Question about Huber loss when k=0 in Casella and Berger In Casella and Berger (page484), the following Huber loss is defined.

Then on the next page, Table 10.2.1 shows the Huber estimator for different $k$:

In particular, $k=0$ gives the median, which doesn't make any sense. As when $k=0$, the Huber loss according to (10.2.2) is identically 0. But in order to get the median the huber loss has to be the absolute value (possibly with a shift). I guess the form of Huber loss that he gives here is wrong. What's the correct form of Huber loss then?
Thanks!

Equation 10.2.2 in Casella & Berger is correct. You are right of course that $\rho(x)$ is identically zero when $k=0$ but the estimate is nevertheless defined at $k=0$ in a limiting sense. The value of $a$ that minimizes the criterion function converges to the median as $k$ converges to zero from above. The estimate can be made arbitrarily close to the median by setting $k$ to a sufficiently small positive value.
