One-step estimator
= One-step estimator
A <one-step estimator> starts with an initial <estimator> $\widetilde\theta$ and applies one correction derived from an estimated <influence function>:
$$
\widehat\theta=\widetilde\theta+\frac1n\sum_{i=1}^n\widehat\psi(X_i).
$$
The correction aims to remove the leading discrepancy between the initial answer and the desired parameter. For the <mean>, use $\widehat\psi(x)=x-\widetilde\theta$. Then the corrected answer is exactly the <sample mean>, whatever initial value was chosen.
For a smooth <estimating equation> with empirical average $U_n(\theta)$ and nonsingular <Jacobian matrix> $A_n(\theta)$, one <Newton method> step is $\widetilde\theta-A_n(\widetilde\theta)^{-1}U_n(\widetilde\theta)$. Its correction has the same form with $\widehat\psi(x)=-A_n(\widetilde\theta)^{-1}\psi(x,\widetilde\theta)$. An <asymptotic linear representation> follows only when the initial error, <derivative> approximation and remaining terms are sufficiently controlled. The formula alone is not a general guarantee of improved accuracy.