Seminar on Probability and Statistics

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Organizer(s) Nakahiro Yoshida, Hiroki Masuda, Teppei Ogihara, Yuta Koike

2021/12/15

14:30-16:00   Room # (Graduate School of Math. Sci. Bldg.)
Estate Khmaladze (Victoria University of Wellington)
Theory of Distribution-free Testing
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)
https://sites.google.com/view/apsps/home

The aim of the talk is to introduce transformations of empirical-type processes by a group of unitary operators. Recall that if v_{nP} is empirical process on real line, based on a sample from P, it can be mapped into empirical process v_{nQ} by appropriate change of time

v_{nP}(h(x))=v_{nQ}(x)

where h(x) is continuous and increasing. This is the basis for distribution-free theory of goodness of fit testing. If w(\phi) is a function-parametric “empirical-type” process (i.e. has functions \phi from a space L as a time) and if K* is a unitary operator on L, then transformed process Kw we define as
Kw(\phi) = w(K*\phi)

These two formulas have good similarity, but one transformation in on the real line, while the other transformation in on functional space.This later one turns out to be of very broad use, and allows to base distribution-free theory upon it. Examples, we have specific results for, are parametric empirical
processes in R^d, regression empirical processes, those in GLM, parametric models for point processes and for Markov processes in discrete time. Hopefully, further examples will follow.
[ Reference URL ]
https://docs.google.com/forms/d/e/1FAIpQLSdFj1XF8WJSPRmE0GFKY2QxscaGxC9msM6GkEsAf0TgD9yv2g/viewform