Point Estimate and Interval Estimation (Maximum Likelihood Method)

Introduction

In statistical inference, estimation is a fundamental process aimed at quantifying uncertainty around a population parameter based on sample data. A point estimate is a single numerical value that serves as the best approximation of the unknown population parameter. This estimate is derived from the data and is often obtained using the maximum likelihood method, a powerful approach for parameter estimation. The maximum likelihood method involves finding the parameter values that maximize the likelihood function, which measures the probability of observing the given sample under the assumed model.

Interval estimation extends point estimation by providing a range of values (an interval) that is likely to contain the true parameter with a certain level of confidence. This interval is constructed using the properties of the maximum likelihood estimator (MLE) and its distribution. The MLE is a pivotal concept in statistical inference, as it ensures consistency, efficiency, and asymptotic normality under regularity conditions. This article explores the theoretical foundations of point estimation and interval estimation, focusing on the maximum likelihood method and its applications.

Point Estimation

Point estimation involves determining a single value (a statistic) that best approximates the unknown population parameter. The most common method for point estimation is the maximum likelihood estimation (MLE), which seeks to find the parameter values that maximize the likelihood function. The likelihood function, $ L(\theta) $, is defined as the probability of observing the given sample under the assumed model for the parameter $ \theta $. For a given sample, the MLE is the value of $ \theta $ that maximizes $ L(\theta) $.

The MLE is particularly useful in scenarios where the underlying distribution is unknown, and the sample size is finite. For example, in the case of the normal distribution, the MLE for the mean $ \mu $ is the sample mean $ \bar{x} $, and the MLE for the variance $ \sigma^2 $ is the sample variance $ s^2 $. These estimates are consistent, meaning they converge to the true parameter as the sample size increases. Additionally, the MLE is efficient, achieving the lowest possible variance among all unbiased estimators, which is a key property for optimal statistical inference.

The derivation of the MLE relies on calculus and optimization techniques. For a given parameter $ \theta $, the likelihood function is maximized by solving the equation $ \frac{\partial \ln L(\theta)}{\partial \theta} = 0 $, where $ \ln L(\theta) $ is the natural logarithm of the likelihood function. This approach ensures that the estimated parameter values are the most probable under the assumed model.

Interval Estimation with Maximum Likelihood

Interval estimation provides a range of values for the population parameter, along with a confidence level that quantifies the probability of the interval containing the true parameter. The confidence interval is constructed using the MLE and its distribution, which is typically approximated by the central limit theorem for large sample sizes. The interval is defined as $ \hat{\theta} \pm z_{\alpha/2} \cdot \text{SE}(\hat{\theta}) $, where $ \hat{\theta} $ is the MLE, $ z_{\alpha/