## Electronic Journal of Statistics

### A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation

#### Abstract

We consider Grenander type estimators for monotone functions $f$ in a very general setting, which includes estimation of monotone regression curves, monotone densities, and monotone failure rates. These estimators are defined as the left-hand slope of the least concave majorant $\widehat{F}_{n}$ of a naive estimator $F_{n}$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\widehat{F}_{n}$ and $F_{n}$ is of the order $O_{p}(n^{-1}\logn)^{2/(4-\tau)}$, for some $\tau\in[0,4)$ that characterizes the tail probabilities of an approximating process for $F_{n}$. In typical examples, the approximating process is Gaussian and $\tau=1$, in which case the convergence rate $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the one obtained by Kiefer and Wolfowitz [9] for the special case of estimating a decreasing density. We also obtain a similar result for the primitive of $F_{n}$, in which case $\tau=2$, leading to a faster rate $n^{-1}\log n$, also found by Wang and Woodfroofe [22]. As an application in our general setup, we show that a smoothed Grenander type estimator and its derivative are asymptotically equivalent to the ordinary kernel estimator and its derivative in first order.

#### Article information

Source
Electron. J. Statist., Volume 8, Number 2 (2014), 2479-2513.

Dates
First available in Project Euclid: 9 December 2014

https://projecteuclid.org/euclid.ejs/1418134261

Digital Object Identifier
doi:10.1214/14-EJS958

Mathematical Reviews number (MathSciNet)
MR3285873

Zentralblatt MATH identifier
1309.62065

Subjects
Primary: 62G05: Estimation
Secondary: 62G07: Density estimation 62G08: Nonparametric regression 62N02: Estimation

#### Citation

Durot, Cécile; Lopuhaä, Hendrik P. A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation. Electron. J. Statist. 8 (2014), no. 2, 2479--2513. doi:10.1214/14-EJS958. https://projecteuclid.org/euclid.ejs/1418134261

#### References

• [1] Balabdaoui, F. and Wellner, J. A., A Kiefer-Wolfowitz theorem for convex densities. In, Asymptotics: particles, processes and inverse problems, vol. 55 of IMS Lecture Notes Monogr. Ser., Inst. Math. Statist., Beachwood, OH, 2007, pp. 1–31.
• [2] Barlow, R. E., Bartholomew, D. J., Bremner, J., and Brunk, H., Statistical Inference Under Order Restrictions: The Theory and Application of Isotonic Regression. Wiley New York, 1972.
• [3] Cybakov, A. B., Introduction à l’estimation non paramétrique, vol. 41. Springer, 2003.
• [4] Dümbgen, L. and Rufibach, K., Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency., Bernoulli 15, 1 (2009), 40–68.
• [5] Durot, C., On the $\mathbbL_p$-error of monotonicity constrained estimators., Ann. Statist. 35, 3 (2007), 1080–1104.
• [6] Durot, C., Groeneboom, P., and Lopuhaä, H. P., Testing equality of functions under monotonicity constraints., Journal of Nonparametric Statistics 25, 6 (2013), 939–970.
• [7] Durot, C. and Tocquet, A.-S., On the distance between the empirical process and its concave majorant in a monotone regression framework. In, Annales de l’Institut Henri Poincare (B) Probability and Statistics (2003), vol. 39, Elsevier, pp. 217–240.
• [8] Grenander, U., On the theory of mortality measurement., Scandinavian Actuarial Journal 1956, 2 (1956), 125–153.
• [9] Kiefer, J. and Wolfowitz, J., Asymptotically minimax estimation of concave and convex distribution functions., Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34, 1 (1976), 73–85.
• [10] Kochar, S. C., Mukerjee, H., and Samaniego, F. J., Estimation of a monotone mean residual life., The Annals of Statistics 28, 3 (2000), 905–921.
• [11] Komlós, J., Major, P., and Tusnády, G., An approximation of partial sums of independent $\mathrmRV$’s and the sample $\mathrmDF$. I., Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 111–131.
• [12] Kulikov, V. N. and Lopuhaä, H. P., The limit process of the difference between the empirical distribution function and its concave majorant., Statist. Probab. Lett. 76, 16 (2006), 1781–1786.
• [13] Lepski, O. V. and Spokoiny, V., Optimal pointwise adaptive methods in nonparametric estimation., The Annals of Statistics 25, 6 (1997), 2512–2546.
• [14] Mammen, E., Estimating a smooth monotone regression function., The Annals of Statistics 19, 2 (1991), 724–740.
• [15] Mukerjee, H., Monotone nonparametric regression., The Annals of Statistics 16, 2 (1988), 741–750.
• [16] Pal, J. K. and Woodroofe, M., On the distance between cumulative sum diagram and its greatest convex minorant for unequally spaced design points., Scandinavian journal of statistics 33, 2 (2006), 279–291.
• [17] Revuz, D. and Yor, M., Continuous martingales and Brownian motion, vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1991.
• [18] Sen, B., Banerjee, M., and Woodroofe, M., Inconsistency of bootstrap: The Grenander estimator., The Annals of Statistics 38, 4 (2010), 1953–1977.
• [19] van der Vaart, A. W. and van der Laan, M. J., Smooth estimation of a monotone density., Statistics 37, 3 (2003), 189–203.
• [20] Wand, M. P. and Jones, M. C., Kernel smoothing, vol. 60 of Monographs on Statistics and Applied Probability. Chapman and Hall Ltd., London, 1995.
• [21] Wang, J.-L., Asymptotically minimax estimators for distributions with increasing failure rate., The Annals of Statistics (1986), 1113–1131.
• [22] Wang, X. and Woodroofe, M., A Kiefer–Wolfowitz comparison theorem for Wicksell’s problem., The Annals of Statistics 35, 4 (2007), 1559–1575.
• [23] Wang, Y., The limit distribution of the concave majorant of an empirical distribution function., Statistics and Probability Letters (1994), 81–84.