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\begin{document} 
\title{Assignment 6}
\section{assignment 6}
\subsection{Problem 1}
Suppose $X=(X_1,X_2,\ldots,X_n)$ is a vector of independent, identically distributed random variables having an exponential distribution with parameter $\theta_1$ and $Y=(Y_1,Y_2,\ldots,Y_m)$ is a vector of independent identically distributed random variables having an exponential distribution with parameter $\theta_2$.  $X$ and $Y$ are independently distributed.
\par\vspace{0.3 cm}\noindent
a) Find the value of $\theta_1$ that maximizes the likelihood function $L(X,\theta_1)$
\par\vspace{0.3 cm}\noindent
b) Find the value of $\theta_2$ that maximizes the likelihood function $L(Y,\theta_2)$
\par\vspace{0.3 cm}\noindent
c) Considering the concatenation of $X$ and $Y$ into a single vector of length $n+m$ given by $U=(X_1,\ldots,X_n,Y_1,\ldots,Y_m)$, what are the values of $\theta_1$ and $\theta_2$ that maximize the likelihood function $L(U,\theta_1,\theta_2)=L(X,\theta_1)\cdot L(Y,\theta_2)$? 
\par\vspace{0.3 cm}\noindent
d) Again considering the concatenation of $X$ and $Y$ into a single vector of length $n+m$ given by $U=(X_1,\ldots,X_n,Y_1,\ldots,Y_m)$, with the restriction that $\theta_1=\theta_2=\theta$, what is the value of $\theta$ that maximize the likelihood function $L(U,\theta)$? 
\subsection{Problem 2}
In the same setting as Problem 1, suppose 
\[
\overline{X}=\frac{\sum_{i=1}^nX_i}{n}=13.6\quad\mbox{and}\quad\overline{Y}=\frac{\sum_{i=1}^mY_i}{m}=14.1
\]
with $n=m=100$.  We are interested in whether the data supports the claim that $\theta_1=\theta_2$.  
\par\vspace{0.3 cm}\noindent
a) Find the likelihood ratio
\[
\lambda=\frac{\max L(U,\theta)}{\max L(U,\theta_1,\theta_2)}
\]
\par\vspace{0.3 cm}\noindent
b) Using the approximate distribution of $-2\ln\lambda$ (Theorem 10.2), construct an acceptance region for testing $H_0:\theta_1=\theta_2$ against the alternative $H_a:\theta_1\neq\theta_2$ of size $1-\alpha=.95$. (Hint: The number of parameters needed to fully specify the likelihood is two.  The number of free parameters is two minus the number of parameter values specified). 
\par\vspace{0.3 cm}\noindent
c) Does the value of $\lambda$ computed in part a) fall inside the acceptance region?
\subsection{Problem 3}
\par\vspace{0.3 cm}
A linear model with a single continuous predictor is called a \textbf{simple regression} model and has the form
\[
Y=X\beta+e
\]
where
\[
Y=\left[\begin{array}{c}
y_1\\
y_2\\
\vdots\\
y_n
\end{array}\right]
\quad
X=\left[\begin{array}{cc}
1 & x_1\\
1 & x_2\\
\vdots & \vdots\\
1 & x_n
\end{array}\right]
\quad
e=\left[\begin{array}{c}
e_1\\
e_2\\
\vdots\\
e_n
\end{array}\right]
\quad
\beta=\left[\begin{array}{c}
\beta_1\\
\beta_2\\
\end{array}\right]
\]
\par\vspace{0.3 cm}
a) Write an expression for $X'X$
\par\vspace{0.3 cm}
b) Write an expression for $X'Y$
\par\vspace{0.3 cm}
c) Recall that 
\[
\left[\begin{array}{cc}
a & b\\
c & d\\
\end{array}\right]^{-1}
\quad=\quad\frac{1}{ad-bc}\left[\begin{array}{rr}
d & -b\\
-c & a\\
\end{array}\right]
\]
\par\vspace{0.3 cm}
Use this to write expressions for $\hat{\beta_1}$ and $\hat{\beta_2}$ where 
\[
\hat{\beta}=\left[\begin{array}{c}
\hat{\beta_1}\\
\hat{\beta_2}\\
\end{array}\right]\;=\;(X'X)^{-1}X'Y
\]
\subsection{Problem 4}
\par\vspace{0.3 cm}
In some simple regression models, the assumption that the regression line passes through the origin can be justified.  This is called a \textbf{no intercept} model and has the form
\[
Y=X\beta+e
\]
where
\[
Y=\left[\begin{array}{c}
y_1\\
y_2\\
\vdots\\
y_n
\end{array}\right]
\quad
X=\left[\begin{array}{c}
x_1\\
x_2\\
\vdots\\
x_n
\end{array}\right]
\quad
e=\left[\begin{array}{c}
e_1\\
e_2\\
\vdots\\
e_n
\end{array}\right]
\quad
\beta=\left[\begin{array}{c}
\beta_1\\
\end{array}\right]
\]
\par\vspace{0.3 cm}
a) Write an expression for $X'X$
\par\vspace{0.3 cm}
b) Write an expression for $X'Y$
\par\vspace{0.3 cm}
c) Write an expression for $\hat{\beta_1}$ where 
\[
\hat{\beta}=\left[\begin{array}{c}
\hat{\beta_1}\\
\end{array}\right]\;=\;(X'X)^{-1}X'Y
\]
\end{document}



