Question

For any \(n \times 1\) vector \(\mathbf{v}\), define the function \(\|\mathbf{v}\|_{W}\) by $$ \|\mathbf{v}\|_{W}=\sum_{i=1}^{n} a_{W}\left(R\left(v_{i}\right)\right) v_{i} $$ where \(R\left(v_{i}\right)\) denotes the rank of \(v_{i}\) among \(v_{1}, \ldots, v_{n}\) and the Wilcoxon scores are given by \(a_{W}(i)=\varphi_{W}[i /(n+1)]\) for \(\varphi_{W}(u)=\sqrt{12}[u-(1 / 2)] .\) By using the correspondence between order statistics and ranks, show that $$ \|\mathbf{v}\|_{W}=\sum_{i=1}^{n} a(i) v_{(i)}, $$ where \(v_{(1)} \leq \cdots \leq v_{(n)}\) are the ordered values of \(v_{1}, \ldots, v_{n} .\) Then, by establishing the following properties, show that the function \((10.9 .53)\) is a pseudo-norm on \(R^{n} .\) (a) \(\|\mathbf{v}\|_{W} \geq 0\) and \(\|\mathbf{v}\|_{W}=0\) if and only if \(v_{1}=v_{2}=\cdots=v_{n}\). Hint: First, because the scores \(a(i)\) sum to 0, show that $$ \sum_{i=1}^{n} a(i) v_{(i)}=\sum_{ij} a(i)\left[v_{(i)}-v_{(j)}\right] $$ where \(j\) is the largest integer in the set \(\\{1,2, \ldots, n\\}\) such that \(a(j)<0\). (b) \(\|c \mathbf{v}\|_{W}=|c|\|\mathbf{v}\|_{W}\), for all \(c \in R\). (c) \(\|\mathbf{v}+\mathbf{w}\|_{W} \leq\|\mathbf{v}\|_{W}+\|\mathbf{w}\|_{W}\), for all \(\mathbf{v}, \mathbf{w} \in R^{n}\) Hint: Determine the permutations, say, \(i_{k}\) and \(j_{k}\) of the integers \(\\{1,2, \ldots, n\\}\), which maximize \(\sum_{k=1}^{n} c_{i_{k}} d_{j_{k}}\) for the two sets of numbers \(\left\\{c_{1}, \ldots, c_{n}\right\\}\) and \(\left\\{d_{1}, \ldots, d_{n}\right\\} .\)

Short answer

The essential steps to solve this problem involve breaking the problem into smaller parts to apply algebra and mathematical reasoning, starting by proving the correspondence between rank and order statistics, then proving properties (a), (b), and (c) using the definitions, inequalities, and algebraic manipulations provided, which results in showing that the defined function is indeed a pseudo-norm on \(R^n\).

Step-by-step solution

Proving property (a) part 1

Using the definition of the function \(\|\mathbf{v}\|_{W}\), the assumption that scores \(a(i)\) sum to zero, and replacing the rank order \(R\left(v_{i}\right)\) with its corresponding ordering \(v_{(i)}\), obtain the expression \(\sum_{i=1}^{n} a(i) v_{(i)}\). Then, break the summation into two summations on either side of \(j\) to prove that \(\sum_{i=1}^{n} a(i) v_{(i)}=\sum_{ij} a(i)\left[v_{(i)}-v_{(j)}\right]\) where \(j\) is the highest number in the set \{1, 2, ..., n\} such that \(a(j)<0\).

Proving property (a) part 2

Next, using the inequalities and conditions presented in the exercise, one must prove that \(\|\mathbf{v}\|_{W} \geq 0\) where equal to zero if and only if \(v_{1}=v_{2}=\cdots=v_{n}\). This can be achieved using the rules and relationships derived in the previous step.

Proving property (b)

To prove that \(\|c \mathbf{v}\|_{W}=|c|\|\mathbf{v}\|_{W}\) replace \(\mathbf{v}\) with \(c \mathbf{v}\) in the given function. This will yield a function where all the terms of the function are multiplied by \(c\). This implies that the pseudo-norm of \(c \mathbf{v}\) is \(|c|\) times the pseudo-norm of \(\mathbf{v}\). Thus property (b) is proven.

Proving property (c)

To prove \(\|\mathbf{v}+\mathbf{w}\|_{W} \leq\|\mathbf{v}\|_{W}+\|\mathbf{w}\|_{W}\), use the provided hint that involves obtaining permutations of the integers \(\{1, 2, ..., n\}\) which optimally order the set of numbers \(\{c_{1}, \ldots, c_{n}\}\) and \(\{d_{1}, \ldots, d_{n}\}\). Demonstrate that the sum of the pseudo-norms of \(\mathbf{v}\) and \(\mathbf{w}\) is greater than or equal to the pseudo-norm of \(\mathbf{v}+\mathbf{w}\), thus proving the triangle inequality and property (c).