Problem 8.2 (4 points)

We are given \(E[\hat{\theta}_1] = E[\hat{\theta}_2] = \theta\), \(V[\hat{\theta}_1]=\sigma_1^2\), \(V[\hat{\theta}_2]=\sigma_2^2\), and \(\hat{\theta}_1\) and \(\hat{\theta}_2\) are independent.

Let \(\hat{\theta}_3 = a\cdot \hat{\theta}_1 + (1-a)\cdot \hat{\theta}_2\).
  1. Since \(E[\hat{\theta}_3] = E[a\cdot \hat{\theta}_1 + (1-a)\cdot \hat{\theta}_2] = a\cdot E[\hat{\theta}_1] + (1-a)\cdot E[\hat{\theta}_2] = a\cdot \theta + (1-a)\cdot \theta = \theta\), we see that \(\hat{\theta}_3\) is an unbiased estimator for \(\theta\) for all \(a\in\mathbb{R}\).
  2. The variance of \(\hat{\theta}_3\) is given by the quadratic function \[ f(a) = V[\hat{\theta}_3] = a^2\cdot \sigma_1^2 + (1 - a)^2\cdot \sigma_2^2. \] The first derivative \(f'(a) = 2\cdot [(\sigma_1^2 + \sigma_2^2)\cdot a - \sigma_2^2]\) equals zero at \(a = \sigma_2^2 / (\sigma_1^2 + \sigma_2^2)\), and since the second derivative \(f''(a) = 2\cdot (\sigma_1^2 + \sigma_2^2) > 0\), we see that the critical point is a global minimum for the quadratic function. Thus, of all possible weighted averages of the unbiased estimators \(\hat{\theta}_1\), \(\hat{\theta}_2\), \[ \hat{\theta}_3 = \frac{\sigma_2^2}{\sigma_1^2 + \sigma_2^2}\cdot \hat{\theta}_1 + \frac{\sigma_1^2}{\sigma_1^2 + \sigma_2^2}\cdot \hat{\theta}_2 \] has the smallest variance \(\sigma_3^2 = V[\hat{\theta}_3] = \sigma_1^2\cdot \sigma_2^2 / (\sigma_1^2 + \sigma_2^2)\). Observe that the variance of \(\hat{\theta}_3\) is less than the variance of either \(\hat{\theta}_1\) or \(\hat{\theta}_2\). This fact becomes apparant when we see that \(1/\sigma_3^2 = 1/\sigma_1^2 + 1/\sigma_2^2\) and noting that both terms in the sum are positive.

    This problem shows that a suitable weighted average of independant, unbiased estimators for \(\theta\) produces a new unbiased estimator for \(\theta\) that has smaller variance than either of the original estimators.

Problem 8.4 (4 points)

Suppose \(Y_1, Y_2, Y_3\) denotes a random sample from an exponential distribution with probability density function \[ f(y) = \begin{cases} \frac{1}{\theta}\cdot e^{-y/\theta} &, y > 0 \\ 0 &, y \le 0. \end{cases} \] Note that \(E[Y_i]=\theta\) and \(V[Y_i]=\theta^2\), \(i=1,2,3\).

  1. Consider: \(\hat{\theta}_1 = Y_1\), \(\hat{\theta}_2 = \frac{1}{2}(Y_1 + Y_2)\), \(\hat{\theta}_3 = \frac{1}{3}Y_1 + \frac{2}{3}Y_2\) \(\hat{\theta}_4 = \min{(Y_1, Y_2, Y_3)}\), \(\hat{\theta}_5 = \overline{Y}\).

