Approaches to the Null Distribution of$\sqrt b_1

A comparative study of approximations to the null distribution of √b1is given for small sample sizes, i.e. n ⩽ 35. The approximations studied are the approximations by the normal distribution, the Johnson SUdistributions, the t or Pearson Type VII distributions, the Cornish-Fisher expansions using moments up to the eighth, with and without Geary's adjustment, and, finally, Monte-Carlo simulations. For the usual levels of significance all the approximations, except the normal approximation, appear satisfactory for n ⩾ 8. Because the SUand t distributions allow approximations to the probability integrals, they can be used to combine the results of independently drawn samples.