n - 1i=1 Let X1, ... , Xn be iid N (u, o) and suppose we are interested in estimating t(µ, 0) = o“. The commonly used estimator is the sample variance: 1 T = s? = (Xi – X)? For this case, recall we have shown that (n – 1)s²/o² ~ xản-1): a) It is well known that T, is unbiased. Show that it is also a consistent estimator of o?. An alternative estimator is the MLE, which in this case is X(n-1). 1 T2 = ô? = (Xi– X)? n i=D1 b) Find the MSE of T2 in terms of n and o. Briefly comment. [HINT: T, = (n – 1)T,/n.] A bit of algebra shows that E,(X; – X)² = E-1 X? – nX2. Furthermore, it is readily be shown that S = (X,E-,X?) is a sufficient statistic for iid normal observations. Define a new estimator of o? as %3D i3D1 T3 n – 1 i=1 where N - Bin(n, 0.5) is independent of all the X;'s c) Show that T3 is unbiased but argue that its MSE is larger than that of T1.
n - 1i=1 Let X1, ... , Xn be iid N (u, o) and suppose we are interested in estimating t(µ, 0) = o“. The commonly used estimator is the sample variance: 1 T = s? = (Xi – X)? For this case, recall we have shown that (n – 1)s²/o² ~ xản-1): a) It is well known that T, is unbiased. Show that it is also a consistent estimator of o?. An alternative estimator is the MLE, which in this case is X(n-1). 1 T2 = ô? = (Xi– X)? n i=D1 b) Find the MSE of T2 in terms of n and o. Briefly comment. [HINT: T, = (n – 1)T,/n.] A bit of algebra shows that E,(X; – X)² = E-1 X? – nX2. Furthermore, it is readily be shown that S = (X,E-,X?) is a sufficient statistic for iid normal observations. Define a new estimator of o? as %3D i3D1 T3 n – 1 i=1 where N - Bin(n, 0.5) is independent of all the X;'s c) Show that T3 is unbiased but argue that its MSE is larger than that of T1.
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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![n- 1Li=1
Let X1, . , Xn be iid N (u, o) and suppose we are interested in estimating t(µ,0) = o². The commonly
used estimator is the sample variance:
T1 = s2 =
(Xi – X)2
For this case, recall we have shown that (n – 1)s²/o2 - xổn-1):
a) It is well known that T, is unbiased. Show that it is also a consistent estimator of o?.
An alternative estimator is the MLE, which in this case is
n2 (Xi – X)2
i=1
T2 = ô?
b) Find the MSE of T2 in terms ofn and o. Briefly comment. [HINT: T2 = (n – 1)T,/n.]
A bit of algebra shows that E,(X; – X)2 = E-1 X? - nX2. Furthermore, it is readily be shown that
S = (X, E-1 X?) is a sufficient statistic for iid normal observations. Define a new estimator of o? as
%3D
2
T3
n -
%3D
i=1
where N - Bin(n, 0.5) is independent of all the X;'s
c) Show that T3 is unbiased but argue that its MSE is larger than that of T1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f106119-ec29-40b4-bdd9-79633e0d2dfb%2Fe9d4ba82-abf1-44d8-8a56-19c17699a07b%2Fdqvtguv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:n- 1Li=1
Let X1, . , Xn be iid N (u, o) and suppose we are interested in estimating t(µ,0) = o². The commonly
used estimator is the sample variance:
T1 = s2 =
(Xi – X)2
For this case, recall we have shown that (n – 1)s²/o2 - xổn-1):
a) It is well known that T, is unbiased. Show that it is also a consistent estimator of o?.
An alternative estimator is the MLE, which in this case is
n2 (Xi – X)2
i=1
T2 = ô?
b) Find the MSE of T2 in terms ofn and o. Briefly comment. [HINT: T2 = (n – 1)T,/n.]
A bit of algebra shows that E,(X; – X)2 = E-1 X? - nX2. Furthermore, it is readily be shown that
S = (X, E-1 X?) is a sufficient statistic for iid normal observations. Define a new estimator of o? as
%3D
2
T3
n -
%3D
i=1
where N - Bin(n, 0.5) is independent of all the X;'s
c) Show that T3 is unbiased but argue that its MSE is larger than that of T1.
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