BoundedIsoMeanTwoDykstra: Compute solution to the problem of two ordered isotonic or antitonic curves

Description

See details below.

Usage

BoundedIsoMeanTwoDykstra(g1, w1, g2, w2, K1 = 1000,  delta = 10^(-8), output = TRUE)

Arguments

Vector in $R^n$, measurements of upper function.

Vector in $R^n$, weights for upper function.

Vector in $R^n$, measurements of lower function.

Vector in $R^n$, weights for lower function.

Upper bound on number of iterations.

delta

Upper bound on the error, defines stopping criterion.

output

Should intermediate results be output?

Value

g1: The estimated function $\hat g_1^\circ$.
g2: The estimated function $\hat g_2^\circ$.
L: Value of the least squares criterion at the minimum.
error: Value of error (norm of difference two consecutive projections).
k: Number of iterations performed.

Warning

Note that we have chosen a very simply stopping criterion here, namely the algorithm stops if the norm of two consecutive projections is smaller than $\delta$. If $n$ is very small, it may happen that two consecutive projections are equal although $L$ is not yet minimal (note that this typically happens if g1 = g2). If that is the case, we suggest to set $\delta < 0$ and let the algorithm run a sufficient number of iterations (specified by K1) to verify that the least squares criterion value can not be decreased anymore.

Details

See BoundedIsoMeanTwo for a description of the problem. This function computes the estimates via Dykstra's (see Dykstra, 1983) cyclical projection algorithm.

The algorithm is implemented for isotonic curves.

References

Balabdaoui, F., Rufibach, K., Santambrogio, F. (2009). Least squares estimation of two ordered monotone regression curves. Preprint.

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 file of the main function of this package:
?BoundedIsoMeanTwo

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