3.10 Estimation theory: problems  (Page 2/3)

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?

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 ?

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.

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.

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 $m\neq 0$ .