# Unbiased Estimator for Mean Response to Treatment

$$\newcommand{\eps}{\varepsilon}\newcommand{\szdp}[1]{\!\left(#1\right)}$$ Problem Statement: Consider the following model for the responses measured in a randomized block design containing $$b$$ blocks and $$k$$ treatments: $$Y_{ij}=\mu+\tau_i+\beta_j+\eps_{ij}$$ \begin{align*} Y_{ij}&=\text{response to treatment i in block j}\\ \mu&=\text{overall mean}\\ \tau_i&=\text{nonrandom effect of treatment i, where \displaystyle\sum_{i=1}^k\tau_i=0}\\ \beta_j&=\text{random effect of block j, where \beta_j are independent, normally distributed random variables}\\ &\phantom{=}\text{with E(\beta_j)=0 and V(\beta_j)=\sigma_\beta^2, for j=1,2,\dots,b.}\\ \eps_{ij}&=\text{random error terms where \eps_{ij} are independent, normally distributed random variables}\\ &\phantom{=}\text{with E(\eps_{ij})=0 and V(\eps_{ij})=\sigma_\eps^2, for i=1,2,\dots,k and j=1,2,\dots,b.} \end{align*} Assume that the $$\beta_j$$ and $$\eps_{ij}$$ are independent, and that $$\mu$$ and $$\tau_i$$ are fixed but unknown constants, while the $$\beta_j$$ and $$\eps_{ij}$$ are random variables. Let $$\overline{Y}_{i\bullet}$$ denote the average of all of the responses to treatment $$i.$$ Is $$\overline{Y}_{i\bullet}$$ an unbiased estimator for the mean response to treatment $$i?$$

Note: This is essentially Exercise 13.80b in Mathematical Statistics with Applications, 5th. Ed., by Wackerly, Mendenhall, and Scheaffer, and is in the context of ANOVA.

My Work So Far: From the model equations, we have \begin{align*} Y_{ij} &=\mu+\tau_i+\beta_j+\eps_{ij}\\ \overline{Y}_{i\bullet} &=\frac1b\sum_{j=1}^bY_{ij}\\ &=\mu+\tau_i+\overline\beta+\overline\eps_{i\bullet}\\ E\szdp{\overline{Y}_{i\bullet}} &=\mu+\tau_i+E\szdp{\overline\beta}+E\szdp{\overline\eps_{i\bullet}}\\ &=\mu+\tau_i. \end{align*}

My Question: To show that $$\overline{Y}_{i\bullet}$$ is an unbiased estimator for the mean response to treatment $$i,$$ I must show that its expected value is equal to the parameter value. But I don't know what the parameter value is. It feels like this is a simple matter of interpretation: what is the parameter corresponding to the mean response to treatment $$i?$$ Why?

Indeed, $$\bar{Y}_{i.}$$ is an unbiased estimator for the mean response to treatment $$i$$:
$$E\left[ \bar{Y}_{i.} \right]=\frac{1}{b}E\left[ \sum_{j=1}^b{{Y}_{ij}} \right]=\frac{1}{b} \sum_{j=1}^b{E\left[{Y}_{ij} \right]}=\frac{b}{b}E[Y_{ij}]=E[\mu]+E[\tau_i]+E[\beta_j]+E[\epsilon_{ij}]\\=\mu+\tau_i+0+0=\mu+\tau_i$$
That's the mean response to treatment $$i$$ according to the setting of $$\mu$$ and $$\tau_i$$, therefore $$\bar{Y}_{i.}$$ is an unbiased estimator.