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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 . r r 0 0 p r r 2 -3 2 r 2 1 2 r 2 0 p

Find the maximum a posteriori estimate of the variance 2 from L statistically independent observations having the exponential density r r 0 p r r 1 2 r 2 where the variance is uniformly distributed over the interval 0 max 2 .

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 K r , where the normalized covariance matrix has trace tr 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 th estimate. Show that 1 l 2 1 2 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 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, l l r l L . Show that it minimizes the mean-square error l l r l m 2 .

Equating the sample average to m 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

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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