Eestimator

Eestimator is a term used in statistics to refer to estimators that target the expectation of a random variable under a specified model. While the simplest and most common E-estimator is the sample mean, the idea encompasses a broader class defined by estimating equations or objective functions that involve the expectation operator E.

Construction and scope

An E-estimator is typically constructed to satisfy equations or optimize criteria that involve expected values. The most familiar example is the sample mean, an unbiased and consistent estimator of the population mean E[X]. More generally, estimators can be formed through weighted means, shrinkage methods, or solving estimating equations of the form E[ψ(X, θ)]=0, where θ is the parameter of interest. This framework includes M-estimators and, in a Bayesian setting, posterior expectations as a form of E-estimation.

Properties

Common desiderata for E-estimators include unbiasedness (when appropriate), consistency (convergence to the true parameter as sample size grows), and efficiency (minimizing variance among unbiased or admissible estimators). Robust variants seek insensitivity to outliers, while Bayesian approaches emphasize incorporating prior information.

Applications

E-estimators are used to estimate population means and other parameters expressible as expectations, such as risk measures, average treatment effects, or long-run averages in econometrics and quality control. They provide a unifying viewpoint for methods that rely on estimating expected values rather than direct likelihood-based inference.

Limitations

The performance of E-estimators depends on model assumptions and data quality. Estimators based on incorrect models or sensitive to outliers may be biased or inefficient, highlighting the need for diagnostic checks and robust alternatives.

See also

Estimator, M-estimator, method of moments, maximum likelihood estimator, Bayesian estimator.