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.

Key concepts

Two-Way Classification

Two-way classification is a statistical method used to analyze data by categorizing observations into different groups based on two independent criteria. This method is particularly useful when we have experiments with factors that can be observed in multiple ways. For example, in an experiment where we are measuring yields from different types of fertilizers and across different weather conditions, the groups could be formed based on these two criteria: fertilizer type and weather condition. In the context of maximum likelihood estimators (MLEs), two-way classification helps to identify the effects that different levels of two factors have on a response variable. **Key Points about Two-Way Classification** - It involves arranging data into a matrix format, called a contingency table, where each "cell" in the table corresponds to unique combinations of the factor levels. - Each cell should ideally have the same number of observations for the method to work effectively, which is represented as 'c' in the formulation. - The MLEs allow us to estimate the contribution of each factor and their combination on the response variable. Using the two-way classification approach ensures that the analysis can differentiate between the effects due to individual factors and the combined interaction. This provides insight into how factors work together or independently.

Unbiased Estimators

An unbiased estimator is a statistical technique used to ensure that on average the estimate matches the actual parameter value. This is crucial when making predictions or inferences based on sample data.Unbiasedness guarantees that you are not systematically overestimating or underestimating the parameter.For the unbiasedness of the MLEs in our problem, each estimator's expected value equals the true parameter value. **Unbiasedness in Estimators**- **For \(\hat{\mu}\):** Since \(\hat{\mu}\) is the overall mean \(\bar{X}_{\ldots}\), its expected value is exactly the population mean \(\mu\). Thus, \(\hat{\mu}\) is unbiased.- **For \(\hat{\alpha}_{i}\):** This measures the effect of level 'i' of the first factor, adjusting for the overall mean. Its expected value is \(\alpha_i\), verifying its unbiasedness.- **For \(\hat{\beta}_{j}\):** Similarly, this adjusts for variation in the second factor, with an expected value matching \(\beta_j\).- **For \(\hat{\gamma}_{ij}\):** This captures the interaction between both factors. Again, it's unbiased as it aligns with \(\gamma_{ij}\).Overall, using unbiased estimators gives a precise and honest assessment of the parameters by aligning estimates closely with reality.

Variance Computation

Variance computation is an essential step in understanding the reliability or dispersion of an estimator's predictions around the actual parameter.The variance indicates the degree to which the estimator can vary from sample to sample, thus providing insight into its stability. Calculating the variance involves the formula for a random variable: \[ VAR(X) = E[(X - E[X])^2] \]Using this formula, we compute the variance for each MLE:- **Variance of \(\hat{\mu}\):** As it's based on a mean of all observations, its variance is influenced by the spread of the overall data.- **Variance of \(\hat{\alpha}_{i}\):** It depends on the variance among observations at level 'i' of factor one, adjusted for the total observations.- **Variance of \(\hat{\beta}_{j}\):** Similarly, this depends on variability at level 'j' of factor two.- **Variance of \(\hat{\gamma}_{ij}\):** The most complex, it covers the interaction effects, requiring the variance across both factor levels.Understanding variance helps in assessing how much the estimation might deviate in repeated experiments, highlighting the precision of the MLE in the given classification.