Many practical problems involve detecting trends.
By considering differences in adjacent levels, an increasing sequence of three may be transformed to an equivalent problem of detecting the positive quadrant. However, because these distributions depend on the unknown covariance matrix, the usable critical points have only been bounds. For important cases the resulting one-sided tests are biased. The disappointing performances of these approximate LRT have been the subject of several critical articles over the years. The bootstrap can rescue the LRT by estimating the approximate critical point.