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Let X1, ..., Xn be iid N (µ, o) and suppose we are interested in estimating t(µ, 0) = o². The commonly used estimator is the sample variance: 1 T; = s? = (X – X)² %3D п — 1 %3D i=1 For this case, recall we have shown that (n – 1)s2/o2 ~ xn-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 u T2 = ô2: (X; – X)² = -

Let X1, ..., Xn be iid N (µ, o) and suppose we are interested in estimating t(µ, 0) = o². The commonly used estimator is the sample variance: 1 T; = s? = (X – X)² %3D п — 1 %3D i=1 For this case, recall we have shown that (n – 1)s2/o2 ~ xn-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 u T2 = ô2: (X; – X)² = -

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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.
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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