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An Isotonic Regression Problem for Infinite-Dimensional Parameters

Author

Listed:
  • R. Khattree

    (Oakland University)

  • D. P. Schmidt

    (Oakland University)

  • I. E. Schochetman

    (Oakland University)

Abstract

We consider an infinite-dimensional isotonic regression problem which is an extension of the suitably revised classical isotonic regression problem. Given p-summable data, for p finite and at least one, there exists an optimal estimator to our problem. For p greater than one, this estimator is unique and is the limit in the p-norm of the sequence of unique estimators in canonical finite-dimensional truncations of our problem. However, for p equal to one, our problem, as well as the finite-dimensional truncations, admit multiple optimal estimators in general. In this case, the sequence of optimal estimator sets to the truncations converges to the optimal estimator set of the infinite problem in the sense of Kuratowski. Moreover, the selection of natural best optimal estimators to the truncations converges in the 1-norm to an optimal estimator of the infinite problem.

Suggested Citation

  • R. Khattree & D. P. Schmidt & I. E. Schochetman, 1999. "An Isotonic Regression Problem for Infinite-Dimensional Parameters," Journal of Optimization Theory and Applications, Springer, vol. 103(2), pages 359-384, November.
    • Handle: RePEc:spr:joptap:v:103:y:1999:i:2:d:10.1023_a:1021704903284
    DOI: 10.1023/A:1021704903284
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    References listed on IDEAS

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    1. Menendez, J. A. & Salvador, B., 1987. "An algorithm for isotonic median regression," Computational Statistics & Data Analysis, Elsevier, vol. 5(4), pages 399-406, September.
    2. Chakravarti, Nilotpal, 1989. "Bounded isotonic median regression," Computational Statistics & Data Analysis, Elsevier, vol. 8(2), pages 135-142, July.
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