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October, 1972 Monotone Median Regression
J. D. Cryer, Tim Robertson, F. T. Wright, Robert J. Casady
Ann. Math. Statist. 43(5): 1459-1469 (October, 1972). DOI: 10.1214/aoms/1177692378


Suppose that for each real number $t$ in [0, 1] we have a distribution with distribution function $F_t(\bullet)$, mean $\mu(t)$ and median $m(t) (\mu(t)$ and $m(t)$ are referred to as regression functions). Consider the problems of estimating $\mu(\bullet)$ and $m(\bullet)$. In this paper we propose and discuss an estimator, $\hat{m}(\bullet)$, of $m(\bullet)$ which is monotone. This estimator is analogous to the estimator $\hat{\mu}(\bullet)$ of $\mu(\bullet)$ which was explored by Brunk (1970) (Estimation of isotonic regression in Nonparametric Techniques in Statistical Inference, Cambridge University Press, 177-195). Rates for the convergence of $\hat{m}(\bullet)$ to $m(\bullet)$ are given and a simulation study, where $\hat{m}(\bullet), \hat{\mu}(\bullet)$ and the least squares linear estimator are compared, is discussed.


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J. D. Cryer. Tim Robertson. F. T. Wright. Robert J. Casady. "Monotone Median Regression." Ann. Math. Statist. 43 (5) 1459 - 1469, October, 1972.


Published: October, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0251.62025
MathSciNet: MR370903
Digital Object Identifier: 10.1214/aoms/1177692378

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 5 • October, 1972
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