



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 meansquared error for each estimate.
Consider an alternate procedure based on the same observations
${r}_{l}$ . Using the MMSE criterion, we estimate
$$ immediately after each observation. This procedure yieldsthe sequence of estimates
$({}_{1}({r}_{1}))$ ,
$({}_{2}({r}_{1}, {r}_{2}))$ ,,
$({}_{L}({r}_{1}, , {r}_{L}))$ . Express
$({}_{1})$ as a function of
$({}_{l1})$ ,
${}_{l1}^{2}$ , and
${r}_{l}$ . Here,
${}_{l}^{2}$ denotes the variance of the estimation error of the
${l}^{\mathrm{th}}$ estimate. Show that
$$\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
$\{{r}_{0}, , {r}_{L1}\}$ 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
$p(rm)$ ?
The sample average is, of course,
$\sum \frac{{r}_{l}}{L}$ .
Show that it minimizes the meansquare 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 meansquared
error and the Gaussian density were closely linked,presaging ideas from modern robust estimation theory.
Questions & Answers
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
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
Do somebody tell me a best nano engineering book for beginners?
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
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
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
what's the easiest and fastest way to the synthesize AgNP?
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is system testing?
AMJAD
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field .
1Electronicsmanufacturad IC ,RAM,MRAM,solar panel etc
2Helth and MedicalNanomedicine,Drug Dilivery for cancer treatment etc
3 Atomobile MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:
OpenStax, Statistical signal processing. OpenStax CNX. Dec 05, 2011 Download for free at http://cnx.org/content/col11382/1.1
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