Archive for April, 2011

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

Author: Panagiotis

25 Apr

In [1, p. 61 exercise 3.7] we are asked to find the MLE of $\mu_{x}$ and $\sigma_x^2$ . 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.
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Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.6

Author: Panagiotis

19 Apr

In [1, p. 61 exercise 3.6] we are asked to assume that the variance is to be estimated as well as the mean for the conditions of [1, p. 60 exercise 3.4] (see also [2, solution of exercise 3.4]) . We are asked to prove for the vector parameter $\mathbf{\theta}=\left[\mu_x \; \sigma^2_x\right]^T$ , that the Fisher information matrix is

$\mathbf{I}_{\theta}=\left[\begin{array}{cc} \frac{N}{\sigma^2_x} & 0 \\ 0 & \frac{N}{2\sigma^4_x} \end{array}\right]$

Furthermore we are asked to find the CR bound and to determine if the sample mean $\hat{\mu}_x$ is efficient. If additionaly the variance is to be estimated as

$\hat{\sigma}^2_x=\frac{1}{N-1}\sum\limits_{n=0}^{N-1}(x[n]-\hat{\mu}_x)^2$