Trigonometric%20scores%20rank%20statistics

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Trigonometric scores rank statistics

• Olena Kravchuk
• (supervisor Phil Pollett)
• Department of Mathematics, UQ

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Ranks and anti-ranks
First sample First sample First sample Second sample Second sample Second sample
Index 1 2 3 4 5 6
Data 5 7 0 3 1 4
Rank 5 6 1 3 2 4
Antirank 3 5 4 6 1 2
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Simple linear rank statistic

• Let us consider the two-sample location problem.
  Assume that the distributions are continuous of
  the same location family, f, and may differ in
  location, µ, only. The inference is made from
  two random samples of size m and n, Nmn, drawn
  from the distributions.

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Random walk model

• Let us start a random walk at the origin and walk
  on the pooled data sample moving up every time we
  see an observation from the first sample and
  down every time we see an observation from the
  second sample. Let us pin the walk T down by
  assigning the appropriate up/down steps, cs.

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Brownian Bridge
Cramer-von Mises statistic

• One of the common form of the statistic is given
  below. There di is the difference between the
  sample distribution functions at the ith point in
  the pooled sample.

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First components of CM

• Durbin and Knott Components of Cramer-von mises
  Statistics

The random variable cos(jpx) is the projection of
a unit vector on a fixed vector where the angle
between the two vectors is distributed uniformly
between 0 and jp. Evidently for testing the
significance of individual components we only
need significance points for the first component.
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Percentage points for the first component
(one-sample)

• Durbin and Knott Components of Cramer-von mises
  Statistics

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Percentage points for the first component
(two-sample)

• Kravchuk Rank test of location optimal for HSD

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Hyperbolic secant distribution
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Some tests of location
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Random walks under the alternative
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Small-sample power
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Small-sample power
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Trigonometric scores rank estimators

• Location estimator of the HSD

Scale estimator of the Cauchy
Trigonometric scores rank estimator