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The maximum likelihood estimation (MLE) for a uniform distribution is a statistical method used to determine the most likely values for the parameters of a uniform distribution, such as the minimum and maximum values. This process involves finding the values of the parameters that maximize the likelihood of the observed data being generated from a uniform distribution. This is typically done by calculating the probability density function for the uniform distribution and using it to create a likelihood function. The parameters are then estimated by finding the values that maximize this likelihood function, often through the use of optimization techniques such as gradient descent. The resulting estimates are considered the most likely values for the parameters of the uniform distribution, providing valuable insights into the underlying data and allowing for more accurate modeling and analysis.
Maximum Likelihood Estimation (MLE) for a Uniform Distribution
A 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.
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)