Refer to Exercise 9.5. Is σ ^ 2 2 a consistent estimator of σ 2 ? 9.5 Suppose that Y 1 , Y 2 , ... , Y n is a random sample from a normal distribution with mean µ and variance σ 2 . Two unbiased estimators of σ 2 are σ ^ 1 2 = S 2 = 1 n − 1 ∑ i = 1 n ( Y i − Y ¯ ) 2 and σ ^ 2 2 = 1 2 ( Y 1 − Y 2 ) 2 . Find the efficiency of σ ^ 1 2 relative to σ ^ 2 2 .
Refer to Exercise 9.5. Is σ ^ 2 2 a consistent estimator of σ 2 ? 9.5 Suppose that Y 1 , Y 2 , ... , Y n is a random sample from a normal distribution with mean µ and variance σ 2 . Two unbiased estimators of σ 2 are σ ^ 1 2 = S 2 = 1 n − 1 ∑ i = 1 n ( Y i − Y ¯ ) 2 and σ ^ 2 2 = 1 2 ( Y 1 − Y 2 ) 2 . Find the efficiency of σ ^ 1 2 relative to σ ^ 2 2 .
Solution Summary: The author explains that an estimator stackreltheta_n is not a consistent estimate of
Refer to Exercise 9.5. Is
σ
^
2
2
a consistent estimator of σ2?
9.5 Suppose that Y1, Y2,..., Yn is a random sample from a normal distribution with meanµ and variance σ2. Two unbiased estimators of σ2 are
σ
^
1
2
=
S
2
=
1
n
−
1
∑
i
=
1
n
(
Y
i
−
Y
¯
)
2
and
σ
^
2
2
=
1
2
(
Y
1
−
Y
2
)
2
.
Find the efficiency of
σ
^
1
2
relative to
σ
^
2
2
.
Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.
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