A problem with the LAD regression line is that it is not always uniquely defined.
(a) Show that, for a data set with three observations, (x1, y1), (x1, y2), and (x3, y3) (note the first two xs are the same), any line that goes through (x3, y3) and lies between (x1, y1) and (x1, y2) is a least absolute deviation line.
(b) For three individuals, measurements are taken on heart rate (x, in beats per minute) and oxygen consumption (y, in ml/kg). The (x, y) pairs are (127, 14.4), (127, 11.9), and (136, 17.9). Calculate the slope and intercept of the least squares line and the range of the least absolute deviation lines.
There seems to be some disagreement over the value of the least absolute deviation line. It is certainly more robust than least squares but can be very difficult to compute (but see Portnoy and Koenker 1997 for an efficient computing algorithm). It also seems that Ellis (1998) questions its robustness, and in a discussion Portnoy and Mizera (1998) question Ellis.