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