The display below shows a variety of shrinkage path **Q-shapes**
for the rank(X) = p = 2 case.

The best known special case of a Q-shaped path is probably Q =
0 for Hoerl-Kennard(1970) **"ordinary" ridge
regression**. This path has a dual "characteristic
property," illustrated in the figure below. Namely, the Q =
0 path contains not only the shortest b-vector of beta coefficient
estimates with any given Residual Sum-of-Squares [RSS] but also
the smallest RSS = (y - Xb)'(y - Xb) for any given b-vector length.

Another well known special case of a Q-shaped path is Q = +1
for **uniform** shrinkage. The coefficient TRACE and shrinkage
factor TRACE for this path are both rather "dull," but
the estimated risk and inferior direction TRACES can still be
quite interesting even when Q = +1.

An extremely important limiting case is Q = minus infinity for
**principal components regression**.
[Marquardt(1970) calls this limit "assigned rank"
regression.] My experience is that the Q = -5 path is frequently
quite close, numerically, to this limiting case. Note in the top
figure on this page that the path with Q = -1 shape is already
near the limit in the p = 2 dimensional case being depicted here.

As a general rule-of-thumb, paths with Q-shapes in the [-1,+2]
range generally tend to be fairly **smooth** ...i.e.
have "rounded" corners. Paths with Q-shapes greater
that +2 or less than -1 can display quite "sharp"
corners. In fact, the paths with limiting shapes of +/- infinity
are actually __linear splines__ with join points at integer
MCAL values!

My computing algorithms provide strong, objective guidance on the choice
of the Q-shape that is __best for your data__. For example,
they implement the methods of Obenchain(1975b, 1981,
1997a) to identify the path Q-shape (and the MCAL-extent of
shrinkage along that path) which have **maximum likelihood **(under
a classical, fixed coefficient, normal-theory model) of achieving
overall minimum MSE risk in estimation of regression
coefficients.