Confidence Coefficients of Interpolated Nonparametric Sign Intervals for Medians Under No or Weak Shape Assumptions

Abstract

Non-parametric “sign” intervals for a parent median based on order statistics have the important property of being generally valid. With small sample sizes, the available confidence coefficients (CCs) are sparse, however, and it is natural to try to interpolate between adjacent sign intervals to attain intermediate levels. This chapter provides the CC associated with weighted means of adjacent sign intervals over some interesting classes of parent distributions, including: (a) all distributions, (b) all symmetric distributions, and (c) all symmetric and unimodal distributions. The behavior of these CCs as functions of the weight is simple but intuitively quite surprising, with certain discontinuities and intervals of constancy. Some unexpected domination relations among weighted means of adjacent sign intervals follow from these results. The resulting nondominated intervals constitute a considerable extension of the sign intervals, with substantially more confidence-coefficient levels; and they are valid under no or weak shape assumptions about the parent distribution.