# 3.10 Estimation theory: problems

Estimates for identical parameters are heavily dependent on the assumed underlying probability densities. To understand thissensitivity better, consider the following variety of problems, each of which asks for estimates of quantitiesrelated to variance. Determine the bias and consistency in each case.

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 $\left[0 , {}_{\mathrm{max}}^{2}\right)$ .

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 $\mathrm{tr}(\stackrel{}{K}_{r})=L$

Imagine yourself idly standing on the corner in a large city when you note the serial number of a passing beer truck.Because you are idle, you wish to estimate (guess may be more accurate here) how many beer trucks the city has fromthis single operation

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?

We make $L$ observations ${r}_{1},,{r}_{L}$ of a parameter  corrupted by additive noise ( ${r}_{l}=+{n}_{l}$ ). The parameter  is a Gaussian random variable [ $(, (0, {}_{}^{2}()))$ ] and ${n}_{l}$ are statistically independent Gaussian random variables [ $({n}_{l}, (0, {}_{n}^{2}()))$ ].

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 $\frac{1}{{}_{l}^{2}}=\frac{1}{{}_{}^{2}}+\frac{1}{{}_{n}^{2}}$

Although the maximum likelihood estimation procedure was not clearly defined until early in the 20th century, Gaussshowed in 1905 that the Gaussian density

It wasn't called the Gaussian density in 1805; this result is one of the reasons why it is.
was the sole density for which the maximum likelihood estimate of the mean equaledthe sample average. Let $\{{r}_{0}, , {r}_{L-1}\}$ be a sequence of statistically independent, identically distributed random variables.

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.

Because both equations equal 0, they can be equated. Use the fact that they must hold for all $L$ to derive the result. Gauss thus showed that mean-squared error and the Gaussian density were closely linked,presaging ideas from modern robust estimation theory.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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