02.3.2. Badly Weighted Least Squares--Solution

Let X_{n ×p} have full column rank p < n and let Ω be a diagonal matrix whose diagonal elements {ω_i }_{i =1}ⁿ are positive and not all equal. The discussion by Professors Koenker and Portnoy suggested that, for any (nonzero)

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, the function

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satisfies e_{0, Δ} ≥ e_{0.5, Δ} and more generally that e_{a , Δ} is a decreasing function of a ∈ [0,1] .

We shall show that the second suggestion, and hence the first also, is true for p = 1. Both can fail if p > 1; we illustrate this by an example in which e_{a , Δ} is not only nonmonotone but is maximized at a = 0.5.

First let p = 1, so that X = x