Beta approximation and its applications

ABSTRACT

During the past 30 years, many statistics have been proposed for tackling change-point problems. However, the null distributions of these statistics are not easy to obtain in both theory and practice. In this paper, we consider to use the beta distribution to approximate a standard distribution by matching their first four moments. Sufficient conditions are derived for the beta approximation to any distribution. The $L_{\mathrm{\infty }}$ distance is used to evaluate the beta approximation to a distribution. To demonstrate the advantage of this approximation over the Rayleigh approximation, the Gumbel approximation and the empirical saddlepoint approximation in change-point analysis, three applications and one real data example are provided for illustration.

KEYWORDS:

• Beta approximation
• change-point
• sufficient conditions
• divergence measure
• Rayleigh approximation
• Gumbel approximation
• empirical saddlepoint approximation

Acknowledgements

The authors would like to thank Professor Nancy Reid for her helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was partially supported by Natural Sciences and Engineering Research Council of Canada.