    \[\begin{align*} E[\hat{\theta}_1] &= E[Y_1] = \theta. \\ E[\hat{\theta}_2] &= \frac{1}{2}\left(E[Y_1] + E[Y_2]\right) = \frac{1}{2}(\theta + \theta) = \theta. \\ E[\hat{\theta}_3] &= \frac{1}{3}E[Y_1] + \frac{2}{3}E[Y_2] = \frac{1}{3}\theta + \frac{2}{3}\theta = \theta. \\ E[\hat{\theta}_4] &= \frac{1}{3}\theta \text{ (see exercise 6.65(a), page 324).} \\ E[\hat{\theta}_5] &= E[\overline{Y}] = \theta. \end{align*}\] So, \(\hat{\theta}_1\), \(\hat{\theta}_2\), \(\hat{\theta}_3\), \(\hat{\theta}_5\) are unbiased estimators for \(\theta\).
  2. We calculate the variances of the unbiased estimators. \[\begin{align*} V[\hat{\theta}_1] &= V[Y_1] = \theta^2 \\ V[\hat{\theta}_2] &= \frac{1}{4}(\theta^2 + \theta^2) = \frac{1}{2}\theta^2 \\ V[\hat{\theta}_3] &= \frac{1}{9}\theta^2 + \frac{4}{9}\theta^2 = \frac{5}{9}\theta^2 \\ V[\hat{\theta}_5] &= \frac{1}{3}\theta^2, \end{align*}\] so \(\hat{\theta}_5 = \overline{Y}\) has the least variance.

Problem 8.8 (6 points)

We are given that \(Y_1, \dotsc, Y_n\) represents a random sample from a uniform distribution on the interval \((\theta,\theta + 1)\). Therefore, \(E[Y_i] = \theta + \frac{1}{2}\), \(V[Y_i] = 1/12\), \(i=1,\dotsc,n\).
  1. The bias for \(\overline{Y}\) (as an estimator for \(\theta\)) is \[ B = E[\overline{Y}] - \theta = (\theta + 1/2) - \theta = 1/2. \]
  2. Let \(\hat{\theta} = \overline{Y} - 1/2\), then \(E[\hat{\theta}] = E[\overline{Y} - 1/2] = (\theta + 1/2) - 1/2 = \theta\), so \(\overline{Y} - 1/2\) is an unbiased estimator for \(\theta\).
  3. \(\text{MSE}[\overline{Y}] = V[\overline{Y}] + B^2 = \frac{1}{12n} + \frac{1}{4} = \frac{3n+1}{12n}\).

Problem 8.10 (6 points)

Let \(Y_1, \dotsc, Y_n\) denote a random sample from a population whose density and distribution functions are \[\begin{align*} f(y) &= \begin{cases} \alpha\cdot y^{\alpha-1}/\theta^\alpha & , 0 \le y \le \theta \\ 0 & , \text{ otherwise} \end{cases} \\ F(y) &= \begin{cases} 1 & , y > \theta \\ (y/\theta)^\alpha & , 0 \le y \le \theta \\ 0 & , y < 0. \end{cases} \end{align*}\] Let \(\hat{\theta} = \max\{Y_1, \dotsc, Y_n\} = Y_{(n)}\). Then \(\hat{\theta}\) has density \[ g_{(n)}(y) = \begin{cases} n \alpha\cdot y^{n \alpha - 1} / \theta^{n \alpha} & , 0 \le y \le \theta \\ 0 & , \text{ otherwise.} \end{cases} \]
  1. We use \(g_{(n)}\) to find the expected value of \(\hat{\theta}\). \[ E[\hat{\theta}] = \int_{0}^{\theta} n\alpha\cdot y\cdot y^{n\alpha - 1}/\theta^{n\alpha}\,dy = \left.\frac{n\alpha}{n\alpha + 1}\cdot \frac{y^{n\alpha + 1}}{\theta^{n\alpha}}\right|_0^{\theta} = \frac{n\alpha}{n\alpha + 1}\cdot \theta. \] Since \(E[\hat{\theta}] \ne \theta\), we see that \(\hat{\theta}\) is a biased estimator for \(\theta\).
  2. \(\frac{n\alpha + 1}{n\alpha}\cdot \hat{\theta}\) is an unbiased estimator for \(\theta\).
  3. The bias for \(\hat{\theta}\) is \(B = \frac{n\alpha}{n\alpha+1}\cdot\theta - \theta = -\theta / (n\alpha + 1)\). We calculate the second moment \[ E\left[\hat{\theta}^2\right] = \int_{0}^{\theta} n\alpha\cdot y^{n\alpha + 1} / \theta^{n\alpha}\,dy = \frac{n\alpha}{n\alpha + 2}\cdot \theta^2. \] Finally, we calculate the mean square error \[\begin{align*} \text{MSE}[\hat{\theta}] &= V[\hat{\theta}] + B^2 \\ &= \frac{n\alpha}{n\alpha + 2}\cdot \theta^2 - \left(\frac{n\alpha}{n\alpha + 1}\right)^2\cdot \theta^2 + \frac{1}{(n\alpha + 1)^2}\cdot \theta^2 \\ &= \frac{2\cdot \theta^2}{(n\alpha + 1)(n\alpha + 2)}. \end{align*}\]

