Advertisement

Linear Rank and Signed Rank Statistics

  • Hira L. Koul
Part of the Lecture Notes in Statistics book series (LNS, volume 166)

Abstract

Let {X ni , F ni } be as in (2.2.23) and {c ni } be p × 1 real vectors. The rank and the absolute rank of the i th residual for 1 ≤ in, u ∈ ℝ p , are defined, respectively, as
$$ \begin{array}{l} R_{iu} = \sum\limits_{j = 1}^n {I(X_{nj} - u'c_{nj} \le X_{ni} - u'c_{ni} )} , \\ R_{iu}^ + = \sum\limits_{j = 1}^n {I(|X_{nj} - u'c_{nj} | \le |X_{ni} - u'c_{ni} |)} \\ \end{array} $$
(3.1.1)
% MathType!End!2!1! Let ϕ be a nondecreasing real valued function on [0,1] and define
$$ \begin{gathered} Td(\phi ,u) = \sum\limits_{i = 1}^n {d_{ni} \phi \left( {\frac{{R_{iu} }} {{n + 1}}} \right),} \hfill \\ T_d^ + (\phi ,u) = \sum\limits_{i = 1}^n {d_{ni} \phi ^ + \left( {\frac{{R_{iu}^ + }} {{n + 1}}} \right)s(X_{ni} - u'c_{ni} ),} \hfill \\ \end{gathered} $$
(3.1.2)
for u ∈ ℝ p , where
$$ \phi ^ + (s) = \phi ((s + 1)/2),0 \le s \le 1;s(x) = I(x > 0) - I(x < 0). $$

Keywords

Weak Convergence Score Function Asymptotic Normality Symmetric Case Signed Rank 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Hira L. Koul
    • 1
  1. 1.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

Personalised recommendations