% 260s20Assignment2.tex                 Unbiased, consistent
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{\Large \textbf{STA 260s20 Assignment Two: Unbiasedness and Consistency}}%\footnote{Copyright information is at the end of the last page.}
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\noindent
These homework problems are not to be handed in.  They are preparation for Quiz 2 (Week of Jan.~20) and Term Test 1. \textbf{Please try each question before looking at the solution}.

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\begin{enumerate} 

\item Let $X_1, \ldots X_n$ be independent Binomial random variables with parameters $m=3$ (known) and $\theta$ (unknown); see the formula sheet.  Let $\widehat{\Theta}_n = \frac{1}{3}\overline{X}_n$. 
    \begin{enumerate}
        \item What is the parameter space $\Omega$ for this problem?
        \item Show that $\widehat{\Theta}_n$ is unbiased.
        \item Show that $\widehat{\Theta}_n$ is consistent.
    \end{enumerate}

\item Let $X_1, \ldots X_n$ be a random sample from a distribution with density $f(x|\theta) = \theta x^{\theta-1} \, I(0<x<1)$, where the parmeter $\theta>0$. 
    \begin{enumerate}
        \item What is the parameter space $\Omega$ for this problem?
        \item Is $\overline{X}_n$ an unbiased estimator of $\theta$? Answer Yes or No and prove your answer.
        \item Is $\overline{X}_n$ a consistent estimator of $\theta$? Answer Yes or No and prove your answer.
    \end{enumerate}

\item Let $X_1, \ldots X_n$ be independent random variables with expected value $\mu$ and variance $\sigma^2$. Other than that, the distributions of the $X_i$ are unspecified.
    \begin{enumerate}
        \item Show that $S^2 = \frac{\sum_{i=1}^n(X_i-\overline{X}_n)^2}{n-1}$ is an unbiased estimator of $\sigma^2$.
        \item Suppose that $\mu$ is known. Is $\widehat{\sigma}^2_n = \frac{1}{n} \sum_{i=1}^n(X_i-\mu)^2$ a biased estimator of $\sigma^2$, or is it unbiased? Show your work.
        \item Why does the Law of Large Numbers imply that $\widehat{\sigma}^2_n$ is consistent?
        \item There is one little hole in the argument for consistency. What is it?
    \end{enumerate}

\item Let $X_1, \ldots X_n$ be independent Poisson random variables with unknown parameter $\lambda$. 
    \begin{enumerate}
        \item What is the parameter space $\Omega$ for this problem?
        \item Suggest an estimator of $\lambda$ that is unbiased and consistent. 
        \item Suggest another estimator of $\lambda$. Is it also unbiased? How do you know?
        \item Using the definition of a limit, it may easily be shown that if the sequence of constants $a_n \rightarrow a$ as an ordinary limit as $n \rightarrow \infty$, then $a_n \stackrel{p}{\rightarrow} a$ as a sequence of degenerate random variables. Using this fact and the multivariable version of continuous mapping for convergence in probability, show that $S^2$ is consistent for $\lambda$.
        \item Finally, here is a silly estimator: $\widehat{\lambda} = (X_1+X_2)/2$. 
                \begin{enumerate}
                    \item Is $\widehat{\lambda}$ unbiased? Why or why not?
                    \item Is $\widehat{\lambda}$ consistent? Why or why not?
                    \item Why is $\widehat{\lambda}$ silly?
                \end{enumerate}         
    \end{enumerate}

\pagebreak

\item Let $X_1, \ldots X_n$ be independent Uniform $(0,\theta)$ random variables.
    \begin{enumerate}
        \item What is the parameter space $\Omega$ for this problem?
        \item Write the cumulative distribution function $F_{_{X_i}}(x|\theta)$ using indicator functions. Show your work.
        \item Let $T_n = \max(X_i)$. Find the cumulative distribution function of $T_n$. Show your work. Write the final answer using indicator functions.
        \item Find the density function of $T_n$. Write it using indicator functions.
        \item Is $T_n$ unbiased for $\theta$? Answer Yes or No and show your work. 
        \item Show that $T_n$ is consistent for $\theta$ using the definition.
        \item Show that $T_n$ is consistent for $\theta$ using the variance rule.
        \item Give an unbiased estimator of $\theta$ based on $T_n$. That is, fix up $T_n$ a bit so it's unbiased. Call the new estimator $\widehat{\Theta}_1$. 
        \item Let $\widehat{\Theta}_2 = 2 \overline{X}_n$. Show that $\widehat{\Theta}_2$ is unbiased and consistent. 
        \item In terms of variance, which is preferable, $\widehat{\Theta}_1$ or $\widehat{\Theta}_2$?
    \end{enumerate}

