# Maximum likelihood estimation

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This module introduces the maximum likelihood estimator. We show how the MLE implements the likelihood principle. Methods for computing th MLE are covered. Properties of the MLE are discussed including asymptotic efficiency and invariance under reparameterization.

The maximum likelihood estimator (MLE) is an alternative to the minimum variance unbiased estimator (MVUE).For many estimation problems, the MVUE does not exist. Moreover, when it does exist, there is no systematic procedure forfinding it. In constrast, the MLE does not necessarily satisfy any optimality criterion, but it can almost always be computed,either through exact formulas or numerical techniques. For this reason, the MLE is one of the most common estimation procedures used in practice.

The MLE is an important type of estimator for the following reasons:

• The MLE implements the likelihood principle.
• MLEs are often simple and easy to compute.
• MLEs have asymptotic optimality properties (consistency and efficiency).
• MLEs are invariant under reparameterization.
• If an efficient estimator exists, it is the MLE.
• In signal detection with unknown parameters (composite hypothesis testing), MLEs are used in implementing thegeneralized likelihood ratio test (GLRT).
This module will discuss these properties in detail, with examples.

## The likelihood principle

Supposed the data $X$ is distributed according to the density or mass function $p(, x)$ . The likelihood function for  is defined by $l(x, )\equiv p(, x)$ At first glance, the likelihood function is nothing new - it is simply a way of rewriting the pdf/pmf of $X$ . The difference between the likelihood and the pdf or pmf is what is held fixed and whatis allowed to vary. When we talk about the likelihood, we view the observation $x$ as being fixed, and the parameter  as freely varying.

It is tempting to view the likelihood function as a probability density for  , and to think of $l(x, )$ as the conditional density of  given $x$ . This approach to parameter estimation is called fiducial inference , and is not accepted by most statisticians.One potential problem, for example, is that in many cases $l(x, )$ is not integrable ( $\int l(x, )\,d \to$ ) and thus cannot be normalized. A more fundamental problem is that  is viewed as a fixed quantity, as opposed to random. Thus, it doesn't make senseto talk about its density. For the likelihood to be properly thought of as a density, a Bayesian approach is required.
The likelihood principle effectively states that all information we haveabout the unknown parameter  is contained in the likelihood function.

## Likelihood principle

The information brought by an observation $x$ about  is entirely contained in the likelihood function $p(, x)$ . Moreover, if ${x}_{1}$ and ${x}_{2}$ are two observations depending on the same parameter  , such that there exists a constant $c$ satisfying $p(, {x}_{1})=cp(, {x}_{2})$ for every  , then they bring the same information about  and must lead to identical estimators.

In the statement of the likelihood principle, it is not assumed that the two observations ${x}_{1}$ and ${x}_{2}$ are generated according to the same model, as long as the model is parameterized by  .

#### Questions & Answers

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
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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