<< Chapter < Page Chapter >> Page >

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 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.
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
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Statistical signal processing. OpenStax CNX. Dec 05, 2011 Download for free at http://cnx.org/content/col11382/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Statistical signal processing' conversation and receive update notifications?