A question in proof from “Regression Quantiles” by Roger Koenker and Gilbert Bassett(Econometrica, 1978)

I have a problem in verifying the conclusion (3.5) in proving Theorem 3.3 as attached here. Particularly, from (3.4)

\begin{align} 0&<\sum_{k=1}^{K}[(1/2-\theta)v_{k}+|v_{k}|]\\ &+\sum_{t\in\overline{h}}[1/2-1/2 sgn^{\ast}(y_{t}-x_{t}\beta^{\ast};x_{t}X(h)^{-1}v)-\theta]x_{t}X(h)^{-1}v \end{align}

But how is this equivalent to (3.5)

\begin{align} (\theta-1)\iota^{'}_{K} &<\sum_{t\in\overline{h}}[1/2-1/2 sgn^{\ast}(y_{t}-x_{t}\beta^{\ast};x_{t}X(h)^{-1}v)-\theta]x_{t}X(h)^{-1}\\ & < \theta\iota^{'}_{K} \end{align}

For me, I can only get that the sum ranges over $((\theta-3/2)\iota^{'}_{K},(\theta+1/2)\iota^{'}_{K})$, which is wider than that of (3.5).

• You should define what is $\iota_k$ and what is $\theta$. Also what is the problem P? – mpiktas Jan 21 '15 at 12:45