# 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.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
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