# 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  .

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
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?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
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.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
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
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
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?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
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
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
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