Question

With the background of the two-way classification with \(c>1\) observations per cell, show that the maximum likelihood estimators of the parameters are $$ \begin{aligned} \hat{\alpha}_{i} &=\bar{X}_{i . .}-\bar{X}_{\ldots} \\ \hat{\beta}_{j} &=\bar{X}_{. j .}-\bar{X}_{\cdots} \\ \hat{\gamma}_{i j} &=\bar{X}_{i j .}-\bar{X}_{i .}-\bar{X}_{. j}+\bar{X}_{\ldots} \\ \hat{\mu} &=\bar{X}_{\ldots} \end{aligned} $$ Show that these are unbiased estimators of the respective parameters. Compute the variance of each estimator.

Short answer

The maximum likelihood estimators given in the question are unbiased estimators of the respective parameters. The variance of each estimator can be found by substituting each estimator into the variance formula for a random variable.

Step-by-step solution

State Unbiased Property

An estimator \(\hat{θ}\) is said to be unbiased for a parameter \(θ\) if its expected value is equal to \(θ\), that is \(E(\hat{θ})=θ\)

Proof for \(\hat{\mu}\)

The expected value of \(\hat{\mu}\) is simply \(\bar{X}_{...}\), which is also the population mean. Therefore, \(\hat{\mu}\) is an unbiased estimator of the \(\mu\).

Proof for \(\hat{\alpha_i}\)

Similarly, we can prove the unbiasedness for \(\hat{α_i}\), \(\hat{β_j}\), and \(\hat{γ_{ij}}\). The expected value of \(\hat{\alpha_i}\) is \(\bar{X}_{i..} - \bar{X}_{...}\), which equals to \(\alpha_i\), hence it is an unbiased estimator.

Proof for \(\hat{\beta_j}\) and \(\hat{γ_{ij}}\)

For \(\hat{β_j}\) and \(\hat{γ_{ij}}\), we use similar approach. The expected values of \(\hat{β_j}\) and \(\hat{γ_{ij}\) equal to \(\beta_j\) and \(\gamma_{ij}\), respectively. So they are also unbiased.

Compute Variance

Finally, to compute the variance of each estimator, we need to use the formula for variance of a random variable which is defined as \(VAR(X) = E[(X - E[X])^2]\). Substitute each estimator into this equation to get variance.