OrdMonReg-package: Compute least squares estimates of one bounded or two ordered antitonic regression curves

Description

We consider the problem of estimating two isotonic regression curves $g^\circ_1$ and $g^\circ_2$ under the constraint that $g^\circ_1 \le g^\circ_2$. Given two sets of $n$ data points $y_1, \ldots, y_n$ and $z_1, \ldots, z_n$ that are observed at (the same) deterministic design points $x_1, \ldots, x_n$, the estimates are obtained by minimizing the Least Squares criterion $$L(a, b) = \sum_{i=1}^n (y_i - a_i)^2 w_1(x_i) + \sum_{i=1}^n (z_i - b_i)^2 w_2(x_i)$$ over the class of pairs of vectors $(a, b)$ such that $a$ and $b$ are isotonic and $a_i \le b_i$ for all $i = {1, \ldots, n}$. We offer two different approaches to compute the estimates: a projected subgradient algorithm where the projection is calculated using a pool-adjacent-violaters algorithm (PAVA) as well as Dykstra's cyclical projection algorithm.. Additionally, functions to solve the bounded isotonic regression problem described in Barlow et al. (1972, p. 57) are provided.

Arguments

docType

package

Details

ll{ Package: OrdMonReg Type: Package Version: 1.0.3 Date: 2011-11-30 License: GPL (>=2) }

References

Balabdaoui, F., Rufibach, K., Santambrogio, F. (2009). Least squares estimation of two ordered monotone regression curves. Preprint. Barlow, R. E., Bartholomew, D. J., Bremner, J. M., Brunk, H. D. (1972). Statistical inference under order restrictions. The theory and application of isotonic regression. John Wiley and Sons, London - New York - Sydney. Dykstra, R.L. (1983). An Algorithm for Restricted Least Squares Regression. J. Amer. Statist. Assoc., 78, 837--842.

Examples

Run this code

## examples are provided in the help files of the main functions of this package:
?BoundedAntiMean
?BoundedAntiMeanTwo