Studentized range

A statistic from the class of so-called Studentized statistics, obtained as a result of a special normalization of a linear combination of order statistics constructed from a normal sample.

Let $ X _ {1} \dots X _ {n} $ be independent normally $ N( a, \sigma ^ {2} ) $- distributed random variables, and let $ X ^ {(n)} = ( X _ {(n1)} \dots X _ {(nn)} ) $ be the vector of order statistics constructed from the observations $ X _ {1} \dots X _ {n} $. Moreover, let the statistic $ \sum _ {i=1} ^ {n} a _ {i} X _ {(ni)} $, which is a linear combination of the order statistics $ X _ {(n1)} \dots X _ {(nn)} $, be independent of some "chi-squared" distribution $ V/ \sigma ^ {2} $ of $ f $ degrees of freedom. Let $ s _ {f} ^ {2} = f ^ { - 1 } V $. In this case, one says that

$$ \frac{1}{s _ {f} } \sum _ { i=1 } ^ { n } a _ {i} X _ {(ni)} $$

is a Studentized statistic.

The Studentized range is the Studentized statistic for which $ \sum _ {i=1} ^ {n} a _ {i} X _ {(ni)} $ is the range of the sample $ X _ {1} \dots X _ {n} $, i.e. if

$$ \sum _ { i=1 } ^ { n } a _ {i} X _ {(ni)} = X _ {(nn)} - X _ {(n1)} ; $$

consequently, the Studentized range takes the form

$$ \frac{X _ {(nn)} - X _ {(n1)} }{s _ {f} } . $$

References

[1]	H. David, "Order statistics" , Wiley (1970)
[2]	S.S. Wilks, "Mathematical statistics" , Wiley (1962)

Comments

The case

$$ s _ {f} ^ {2} = \frac{1}{n-1} \sum ( X _ {i} - \overline{X}\; ) ^ {2} $$

is used for tests of normality and outlying observations, cf. [1], Chapt. 8. For a table of the quantiles of the Studentized range see [a2].

References

[a1]	A.M. Mood, F.A. Graybill, "Introduction to the theory of statistics" , McGraw-Hill (1963) pp. 243
[a2]	P.H. Müller, P. Neumann, R. Storm, "Tafeln der mathematischen Statistik" , C. Hauser (1977) pp. 166–169