A note on bias reduction

Let $\widehat{w}$ be an unbiased estimate of an unknown $w\in R$. Given a function $t(w)$, we show how to choose a function $f_n(w)$ such that for $w^*=\widehat{w} + f_n(w)$, $E\ t\left(w^*\right) =t(w)$. We illustrate this with $t(w)=w^a$ for a given constant $a$. For $a=2$ and $\widehat{w}$ normal, this leads to the convolution equation $c_r=c_r\otimes c_r$

A note on bias reduction

Let $\widehat{w}$ be an unbiased estimate of an unknown $w\in R$. Given a function $t(w)$, we show how to choose a function $f_n(w)$ such that for $w^*=\widehat{w} + f_n(w)$, $E\ t\left(w^*\right) =t(w)$. We illustrate this with $t(w)=w^a$ for a given constant $a$. For $a=2$ and $\widehat{w}$ normal, this leads to the convolution equation $c_r=c_r\otimes c_r$