<< Chapter < Page Chapter >> Page >
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 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 ( l x ) 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 c p 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 is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
what is the coefficient of -4×
Mehri Reply
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
I'm 13 and I understand it great
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
I rally confuse this number And equations too I need exactly help
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
greetings from Iran
salut. from Algeria
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. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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?