Studentization or the Elimination of the Standard Deviation of the Parent Population from the Random Sample-Distribution of Statistics

Abstract

Let [chi]i, . . . , [chi]n be a sample of n observations drawn from a normal population with standard deviation. Let W be a general statistic calculated from this sample, where W > O and W is proportional to [sigma]. Let f(W) be the distribution function of W when [sigma]=l and let [image], the probability integral. Let S be an independent estimate of [sigma] based upon n degrees of freedom. Let S [image] and r=W/S. Let fn(r) be the random sample distribution of r and let its probability integral be [image] which is called the studentized integral. A partial differential equation of pn(R) is set up, which is solved approximately by iteration. The solution leads to formulas from which pn(R) can be computed. The evaluations of the integral pn are approximations of the B-functions.