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When the a priori density of a parameter is not known or the parameter itself is inconveniently described asa random variable, techniques must be developed that make no presumption about the relative possibilities of parametervalues. Lacking this knowledge, we can expect the error characteristics of the resulting estimates to be worse thanthose which can use it.

The maximum likelihood estimate ML r of a nonrandom parameter is, simply, that value which maximizes the likelihood function (the a priori density of the observations). Assuming that the maximum can be found by evaluating a derivative, ML r is defined by

ML p r 0
The logarithm of the likelihood function may also be used in this maximization.

Let r l be a sequence of independent, identically distributed Gaussian random variables having an unknown mean but a known variance n 2 . Often, we cannot assign a probability density to a parameter of a random variable's density; we simply do not knowwhat the parameter's value is. Maximum likelihood estimates are often used in such problems. In the specific case here, thederivative of the logarithm of the likelihood function equals p r 1 n 2 l 0 L 1 r l The solution of this equation is the maximum likelihood estimate, which equals the sample average. ML 1 L l 0 L 1 r l The expected value of this estimate ML equals the actual value , showing that the maximum likelihood estimate is unbiased. Themean-squared error equals n 2 L and we infer that this estimate is consistent.

Parameter vectors

The maximum likelihood procedure (as well as the others being discussed) can be easily generalized to situations where morethan one parameter must be estimated. Letting denote the parameter vector, the likelihood function is now expressed as p r . The maximum likelihood estimate ML of the parameter vector is given by the location of the maximum of the likelihood function (or equivalently of itslogarithm). Using derivatives, the calculation of the maximum likelihood estimate becomes

ML p r 0
where denotes the gradient with respect to the parameter vector. This equation means that we must estimate all of theparameter simultaneously by setting the partial of the likelihood function with respect to each parameter to zero. Given P parameters, we must solve in most cases a set of P nonlinear, simultaneous equations to find the maximum likelihoodestimates.

Let's extend the previous example to the situation where neither the mean nor the variance of a sequence of independentGaussian random variables is known. The likelihood function is, in this case, p r l 0 L 1 1 2 2 1 2 2 r l 1 2 Evaluating the partial derivatives of the logarithm of this quantity, we find the following set of two equations to solvefor 1 , representing the mean, and 2 , representing the variance.

The variance rather than the standard deviation is represented by 2 . The mathematics is messier and the estimator has less attractive properties in the latter case. This problem illustrates this point.
1 2 l 0 L 1 r l 1 0 L 2 2 1 2 2 2 l 0 L 1 r l 1 2 0 The solution of this set of equations is easily found to be 1 ML 1 L l 0 L 1 r l 2 ML 1 L l 0 L 1 r l 1 ML 2

The expected value of 1 ML equals the actual value of 1 ; thus, this estimate is unbiased. However, the expected value of the estimate of the variance equals 2 L 1 L . The estimate of the variance is biased, but asymptotically unbiased. This bias can be removed by replacingthe normalization of L in the averaging computation for 2 ML by L 1 .

Questions & Answers

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
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
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 .
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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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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