hattheta

Hat theta, or θ̂, is the standard notation in statistics for an estimator of a parameter θ. It refers to a function of the observed data that is used to approximate the true, but unknown, value of θ. Because θ̂ depends on the particular sample drawn, it is a random variable and can vary across samples.

Key concepts surrounding θ̂ include bias, variance, and accuracy. The bias of an estimator is the difference between its expected value and the true parameter, Bias(θ̂) = E[θ̂] − θ. An estimator is unbiased if this quantity is zero. The precision of θ̂ is summarized by its variance. A common measure of overall accuracy is the mean squared error, MSE(θ̂) = E[(θ̂ − θ)²] = Var(θ̂) + Bias(θ̂)². Consistency is another important property: θ̂ is consistent if it converges in probability to θ as the sample size grows. Asymptotic normality often holds, meaning, after appropriate scaling, θ̂ has an approximately normal distribution for large samples. Efficiency concerns comparison among estimators; among unbiased estimators, an efficient one has the smallest variance, and the Cramér-Rao bound provides a lower limit for variance in many regular models.

Common examples illustrate θ̂ in practice. The sample mean X̄ is an estimator for the population mean μ. The sample proportion p̂ estimates a population proportion p. In maximum likelihood estimation, θ̂ is the parameter value that maximizes the likelihood given the data. In linear regression, θ̂ typically denotes the ordinary least squares estimates of the coefficient vector β.

In practice, the method of obtaining θ̂ (analytical solutions, numerical optimization, or resampling) depends on the model, and standard errors are used to quantify uncertainty around θ̂.