(10) = {01²-¹0 0, elsewhere. a Show that this density function is in the (one-parameter) exponential family and that -In(Y) is sufficient for 9. (See Exercise 9.45.) b If W₁ = -In(Y), show that W, has an exponential distribution with mean 1/0. c Use methods similar to those in Example 9.10 to show that 20 with 2n df. W, has a x² distribution
(10) = {01²-¹0 0, elsewhere. a Show that this density function is in the (one-parameter) exponential family and that -In(Y) is sufficient for 9. (See Exercise 9.45.) b If W₁ = -In(Y), show that W, has an exponential distribution with mean 1/0. c Use methods similar to those in Example 9.10 to show that 20 with 2n df. W, has a x² distribution
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Answer 9.60
![**9.45**
Suppose that \( Y_1, Y_2, \ldots, Y_n \) is a random sample from a probability density function in the (one-parameter) exponential family so that
\[
f(y \mid \theta) =
\begin{cases}
a(\theta)b(y)e^{-[c(\theta)d(y)]}, & a \leq y \leq b, \\
0, & \text{elsewhere},
\end{cases}
\]
where \( a \) and \( b \) do not depend on \(\theta\). Show that \(\sum_{i=1}^n d(Y_i)\) is sufficient for \(\theta\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2Fdff90ca2-eafa-49c4-a8a1-2828e913deeb%2F9w56p08_processed.png&w=3840&q=75)
Transcribed Image Text:**9.45**
Suppose that \( Y_1, Y_2, \ldots, Y_n \) is a random sample from a probability density function in the (one-parameter) exponential family so that
\[
f(y \mid \theta) =
\begin{cases}
a(\theta)b(y)e^{-[c(\theta)d(y)]}, & a \leq y \leq b, \\
0, & \text{elsewhere},
\end{cases}
\]
where \( a \) and \( b \) do not depend on \(\theta\). Show that \(\sum_{i=1}^n d(Y_i)\) is sufficient for \(\theta\).
![**Exercise 9.60**
Let \( Y_1, Y_2, \ldots, Y_n \) denote a random sample from the probability density function
\[
f(y \mid \theta) =
\begin{cases}
\theta y^{\theta - 1}, & 0 < y < 1, \, \theta > 0, \\
0, & \text{elsewhere}.
\end{cases}
\]
**a.** Show that this density function is in the (one-parameter) exponential family and that \( \sum_{i=1}^{n} -\ln(Y_i) \) is sufficient for \( \theta \). (See Exercise 9.45.)
**b.** If \( W_i = -\ln(Y_i) \), show that \( W_i \) has an exponential distribution with mean \( 1 / \theta \).
**c.** Use methods similar to those in Example 9.10 to show that \( 2\theta \sum_{i=1}^{n} W_i \) has a \(\chi^2\) distribution with \( 2n \) degrees of freedom.
**d.** Show that
\[
E \left( \frac{1}{2\theta \sum_{i=1}^{n} W_i} \right) = \frac{1}{2(n-1)}.
\]
[Hint: Recall Exercise 4.112.]
**e.** What is the MVUE for \( \theta \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2Fdff90ca2-eafa-49c4-a8a1-2828e913deeb%2F1qfsxsk_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 9.60**
Let \( Y_1, Y_2, \ldots, Y_n \) denote a random sample from the probability density function
\[
f(y \mid \theta) =
\begin{cases}
\theta y^{\theta - 1}, & 0 < y < 1, \, \theta > 0, \\
0, & \text{elsewhere}.
\end{cases}
\]
**a.** Show that this density function is in the (one-parameter) exponential family and that \( \sum_{i=1}^{n} -\ln(Y_i) \) is sufficient for \( \theta \). (See Exercise 9.45.)
**b.** If \( W_i = -\ln(Y_i) \), show that \( W_i \) has an exponential distribution with mean \( 1 / \theta \).
**c.** Use methods similar to those in Example 9.10 to show that \( 2\theta \sum_{i=1}^{n} W_i \) has a \(\chi^2\) distribution with \( 2n \) degrees of freedom.
**d.** Show that
\[
E \left( \frac{1}{2\theta \sum_{i=1}^{n} W_i} \right) = \frac{1}{2(n-1)}.
\]
[Hint: Recall Exercise 4.112.]
**e.** What is the MVUE for \( \theta \)?
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