Research Topics in Empirical Processes

<p>The first chapter consists of an overview of the theory of empirical processes, covering an introduction to empirical processes in R, uniform empirical processes and function parametric empirical processes in Section 1.1. Section 1.2 contains an overview of the theory related to the law of the iterated logarithm for Brownian motion and the modulus of continuity for Brownian motion. Section 1.3 contains the theory of the limiting processes for the empirical process, most importantly Brownian motion, Brownian bridge and the connections and relationships between them, with distributions of selected statistics of Brownian motion and Brownian bridge derived from reflection principles. Section 1.4 contains an overview of the theory required to prove central limit results for the empirical processes, covering the theory of the space C and Donsker’s theorem. The second chapter covers research topics, starting with Fourier analysis of mixture distributions and associated theory in Section 2.1. Section 2.2 covers findings in a research problem about non-linear autoregressive processes. Section 2.3 introduces a martingale approach to testing a regression model. Section 2.4 links the theory of ranks and sequential ranks to the theory of empirical processes.</p>

Research Topics in Empirical Processes

<p>The first chapter consists of an overview of the theory of empirical processes, covering an introduction to empirical processes in R, uniform empirical processes and function parametric empirical processes in Section 1.1. Section 1.2 contains an overview of the theory related to the law of the iterated logarithm for Brownian motion and the modulus of continuity for Brownian motion. Section 1.3 contains the theory of the limiting processes for the empirical process, most importantly Brownian motion, Brownian bridge and the connections and relationships between them, with distributions of selected statistics of Brownian motion and Brownian bridge derived from reflection principles. Section 1.4 contains an overview of the theory required to prove central limit results for the empirical processes, covering the theory of the space C and Donsker’s theorem. The second chapter covers research topics, starting with Fourier analysis of mixture distributions and associated theory in Section 2.1. Section 2.2 covers findings in a research problem about non-linear autoregressive processes. Section 2.3 introduces a martingale approach to testing a regression model. Section 2.4 links the theory of ranks and sequential ranks to the theory of empirical processes.</p>

A three-player gambler's ruin problem: some extensions

For the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. For the case of asymmetric game, the martingale approach is not valid to determine the ruin time. In general, the ruin probabilities for both cases, i.e. symmetric and asymmetric game and expected ruin time for asymmetric game are still awaiting to be solved for this game. The current work is also about three-player gambler’s ruin problem with some extensions as well. We provide expressions for the ruin time with (without) ties when all the players have equal (unequal) initial fortunes. Finally, the validity of asymmetric game is also tested through a Monte Carlo simulation study.

Efficiency Testing of Prediction Markets: Martingale Approach, Likelihood Ratio and Bayes Factor Analysis

This paper studies efficient market hypothesis in prediction markets and the results are illustrated for the in-play football betting market using the quoted odds for the English Premier League. Our analysis is based on the martingale property, where the last quoted probability should be the best predictor of the outcome and all previous quotes should be statistically insignificant. We use regression analysis to test for the significance of the previous quotes in both the time setup and the spatial setup based on stopping times, when the quoted probabilities reach certain bounds. The main contribution of this paper is to show how a potentially different distributional opinion based on the violation of the market efficiency can be monetized by optimal trading, where the agent maximizes logarithmic utility function. In particular, the trader can realize a trading profit that corresponds to the likelihood ratio in the situation of one market maker and one market taker, or the Bayes factor in the situation of two or more market takers.

A CUMULATIVE RESIDUAL INACCURACY MEASURE FOR COHERENT SYSTEMS AT COMPONENT LEVEL AND UNDER NONHOMOGENEOUS POISSON PROCESSES

Inaccuracy and information measures based on cumulative residual entropy are quite useful and have attracted considerable attention in many fields including reliability theory. Using a point process martingale approach and a compensator version of Kumar and Taneja's generalized inaccuracy measure of two nonnegative continuous random variables, we define here an inaccuracy measure between two coherent systems when the lifetimes of their common components are observed. We then extend the results to the situation when the components in the systems are subject to failure according to a double stochastic Poisson process.