Problem 8.12 (6 points)

Suppose \(Y_1, \dotsc, Y_n\) constitute a random sample from a normal distribution with mean \(\mu\) and variance \(\sigma^2\).

Let \(Y = (n - 1)\cdot S^2 / \sigma^2\), which we know has the \(\chi^2\) probability distribution with \(\nu = n - 1\) degrees of freedom. By exercise 4.90(c), \(E[\sqrt{Y}] = \sqrt{2}\cdot \Gamma(n/2) / \Gamma((n-1)/2)\).
  1. Now, \(S = {\sigma}\cdot \sqrt{Y} / {\sqrt{n-1}}\) and therefore \[ E[S] = \frac{\sigma}{\sqrt{n-1}}\cdot E\left[\sqrt{Y}\right] = \sqrt{\frac{2}{n-1}}\cdot \frac{\Gamma(n/2)}{\Gamma\left((n-1)/2)\right)}\cdot \sigma. \] We see that \(S\) is a biased estimator for \(\sigma\), even though \(S^2\) is an unbiased estimator for \(\sigma^2\).
  2. We can adjust \(S\) by a multiplicative constant to create the unbiased estimator for \(\sigma\) \[ \hat{\sigma} = \sqrt{\frac{n-1}{2}}\cdot \frac{\Gamma\left((n-1)/2)\right)}{\Gamma(n/2)}\cdot S. \]
  3. An unbiased estimator for \(\mu - z_{\alpha}\cdot \sigma\) is \(\overline{Y} - z_{\alpha}\cdot \hat{\sigma}\), where \(\overline{Y} = \frac{1}{n}\sum_{i=1}^{n} Y_{i}\) and \(\hat{\sigma}\) is the estimator defined in part (b).

Problem 8.14 (2 points)

Let \(Y_1, \dotsc, Y_n\) denote a random sample from a population with a uniform distribution on the interval \((0,\theta)\). Let \(Y_{(1)} = \min\{Y_1, \dotsc, Y_n\}\). Recall that the density and distribution functions for this uniform distribution are, respectively, \[\begin{align*} f(y) &= \begin{cases} \frac{1}{\theta} & , 0 < y < \theta \\ 0 & , \text{ otherwise}, \end{cases} \\ F(y) &= \begin{cases} 1 & , y \ge \theta \\ \frac{y}{\theta} & , 0 < y < \theta \\ 0 & , y \le 0. \end{cases} \end{align*}\]

Thus, the density function for \(Y_{(1)}\) is \[ g_{(1)}(y) = n\cdot \left[ 1 - F(y)\right]^{n-1}\cdot f(y) = \begin{cases} \frac{n}{\theta} \cdot \left( 1 - \frac{y}{\theta}\right)^{n-1} & , 0 < y < \theta \\ 0 & , \text{ otherwise} \end{cases}. \] Now that we have the density for \(Y_{(1)}\), we can calculate \[ E[Y_{(1)}] = \int_{0}^{\theta} \frac{ny}{\theta}\cdot \left( 1 - \frac{y}{\theta}\right)^{n-1}\,dy = \frac{\theta}{(n + 1)}. \] We see that \(Y_{(1)}\) is a biased estimator for \(\theta\), but we may adjust it by a multiplicative constant to obtain the unbiased estimator \(\hat{\theta} = (n + 1)\cdot Y_{(1)}\).

