$\newcommand{\eps}{\varepsilon}\newcommand{\szdp}[1]{\!\left(#1\right)}$
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?
Best Answer
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.