Monotonic improved critical values for two χ2 asymptotic criteria

Size-corrected critical values are usually obtained by inverting the Edgeworth expansion for the null distribution of a test statistic, e.g., the likelihood ratio, score or Wald statistic. A potentially serious problem with this approach is that the size-corrected critical value is not typically a monotonic function of the original critical value, and hence it can even be negative in cases where the test statistic is always non-negative. We address this problem by deriving three different size-corrected critical values which are always monotonic (non-decreasing) functions of the original critical value.