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Abstract:
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Shape constrained nonparametric estimation dates back to the 1950s. In his milestone paper in 1956, Grenander derived the maximum likelihood estimator of a nonincreasing density, whereas Brunk (1958) obtained the least squares estimator of a monotone regression function. The last decades, similar estimators have been proposed in other statistical models, including Cox's proportional hazards model with a monotone baseline hazard. Typically, these isotonic estimators are step functions that exhibit a non normal limit distribution at cube-root n rate. On the other hand, a long stream of research has shown that, if one is willing to assume more regularity on the function of interest, smooth estimators can be used to achieve a faster rate of convergence to a Gaussian distributional law and to estimate derivatives. The asymptotic behavior of global distances between the estimator and the function of interest, such as L_p distances and supremum distance, has been studied for traditional isotonic estimators. In this talk, similar results are presented for smooth isotonic methods, e.g., right-censored data.
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