Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a statistical model. The main aim is to find the parameter values that maximize the likelihood of making the observations given the model. MLE operates on the principle that the observed data is most probable under the true set of parameters. In our exercise, MLE is employed to estimate \(\theta\) in a location model where \(X_i = \theta + W_i\) with \(W_i\) following a specified distribution. Here's how it works:
- The likelihood function is composed using observed data points.
- This function is subsequently maximized concerning the model parameters.
- Tools such as calculus often assist in finding where this maximum occurs.
For symmetric distributions like the logistic, it often turns out that the maximum likelihood estimator for location parameters (like \(\theta\)) is the sample median of the observed data. This is because, in symmetric settings, the median balances the data to minimize the sum of absolute deviations from the central value, thus providing the most likely estimate. Understanding MLE's robust and consistent nature helps ensure the parameter estimates are reliable and valid across different contexts.