sqrtnthetahat

sqrtnthetahat is a notational construct found in statistical theory to describe a quantity that combines the sample size n with an estimator theta_hat of a population parameter theta. The most common interpretation is that sqrtnthetahat denotes the product sqrt(n) times the estimator, that is, sqrtnthetahat = sqrt(n) * theta_hat. In some contexts, authors may also encounter interpretations such as sqrt(n * theta_hat) when theta_hat estimates a nonnegative parameter and a square-root scaling is required; however, this usage is less standard and depends on the specific modeling situation.

In asymptotic analysis, scaling estimators by sqrt(n) is standard practice because many estimators are consistent and asymptotically normal with variance that decreases at rate 1/n. For example, if theta_hat is a consistent estimator of theta with asymptotic variance V/n, then the centered and scaled form sqrt(n)(theta_hat − theta) converges in distribution to a normal distribution with mean 0 and variance V. The quantity sqrtnthetahat = sqrt(n) * theta_hat is related to this framework through a decomposition: sqrt(n) * theta_hat = sqrt(n)(theta_hat − theta) + sqrt(n) * theta. If theta ≠ 0, the second term grows with sqrt(n), so sqrtnthetahat is not typically used for the limiting distribution of the centered estimator; instead, centered and standardized statistics are used.

Applications of the concept appear in large-sample theory, construction of confidence intervals, and hypothesis testing, where the behavior of scaled estimators under increasing n informs inference. The exact meaning of sqrtnthetahat can vary by text, so it is important to consult the specific notation defined in a given paper or book.

See also: asymptotic normality, central limit theorem, estimator, large-sample theory.