91Ó°ÊÓ

• Probability is concerned with forecasting the possibility of future events, whereas statistics is concerned with analysing the frequency of previous events.
• Probability is largely a theoretical discipline of mathematics that investigates the implications of mathematical notions.

We need to determine an unbiased estimator for \({\sigma ^2}\)in a two-way layout that contains \(K\)observations in each cell.

\(K \ge 2\)

The maximum likelihood estimator of \({\sigma ^2}\)is given in the textbook as:

\({\hat \sigma ^2} = \frac{1}{{IJK}}\sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {\sum\limits_{k = 1}^K {{{\left( {{Y_{ijk}} - {{\bar Y}_{ij + }}} \right)}^2}} } } \)

Let us first determine the expected value of the estimator \({\hat \sigma ^2}\)using that \({Y_{ijk}}\)has a normal distribution with mean \({\theta _{ij}}\)and variance \({\sigma ^2}\)and \({\bar Y_{ij + }}\)has a normal distribution with mean \({\theta _{ij}}\)and variance\(\frac{{{\sigma ^2}}}{K}\).

\(\begin{array}{*{20}{c}}{}\\{}\end{array}\)

\(\)

Combine like terms

An unbiased estimator \(x\)of \(\theta \)requires\(E(x) = \theta \):

\(\begin{array}{c}E\left( {\frac{K}{{K - 1}}{{\hat \sigma }^2}} \right) = \frac{K}{{K - 1}}\\E\left( {{{\hat \sigma }^2}} \right) = \frac{K}{{K - 1}}\frac{{(K - 1){\sigma ^2}}}{K}\\ = {\sigma ^2}\end{array}\)

We then note that \(\frac{K}{{K - 1}}{\hat \sigma ^2}\)is an unbiased estimator for\({\sigma ^2}\).

\(\frac{K}{{K - 1}}{\hat \sigma ^2} = \frac{1}{{IJ(K - 1)}}\sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {\sum\limits_{k = 1}^K {{{\left( {{Y_{ijk}} - {{\bar Y}_{ij + }}} \right)}^2}} } } .\)

\(\frac{1}{{IJ(K - 1)}}\sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {\sum\limits_{k = 1}^K {{{\left( {{Y_{ijk}} - {{\bar Y}_{ij + }}} \right)}^2}} } } \)is an unbiased estimator for \({\sigma ^2}.\)