3. (Problem 10.41 of Freund's) To show that an estimator can be consistent with- out being unbiased nor asymptotically unbiased, consider the following estimation procedure. Let X₁,..., Xn be a random sample from a population with mean and finite variance o2. Consider the following estimation procedure: First, we ran- domly draw one slip of n papers numbered from 1 to n. If we get 2,3,..., n, we use X = n¹1 as our estimate for μ; otherwise, we use n² as the estimate for -1 i=1 u. Show that this estimation procedure is (a) consistent. (b) neither unbiased nor asymptotically unbiased.
3. (Problem 10.41 of Freund's) To show that an estimator can be consistent with- out being unbiased nor asymptotically unbiased, consider the following estimation procedure. Let X₁,..., Xn be a random sample from a population with mean and finite variance o2. Consider the following estimation procedure: First, we ran- domly draw one slip of n papers numbered from 1 to n. If we get 2,3,..., n, we use X = n¹1 as our estimate for μ; otherwise, we use n² as the estimate for -1 i=1 u. Show that this estimation procedure is (a) consistent. (b) neither unbiased nor asymptotically unbiased.
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
Transcribed Image Text:3. (Problem 10.41 of Freund's) To show that an estimator can be consistent with-
out being unbiased nor asymptotically unbiased, consider the following estimation
procedure. Let X₁,..., Xn be a random sample from a population with mean μ
and finite variance o2. Consider the following estimation procedure: First, we ran-
domly draw one slip of n papers numbered from 1 to n. If we get 2,3,...,n, we
-1 n
use X = n ¹₁ as our estimate for u; otherwise, we use n² as the estimate for
i=1
μ. Show that this estimation procedure is
(a) consistent.
(b) neither unbiased nor asymptotically unbiased.
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