An Analogue to Fieller's Theorem Using Scheffé's Solution to the Fisher-Behrens Problem
R. C. Elston · The American Statistician · 1969
Suppose xi and X2 are random variables and E(xl)/ E(x2) = /u. We shall consider here the problem of estimating a confidence interval for ,i under -normality assumptions. Should it happen that log xi and log X2 are normally distributed with the same variance, as occurs for example in the case of a dilution bioassay, it is a simple matter to find confidence limits for ,u (Finney, 1952, section 2.10). We shall here, however, assume x1 and X2 are normally distributed, and not necessarily with the same variance. Finney (1952, section 2.7) shows how in this case, fiducial limits can be obtained by using the Sukhatme d-statistic and the Fisher-Behrens distribution. In this particular case, the fiducial limits obtained are not at the same time confidence limits, and do not have the property that the assertion as to the parameter lying within the given limits will be true in a preassigned proportion of cases; furthermore, the procedure is computationally laborious since an iterative solution is necessary. It does not seem to have been pointed out that confidence limits can be obtained for this particular problem, and so various methods of doing so are presented here; it will be seen that a particular method, which is due to Scheff6 (1943), not only gives an exact confidence interval but also does not require iteration for its solution. This method involves randomization, so that two statisticians using it on the same set of data would not necessarily arrive at the same limits. The method can be modified to lessen or completely eliminate the effects of randomization, however, if this is considered desirable; the intervals then obtained will not on an average differ greatly from that obtained using Scheff6's method unmodified.