When Outliers Are Different. (arXiv:2106.05212v1 [astro-ph.HE])

When Outliers Are Different. (arXiv:2106.05212v1 [astro-ph.HE])
<a href="http://arxiv.org/find/astro-ph/1/au:+Katz_J/0/1/0/all/0/1">J. I. Katz</a>

When does the presence of an outlier in some measured property indicate that
the outlying object differs qualitatively, rather than quantitatively, from
other members of its apparent class? Historical examples include the many types
of supernovae and short {it vs./} long Gamma Ray Bursts. There may be only
one parameter and one outlier, so that principal component analyses are
inapplicable. A qualitative difference implies that some parameter has a
characteristic scale, and hence its distribution cannot be a power law (that
can have no such scale). If the distribution is a power law the objects differ
only quantitatively. The applicability of a power law to an empirical
distribution may be tested by comparing the most extreme member to its
next-most extreme. The probability distribution of their ratio is calculated,
and compared to data for stars, radio and X-ray sources, and the fluxes,
fluences and rotation measures of Fast Radio Bursts.

When does the presence of an outlier in some measured property indicate that
the outlying object differs qualitatively, rather than quantitatively, from
other members of its apparent class? Historical examples include the many types
of supernovae and short {it vs./} long Gamma Ray Bursts. There may be only
one parameter and one outlier, so that principal component analyses are
inapplicable. A qualitative difference implies that some parameter has a
characteristic scale, and hence its distribution cannot be a power law (that
can have no such scale). If the distribution is a power law the objects differ
only quantitatively. The applicability of a power law to an empirical
distribution may be tested by comparing the most extreme member to its
next-most extreme. The probability distribution of their ratio is calculated,
and compared to data for stars, radio and X-ray sources, and the fluxes,
fluences and rotation measures of Fast Radio Bursts.

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