Lysario – by Panagiotis Chatzichrisafis
"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.10
Author: Panagiotis23 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
to minimize the MSE or prediction error power
We are asked to find the optimal prediction parameter and the minimum prediction error power by using the orthogonality principle.
read the conclusion >
by using a linear predictor
![\hat{x}[n]=-\alpha_{1}x[n-1]](https://lysario.de/wp-content/cache/tex_3edb3496c7a3a354e025c6fc6946e035.png) | (1) | ||
is expressed in [1, p. 61 exercise 3.10]. It is asked to chose
to minimize the MSE or prediction error power
![MSE = \mathcal{E}\left\{\left| x[n] -\hat{x}[n] \right|^{2} \right\}.](https://lysario.de/wp-content/cache/tex_108f20c754ba9aebb9393528fd95ffae.png) | (2) | ||
We are asked to find the optimal prediction parameter
and the minimum prediction error power by using the orthogonality principle. read the conclusion >
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.9
Author: Panagiotis3 Jan
In [1, p. 61 exercise 3.9] we are asked to consider the real linear model
and find the MLE of the slope
and the intercept 
by assuming that ![z[n]](https://lysario.de/wp-content/cache/tex_8a8c996b9e9d1294c8f815911479257f.png)
is real white Gaussian noise with mean zero and variance 
.
Furthermore it is requested to find the MLE of if in the linear model we set 
.
read the conclusion >
![x[n]=\alpha + \beta n + z[n] \; n=0,1,...,N-1](https://lysario.de/wp-content/cache/tex_b693a80228f813c59eb716ad50e93b09.png) | (1) | ||
and find the MLE of the slope
and the intercept by assuming that is real white Gaussian noise with mean zero and variance .
Furthermore it is requested to find the MLE of if in the linear model we set .
read the conclusion >
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.8
Author: Panagiotis22 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 
is known. We are asked to find the MLE of the mean by maximizing 
.
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is known. We are asked to find the MLE of the mean by maximizing .
read the conclusion >
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.7
Author: Panagiotis25 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.
read the conclusion >
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. read the conclusion >
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 61 exercise 3.6
Author: Panagiotis19 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
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
read the conclusion >
, 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]](https://lysario.de/wp-content/cache/tex_42789b24f24a6a32db03314ccac291c8.png) | |||
Furthermore we are asked to find the CR bound and to determine if the sample mean
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](https://lysario.de/wp-content/cache/tex_13d6ef5f9b8f0b1d9832efd9b58af88b.png) | |||
then we are asked to determine if this estimator is unbiased and efficient. Hint: We are instructed to use the result that
 | |||
read the conclusion >