Estimation Under Communication Constraints

Abstract

We analyze the following model: One person, called “helper” observes an outcome$x^n=(x_1,\dots ,x_n)\in {\mathcal X}^n$ of the sequence $X^n=(X_1,\dots ,X_n)$ of i.i.d. RV’s and the statistician gets a sample$y^n=(y_1,\dots ,y_n)$ of the sequence $Y^n(\theta ,x^n)$ of RV’s with a density $\prod _{t=1}^n f(y_t|\theta , x_t)$. The helper can give some (side) information about $x^n$ to the statistician via an encoding function $s^n:{\mathcal X}^n\rightarrow \mathbb {N}$ with $\text {rate} (s_n)\triangleq (1/n)\log \# \text {range}(s_n)\le R$. Based on the knowledge of $s_n(x^n)$ and $y^n$ the statistician tries to estimate $\theta$ by an estimator $\hat{\theta }_n$.