The scaled inverse chi-squared distribution also has a particular use in Bayesian statistics, somewhat unrelated to its use as a predictive distribution for x = 1/s2.
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Analyse-it provides a range of standard parametric and non-parametric procedures, including Descriptive statistics, ANOVA, Mann–Whitney, Wilcoxon, chi-square, correlation, linear regression, logistic regression, polynomial regression and advanced model fitting.
Brown and Forsythe performed Monte Carlo studies that indicated that using the trimmed mean performed best when the underlying data followed a Cauchy distribution (a heavy-tailed distribution) and the median performed best when the underlying data followed a Chi-squared distribution with four degrees of freedom (a heavily skewed distribution).
Brown and Forsythe performed Monte Carlo studies that indicated that using the trimmed mean performed best when the underlying data followed a Cauchy distribution (a heavy-tailed distribution) and the median performed best when the underlying data followed a Chi-squared distribution with four degrees of freedom (a heavily skewed distribution).
The other reason is that the Wald test uses two approximations (that we know the standard error, and that the distribution is chi-squared), whereas the likelihood ratio test uses one approximation (that the distribution is chi-squared).