Problem 8.16 (4 points)

Suppose that \(Y_1, Y_2, Y_3, Y_4\) denotes a random sample from a population with an exponential distribution whose density is given by \[ f(y) = \begin{cases} \frac{1}{\theta}\cdot e^{-y/\theta} & , y > 0 \\ 0 & , \text{ otherwise} \end{cases}. \] Recall that \(Y\) is a Gamma distribution with parameters \(\alpha = 1\) and \(\beta = \theta\). By exercise 4.89(a), \(E[Y^a] = \theta^{a}\cdot \Gamma(1 + a)/\Gamma(1) = \theta^{a}\cdot a\cdot \Gamma(a)\), therefore \[ E[ \sqrt{Y} ] = \frac{1}{2} \Gamma\left(\frac{1}{2}\right)\cdot \sqrt{\theta} = \frac{1}{2}\sqrt{\pi\cdot \theta}. \]
  1. Let \(X = \sqrt{Y_1\cdot Y_2}\). Since \(Y_1\) and \(Y_2\) are independent, \[ E[X] = E[Y_1]\cdot E[Y_2] = \frac{1}{4}\pi\cdot \theta, \] so an unbiased estimator for \(\theta\) is \(\frac{4}{\pi}\sqrt{Y_1\cdot Y_2}\).
  2. Let \(W = \sqrt{Y_1\cdot Y_2\cdot Y_3\cdot Y_4}\), then \[ E[W] = E[Y_1]\cdot E[Y_2]\cdot E[Y_3]\cdot E[Y_4] = \frac{1}{16}\pi^2\cdot \theta^2. \] Thus, an unbiased estimator for \(\theta^2\) is \(\frac{16}{\pi^2}\sqrt{Y_1\cdot Y_2\cdot Y_3\cdot Y_4}\).
---
title: "Solutions to Homework Assignment 3"
output: html_notebook
---

## Problem 8.2 (4 points)
We are given \(E[\hat{\theta}_1] = E[\hat{\theta}_2] = \theta\), \(V[\hat{\theta}_1]=\sigma_1^2\), \(V[\hat{\theta}_2]=\sigma_2^2\), and \(\hat{\theta}_1\) and \(\hat{\theta}_2\) are independent.

Let \(\hat{\theta}_3 = a\cdot \hat{\theta}_1 + (1-a)\cdot \hat{\theta}_2\).
<ol type="a">
<li>
Since \(E[\hat{\theta}_3] = E[a\cdot \hat{\theta}_1 + (1-a)\cdot \hat{\theta}_2]
= a\cdot E[\hat{\theta}_1] + (1-a)\cdot E[\hat{\theta}_2] = a\cdot \theta + (1-a)\cdot \theta 
= \theta\), we see that \(\hat{\theta}_3\) is an unbiased estimator for \(\theta\) for all \(a\in\mathbb{R}\).
</li>
<li>
The variance of \(\hat{\theta}_3\) is given by the quadratic function
\[
f(a) = V[\hat{\theta}_3] = a^2\cdot \sigma_1^2 + (1 - a)^2\cdot \sigma_2^2.
\]
The first derivative \(f'(a) = 2\cdot [(\sigma_1^2 + \sigma_2^2)\cdot a - \sigma_2^2]\) equals zero at \(a = \sigma_2^2 / (\sigma_1^2 + \sigma_2^2)\), and since the second derivative \(f''(a) = 2\cdot (\sigma_1^2 + \sigma_2^2) > 0\), we see that the critical point is a global minimum for the quadratic function. Thus, of all possible weighted averages of the unbiased estimators \(\hat{\theta}_1\), \(\hat{\theta}_2\), 
\[
\hat{\theta}_3 = \frac{\sigma_2^2}{\sigma_1^2 + \sigma_2^2}\cdot \hat{\theta}_1
  + \frac{\sigma_1^2}{\sigma_1^2 + \sigma_2^2}\cdot \hat{\theta}_2
\]
has the smallest variance \(\sigma_3^2 = V[\hat{\theta}_3] = \sigma_1^2\cdot \sigma_2^2 / (\sigma_1^2 + \sigma_2^2)\). Observe that the variance of \(\hat{\theta}_3\) is less than the variance of either \(\hat{\theta}_1\) or \(\hat{\theta}_2\). This fact becomes apparant when we see that \(1/\sigma_3^2 = 1/\sigma_1^2 + 1/\sigma_2^2\) and noting that both terms in the sum are positive. 

