Estymatory

Estymatory, in Polish statistical literature, refer to the estimators used to infer unknown parameters from data. In English-language texts the term estimators is standard. An estymator is typically a function of a sample X1, …, Xn and is thus a random variable whose realized value serves as the estimate of a parameter θ.

Key concepts describe the quality of estymators. Bias measures whether the expected value of the estimator equals the true parameter; an unbiased estimator satisfies E[T(X1, …, Xn)] = θ. Variance quantifies dispersion of the estimator across samples. Consistency means Tn converges in probability to θ as the sample size n grows. Efficiency concerns achieving the smallest possible variance among unbiased estimators, and regularity conditions often yield asymptotic normality and, via the Cramér–Rao bound, asymptotic efficiency.

Common classes include point estimators such as those obtained by maximum likelihood (MLE) or the method of moments. The sample mean is the classic estimator of the population mean, while the sample variance (with denominator n−1) estimates the population variance. Bayesian estimators, such as posterior means or medians, are also used as point estimates within a Bayesian framework.

Estymators are central to statistical inference, informing confidence intervals, hypothesis tests, and model comparison. Their properties depend on the underlying model and sampling scheme, and choosing an appropriate estimator involves trade-offs among bias, variance, and robustness. See also Estimator; Bias; Consistency; Efficiency; Maximum likelihood estimation.