# Maximum Likelihood Function

Definition 1: Suppose a random variable x has a frequency function f(x; θ) that depends on parameters θ = {θ1, θ2, …, θk}. For a sample {x1, x2, …, xn} the likelihood function is defined by

Here we treat x1, x2, …, xn as fixed. The maximum likelihood estimator of θ is the value of θ that maximizes L(θ). We can then view the maximum likelihood estimator of θ as a function of the sample x1, x2, …, xnWe will commonly represent the maximum likelihood estimator of θ as θ-hat, written

Definition 2: Let θ-hat be the maximum likelihood estimator of θ for the sample {x1, x2, …, xn}. Suppose we want to test a null hypothesis regarding the parameters in θ and let θ’-hat be the maximum likelihood estimator of θ for the sample {x1, x2, …, xn} when the null hypothesis is true. Now define

Observation: Here λ can be viewed as a function of x1, x2, …, xn.. Clearly

and so 0 ≤ λ ≤ 1. For any given x1, x2, …, xn., values of λ is near 1 correspond to the null hypothesis being true. The closer that  is to 1 the more reasonable it is to accept the null hypothesis, while the farther away from 1 the more reasonable it is to reject the null hypothesis. Thus λ can be used as a statistic for testing the validity of the null hypothesis.

Suppose g(λ) is the frequency function for λ where g(λ) doesn’t depend on any unknown parameters, then for any significance level α, there is a value λ0 such that the cumulative distribution function F at λ0 takes the value α. For those familiar with calculus, this means that

λ is the likelihood ratio test statistic for the hypothesis H0. We reject the null hypothesis if and only if the value of λ ≤ λ0 where λ is the value of λ for the sample {x1, x2, …, xn} and λis the value such that F(λ0) = α.

### 10 Responses to Maximum Likelihood Function

1. Denis Patry says:

All my appreciation.
_Excel has more to offer then i tough.
_College Stat-Math concept were far gone and now i need those for employment.

Thank you for presenting quite a body of knowledge and so reachable.

Denis

2. Víctor Gabriel Baldovino Medrano says:

Question:
What is the reason for having a #value answer for the Max Likelihood part of the Chi-test?

• Charles says:

Victor,
It is an error of some sort, but I would have to see the data to give you a more specific answer. If you send me an Excel file with your data, I could answer more specifically (my email address is available via Contact Us).
Charles

3. Daniel Jacob says:

good evening, i have been following your work lately especially on the above subject matter.
Please i need your help as i need to calculate the maximum livelihood estimate for forest officers in my state.
Please let me if you will be chance so i can send the data to you to help me with the analysis.

Thanks

• Charles says:

Daniel,
You can sends an Excel file with your data and analysis to me. Please clearly state what you are trying to accomplish. You can find my email address at Contact Us.
Charles

4. Nejood says:

Hi, pleas i need your help.
how i can drive the equation of Likelihood maximization.
and log likelihood maximization with respect the to the parameters THETA

5. Joe says:

Thanks for explaining the function. I have another question: What ist alpha? I don’t see an explanation of alpha anywhere. Could you please, please help me with that? Thank you in advance!!!

Joe

• Charles says:

Joe,
Alpha is the significance level of the test, generally set to .05.
Charles

• Joe says:

Oh, I see! Thank you! Sorry… I forgot.