This problem shows that a suitable weighted average of independant, unbiased estimators for \(\theta\) produces a new unbiased estimator for \(\theta\) that has smaller variance than either of the original estimators. 

##Problem 8.4 (4 points)
Suppose \(Y_1, Y_2, Y_3\) denotes a random sample from an exponential distribution with probability density function
\[
f(y) = 
\begin{cases}
\frac{1}{\theta}\cdot e^{-y/\theta} &, y > 0 \\
0 &, y \le 0.
\end{cases}
\]
Note that \(E[Y_i]=\theta\) and \(V[Y_i]=\theta^2\), \(i=1,2,3\).

<ol type="a">
<li>
Consider: 
\(\hat{\theta}_1 = Y_1\), 
\(\hat{\theta}_2 = \frac{1}{2}(Y_1 + Y_2)\),
\(\hat{\theta}_3 = \frac{1}{3}Y_1 + \frac{2}{3}Y_2\)
\(\hat{\theta}_4 = \min{(Y_1, Y_2, Y_3)}\),
\(\hat{\theta}_5 = \overline{Y}\).

\begin{align*}
E[\hat{\theta}_1] &= E[Y_1] = \theta. \\
E[\hat{\theta}_2] &= \frac{1}{2}\left(E[Y_1] + E[Y_2]\right) = \frac{1}{2}(\theta + \theta) = \theta. \\
E[\hat{\theta}_3] &= \frac{1}{3}E[Y_1] + \frac{2}{3}E[Y_2] = \frac{1}{3}\theta + \frac{2}{3}\theta = \theta. \\
E[\hat{\theta}_4] &= \frac{1}{3}\theta \text{ (see exercise 6.65(a), page 324).} \\
E[\hat{\theta}_5] &= E[\overline{Y}] = \theta.
\end{align*}
So, \(\hat{\theta}_1\), \(\hat{\theta}_2\), \(\hat{\theta}_3\), \(\hat{\theta}_5\) are unbiased estimators for \(\theta\).
</li>
<li>
We calculate the variances of the unbiased estimators.
\begin{align*}
V[\hat{\theta}_1] &= V[Y_1] = \theta^2 \\
V[\hat{\theta}_2] &= \frac{1}{4}(\theta^2 + \theta^2) = \frac{1}{2}\theta^2 \\
V[\hat{\theta}_3] &= \frac{1}{9}\theta^2 + \frac{4}{9}\theta^2 = \frac{5}{9}\theta^2 \\
V[\hat{\theta}_5] &= \frac{1}{3}\theta^2,
\end{align*}
so \(\hat{\theta}_5 = \overline{Y}\) has the least variance.
</li>
</ol>

##Problem 8.8 (6 points)
We are given that \(Y_1, \dotsc, Y_n\) represents a random sample from a uniform distribution on the interval \((\theta,\theta + 1)\). Therefore, \(E[Y_i] = \theta + \frac{1}{2}\), \(V[Y_i] = 1/12\), \(i=1,\dotsc,n\).
<ol type="a">
<li>
The bias for \(\overline{Y}\) (as an estimator for \(\theta\)) is
\[
B = E[\overline{Y}] - \theta = (\theta + 1/2) - \theta = 1/2.
\]
</li>
<li>
Let \(\hat{\theta} = \overline{Y} - 1/2\), then \(E[\hat{\theta}] = E[\overline{Y} - 1/2] = (\theta + 1/2) - 1/2 = \theta\), so \(\overline{Y} - 1/2\) is an unbiased estimator for \(\theta\).
</li>
<li>
\(\text{MSE}[\overline{Y}] = V[\overline{Y}] + B^2 = \frac{1}{12n} + \frac{1}{4} = \frac{3n+1}{12n}\).
</li>
</ol>