\item For $i=1, \ldots, n$, let $Y_i = \beta x_i + \epsilon_i$, where
        \begin{itemize}
            \item[] $x_1, \ldots, x_n$ are fixed, known constants
            \item[] $\epsilon_1, \ldots, \epsilon_n$ are independent and identically distributed Normal(0,$\sigma^2$) random variables; the parameters $\beta$ and $\sigma^2$ are unknown.
        \end{itemize}
This is a very simple regression model. For example, the $x_i$ values could be drug doses, and the $Y_i$ could be response to the drug. Naturally, the main interest is in $\beta$, because $\beta$ is the connection between dose and response.
    \begin{enumerate}
        \item A suggested estimator is $\widehat{\beta_1} = \frac{\sum_{i=1}^nx_iY_i}{\sum_{i=1}^n x_i^2}$.
                \begin{enumerate}
                    \item Is $\widehat{\beta_1}$ unbiased for $\beta$? Answer Yes or No and show your work.
                    \item Assume that $\lim_{n \rightarrow \infty}\frac{1}{\sum_{i=1}^n x_i^2} = 0$, which is reasonable for drug doses. Is $\widehat{\beta_1}$ consistent for $\beta$? Answer Yes or No and show your work.
                \end{enumerate}
        \item Another suggested estimator is $\widehat{\beta_2} = \frac{\overline{Y}_n}{\overline{x}_n}$. 
                \begin{enumerate}
                    \item Is $\widehat{\beta_2}$ unbiased for $\beta$? Answer Yes or No and show your work.
                    \item Is $\widehat{\beta_2}$ consistent for $\beta$? Answer Yes or No and show your work. Note that you can't use the Law of Large Numbers, because the $Y_i$ don't have the same expected value. However, you may assume that $\lim_{n \rightarrow \infty}\overline{x}_n = c \neq 0 $, which is reasonable for drug doses.
                \end{enumerate}
    \item It is tough to show, but $Var(\widehat{\beta_1}) \leq Var(\widehat{\beta_2})$. Do you feel like giving it a try? This will not be on any test or exam.
    \end{enumerate} % End of regression questions

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\item Let $X_1, \ldots X_n$ be independent Exponential $(\lambda)$ random variables.
    \begin{enumerate}
        \item Suggest a reasonable estimator for $\lambda$. 
        \item It is easy to see that your estimator is consistent. Why?
        \item Unbiasedness is another issue. First, derive the distribution of $\overline{X}_n$ and write the density $f_{_{\overline{X}_n}}(\overline{x}|\lambda)$.
        \item Now directly calculate $E\left( 1/\overline{X}_n \right)$. Is this estimator unbiased for $\lambda$?
        \item Show that $\frac{n-1}{\sum_{i=1}^n X_i}$ is unbiased for $\lambda$.
        \item Show that $\frac{n-1}{\sum_{i=1}^n X_i}$ is consistent for $\lambda$.
    \end{enumerate}

\item Let $X_1, \ldots X_n$ be independent random variables with expected value $\mu$ and variance $\sigma^2$. Other than that, the distributions of the $X_i$ are unspecified. We seek to estimate $\mu$ with the linear combination $L = a_1X_1 + \cdots + a_nX_n = \sum_{i=1}^n a_iX_i$, where $a_1, \ldots, a_n$ are constants.
    \begin{enumerate}
        \item What condition on $a_1, \ldots, a_n$ is required for $L$ to be an unbiased estimator of $\mu$? Show your work.
        \item $\overline{X}_n$ is one such linear combination. What are the coefficients $a_1, \ldots, a_n$?
        \item Show that the variance of $\overline{X}_n$ is less than the variance of any other unbiased linear combination $L$. That is, $\overline{X}_n$ is the Best Linear Unbiased Estimator (BLUE).
    \end{enumerate}
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\end{enumerate}

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\noindent
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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/260s20} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/260s20}}
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Maybe put on Assignment 2