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The Isotonic Regression Problem and Its Dual

R. E. Barlow and H. D. Brunk
Journal of the American Statistical Association
Vol. 67, No. 337 (Mar., 1972), pp. 140-147
DOI: 10.2307/2284712
Stable URL: http://www.jstor.org/stable/2284712
Page Count: 8
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The Isotonic Regression Problem and Its Dual
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Abstract

The isotonic regression problem is to minimize ∑ki = 1[ gi - xi]2wi subject to xi ≤ xj when $i \preceq i$ where $w_i > 0$ and gi (i = 1, 2, ⋯, k) are given and $\preceq$ is a specified partial ordering on {1, 2, ⋯, k}. The solution is called the isotonic regression on g. We formulate a generalization of this problem and calculate its Fenchel dual. A function of the isotonic regression also solves these problems. Problems in inventory theory and statistics are identified as dual isotonic regression problems.

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