astar_1

bstar_n

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.

These functions compute the values $a_1^*$ and $b_n^*$, the value of the estimate of the 
upper function at $x_1$ and the value of the lower estimated function at $x_n$ in the two ordered
antitonic regression functions problem. These values can be computed via explicit formulas, unlike the values at
$x \in {x_2, \ldots, x_{n-1}}$, which are received via a projected subgradient algorithm. However, 
$b_n^*$ enters this algorithm as an auxiliary quantity.

regression

nonparametric

astar_1, bstar_n

We consider the problem of estimating two isotonic regression curves g1* and g2* under the constraint that they are ordered, i.e. g1* <= g2*. Given two sets of n data points y_1, ..., y_n and z_1, ..., z_n that are observed at (the same) deterministic design points x_1, ..., x_n, the estimates are obtained by minimizing the Least Squares criterion L(a, b) = sum_{i=1}^n (y_i - a_i)^2 w1(x_i) + sum_{i=1}^n (z_i - b_i)^2 w2(x_i) over the class of pairs of vectors (a, b) such that a and b are isotonic and a_i <= b_i for all i = 1, ..., n. We offer two different approaches to compute the estimates: a projected subgradient algorithm where the projection is calculated using a PAVA as well as Dykstra's cyclical projection algorithm.

astar_1, bstar_n: Computes explicitly known values of the estimates in the two ordered functions antitonic regression problem

Description

Usage

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Value

References

See Also