# 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.