Computing the exact distribution of the Bartlett's test statistic by numerical inversion of its characteristic function

Abstract

Application of the exact statistical inference frequently leads to non-standard probability distributions of the considered estimators or test statistics. The exact distributions of many estimators and test statistics can be specified by their characteristic functions, as is the case for the null distribution of the Bartlett's test statistic. However, analytical inversion of the characteristic function, if possible, frequently leads to complicated expressions for computing the distribution function and the corresponding quantiles. An efficient alternative is the well-known method based on numerical inversion of the characteristic functions, which is, however, ignored in popular statistical software packages. In this paper, we present the explicit characteristic function of the corrected Bartlett's test statistic together with the computationally fast and efficient implementation of the approach based on numerical inversion of this characteristic function, suggested for evaluating the exact null distribution used for testing homogeneity of variances in several normal populations, with possibly unequal sample sizes.

Keywords:

• Bartlett's test statistic
• exact distribution
• characteristic function
• numerical inversion

2010 Mathematics Subject Classifications:

• 62H10
• 62E15

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The work was supported by the Slovak Research and Development Agency (Agentúra na podporu výskumu a vývoja), project APVV-15-0295, and by the Scientific Grant Agency VEGA of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences (Vedecká grantová agentúra MŠVVaŠ SR a SAV), projects VEGA 2/0054/18 and VEGA 2/0081/19.