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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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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