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\begin{document} 
\title{Assignment 4}
\section{assignment 4}
\subsection{Problem 1}
If a probability distribution depends on a single parameter $\theta$ where $\theta\in\Theta$ for some set $\Theta$ of possible parameter values, the collection of distributions taken over all possible values of $\theta$ is said to form 
a \textbf{one-parameter exponential family} if there exist real-valued functions $\eta,B:\Theta\rightarrow\mathbb{R}$ and $T,h:\mathbb{R}^q\rightarrow\mathbb{R}$ such that the density function $f(x;\theta)$ or probability mass function $p(x;\theta)$ of any distribution in the collection has the form
\[
p(x;\theta)\mbox{  or  }f(x;\theta) = h(x)\exp\left[\eta(\theta)T(x)-B(\theta)\right]
\]
\par\vspace{0.3 cm}\noindent
a) Show that for values of $\theta\in\Theta=(0,1)$ and a given value of $n$, the binomial distributions $B(n;\theta)$ 
\[
p(x;\theta)\;=\;{n\choose x}\theta^x(1-\theta)^{n-x}
\]
form a one-parameter exponential family with $q=1$ by identifying $h(x),\eta(\theta),T(x)$, and $B(\theta)$.
\par\vspace{0.3 cm}\noindent
b) The function $T(x)$ is called the \textbf{natural sufficient statistic} of the family.  Use the Neyman factorization theorem to show that $T(x)$ is sufficient for $\theta$ in any distribution that belongs to a one-parameter exponential family.
\subsection{Problem 2}
\par\vspace{0.3 cm}
Suppose $Y=(Y_1,\ldots,Y_n)$ is a vector of $n$ IID random variables from a one-parameter exponential family with $q=1$. 
\[
f(y_i,\theta)\;=\;h(y_i)\exp\left[\eta(\theta)T(y_i)-B(\theta)\right]
\]
Show that the joint density function of $Y$, $f(Y;\theta)$ belongs to a one-parameter exponential family with $q=n$ by identifying $h(x),\eta(\theta),T(x)$, and $B(\theta)$.
\subsection{Problem 3} If $y=(y_1,\ldots,y_n)$ is a random sample from a population with density function
\[
f(y;\theta)=(1+\theta)y^{\theta}\quad 0<1<1,\quad \theta>-1
\]
find the maximum likelihood estimator of $\theta$. (Hint: It is sometimes easier to find the value of $\theta$ that maximizes the log of the likelihood function)
\subsection{Problem 4}
\par\vspace{0.3 cm}
Let $y=(y_1,\ldots,y_n)$ be a random sample of size $n$ from an exponential distribution with parameter $\theta$. 
\par\vspace{0.3 cm}\noindent
a) Show that 
\[
\overline{y}=\frac{1}{n}\sum_{i=1}^n
\]
is an efficient estimator for $\theta$
\par\vspace{0.3 cm}\noindent
b) Show that $\overline{y}$ is also the maximum likelihood estimator for $\theta$
\end{document}



