# 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

Covariance Hull Dition Balan diCi