Lysario – by Panagiotis Chatzichrisafis

"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"

The desire to predict the complex WSS random process based on the sample x[n-1] by using a linear predictor
\hat{x}[n]=-\alpha_{1}x[n-1] (1)

is expressed in [1, p. 61 exercise 3.10]. It is asked to chose \alpha_{1} to minimize the MSE or prediction error power
MSE = \mathcal{E}\left\{\left| x[n] -\hat{x}[n] \right|^{2} \right\}. (2)

We are asked to find the optimal prediction parameter \alpha_{1} 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
x[n]=\alpha + \beta n + z[n]  \; n=0,1,...,N-1 (1)

and find the MLE of the slope \beta and the intercept \alpha by assuming that z[n] is real white Gaussian noise with mean zero and variance \sigma_{z}^{2}. Furthermore it is requested to find the MLE of \alpha if in the linear model we set \beta=0. read the conclusion >
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} is known. We are asked to find the MLE of the mean by maximizing p(\hat{\mu}_{x},\mu_{x}). read the conclusion >
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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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

then we are asked to determine if this estimator is unbiased and efficient. Hint: We are instructed to use the result that
\frac{(N-1)\hat{\sigma}^2_x}{\sigma^2_x} \sim \chi^2_{N-1}

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