9.43 Let Y₁, Y₂,..., Y₁ denote independent and identically distributed random variables from a power family distribution with parameters a and 0. Then, by the result in Exercise 6.17, if a, 0 > 0, f(y|a, 0) = [aya-1/0°, 0₁. If is known, show that I Y; is sufficient for a. 0≤ y ≤0, elsewhere.
9.43 Let Y₁, Y₂,..., Y₁ denote independent and identically distributed random variables from a power family distribution with parameters a and 0. Then, by the result in Exercise 6.17, if a, 0 > 0, f(y|a, 0) = [aya-1/0°, 0₁. If is known, show that I Y; is sufficient for a. 0≤ y ≤0, elsewhere.
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Answer 9.52
![**Exercise 9.43**
Consider the random variables \( Y_1, Y_2, \ldots, Y_n \) which are independent and identically distributed following a power family distribution with parameters \( \alpha \) and \( \theta \). According to the result from Exercise 6.17, for \( \alpha, \theta > 0 \), the probability density function is given by:
\[
f(y \mid \alpha, \theta) =
\begin{cases}
\alpha y^{\alpha-1}/\theta^{\alpha}, & \text{if } 0 \leq y \leq \theta, \\
0, & \text{elsewhere.}
\end{cases}
\]
The task is to prove that if \( \theta \) is known, then the product \( \prod_{i=1}^n Y_i \) is a sufficient statistic for \( \alpha \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2F9d6b38f0-5a90-4386-a7d3-2e625728e07c%2F102ii8_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 9.43**
Consider the random variables \( Y_1, Y_2, \ldots, Y_n \) which are independent and identically distributed following a power family distribution with parameters \( \alpha \) and \( \theta \). According to the result from Exercise 6.17, for \( \alpha, \theta > 0 \), the probability density function is given by:
\[
f(y \mid \alpha, \theta) =
\begin{cases}
\alpha y^{\alpha-1}/\theta^{\alpha}, & \text{if } 0 \leq y \leq \theta, \\
0, & \text{elsewhere.}
\end{cases}
\]
The task is to prove that if \( \theta \) is known, then the product \( \prod_{i=1}^n Y_i \) is a sufficient statistic for \( \alpha \).
![### Exercise 9.54
Consider a set of independent and identically distributed random variables \( Y_1, Y_2, \ldots, Y_n \) drawn from a power family distribution characterized by parameters \( \alpha \) and \( \theta \).
Given:
- If \( \alpha, \theta > 0 \), the probability density function is defined as:
\[
f(y \mid \alpha, \theta) =
\begin{cases}
\alpha y^{\alpha - 1} / \theta^\alpha, & 0 \leq y \leq \theta, \\
0, & \text{elsewhere}.
\end{cases}
\]
**Task:**
Demonstrate that the maximum value among the observations \( \max(Y_1, Y_2, \ldots, Y_n) \) and the product of all observations \( \prod_{i=1}^{n} Y_i \) are jointly sufficient statistics for the parameters \( \alpha \) and \( \theta \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2F9d6b38f0-5a90-4386-a7d3-2e625728e07c%2F2zjw8up_processed.png&w=3840&q=75)
Transcribed Image Text:### Exercise 9.54
Consider a set of independent and identically distributed random variables \( Y_1, Y_2, \ldots, Y_n \) drawn from a power family distribution characterized by parameters \( \alpha \) and \( \theta \).
Given:
- If \( \alpha, \theta > 0 \), the probability density function is defined as:
\[
f(y \mid \alpha, \theta) =
\begin{cases}
\alpha y^{\alpha - 1} / \theta^\alpha, & 0 \leq y \leq \theta, \\
0, & \text{elsewhere}.
\end{cases}
\]
**Task:**
Demonstrate that the maximum value among the observations \( \max(Y_1, Y_2, \ldots, Y_n) \) and the product of all observations \( \prod_{i=1}^{n} Y_i \) are jointly sufficient statistics for the parameters \( \alpha \) and \( \theta \).
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