Exact confidence regions for scale and location parameters based on sample quartiles

In this paper we present relatively simple (ruler, paper, and pencil) nonparametric procedures for constructing joint confidence regions for (i) the median and the inner quartile range for the symmetric one-sample problem and (ii) the shift and ratio of scale parameters for the two-sample case. Both procedures are functions of the sample quartiles and have exact confidence levels when the populations are continuous. The one-sample case requires symmetry of first and third quartiles about the median.

The confidence regions we propose are always convex, nested for decreasing confidence levels and are compact for reasonably large sample sizes. Both exact small sample and approximate large sample distributions are given.