Adaptive Estimation of Directional Trend

Consider a one-way layout with one directional observation per factor level. Each observed direction is a unit vector in R measured with random error. Information accompanying the measurements suggests that the mean directions, normalized to unit length, follow a trend: the factor levels are ordinal and mean directions at nearby factor levels may be close. Measured positions of the paleomagnetic north pole in time illustrate this design. The directional trend estimators studied in this paper stem from penalized least squares (PLS) fits in which the penalty function is the squared norm of first-order or second-order differences of mean vectors at adjacent factor levels. Expressed in spectral form, such PLS estimators suggest a much larger class of monotone shrinkage estimators that use the orthogonal basis implicit in PLS. Penalty weights and, more generally, monotone shrinkage factors are selected to minimize estimated risk. The possibly large risk reduction achieved by such adaptive monotone shrinkage estimators reflects the economy of the PLS orthogonal basis in representing the actual trend and the flexibility of unconstrained monotone shrinkage.