## Bernoulli

• Bernoulli
• Volume 12, Number 3 (2006), 469-490.

### A simple nonparametric estimator of a strictly monotone regression function

#### Abstract

A new method for monotone estimation of a regression function is proposed, which is potentially attractive to users of conventional smoothing methods. The main idea of the new approach is to construct a density estimate from the estimated values $m̂ (i/N)$ ($i =1,…,N$) of the regression function and to use these `data' for the calculation of an estimate of the inverse of the regression function. The final estimate is then obtained by a numerical inversion. Compared to the currently available techniques for monotone estimation the new method does not require constrained optimization. We prove asymptotic normality of the new estimate and compare the asymptotic properties with the unconstrained estimate. In particular, it is shown that for kernel estimates or local polynomials the bandwidths in the procedure can be chosen such that the monotone estimate is first-order asymptotically equivalent to the unconstrained estimate. We also illustrate the performance of the new procedure by means of a simulation study.

#### Article information

Source
Bernoulli, Volume 12, Number 3 (2006), 469-490.

Dates
First available in Project Euclid: 28 June 2006

https://projecteuclid.org/euclid.bj/1151525131

Digital Object Identifier
doi:10.3150/bj/1151525131

Mathematical Reviews number (MathSciNet)
MR2232727

Zentralblatt MATH identifier
1100.62045

#### Citation

Dette, Holger; Neumeyer, Natalie; Pilz, Kay F. A simple nonparametric estimator of a strictly monotone regression function. Bernoulli 12 (2006), no. 3, 469--490. doi:10.3150/bj/1151525131. https://projecteuclid.org/euclid.bj/1151525131

#### References

• [1] Brunk, H.D. (1955) Maximum likelihood estimates of monotone parameters. Ann. Math. Statist., 26, 607-616.
• [2] Bennett, C. and Sharpley, R. (1988) Interpolation of Operators, Pure Appl. Math. 129. Boston: Academic Press.
• [3] Cheng, K.F. and Lin, P.E. (1981) Nonparametric estimation of a regression function. Z. Wahrscheinlichkeitstheorie Verw. Geb., 57, 223-233.
• [4] Delecroix, M. and Thomas-Agnan, C. (2000) Spline and kernel regression under shape restrictions. In M.G. Schimek (ed.), Smoothing and Regression. Approaches, Computation and Application. New York: Wiley.
• [5] Dette, H. and Pilz, K.F. (2004) A comparative study of monotone nonparametric kernel estimates. Technical report. http://www.ruhr-uni-bochum.de/mathematik3/preprint.htm
• [6] Fan, J. and Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. London: Chapman & Hall.
• [7] Friedman, J. and Tibshirani, R. (1984) The monotone smoothing of scatterplots. Technometrics, 26, 243-250.
• [8] Gasser, T. and Mü ller, H.G. (1979) Kernel estimates of regression functions. In T. Gasser and M. Rosenblatt (eds), Smoothing Techniques for Curve Estimation, Lecture Notes in Math. 757. Berlin: Springer-Verlag.
• [9] Gijbels, I. (2005) Monotone regression. In N. Balakrishnan, S. Kotz, C.B. Read and B. Vadakovic (eds), The Encyclopedia of Statistical Sciences, 2nd edition. Hoboken, NJ: Wiley.
• [10] Hall, P. and Huang, L.-S. (2001) Nonparametric kernel regression subject to monotonicity constraints. Ann. Statist., 29, 624-647.
• [11] Kelly, C. and Rice, J. (1990) Monotone smoothing with application to dose response curves and the assessment of synergism. Biometrics, 46, 1071-1085.
• [12] Mack, Y.P. and Silverman, B.W. (1982) Weak and strong uniform consistency of kernel regression estimates. Z. Wahrscheinlichkeitstheorie Verw. Geb., 61, 405-415.
• [13] Mammen, E. (1991) Estimating a smooth monotone regression function. Ann. Statist., 19, 724-740.
• [14] Mammen, E. and Thomas-Agnan, C. (1999) Smoothing splines and shape restrictions. Scand. J. Statist., 26, 239-252.
• [15] Mammen, E., Marron, J.S., Turlach, B.A. and Wand, M.P. (2001) A general projection framework for constrained smoothing. Statist. Sci., 16, 232-248.
• [16] Mü ller, H.G. (1985) Kernel estimators of zeros and of location and size of extrema of regression functions. Scand. J. Statist., 12, 221-232.
• [17] Mukerjee, H. (1988) Monotone nonparametric regression. Ann. Statist., 16, 741-750.
• [18] Ramsay, J.O. (1988) Monotone regression splines in action (with comments). Statist. Sci., 3, 425-461.
• [19] Ramsay, J.O. (1998) Estimating smooth monotone functions. J. Roy. Statist. Soc. Ser. B, 60, 365-375.
• [20] Rice, J. (1984) Bandwidth choice for nonparametric regression. Ann. Statist., 12, 1215-1230.
• [21] Ryff, J.V. (1965) Orbits of L1-functions under doubly stochastic transformations. Trans. Amer. Math. Soc., 117, 92-100.
• [22] Ryff, J.V. (1970) Measure preserving transformations and rearrangements. J. Math. Anal. Appl., 31, 449-458.
• [23] Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. New York: Wiley.
• [24] Wand, M.P. and Jones, M.G. (1995) Kernel Smoothing. London: Chapman & Hall.
• [25] Wright, F.T. (1981) The asymptotic behavior of monotone regression estimates. Ann. Statist., 9, 443-448.