Deltahat

Deltahat is a common notation in statistics and related disciplines for an estimator of a parameter denoted by Delta (Δ). The hat indicates an estimate derived from sample data, as opposed to the unknown true value Δ.

In practice, deltahat is used whenever the quantity of interest is a specific effect or difference that researchers aim to quantify from data. A widely used case is estimating a treatment effect. If Δ represents the true difference in population means between two groups (μ1 − μ0), a typical estimator is deltahat = X̄1 − X̄0, the difference between sample means. The standard error of this estimator is used to form confidence intervals and to conduct hypothesis tests, for example se(Δ̂) = sqrt(s1^2/n1 + s0^2/n0) under certain assumptions.

Another common context is regression analysis, where deltahat is the coefficient on a treatment indicator in a model such as Y = α + Δ T + Xβ + ε. Here Δ̂ estimates the conditional average treatment effect, adjusting for covariates, under standard regression assumptions.

Deltahat also appears in meta-analysis as study-specific estimated effects that are subsequently combined, and in causal inference as estimators of various estimands like the average treatment effect (ATE) or the average treatment effect on the treated (ATT).

Properties of deltahat depend on the estimator and data design. They include bias, variance, consistency, and asymptotic normality under regularity conditions. The interpretation of deltahat relies on the study design (randomized vs observational) and on model assumptions; mis-specification or unaddressed confounding can distort the estimated quantity.