# 3.4 Maximum likelihood estimators of parameters

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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 $({}_{\mathrm{ML}})(r)$ 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, $({}_{\mathrm{ML}})(r)$ is defined by

$(, , \frac{\partial^{1}p(, r)}{\partial })=0$
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 $\frac{\partial^{1}\ln p(, r)}{\partial }=\frac{1}{{}_{n}^{2}}\sum_{l=0}^{L-1} r(l)-$ The solution of this equation is the maximum likelihood estimate, which equals the sample average. $({}_{\mathrm{ML}})=\frac{1}{L}\sum_{l=0}^{L-1} r(l)$ The expected value of this estimate $(, ({}_{\mathrm{ML}}))$ equals the actual value  , showing that the maximum likelihood estimate is unbiased. Themean-squared error equals $\frac{{}_{n}^{2}}{L}$ 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 $({}_{\mathrm{ML}})$ 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

$(, , \frac{d \ln p(, r)}{d }})=0$
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, $p(, r)=\prod_{l=0}^{L-1} \frac{1}{\sqrt{2\pi {}_{2}}}e^{-(\frac{1}{2{}_{2}}(r(l)-{}_{1})^{2})}$ 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.
$\frac{1}{{}_{2}}\sum_{l=0}^{L-1} r(l)-{}_{1}=0$ $-\left(\frac{L}{2{}_{2}}\right)+\frac{1}{2{}_{2}^{2}}\sum_{l=0}^{L-1} (r(l)-{}_{1})^{2}=0$ The solution of this set of equations is easily found to be $({}_{1}^{\mathrm{ML}})=\frac{1}{L}\sum_{l=0}^{L-1} r(l)$ $({}_{2}^{\mathrm{ML}})=\frac{1}{L}\sum_{l=0}^{L-1} (r(l)-({}_{1}^{\mathrm{ML}}))^{2}$

The expected value of $({}_{1}^{\mathrm{ML}})$ equals the actual value of ${}_{1}$ ; thus, this estimate is unbiased. However, the expected value of the estimate of the variance equals ${}_{2}\frac{L-1}{L}$ . 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 $({}_{2}^{\mathrm{ML}})$ by $L-1$ .

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
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