Thetahat

Thetahat, written as θ̂, denotes an estimator of a parameter θ in a statistical model. The hat signals that the value is inferred from observed data rather than known with certainty. An estimator is a function of the sample data that aims to approximate the true parameter.

Estimators have properties used to evaluate their quality. Bias measures systematic error (E[θ̂] = θ indicates unbiasedness). Variance reflects precision. Consistency means θ̂ converges to θ as the sample size grows. The Cramér-Rao bound provides a lower limit on the variance of unbiased estimators under regularity conditions, defining a benchmark for efficiency.

Common examples illustrate the idea. The sample mean x̄ is the estimator for the population mean μ in independent and identically distributed data. The sample proportion p̂ estimates the population proportion p in Bernoulli trials. In regression, β̂ denotes estimated coefficients. Estimators can be obtained by different methods, notably maximum likelihood estimation (MLE), which selects θ̂ to maximize the likelihood function; and the method of moments. In Bayesian analysis, θ̂ may be taken as the posterior mean or mode, depending on the loss function.

Inference built on θ̂ includes confidence intervals and hypothesis tests. With large samples, θ̂ often has an approximately normal distribution (asymptotic normality), enabling standard error estimation and interval construction. Notation varies, but θ̂ is the standard symbol for an estimated parameter; subscripts or alternative forms (such as θ̂_MLE or θ̂_Bayes) indicate the estimation method.

Limitations of θ̂ arise from bias, model misspecification, and finite-sample issues, making robustness and efficiency considerations central to the choice of estimator.