Abstract
We investigate the distribution of some global measures of deviation between the empirical distribution function and its least concave majorant. In the case that the underlying distribution has a strictly decreasing density, we prove asymptotic normality for several L k -type distances. In the case of a uniform distribution, we also establish their limit distribution together with that of the supremum distance. It turns out that in the uniform case, the measures of deviation are of greater order and their limit distributions are different.
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Carolan, C.A.: The least concave majorant of the empirical distribution function. Can. J. Stat. 30, 317–328 (2002)
Dudley, R.M., Norvaiša, R.: Differentiability of Six Operators on Nonsmooth Functions and p-Variation. Lecture Notes in Mathematics, vol. 1703. Springer, Berlin (1999)
Durot, C.: A Kolmogorov-type test for monotonicity of regression. Stat. Probab. Lett. 63, 425–433 (2003)
Durot, C., Tocquet, A.S.: On the distance between the empirical process and its concave majorant in a monotone regression framework. Ann. Inst. H. Poincaré Probab. Stat. 39, 217–240 (2003)
Groeneboom, P.: The concave majorant of Brownian motion. Ann. Probab. 11(4), 1016–1027 (1983)
Groeneboom, P.: Estimating a monotone density. In: Le, L.M. (ed.) Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, vol. II, pp. 539–555. Wadsworth, Belmont (1985)
Groeneboom, P., Hooghiemstra, G., Lopuhaä, H.P.: Asymptotic normality of the L 1 error of the Grenander estimator. Ann. Stat. 27, 1316–1347 (1999)
Kiefer, J., Wolfowitz, J.: Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrsch. Verw. Geb. 34, 73–85 (1976)
Kómlos, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV’s and the sample DF. Z. Wahrsch. Verw. Geb. 32, 111–131 (1975)
Kulikov, V.N.: Direct and indirect use of maximum likelihood. PhD Thesis, TU Delft (2003)
Kulikov, V.N., Lopuhaä, H.P.: Asymptotic normality of the L k -error of the Grenander estimator. Ann. Stat. 33(5), 2228–2255 (2005)
Kulikov, V.N., Lopuhaä, H.P.: The limit process of the difference between the empirical distribution function and its concave majorant. