Parametric embedding of nonparametric inference problems

ABSTRACT

In 1937, Neyman introduced the notion of smooth tests of the null hypothesis that the sample data come from a uniform distribution on the interval (0,1) against alternatives in a smooth parametric family. This idea can be used to embed various nonparametric inference problems in a parametric family. Focusing on nonparametric rank tests, we show how to derive traditional rank tests by applying this approach. We also show how to use it to obtain simplifying insights and optimality results in complicated settings that involve censored and truncated data, for which it is more convenient to use hazard functions to define the embedded family. We describe an application of the embedding approach to the problem of testing for trend in environmental studies.

KEYWORDS:

• Parametric embedding
• nonparametric inference
• smooth tests
• censored and truncated data
• hazard rank tests

AMS SUBJECT CLASSIFICATION:

• 62G

Notes

¹ Lehmann and Stein considered the case and Hoeffding general , including .

² This is also equivalent to if and only if for any sequence of events .

³ In fact, Gehan introduced a further refinement depending on whether the larger observation is censored or not.

Additional information

Funding

This research is supported by Natural Sciences and Engineering Research Council of Canada OGP0009068 and the Research Grants Council of the Hong Kong Special Administrative Region, China (project 17303515). It is also supported by the National Science Foundation, grant DMS-1407828, and supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (project 17303515).