## Estimating a smooth monotone regression function.(English)Zbl 0737.62038

The problem of estimating a smooth monotone regression function $$m$$ is studied. Two estimators $$m_{SI}$$ and $$m_{IS}$$ are compared. $$m_{SI}$$ consists of two steps: (i) smoothing of the data by the kernel estimator, (ii) isotonisation of the data by the pool adjacent violator algorithm. The estimator $$m_{IS}$$ is constructed by interchanging these two steps.
The author considers the asymptotic behaviour of these estimators at a fixed point $$x_ 0$$ where the function $$m$$ is assumed to be strictly monotone and smooth. It is shown that if the bandwidth of the kernel estimator is chosen in the optimal order $$n^{-1/5}$$, $$m_{SI}(x_ 0)$$ and $$m_{IS}(x_ 0)$$ are of order $$n^{-2/5}$$ and that they are asymptotically equivalent in first order. But $$m_{SI}(x_ 0)- m_{IS}(x_ 0)$$ is of the only slightly lower order $$n^{-8/15}.$$
Theorem 3 of the paper deals with stochastic higher order expansions for $$m_{SI}(x_ 0)$$ and $$m_{IS}(x_ 0)$$. These expansions entail that $$m_{IS}(x_ 0)$$ has always a smaller variance and a larger bias than $$m_{SI}(x_ 0)$$. Furthermore it is shown that the kernel function $$K$$ of the chosen kernel estimator mainly determines whether one should prefer the estimator $$m_{SI}$$ or $$m_{IS}$$. If the bandwidth of the kernel estimator $$m_ S$$ is chosen such that the mean square error is asymptotically minimized, then $$m_{IS}(x_ 0)$$ has asymptotically smaller mean square error than $$m_{SI}(x_ 0)$$ if and only if $\int K^ 2(t)dt [\int t^ 2K(t)dt\int K'(t)^ 2dt ]^{-1}$ is smaller than a universal constant.
For related literature see K. Cheng and P. Lin, Z. Wahrscheinlichkeitstheor. Verw. Geb. 57, 223-233 (1981; Zbl 0443.62029), and R. E. Barlow, D. J. Bartholomew, J. M. Bremner and H. D. Brunk, Statistical inference under order restrictions. The theory and application of isotonic regression (1972; Zbl 0246.62038).

### MSC:

 62G07 Density estimation

### Citations:

Zbl 0459.62030; Zbl 0443.62029; Zbl 0246.62038
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