##Problem 8.10 (6 points)
Let \(Y_1, \dotsc, Y_n\) denote a random sample from a population whose density and distribution functions are
\begin{align*}
f(y) &=
\begin{cases}
\alpha\cdot y^{\alpha-1}/\theta^\alpha & , 0 \le y \le \theta \\
0 & , \text{ otherwise}
\end{cases} \\
F(y) &=
\begin{cases}
1 & , y > \theta \\
(y/\theta)^\alpha & , 0 \le y \le \theta \\
0 & , y < 0.
\end{cases}
\end{align*}
Let \(\hat{\theta} = \max\{Y_1, \dotsc, Y_n\} = Y_{(n)}\). Then \(\hat{\theta}\) has density
\[
g_{(n)}(y) =
\begin{cases}
n \alpha\cdot y^{n \alpha - 1} / \theta^{n \alpha} & , 0 \le y \le \theta \\
0 & , \text{ otherwise.}
\end{cases}
\]
<ol type="a">
<li>
We use \(g_{(n)}\) to find the expected value of \(\hat{\theta}\).
\[
E[\hat{\theta}] =
\int_{0}^{\theta} n\alpha\cdot y\cdot y^{n\alpha - 1}/\theta^{n\alpha}\,dy =
\left.\frac{n\alpha}{n\alpha + 1}\cdot \frac{y^{n\alpha + 1}}{\theta^{n\alpha}}\right|_0^{\theta} =
\frac{n\alpha}{n\alpha + 1}\cdot \theta.
\]
Since \(E[\hat{\theta}] \ne \theta\), we see that \(\hat{\theta}\) is a biased estimator for \(\theta\).
</li>
<li>
\(\frac{n\alpha + 1}{n\alpha}\cdot \hat{\theta}\) is an unbiased estimator for \(\theta\).
</li>
<li>
The bias for \(\hat{\theta}\) is \(B = \frac{n\alpha}{n\alpha+1}\cdot\theta - \theta = -\theta / (n\alpha + 1)\). We calculate the second moment
\[
E\left[\hat{\theta}^2\right] = \int_{0}^{\theta} n\alpha\cdot y^{n\alpha + 1} / \theta^{n\alpha}\,dy
= \frac{n\alpha}{n\alpha + 2}\cdot \theta^2.
\]
Finally, we calculate the mean square error
\begin{align*}
\text{MSE}[\hat{\theta}] &= V[\hat{\theta}] + B^2 \\
&= \frac{n\alpha}{n\alpha + 2}\cdot \theta^2 
- \left(\frac{n\alpha}{n\alpha + 1}\right)^2\cdot \theta^2
+ \frac{1}{(n\alpha + 1)^2}\cdot \theta^2 \\
&= \frac{2\cdot \theta^2}{(n\alpha + 1)(n\alpha + 2)}.
\end{align*}
</li>
</ol>

##Problem 8.12 (6 points)
Suppose \(Y_1, \dotsc, Y_n\) constitute a random sample from a normal distribution with mean \(\mu\) and variance \(\sigma^2\).

Let \(Y = (n - 1)\cdot S^2 / \sigma^2\), which we know has the 
\(\chi^2\) probability distribution with \(\nu = n - 1\) degrees of freedom. By exercise 4.90(c), \(E[\sqrt{Y}] = \sqrt{2}\cdot \Gamma(n/2) / \Gamma((n-1)/2)\). 
<ol type="a">
<li>
Now, \(S = {\sigma}\cdot \sqrt{Y} / {\sqrt{n-1}}\) and therefore
\[
E[S] = \frac{\sigma}{\sqrt{n-1}}\cdot E\left[\sqrt{Y}\right]  = 
\sqrt{\frac{2}{n-1}}\cdot \frac{\Gamma(n/2)}{\Gamma\left((n-1)/2)\right)}\cdot \sigma.
\]
We see that \(S\) is a biased estimator for \(\sigma\), even though \(S^2\) is an unbiased estimator for \(\sigma^2\).
</li>
<li>
We can adjust \(S\) by a multiplicative constant to create the unbiased estimator for \(\sigma\)
\[
\hat{\sigma} = \sqrt{\frac{n-1}{2}}\cdot \frac{\Gamma\left((n-1)/2)\right)}{\Gamma(n/2)}\cdot S.
\]
</li>
<li>
An unbiased estimator for \(\mu - z_{\alpha}\cdot \sigma\) is \(\overline{Y} - z_{\alpha}\cdot \hat{\sigma}\), where \(\overline{Y} = \frac{1}{n}\sum_{i=1}^{n} Y_{i}\) and \(\hat{\sigma}\) is the estimator defined in part (b).
</li>
</ol>

