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denote a random sample from a population having mean µ and
Chapter 7, Problem 27E(choose chapter or problem)
Let \(X_{1}, X_{2}, \ldots, X_{7}\) denote a random sample from a population having mean \(\mu\) and variance \(\sigma^{2}\). Consider the following estimators of \(\mu\):
\(\hat{\Theta}_{1}=\frac{X_{1}+X_{2}+\cdots+X_{7}}{7}\)
\(\hat{\Theta}_{2}=\frac{2 X_{1}-X_{6}+X_{4}}{2}\)
(a) Is either estimator unbiased?
(b) Which estimator is better? In what sense is it better? Calculate the relative efficiency of the two estimators.
Equation Transcription:
Text Transcription:
X_1, X_2,..., X_7
mu
sigma^2
mu
Theta hat_1=X_1+X_2++X_7 over 7
Theta hat_2=2X_1-X_6+X_4 over 2
Questions & Answers
QUESTION:
Let \(X_{1}, X_{2}, \ldots, X_{7}\) denote a random sample from a population having mean \(\mu\) and variance \(\sigma^{2}\). Consider the following estimators of \(\mu\):
\(\hat{\Theta}_{1}=\frac{X_{1}+X_{2}+\cdots+X_{7}}{7}\)
\(\hat{\Theta}_{2}=\frac{2 X_{1}-X_{6}+X_{4}}{2}\)
(a) Is either estimator unbiased?
(b) Which estimator is better? In what sense is it better? Calculate the relative efficiency of the two estimators.
Equation Transcription:
Text Transcription:
X_1, X_2,..., X_7
mu
sigma^2
mu
Theta hat_1=X_1+X_2++X_7 over 7
Theta hat_2=2X_1-X_6+X_4 over 2
ANSWER:
Solution
Step 1 of 2
Let denotes the random sample from a population
with mean and variance
Given that the estimators of are
And
a) We have to check that the estimators are unbiased or not
Now
=
= [Since
]
=
Now
=
=
=
=
Hence both and
are unbiased estimators of