Probstat/notes/anova - ประวัติรุ่นแก้ไข

Jittat: หน้าที่ถูกสร้างด้วย ': ''This is part of probstat.'' == Variance estimations == '''Estimator 1.''' The first estimator for $\sigma^2$ is
2014-12-05T21:14:17Z

หน้าที่ถูกสร้างด้วย ': ''This is part of probstat.'' == Variance estimations == '''Estimator 1.''' The first estimator for <math>\sigma^2</math> is <c...'

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: ''This is part of [[probstat]].''

== Variance estimations ==
'''Estimator 1.''' The first estimator for <math>\sigma^2</math> is

<center>
<math>\frac{1}{m}\sum_{i=1}^m \sum_{j=1}^n \frac{(X_{ij} - \bar{X}_{i\centerdot})^2}{n-1}.</math>
</center>

This estimator is always correct. (Even when <math>H_0</math> is not true.)

'''Estimator 2.''' The second estimator is calculated from the sample means. It will be correct only when <math>H_0</math> is true.

<center>
<math>n \sum_{i=1}^m \frac{(\bar{X}_{i\centerdot} - \bar{X}_{\centerdot\centerdot})^2}{m-1}</math>
</center>

== Hypothesis testing ==
Then compute

<center>
<math>SS_w = \sum_{i=1}^m \sum_{j=1}^n (X_{ij} - \bar{X}_{i\centerdot})^2</math>
</center>

and

<center>
<math>SS_b = n \sum_{i=1}^m (\bar{X}_{i\centerdot} - \bar{X}_{\centerdot\centerdot})^2</math>
</center>

Recall that

<center>
<math>\frac{SS_w}{\sigma^2} \sim \chi_{m(n-1)}^2</math>,
</center>

and

<center>
<math>\frac{SS_b}{\sigma^2} \sim \chi_{m-1}^2</math>.
</center>

Also, since <math>S_i^2</math> is independent of <math>\bar{X}_{i\centerdot}</math>, <math>SS_w</math> and <math>SS_b</math> are independent. Thus, the statistic

<center>
<math>
\frac{SS_b/(m-1)}{SS_w/m(n-1)} \sim F_{m-1,m(n-1)}.
</math>
</center>

Therefore, we can use the table for the F-distribution to test for <math>H_0</math>.