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, … , Xn)
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, … , Xn)