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Showing posts from April, 2015

Parametric Inference: Likelihood Ratio Test by Example

Hypothesis testing have been extensively used on different discipline of science. And in this post, I will attempt on discussing the basic theory behind this, the Likelihood Ratio Test (LRT) defined below from Casella and Berger (2001), see reference 1. Definition . The likelihood ratio test statistic for testing H_0:\theta\in\Theta_0 versus H_1:\theta\in\Theta_0^c is \begin{equation} \label{eq:lrt} \lambda(\mathbf{x})=\frac{\displaystyle\sup_{\theta\in\Theta_0}L(\theta|\mathbf{x})}{\displaystyle\sup_{\theta\in\Theta}L(\theta|\mathbf{x})}. \end{equation}
A likelihood ratio test (LRT) is any test that has a rejection region of the form \{\mathbf{x}:\lambda(\mathbf{x})\leq c\}, where c is any number satisfying 0\leq c \leq 1. The numerator of equation (\ref{eq:lrt}) gives us the supremum probability of the parameter, \theta, over the restricted domain (null hypothesis, \Theta_0) of the parameter space \Theta, that maximizes the joint probability of the sample, $\math...

SAS®: Getting Started with PROC IML

Another powerful procedure of SAS, my favorite one, that I would like to share is the PROC IML (Interactive Matrix Language). This procedure treats all objects as a matrix, and is very useful for doing scientific computations involving vectors and matrices. To get started, we are going to demonstrate and discuss the following: Creating and Shaping Matrices; Matrix Query; Subscripts; Descriptive Statistics; Set Operations; Probability Functions and Subroutine; Linear Algebra; Reading and Creating Data; Above outline is based on the IML tip sheet (see Reference 1). So to begin on the first bullet, consider the following code:

Python and R: Basic Sampling Problem

In this post, I would like to share a simple problem about sampling analysis. And I will demonstrate how to solve this using Python and R. The first two problems are originally from Sampling: Design and Analysis book by Sharon Lohr. Problems Let N=6 and n=3. For purposes of studying sampling distributions, assume that all population values are known. y_1 = 98 y_2 = 102 y_3=154 y_4 = 133 y_5 = 190 y_6=175 We are interested in \bar{y}_U, the population mean. Consider eight possible samples chosen. Sample No. Sample, \mathcal{S} P(\mathcal{S}) 1 \{1,3,5\} 1/8 2 \{1,3,6\} 1/8 3 \{1,4,5\} 1/8 4 \{1,4,6\} 1/8 5 \{2,3,5\} 1/8 6 \{2,3,6\} 1/8 7 \{2,4,5\} 1/8 8 \{2,4,6\} 1/8