On the automatic selection of the tuning parameter appearing in certain families of goodness-of-fit tests

ABSTRACT

The situation, common in the current literature, is that of a whole family of location-scale/scale invariant test statistics, indexed by a parameter $\lambda \in \mathrm{\Lambda }$, is available to test the goodness of fit of F, the underlying distribution function of a set of independent real-valued random variables, to a location-scale/scale family of distribution functions. The power properties of the tests associated with the different statistics usually depend on the parameter λ, called the “tuning parameter”, which is the reason that its choice is crucial to obtain a performing test procedure. In this paper, we address the automatic selection of the tuning parameter when Λ is finite, as well as the calibration of the associated goodness-of-fit test procedure. Examples of existing and new tuning parameter selectors are discussed, and the methodology presented of combining different test statistics into a single test procedure is applied to well known families of test statistics for normality and exponentiality. A simulation study is carried out to access the power of the different tests under consideration, and to compare them with the fixed tuning parameter procedure, usually recommended in the literature.

Keywords:

• Goodness-of-fit tests
• data-based tuning parameter selection
• calibration
• normality tests
• empirical characteristic function
• exponentiality tests
• empirical Laplace transform

AMS 2010 subject classifications:

• 62G10
• 62G20

Acknowledgments

We thank our colleague Alexander Kovacec for drawing our attention to the result on nonconstant analytic functions used in the proof of Theorem 4.1, and the reviewers for their comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Research partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.