Maximum Likelihood Estimation (MLE) for a Uniform Distribution

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A uniform distribution is a probability distribution in which every value between an interval from a to b is equally likely to be chosen.

The probability that we will obtain a value between x₁ and x₂ on an interval from a to b can be found using the formula:

P(obtain value between x₁ and x₂)  =  (x₂ – x₁) / (b – a)

This tutorial explains how to find the maximum likelihood estimate (mle) for parameters a and b of the uniform distribution.

Maximum Likelihood Estimation

Step 1: Write the likelihood function.

For a uniform distribution, the likelihood function can be written as:

Step 2: Write the log-likelihood function. 

Step 3: Find the values for a and b that maximize the log-likelihood by taking the derivative of the log-likelihood function with respect to a and b.

The derivative of the log-likelihood function with respect to a can be written as:

Similarly, the derivative of the log-likelihood function with respect to b can be written as:

Step 4: Identify the maximum likelihood estimators for a and b.

Notice that the derivative with respect to a is monotonically increasing. Thus, the mle for a would be the largest a possible, which would simply be:

min(X₁, X₂, … , X_n)

Also notice that the derivative with respect to b is monotonically decreasing. Thus, the mle for b would be the smallest b possible, which would be:

max(X₁, X₂, … , X_n)