QSpace Collection:http://hdl.handle.net/1974/61582016-08-25T15:01:33Z2016-08-25T15:01:33ZEstimation of Sample Size and Power For Quantile RegressionGong, Zhenxianhttp://hdl.handle.net/1974/147392016-08-25T05:10:50Z2016-08-24T04:00:00ZTitle: Estimation of Sample Size and Power For Quantile Regression
Authors: Gong, Zhenxian
Abstract: Quantile regression (QR) was first introduced by Roger Koenker and Gilbert Bassett in 1978. It is robust to outliers which affect least squares estimator on a large scale in linear regression. Instead of modeling mean of the response, QR provides an alternative way to model the relationship between quantiles of the response and covariates. Therefore, QR can be widely used to solve problems in econometrics, environmental sciences and health sciences.
Sample size is an important factor in the planning stage of experimental design and observational studies. In ordinary linear regression, sample size may be determined based on either precision analysis or power analysis with closed form formulas. There are also methods that calculate sample size based on precision analysis for QR like C.Jennen-Steinmetz and S.Wellek (2005). A method to estimate sample size for QR based on power analysis was proposed by Shao and Wang (2009). In this paper, a new method is proposed to calculate sample size based on power analysis under hypothesis test of covariate effects.
Even though error distribution assumption is not necessary for QR analysis itself, researchers have to make assumptions of error distribution and covariate structure in the planning stage of a study to obtain a reasonable estimate of sample size. In this project, both parametric and nonparametric methods are provided to estimate error distribution. Since the method proposed can be implemented in R, user is able to choose either parametric distribution or nonparametric kernel density estimation for error distribution. User also needs to specify the covariate structure and effect size to carry out sample size and power calculation.
The performance of the method proposed is further evaluated using numerical simulation. The results suggest that the sample sizes obtained from our method provide empirical powers that are closed to the nominal power level, for example, 80%.2016-08-24T04:00:00ZA Review on Repeated Games and Reputations with Incomplete InformationVerlezza, Michaelhttp://hdl.handle.net/1974/146632016-07-21T05:11:09Z2016-07-20T04:00:00ZTitle: A Review on Repeated Games and Reputations with Incomplete Information
Authors: Verlezza, Michael
Abstract: In this project we review the effects of reputation within the context of game theory.
This is done through a study of two key papers. First, we examine a paper from
Fudenberg and Levine: Reputation and Equilibrium Selection in Games with a Patient
Player (1989). We add to this a review Gossner’s Simple Bounds on the Value of a
Reputation (2011). We look specifically at scenarios in which a long-run player faces
a series of short-run opponents, and how the former may develop a reputation. In
turn, we show how reputation leads directly to both lower and upper bounds on
the long-run player’s payoffs.2016-07-20T04:00:00ZThe Chowla Problem and its generalizationsPathak, Siddhihttp://hdl.handle.net/1974/136142015-09-17T05:11:19Z2015-09-15T04:00:00ZTitle: The Chowla Problem and its generalizations
Authors: Pathak, Siddhi
Abstract: In this thesis, we study the vanishing of certain $L$-series attached to periodic arithmetical functions. Throughout the writeup, we let $f$ be an algebraic-valued (at times even rational-valued) arithmetical function, periodic with period $q$. Define
\begin{equation*}
L(s,f) := \sum_{n=1}^{\infty} \frac{f(n)}{n^s}.
\end{equation*}
We discuss a conjecture of Sarvadaman Chowla, made in the early 1960's of the non-vanishing of $L(1,f)$ for rational-valued functions $f$. We further discuss the theorem of Baker-Birch-Wirsing that classifies all odd algebraic-valued, periodic arithmetical functions $f$ with $L(1,f)=0$. We then apply a beautiful result of Bass to give a necessary condition for even algebraic-valued, periodic arithmetical functions $f$, to satisfy $L(1,f)=0$. The sufficiency condition obtained via this method is unfortunately not very clean. This, in view of a theorem of Ram Murty and Tapas Chatterjee, completes the characterization of all algebraic-valued periodic arithmetical functions $f$ with $L(1,f) = 0$.
In the next part of the thesis, we discuss the Lerch-zeta function and its functional equation. We also deduce the transcendence of its certain special values. Keeping this in mind, we define a new $L$-series attached to periodic functions and deduce a necessary condition for the vanishing of this $L$-series at $s=1$.
Finally, we mention some research topics that were stumbled upon during the course of this study and a few partial results regarding these questions.2015-09-15T04:00:00ZThe Saddle Point Method and its Applications to Number TheorySampath, Kannappanhttp://hdl.handle.net/1974/136132015-09-16T04:58:41Z2015-09-15T04:00:00ZTitle: The Saddle Point Method and its Applications to Number Theory
Authors: Sampath, Kannappan
Abstract: In this thesis, we study the classical procedures useful in obtaining asymptotic expansions of functions defined by integrals and their
applications to number theory. The final chapter in the thesis reports on a recent joint work with Ram Murty on the theme of using arithmetic formulas
to obtain asymptotic formulas to study the Fourier coefficients of the j-function and a related
sequence j-sub-m (m ≥ 0) of modular functions.
Description: Masters' Project Report2015-09-15T04:00:00Z