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 be independent normally -distributed random variables, and let be the vector of order statistics constructed from the observations . Moreover, let the statistic , which is a linear combination of the order statistics , be independent of some "chi-squared" distribution of degrees of freedom. Let . In this case, one says that

is a Studentized statistic.

The Studentized range is the Studentized statistic for which is the range of the sample , i.e. if

consequently, the Studentized range takes the form

References

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

Comments

The case

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