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

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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.

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