Confidence Intervals with More Power to Determine the Sign: Two Ends Constrain the Means
针对连续、单峰、对称随机变量的均值，提出两类非等变双侧置信区间，相比传统对称区间，在观测值小时更短且能限制均值符号，适用于需要判断效应方向且不预先指定方向的场景。
Abstract We present two new families of two-sided nonequivariant confidence intervals for the mean θ of a continuous, unimodal, symmetric random variable. Compared with the conventional symmetric equivariant confidence interval, they are shorter when the observation is small, and restrict the sign of θ for smaller observations. One of the families, a modification of Pratt's construction of intervals with minimal expected length when θ = 0, is longer than the conventional symmetric interval when |X| is large and has longer expected length when |θ| is large. The other family gives the conventional symmetric interval when |X| is large, with a change to the proximal endpoint when |X| is small. Its expected length is smaller than that of the conventional symmetric interval when |θ| is small, larger for an intermediate range of |θ|, and approaches that of the conventional interval for large |θ|. This slight modification of the conventional two-sided interval has most of the power advantage of a one-sided interval, but short length. Neither procedure requires that a preferred direction be specified in advance. The constants that determine the procedures can be found for symmetrically distributed statistics using any software package that includes the cumulative distribution function and inverse cumulative distribution function of the statistic, along with a root finder. We present tables of constants needed to apply the procedures for normally and t-distributed test statistics, and give an application to employment discrimination litigation.