first Khmaladze transformation and can be used for an even wider class of hypothesis testing problems and is simpler in implementation. This thesis shows how the approach could be used to create the asymptotically distribution free empirical process in two well-known testing problems. The first problem is the problem of testing independence of two discrete random variables/vectors in a contingency table context. Although this problem has a long history, the use of GOF tests for it has been restricted to only one possible choice -- the chi-square test and its several modifications. We start our approach by viewing the problem as one of parametric hypothesis testing and suggest looking at the marginal distributions as parameters. The crucial difficulty is that when the dimension of the table is large, the dimension of the vector of parameters is large as well. Nevertheless, we demonstrate the efficiency of our approach and confirm by simulations the distribution free property of the new empirical process and the GOF tests based on it. The number of parameters is as big as $30$. As an additional benefit, we point out some cases when the GOF tests based on the new process are more powerful than the traditional chi-square one. The second problem is testing whether a distribution has a regularly varying tail. This problem is inspired mainly by the fact that regularly varying tail distributions play an essential role in characterization of the domain of attraction of extreme value distributions. While there are numerous studies on estimating the exponent of regular variation of the tail, using GOF tests for testing relevant distributions has appeared in few papers. We contribute to this latter aspect a construction of a class of GOF tests for testing regularly varying tail distributions.</p>

Asymptotic methods of testing statistical hypotheses

<p>For a long time, the goodness of fit (GOF) tests have been one of the main objects of the theory of testing of statistical hypotheses. These tests possess two essential properties. Firstly, the asymptotic distribution of GOF test statistics under the null hypothesis is free from the underlying distribution within the hypothetical family. Secondly, they are of omnibus nature, which means that they are sensitive to every alternative to the null hypothesis. GOF tests are typically based on non-linear functionals from the empirical process. The first idea to change the focus from particular functionals to the transformation of the empirical process itself into another process, which will be asymptotically distribution free, was first formulated and accomplished by {\bf Khmaladze} \cite{Estate1}. Recently, the same author in consecutive papers \cite{Estate} and \cite{Estate2} introduced another method, called here the {\bf Khmaladze-2} transformation, which is distinct from the first Khmaladze transformation and can be used for an even wider class of hypothesis testing problems and is simpler in implementation. This thesis shows how the approach could be used to create the asymptotically distribution free empirical process in two well-known testing problems. The first problem is the problem of testing independence of two discrete random variables/vectors in a contingency table context. Although this problem has a long history, the use of GOF tests for it has been restricted to only one possible choice -- the chi-square test and its several modifications. We start our approach by viewing the problem as one of parametric hypothesis testing and suggest looking at the marginal distributions as parameters. The crucial difficulty is that when the dimension of the table is large, the dimension of the vector of parameters is large as well. Nevertheless, we demonstrate the efficiency of our approach and confirm by simulations the distribution free property of the new empirical process and the GOF tests based on it. The number of parameters is as big as $30$. As an additional benefit, we point out some cases when the GOF tests based on the new process are more powerful than the traditional chi-square one. The second problem is testing whether a distribution has a regularly varying tail. This problem is inspired mainly by the fact that regularly varying tail distributions play an essential role in characterization of the domain of attraction of extreme value distributions. While there are numerous studies on estimating the exponent of regular variation of the tail, using GOF tests for testing relevant distributions has appeared in few papers. We contribute to this latter aspect a construction of a class of GOF tests for testing regularly varying tail distributions.</p>

Effects of Climate Change on Precipitation and the Maximum Daily Temperature (Tmax) at Two US Military Bases with Different Present-Day Climates

In this study, the effects of climate change on precipitation and the maximum daily temperature (Tmax) at two USA locations that have different climates—the Travis Airforce Base (AFB) in California [38.27° N, 121.93° W] and Fort Bragg (FBR) in North Carolina [35.14 N, 79.00 W]—are analyzed. The effects of climate change on central tendency, tail distributions, and both auto- and cross-covariance structures in precipitation and Tmax fields for three time periods in the 21st century centered on the years 2020, 2050, and 2100 were analyzed. It was found that, on average, Tmax under the Representative Concentration Pathway (RCP) 4.5 emission scenario is projected to increase for the years 2020, 2050, and 2100 by 1.1, 2.0, and 2.2 °C, respectively, for AFB, and 0.9, 1.2, and 1.6 °C, respectively, for FBR, while under the RCP8.5 emission scenario Tmax will increase by 1.1, 1.9, and 2.7 °C, respectively, for AFB, and 0.1, 1.5, and 2.2 °C, respectively, for FBR. The climate change signal in precipitation is weak. The results show that, under different emission scenarios, events considered to be within 1% of the most extreme events in the past will become ~13–30 times more frequent for Tmax, ~and 0.05–3 times more frequent for precipitation in both locations. Several analytical methods were deployed in a sequence, creating an easily scalable framework for similar analyses in the future.

Adsorption of a Helical Filament Subject to Thermal Fluctuations

We consider semiflexible chains governed by preferred curvature and twist and their flexural and twist moduli. These filaments possess a helical rather than straight three-dimensional (3D) ground state and we call them helical filaments (H-filament). Depending on the moduli, the helical shape may be smeared by thermal fluctuations. Secondary superhelical structures are expected to form on top of the specific local structure of biofilaments, as is documented for vimentin. We study confinement and adsorption of helical filaments utilizing both a combination of numerical simulations and analytical theory. We investigate overall chain shapes, transverse chain fluctuations, loop and tail distributions, and energy distributions along the chain together with the mean square average height of the monomers ⟨ z 2 ⟩ . The number fraction of adsorbed monomers serves as an order parameter for adsorption. Signatures of adsorbed helical polymers are the occurrence of 3D helical loops/tails and spiral or wavy quasi-flat shapes. None of these arise for the Worm-Like-Chain, whose straight ground state can be embedded in a plane.

Aggregation of Dependent Risks with Heavy-Tail Distributions

Straightforward methods to evaluate risks arising from several sources are specially difficult when risk components are dependent and, even more if that dependence is strong in the tails. We give an explicit analytical expression for the probability distribution of the sum of non-negative losses that are tail-dependent. Our model allows dependence in the extremes of the marginal beta distributions. The proposed model is flexible in the choice of the parameters in the marginal distribution. The estimation using the method of moments is possible and the calculation of risk measures is easily done with a Monte Carlo approach. An illustration on data for insurance losses is presented.