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In this example , we derived the maximum likelihood estimate of the mean andvariance of a Gaussian random vector. You might wonder why we chose to estimate the variance $^{2}$ rather than the standard deviation $$ . Using the same assumptions provided in the example, let's explore theconsequences of estimating a function of a parameter ( van Trees: Probs 2.4.9, 2.4.10 ).
Assuming that the mean is known, find the maximum likelihood estimates of first the variance, then thestandard deviation.
Are these estimates biased?
Describe how these two estimates are related. Assuming that $f()$ is a monotonic function, how are $({}_{\mathrm{ML}})$ and $({f\left(\right)}_{\mathrm{ML}})$ related in general? These results suggest a general question. Consider the problem of estimating somefunction of a parameter $$ , say ${f}_{1}()$ . The observed quantity is $r$ and the conditional density $p(r, , r)$ is known. Assume that $$ is a nonrandom parameter.
What are the conditions for an efficient estimate $({f}_{1}())$ to exist?
What is the lower bound on the variance of the error of any unbiased estimate of ${f}_{1}()$ ?
Assume an efficient estimate of ${f}_{1}()$ exists; when can an efficient estimate of some other function ${f}_{2}()$ exist?
Let the observations $r(l)$ consist of statistically independent, identically distributed Gaussian random variables having zero mean butunknown variance. We wish to estimate $^{2}$ , their variance.
Find the maximum likelihood estimate $({}_{\mathrm{ML}}^{2})$ and compute the resulting mean-squared error.
Show that this estimate is efficient.
Consider a new estimate $({}_{\mathrm{NEW}}^{2})$ given by $({}_{\mathrm{NEW}}^{2})=({}_{\mathrm{ML}}^{2})$ , where $$ is a constant. Find the value of $$ that minimizes the mean-squared error for $({}_{\mathrm{NEW}}^{2})$ . Show that the mean-squared error of $({}_{\mathrm{NEW}}^{2})$ is less than that of $({}_{\mathrm{ML}}^{2})$ . Is this result compatible with this previous part ?
Let the observations be of the form $r=H+n$ where $$ and $n$ are statistically independent Gaussian random vectors. $$(, (0, {K}_{}()))$$ $$(n, (0, {K}_{n}()))$$ The vector $$ has dimension $M$ ; the vectors $r$ and $n$ have dimension $N$ .
Derive the minimum mean-squared error estimate of $$ , $({}_{\mathrm{MMSE}})$ , from the relationship $({}_{\mathrm{MMSE}})=(r, )$
Show that this estimate and the optimum linear estimate $({}_{\mathrm{LIN}})$ derived by the Orthogonality Principle are equal.
Find an expression for the mean-squared error when these estimates are used.
To illustrate the power of importance sampling, let's consider a somewhat nave example. Let $r$ have a zero-mean Laplacian distribution; we want to employ importance sampling techniques to estimate $(r> )$ (despite the fact that we can calculate it easily). Let the density for $\stackrel{}{r}$ be Laplacian having mean $$ .
Find the weight ${c}_{l}$ that must be applied to each decision based on the variable $\stackrel{}{r}$ .
Find the importance sampling gain. Show that this gain means that a fixed number of simulations are needed to achieve a given percentageestimation error (as defined by the coefficient of variation). Express this number as a function of thecriterion value for the coefficient of variation.
Now assume that the density for $\stackrel{}{r}$ is Laplacian, but with mean $m$ . Optimize $m$ by finding the value that maximizes the importance sampling gain.
Suppose we consider an estimate of the parameter $$ having the form $()=(r)+C$ , where $r$ denotes the vector of the observables and $()$ is a linear operator. The quantity $C$ is a constant. This estimate is not a linear function of the observables unless $C=0$ . We are interested in finding applications for which it is advantageous to allow $C\neq 0$ . Estimates of this form we term "quasi-linear" .
Show that the optimum (minimum mean-squared error) quasi-linear estimate satisfies $$({}_{}(r)+{C}_{}-\cdot (r)+C)=0$$ for all $()$ and $C$ where $({}_{\mathrm{QLIN}})={}_{}(r)+{C}_{}$ .
Find a general expression for the mean-squared error incurred by the optimum quasi-linear estimate.
Such estimates yield a smaller mean-squared error when the parameter $$ has a nonzero mean. Let $$ be a scalar parameter with mean $m$ . The observables comprise a vector $r$ having components given by ${r}_{l}=+{n}_{l}$ , $l\in \{1, , N\}$ where ${n}_{l}$ are statistically independent Gaussian random variables [ $({n}_{l}, (0, {}_{n}^{2}()))$ ] independent of $$ . Compute expressions for $({}_{\mathrm{QLIN}})$ and $({}_{\mathrm{LIN}})$ . Verify that $({}_{\mathrm{QLIN}})$ yields a smaller mean-squared error when $m\neq 0$ .
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