Von: Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.7

Mon, 25 Apr 2011 19:30:02 +0000

[...] 25 Apr In [1, p. 61 exercise 3.7] we are asked to find the MLE of and . for the conditions of Problem [1, p. 60 exercise 3.4] (see also [2, solution of exercise 3.4]). We are asked if the MLE of the parameters are asymptotically unbiased , efficient and Gaussianly distributed. Solution: The p.d.f of the observations is given by With and . Thus the determinant is given by . Furthermore we can simplify For a given measurement we obtain the likelihood function: (1) Obviously the previous equation is positive and thus the estimator of the mean which will maximize the probability of the observation will provide a local maximum at the points where the derivative in respect to will be zero. Because the function is positive we can also use the natural logarithm of the likelihood function (the log-likelihood function) in order to obtain the maximum likelihood of . This was already derived in [3, relation (3)]: [...]

Von: Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.6

Tue, 19 Apr 2011 19:03:44 +0000

[...] with normal distribution and the natural logarithm of the joint pdf is given by from cite[ relation (3) ]{Chatzichrisafis352010}: From this relation we can find the gradient in respect to the vector [...]

Kommentare zu: Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.5

Von: Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.7

Von: Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.6