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

Likelihood Function

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

Likelihood ratio test statistic

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.


  2. Víctor Gabriel Baldovino Medrano says:

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

    • Charles says:

      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).

  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.


  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!!!


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