  This paper constructs improved estimators of the means in the Gaussian
saturated one-way layout with an ordinal factor. The least squares estimator
for the mean vector in this saturated model is usually inadmissible. The hybrid
shrinkage estimators of this paper exploit the possibility of slow variation in
the dependence of the means on the ordered factor levels but do not assume it
and respond well to faster variation if present. To motivate the development,
candidate penalized least squares (PLS) estimators for the mean vector of a
one-way layout are represented as shrinkage estimators relative to the penalty
basis for the regression space. This canonical representation suggests further
classes of candidate estimators for the unknown means: monotone shrinkage (MS)
estimators or soft-thresholding (ST) estimators or, most generally, hybrid
shrinkage (HS) estimators that combine the preceding two strategies. Adaptation
selects the estimator within a candidate class that minimizes estimated risk.
Under the Gaussian saturated one-way layout model, such adaptive estimators
minimize risk asymptotically over the class of candidate estimators as the
number of factor levels tends to infinity. Thereby, adaptive HS estimators
asymptotically dominate adaptive MS and adaptive ST estimators as well as the
least squares estimator. Local annihilators of polynomials, among them
difference operators, generate penalty bases suitable for a range of numerical
examples.
