The Problem

1. When the error becomes small, that is, as s ® 0, the first and second derivatives of the b's all go to zero. For example, this is the first derivative for s₁:
   
   
   As s ® 0 for i Î Yea,
   X_i^'b ® ¥,
   f (X_i^'b) ® 0, and
   F (X_i^'b) ® 1.
   
   As s ® 0 for i Î Nay,
   X_i^'b ® -¥,
   f (X_i^'b) ® 0, and
   1-F (X_i^'b) ® 1.
   
   
2. However, subject to a small "wiggle", the cutting line and the normal vector are identified! Technically, the b's are identified up to a multiplicative constant.
   
   
3. This problem is known as complete separation in Econometrics (Albert and Anderson, 1984; Silvapulle, 1981).