Description: M-estimators are a broad class of estimators defined as the solution to an optimization problem, where the goal is to minimize or maximize an objective function. This function is typically related to the discrepancy between observed values and values predicted by a statistical model. M-estimators generalize maximum likelihood estimators and least squares estimators, and are used to estimate parameters in statistical models. Their flexibility allows them to adapt to different types of data and distributions, making them valuable tools in statistical inference. One of the most important characteristics of M-estimators is that they can be robust to the presence of outliers, depending on the loss function used in the optimization. This makes them especially useful in situations where data may not follow a normal distribution or may contain measurement errors. In summary, M-estimators are fundamental in modern statistics, providing a solid theoretical framework for parameter estimation in a variety of contexts.
History: M-estimators were introduced in the 1960s by statistician Peter J. Huber, who sought to develop more robust methods for parameter estimation in the presence of outliers. His work laid the groundwork for the development of robust techniques in statistics, expanding the traditional approach of maximum likelihood. Over the years, M-estimators have evolved and adapted to various areas of statistics, including robust regression and multivariate data analysis.
Uses: M-estimators are used in a variety of statistical applications, including robust regression, where the goal is to fit models to data that may contain outliers. They are also useful in multivariate data analysis and in parameter estimation in survival models. Their ability to handle non-normal data and robustness to outliers makes them preferred in situations where traditional methods may fail.
Examples: A practical example of M-estimators is the use of robust least squares estimation, which is applied in linear regression to minimize the impact of outliers on model fitting. Another example is parameter estimation in logistic regression models, where M-estimators are used to obtain more accurate estimates in the presence of biased data.
