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Gordon Smyth
Gordon Smyth
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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 setsetting $k$ to a sufficiently small positive value.

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 set $k$ to a sufficiently small positive value.

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.

Source Link
Gordon Smyth
Gordon Smyth
  • 13.5k
  • 1
  • 40
  • 56

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 set $k$ to a sufficiently small positive value.