##Problem 8.14 (2 points)
Let \(Y_1, \dotsc, Y_n\) denote a random sample from a population with a uniform distribution on the interval
\((0,\theta)\). Let \(Y_{(1)} = \min\{Y_1, \dotsc, Y_n\}\). Recall that the density and distribution functions for this uniform distribution are, respectively,
\begin{align*}
f(y) &= 
\begin{cases}
\frac{1}{\theta} & , 0 < y < \theta \\
0 & , \text{ otherwise},
\end{cases} \\
F(y) &=
\begin{cases}
1 & , y \ge \theta \\
\frac{y}{\theta} & , 0 < y < \theta \\
0 & , y \le 0.
\end{cases}
\end{align*}
Thus, the density function for \(Y_{(1)}\) is
\[
g_{(1)}(y) =
n\cdot \left[ 1 - F(y)\right]^{n-1}\cdot f(y) =
\begin{cases}
\frac{n}{\theta} \cdot \left( 1 - \frac{y}{\theta}\right)^{n-1} & , 0 < y < \theta \\
0 & , \text{ otherwise}
\end{cases}.
\]
Now that we have the density for \(Y_{(1)}\), we can calculate
\[
E[Y_{(1)}] = \int_{0}^{\theta} \frac{ny}{\theta}\cdot \left( 1 - \frac{y}{\theta}\right)^{n-1}\,dy = \frac{\theta}{(n + 1)}.
\]
We see that \(Y_{(1)}\) is a biased estimator for \(\theta\), but we may adjust it by a multiplicative constant to obtain the unbiased estimator \(\hat{\theta} = (n + 1)\cdot Y_{(1)}\).

##Problem 8.16 (4 points)
Suppose that \(Y_1, Y_2, Y_3, Y_4\) denotes a random sample from a population with an exponential distribution whose density is given by
\[
f(y) =
\begin{cases}
\frac{1}{\theta}\cdot e^{-y/\theta} & , y > 0 \\
0 & , \text{ otherwise}
\end{cases}.
\]
Recall that \(Y\) is a Gamma distribution with parameters \(\alpha = 1\) and \(\beta = \theta\).
By exercise 4.89(a), \(E[Y^a] = \theta^{a}\cdot \Gamma(1 + a)/\Gamma(1) = \theta^{a}\cdot a\cdot \Gamma(a)\), 
therefore
\[
E[ \sqrt{Y} ] = \frac{1}{2} \Gamma\left(\frac{1}{2}\right)\cdot \sqrt{\theta} = \frac{1}{2}\sqrt{\pi\cdot \theta}.
\]
<ol type="a">
<li>
Let \(X = \sqrt{Y_1\cdot Y_2}\). Since \(Y_1\) and \(Y_2\) are independent,
\[
E[X] = E[Y_1]\cdot E[Y_2] = \frac{1}{4}\pi\cdot \theta,
\]
so an unbiased estimator for \(\theta\) is \(\frac{4}{\pi}\sqrt{Y_1\cdot Y_2}\).
</li>
<li>
Let \(W = \sqrt{Y_1\cdot Y_2\cdot Y_3\cdot Y_4}\), then
\[
E[W] = E[Y_1]\cdot E[Y_2]\cdot E[Y_3]\cdot E[Y_4] = \frac{1}{16}\pi^2\cdot \theta^2.
\]
Thus, an unbiased estimator for \(\theta^2\) is 
\(\frac{16}{\pi^2}\sqrt{Y_1\cdot Y_2\cdot Y_3\cdot Y_4}\).
</li>
</ol>










