Inference: Small samples

Abstract

The finite sample distribution of the score statistic is considered more closely. Since the other test statistics are derived from this, and the regression coefficient itself a monotonic function of the score, it is enough to restrict attention to the score statistic alone. One direct approach leads to a simple convolution expression which can be evaluated by interated integrals. It is also possible to make improvements to the large sample normal approximation via the use of saddlepoint approximations or Cornish-Fisher expansions. For these we can use the results of Corollaries 7.3 and 7.4. Corrections to the distribution of the score statistic can be particularly useful when the distribution of the explanatory variable is asymmetric. Corrections to the distribution of the score equation have a rather small impact in the case of the fourth moment but can be of significance in the case of the third moment. The calculations themselves are uncomplicated and simplify further in the case of an exponential distribution. Since we can transform an arbitrary marginal distribution to one of the exponential form, while preserving the ranks, we can then consider the results for the exponential case to be of broader generality.