Point Estimation and Properties of Estimators

Lesson 33 of STA301 introduces Statistical Estimation, the branch of inferential statistics concerned with estimating unknown population parameters from observed sample data. Estimation is categorized into Point Estimation, which produces a single numerical value as the best guess for the parameter, and Interval Estimation, which constructs a range of plausible values with an associated confidence level. The lecture clarifies the distinction between an Estimator, which is a mathematical formula, rule, or sample statistic (such as X-bar or s^2) used to estimate a parameter, and an Estimate, which is the specific numerical value obtained when sample data is substituted into the estimator formula. Because multiple competing estimators can be formulated for the same parameter, Sir Ronald A. Fisher established four desirable mathematical criteria to evaluate an ideal point estimator: 1. Unbiasedness: An estimator theta-hat is an unbiased estimator of parameter theta if its expected value equals the parameter: E(theta-hat) = theta. If E(theta-hat) != theta, the estimator is biased, with Bias = E(theta-hat) - theta. Sample mean X-bar is an unbiased estimator of population mean mu (E(X-bar) = mu). However, sample variance with divisor n is downwardly biased (E[sum(X - X-bar)^2 / n] = [(n-1)/n]*sigma^2), which is why Bessel's correction with divisor n - 1 is applied to produce the unbiased sample variance s^2. 2. Consistency: An estimator is consistent if it converges in probability to the true parameter as sample size approaches infinity: plim theta-hat = theta as n -> inf. A sufficient condition for consistency is that the estimator is asymptotically unbiased and its variance approaches zero as n -> inf. 3. Efficiency: Among competing unbiased estimators, the most efficient estimator is the one possessing the minimum variance. The Minimum Variance Unbiased Estimator (MVUE) provides the most concentrated estimates around the parameter. 4. Sufficiency: An estimator is sufficient if it utilizes all the information about the parameter contained in the sample data. In Virtual University final term examinations, proving unbiasedness of X-bar and s^2 and defining Fisher's four criteria are prominent theoretical questions. Consulting sta301 handouts and solved papers ensures thorough preparation.