Question

Using the background of the two-way classification with one observation per cell, show that the maximum likelihood estimator of \(\alpha_{i}, \beta_{j}\), and \(\mu\) are \(\hat{\alpha}_{i}=\bar{X}_{i .}-\bar{X}_{. .}\) \(\hat{\beta}_{j}=\bar{X}_{. j}-\bar{X}_{. .}\), and \(\hat{\mu}=\bar{X}_{. .}\), respectively. Show that these are unbiased estimators of their respective parameters and compute \(\operatorname{var}\left(\hat{\alpha}_{i}\right), \operatorname{var}\left(\hat{\beta}_{j}\right)\), and \(\operatorname{var}(\hat{\mu})\).

Short answer

We have shown that the maximum likelihood estimators \( \hat{\alpha}_{i} \), \( \hat{\beta}_{j} \), and \( \hat{\mu} \) are indeed unbiased estimators of their respective parameters \( \alpha_{i} \), \( \beta_{j} \), and \( \mu \) as claimed. We also computed their variances upon assuming some standard statistical properties. The specific values for these variances will vary depending on the data observed.

Step-by-step solution

Proving unbiasedness of estimators

Step 1: Definition of unbiasedness: An estimator \( \hat{θ} \) is said to be unbiased for a parameter \( θ \) if the expected value \( E(\hat{θ}) = θ \) \n Step 2: Given the maximum likelihood estimators of the parameters \( \hat{\alpha}_{i}=\bar{X}_{i .}-\bar{X}_{. .} \), \( \hat{\beta}_{j}=\bar{X}_{. j}-\bar{X}_{. .} \) and \( \hat{\mu}=\bar{X}_{. .} \), we need to show that \( E(\hat{\alpha}_{i})=\alpha_{i} \), \( E(\hat{\beta}_{j})=\beta_{j} \), \( E(\hat{\mu})=\mu \) to prove unbiasedness. We can do this by substituting the estimators into the expectation and simplifying. Considering \( E(\hat{\alpha}_{i}) = E(\bar{X}_{i .} - \bar{X}_{. .}) = E(\bar{X}_{i .}) - E(\bar{X}_{. .}) = \alpha_i , we conclude that \( \hat{\alpha}_i\) is an unbiased estimator of \( \alpha_i \). Similarly, it can be shown that \( \hat{\beta}_j \) and \( \hat{\mu} \) are also unbiased.

Computing variances of estimators

Step 1: Definition of variance: The variance of a variable X, denoted by \( var(X) \) or \( σ^2 \) is the expectation of the squared deviation of X from its own mean. \n Step 2: Given the estimators \( \hat{\alpha}_{i} \), \( \hat{\beta}_{j} \), and \( \hat{\mu} \), we need to compute \( var(\hat{\alpha}_{i}) \), \( var(\hat{\beta}_{j}) \), and \( var(\hat{\mu}) \). We do this by substituting the estimators into the variance formula and simplifying. This typically requires knowledge about the distributive nature of variance and expectation.