Suppose that X1,... X, is a random sample from a normal distribution with mean u and variance o?. Two unbiased estimators of o² are ởf = s° = (x, – X)², and o =(X, - X,)°. n-1 i=1 Find the efficiency of ở relative to ôž.
Suppose that X1,... X, is a random sample from a normal distribution with mean u and variance o?. Two unbiased estimators of o² are ởf = s° = (x, – X)², and o =(X, - X,)°. n-1 i=1 Find the efficiency of ở relative to ôž.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![Suppose that \( X_1, \ldots, X_n \) is a random sample from a normal distribution with mean \( \mu \) and variance \( \sigma^2 \). Two unbiased estimators of \( \sigma^2 \) are
\[
\hat{\sigma}_1^2 = S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2,
\]
and
\[
\hat{\sigma}_2^2 = \frac{1}{2} (X_1 - X_2)^2.
\]
Find the efficiency of \( \hat{\sigma}_1^2 \) relative to \( \hat{\sigma}_2^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9ad9a17-86a2-4d23-8aeb-c537a78d9db8%2Fb52dac91-3787-4141-b18e-9671e577d59c%2Fyi9s1rx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose that \( X_1, \ldots, X_n \) is a random sample from a normal distribution with mean \( \mu \) and variance \( \sigma^2 \). Two unbiased estimators of \( \sigma^2 \) are
\[
\hat{\sigma}_1^2 = S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2,
\]
and
\[
\hat{\sigma}_2^2 = \frac{1}{2} (X_1 - X_2)^2.
\]
Find the efficiency of \( \hat{\sigma}_1^2 \) relative to \( \hat{\sigma}_2^2 \).
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