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In this example , we derived the maximum likelihood estimate of the mean andvariance of a Gaussian random vector. You might wonder why we chose to estimate the variance 2 rather than the standard deviation . Using the same assumptions provided in the example, let's explore theconsequences of estimating a function of a parameter ( van Trees: Probs 2.4.9, 2.4.10 ).

Assuming that the mean is known, find the maximum likelihood estimates of first the variance, then thestandard deviation.

Are these estimates biased?

Describe how these two estimates are related. Assuming that f is a monotonic function, how are ML and f ( ) ML related in general? These results suggest a general question. Consider the problem of estimating somefunction of a parameter , say f 1 . The observed quantity is r and the conditional density p r r is known. Assume that is a nonrandom parameter.

What are the conditions for an efficient estimate f 1 to exist?

What is the lower bound on the variance of the error of any unbiased estimate of f 1 ?

Assume an efficient estimate of f 1 exists; when can an efficient estimate of some other function f 2 exist?

Let the observations r l consist of statistically independent, identically distributed Gaussian random variables having zero mean butunknown variance. We wish to estimate 2 , their variance.

Find the maximum likelihood estimate ML 2 and compute the resulting mean-squared error.

Show that this estimate is efficient.

Consider a new estimate NEW 2 given by NEW 2 ML 2 , where is a constant. Find the value of that minimizes the mean-squared error for NEW 2 . Show that the mean-squared error of NEW 2 is less than that of ML 2 . Is this result compatible with this previous part ?

Let the observations be of the form r H n where and n are statistically independent Gaussian random vectors. 0 K n 0 K n The vector has dimension M ; the vectors r and n have dimension N .

Derive the minimum mean-squared error estimate of , MMSE , from the relationship MMSE r

Show that this estimate and the optimum linear estimate LIN derived by the Orthogonality Principle are equal.

Find an expression for the mean-squared error when these estimates are used.

To illustrate the power of importance sampling, let's consider a somewhat nave example. Let r have a zero-mean Laplacian distribution; we want to employ importance sampling techniques to estimate r (despite the fact that we can calculate it easily). Let the density for r be Laplacian having mean .

Find the weight c l that must be applied to each decision based on the variable r .

Find the importance sampling gain. Show that this gain means that a fixed number of simulations are needed to achieve a given percentageestimation error (as defined by the coefficient of variation). Express this number as a function of thecriterion value for the coefficient of variation.

Now assume that the density for r is Laplacian, but with mean m . Optimize m by finding the value that maximizes the importance sampling gain.

Suppose we consider an estimate of the parameter having the form r C , where r denotes the vector of the observables and is a linear operator. The quantity C is a constant. This estimate is not a linear function of the observables unless C 0 . We are interested in finding applications for which it is advantageous to allow C 0 . Estimates of this form we term "quasi-linear" .

Show that the optimum (minimum mean-squared error) quasi-linear estimate satisfies r C r C 0 for all and C where QLIN r C .

Find a general expression for the mean-squared error incurred by the optimum quasi-linear estimate.

Such estimates yield a smaller mean-squared error when the parameter has a nonzero mean. Let be a scalar parameter with mean m . The observables comprise a vector r having components given by r l n l , l 1 N where n l are statistically independent Gaussian random variables [ n l 0 n 2 ] independent of . Compute expressions for QLIN and LIN . Verify that QLIN yields a smaller mean-squared error when m 0 .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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 .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
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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