Analysis with Programming

More on Likelihood Ratio Test, the following problem is originally from Casella and Berger (2001), exercise 8.12. Problem For samples of size $n=1,4,16,64,100$ from a normal population with mean $\mu$ and known variance $\sigma^2$, plot the power function of the following LRTs (Likelihood Ratio Tests). Take $\alpha = .05$. $H_0:\mu\leq 0$ versus $H_1:\mu>0$ $H_0:\mu=0$ versus $H_1:\mu\neq 0$ Solution The LRT statistic is given by $$ \lambda(\mathbf{x})=\frac{\displaystyle\sup_{\mu\leq 0}\mathcal{L}(\mu|\mathbf{x})}{\displaystyle\sup_{-\infty<\mu<\infty}\mathcal{L}(\mu|\mathbf{x})}, \;\text{since }\sigma^2\text{ is known}. $$ The denominator can be expanded as follows: $$ \begin{aligned} \sup_{-\infty<\mu<\infty}\mathcal{L}(\mu|\mathbf{x})&=\sup_{-\infty<\mu<\infty}\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x_i-\mu)^2}{2\sigma^2}\right]\\ &=\sup_{-\infty<\mu<\infty}\frac{1}{(2\pi\sigma^2)^{1/n}}\exp\left[-\displaystyle\sum...

Another post for mathematical statistics, the problem below is originally from Casella and Berger (2001) ( see Reference 1), exercise 8.6. Problem Suppose that we have two independent random samples $X_1,\cdots, X_n$ are exponential($\theta$), and $Y_1,\cdots, Y_m$ are exponential($\mu$). Find the LRT (Likelihood Ratio Test) of $H_0:\theta=\mu$ versus $H_1:\theta\neq\mu$. Show that the test in part (a) can be based on the statistic $$ T=\frac{\sum X_i}{\sum X_i+\sum Y_i}. $$ Find the distribution of $T$ when $H_0$ is true. Solution The Likelihood Ratio Test is given by $$ \lambda(\mathbf{x},\mathbf{y}) = \frac{\displaystyle\sup_{\theta = \mu,\mu>0}\mathrm{P}(\mathbf{x},\mathbf{y}|\theta,\mu)}{\displaystyle\sup_{\theta > 0,\mu>0}\mathrm{P}(\mathbf{x}, \mathbf{y}|\theta,\mu)}, $$ where the denominator is evaluated as follows: $$ \sup_{\theta > 0,\mu>0}\mathrm{P}(\mathbf{x}, \mathbf{y}|\theta,\mu)= \sup_{\theta > 0}\mathrm{P}(\mathbf{x}|\theta)\sup_{\m...

In Statistics, we model random phenomenon and make conclusions about its population. For example, in an experiment of determining the true heights of the students in the university. Suppose we take sample from the population of the students, and consider testing the null hypothesis that the average height is 5.4 ft against an alternative hypothesis that the average height is greater than 5.4 ft. Mathematically, we can represent this as $H_0:\theta=\theta_0$ vs $H_1:\theta>\theta_0$, where $\theta$ is the true value of the parameter, and $\theta_0=5.4$ is the testing value set by the experimenter. And because we only consider subset (the sample) of the population for testing the hypotheses, then we expect for errors we commit. To understand these errors, consider if the above test results into rejecting $H_0$ given that $\theta\in\Theta_0$, where $\Theta_0$ is the parameter space of the null hypothesis, in other words we mistakenly reject $H_0$, then in this case we committed a Type ...