3.10 Estimation theory: problems

Compute the maximum a posteriori and maximum likelihood estimates of $$ based on $L$ statistically independent observations of a Maxwellian random variable $r$ . $$\forall r, , (r> 0)\land (> 0)\colon p(r, , r)=\sqrt{\frac{2}{}}^{-3/2}r^{2}e^{-(\frac{1}{2}\frac{r^2}{})}$$ $$\forall , > 0\colon p(, )=e^{-()}$$

Find the maximum a posteriori estimate of the variance $^2$ from $L$ statistically independent observations having the exponential density $$\forall r, r> 0\colon p(r, r)=\frac{1}{\sqrt{^2}}e^{-\left(\frac{r}{\sqrt{^2}}\right)}$$ where the variance is uniformly distributed over the interval $\left[0 , {}_{\max }^2\right)$ .

Find the maximum likelihood estimate of the variance of $L$ identically distributed, but dependent Gaussian random variables. Here, the covariance matrix is written $K_r=^2\stackrel{}{K}_r$ , where the normalized covariance matrix has trace $\mathrm{tr}(\stackrel{}{K}_r)=L$

Making appropriate assumptions, the beer truck's number is drawn from a uniform probability density ranging betweenzero and some unknown upper limit, find the maximum likelihood estimate of the upper limit.

Show that this estimate is biased.

In one of your extraordinarily idle moments, you observe throughout the city $L$ beer trucks. Assuming them to be independent observations, now what is the maximum likelihood estimateof the total?

Is this estimate of $$ biased? asymptotically biased? consistent?

Find the MMSE estimate of $$ .

Find the maximum a posteriori estimate of $$ .

Compute the resulting mean-squared error for each estimate.

Consider an alternate procedure based on the same observations $r_l$ . Using the MMSE criterion, we estimate $$ immediately after each observation. This procedure yieldsthe sequence of estimates $({}_1(r_1))$ , $({}_2(r_1, r_2))$ ,, $({}_L(r_1, , r_L))$ . Express $({}_1)$ as a function of $({}_{l-1})$ , ${}_{l-1}^2$ , and $r_l$ . Here, ${}_l^2$ denotes the variance of the estimation error of the $l^{\mathrm{th}}$ estimate. Show that $$\frac{1}{{}_l^2}=\frac{1}{{}_{}^2}+\frac{1}{{}_n^2}$$

What equation defines the maximum likelihood estimate $(m_{\mathrm{ML}})$ of the mean $m$ when the common probability density function of the data has the form $p(r-m)$ ?

The sample average is, of course, $\sum \frac{r_l}{L}$ . Show that it minimizes the mean-square error $\sum (r_l-m)^2$ .

Equating the sample average to $(m_{\mathrm{ML}})$ , combine this equation with the maximum likelihood equation to show that the Gaussian densityuniquely satisfies the equations.