Author: Panagiotis
23
Jan
The desire to predict the complex WSS random process based on the sample
![x[n-1]](https://lysario.de/wp-content/cache/tex_f14c06cc1753ec7fe07c98dd3f429390.png)
by using a linear predictor
is expressed in
[1, p. 61 exercise 3.10].
It is asked to chose
![\alpha_{1}](https://lysario.de/wp-content/cache/tex_42b65ba9cd07af0a2b5901a7a68770e9.png)
to minimize the MSE or prediction error power
We are asked to find the optimal prediction parameter
![\alpha_{1}](https://lysario.de/wp-content/cache/tex_42b65ba9cd07af0a2b5901a7a68770e9.png)
and the minimum prediction error power by using the orthogonality principle.
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In
[1, p. 61 exercise 3.9] we are asked to consider the real linear model
and find the MLE of the slope
![\beta](https://lysario.de/wp-content/cache/tex_b0603860fcffe94e5b8eec59ed813421.png)
and the intercept
![\alpha](https://lysario.de/wp-content/cache/tex_7b7f9dbfea05c83784f8b85149852f08.png)
by assuming that
![z[n]](https://lysario.de/wp-content/cache/tex_8a8c996b9e9d1294c8f815911479257f.png)
is real white Gaussian noise with mean zero and variance
![\sigma_{z}^{2}](https://lysario.de/wp-content/cache/tex_6e30ec6d29d935a9fb047bf6637bd483.png)
.
Furthermore it is requested to find the MLE of
![\alpha](https://lysario.de/wp-content/cache/tex_7b7f9dbfea05c83784f8b85149852f08.png)
if in the linear model we set
![\beta=0](https://lysario.de/wp-content/cache/tex_e67d6e83066e4bc070af866c01375a4c.png)
.
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Author: Panagiotis
22
Sep
In
[1, p. 61 exercise 3.8] we are asked to prove that the sample mean is a sufficient statistic for the mean under the conditions of
[1, p. 61 exercise 3.4].
Assuming that
![\sigma^{2}_{x}](https://lysario.de/wp-content/cache/tex_635a7e0ec10a5326776e78fee24a17c4.png)
is known. We are asked to find the MLE of the mean by maximizing
![p(\hat{\mu}_{x},\mu_{x})](https://lysario.de/wp-content/cache/tex_d28acd4ebc173a262e89a399921d72af.png)
.
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Author: Panagiotis
25
Apr
In
[1, p. 61 exercise 3.7] we are asked to find the MLE of
![\mu_{x}](https://lysario.de/wp-content/cache/tex_93e9d5d978e97b02f144d987f662ab2f.png)
and
![\sigma_x^2](https://lysario.de/wp-content/cache/tex_08180e574ab04d61ec6fe43cf42bfe44.png)
.
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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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](https://lysario.de/wp-content/cache/tex_b1d973967798bb8525bff936468f9211.png)
, that the Fisher information matrix is
Furthermore we are asked to find the CR bound and to determine if the sample mean
![\hat{\mu}_x](https://lysario.de/wp-content/cache/tex_5268cb94a28fa94eb6e5aa6c54410d6c.png)
is efficient.
If additionaly the variance is to be estimated as
then we are asked to determine if this estimator is unbiased and efficient. Hint: We are instructed to use the result that
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