For i = 1,...,s and j = 1,...,ni , let Xij be independent with Xi,j having distribution Fi , where Fi is
an arbitrary distribution with mean i and finite variances 2 that vary for each Fi......
Robustness of F- statistic under unequal variances
(A separate Excel working is attached for the development of Rejection probabilities and it graph)
The F-statistic that we consider in this exercise is nothing but the ratio of "Mean Between Sum of Squares" and "Mean Within Sum of Squares". If this ratio is sufficiently large, we reject the null hypothesis. The distribution of this ratio is derived based on the fact that its denominator and numerator follow independent Chi-square distributions. Again the assumption of chi-square distribution requires the assumption that the samples are taken from Normal distribution with almost similar variances.
In this exercise we are going to verify how this F-statistic is robust when the assumption of equal variances is violated. To establish this fact we make use of a hypothetical population with the following structure.
Groups Sample size Mean Standard Deviation
1 10 55 10
2 14 80 10
3 16 62 10
The above data consists of three groups with considerably different means (Group means are not equal) and with equal standard deviation. Thus one would expect to reject the ...
Basic large sample theory is examined for distributions. Finite variances are analyzed.