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>

Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

In this paper, we establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge-Ampère equation det ( D 2 u ) = b ( x ) g ( − u ) , u < 0 in Ω and u = 0 on ∂ Ω , $$\mbox{ det}(D^{2} u)=b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and } u=0 \mbox{ on }\partial\Omega, $$ where Ω is a bounded, smooth and strictly convex domain in ℝ N (N ≥ 2), b ∈ C∞(Ω) is positive and may be singular (including critical singular) or vanish on the boundary, g ∈ C 1((0, ∞), (0, ∞)) is decreasing on (0, ∞) with lim t → 0 + g ( t ) = ∞ $ \lim\limits_{t\rightarrow0^{+}}g(t)=\infty $ and g is normalized regularly varying at zero with index −γ(γ>1). Our results reveal the refined influence of the highest and the lowest values of the (N − 1)-th curvature on the second boundary behavior of the unique strictly convex solution to the problem.