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Answer Given that p be an n-dimensional random vector of zero mean and positive definite covariance matrix Ql Given y = WB + ei, where rank of w is m. B^ is the linear minimum variance estimate of B based on y. Covariance of the error ß - B^ Varß = Eß – B^B - BA' = EW'W - 1W'ee'www -1 = wW- 1W'Eee'WW'W – 1 = W'W - iw'62IWW'w – 1= 02W'W – 1 The diagonal elements of this matrix are the variances of the estimators of the individual parameters, and the off-diagonal elements are the covariances between these estimators. Ee'e = n – mo2 so that s2 = e'en - m+1 is an unbiased estimator of o2. e'eo2 = e'Qeo2, where Q is a positive definite matrix of rank m. Therefore, the rank of the covariance of the error ß – B^ is n – m.

Answer Given that p be an n-dimensional random vector of zero mean and positive definite covariance matrix Ql Given y = WB + ei, where rank of w is m. B^ is the linear minimum variance estimate of B based on y. Covariance of the error ß - B^ Varß = Eß – B^B - BA' = EW'W - 1W'ee'www -1 = wW- 1W'Eee'WW'W – 1 = W'W - iw'62IWW'w – 1= 02W'W – 1 The diagonal elements of this matrix are the variances of the estimators of the individual parameters, and the off-diagonal elements are the covariances between these estimators. Ee'e = n – mo2 so that s2 = e'en - m+1 is an unbiased estimator of o2. e'eo2 = e'Qeo2, where Q is a positive definite matrix of rank m. Therefore, the rank of the covariance of the error ß – B^ is n – m.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 13EQ
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Answer Given that p be an n-dimensional random vector of zero mean and positive definite covariance matrix Ql Given y = WB + ei, where rank of w is m. B^ is the linear minimum variance estimate of ß based on y. Covariance of the error ß - B^ Varß = Eß – B^B – BA" = EW'W – 1W'ee'www –1 = w'W – 1W'Eɛ='WW'W – 1 = w'W - 1w'62IWW'w – 1 = 02W'W – 1 The diagonal elements of this matrix are the variances of the estimators of the individual parameters, and the off-diagonal elements are the covariances between these estimators. Ee'e = n – mo2 so that s2 = e'en – m +1 is an unbiased estimator of o2. e'eo2 = e'Qeo2, where Q is a positive definite matrix of rank m. Therefore, the rank of the covariance of the error ß - B^ is n – m.
Transcribed Image Text:Answer Given that p be an n-dimensional random vector of zero mean and positive definite covariance matrix Ql Given y = WB + ei, where rank of w is m. B^ is the linear minimum variance estimate of ß based on y. Covariance of the error ß - B^ Varß = Eß – B^B – BA" = EW'W – 1W'ee'www –1 = w'W – 1W'Eɛ='WW'W – 1 = w'W - 1w'62IWW'w – 1 = 02W'W – 1 The diagonal elements of this matrix are the variances of the estimators of the individual parameters, and the off-diagonal elements are the covariances between these estimators. Ee'e = n – mo2 so that s2 = e'en – m +1 is an unbiased estimator of o2. e'eo2 = e'Qeo2, where Q is a positive definite matrix of rank m. Therefore, the rank of the covariance of the error ß - B^ is n – m.
4. Let B be an n-dimensional random vector of zero mean and positive- definite covariance matrix Q. Suppose measurements of the form y = WB are made where the rank of W is m. If ß is the linear minimum variance estimate of ß based on y, show that the covariance of the error B - B has rank n - m.
Transcribed Image Text:4. Let B be an n-dimensional random vector of zero mean and positive- definite covariance matrix Q. Suppose measurements of the form y = WB are made where the rank of W is m. If ß is the linear minimum variance estimate of ß based on y, show that the covariance of the error B - B has rank n - m.
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