5 - Point, Interval, and Ratio Estimators

Summary

The choice of an estimator depends on criteria for "best estimator". The key estimator properties of consistency, unbiasedness, and robustness are defined. The mean squared error is introduced and used to characterize different unbiased estimators. The Cramér-Rao lower bound on the variance of an estimator as the inverse sample Fisher information is derived. The concept of a sufficient statistic is introduced, and its properties are defined, leading to the minimal and complete forms. Statistical decision theory is outlined, leading to risk functions and admissible estimators.The method of moments that has wide applicability but limted optimality is introduced. The maximum likelihood estimator is derived, and is shown to be asymptotically consitent, asymptotically efficient and asymptotically Gaussian, so its covariance is given by the inverse Fisher information. The confidence interval is defined for discrete and continuous random variables, and the Bonferroni method to control the tail probability when multiple parameters are estimated is elucidated. Estimators for the ratio of two random variables are examined.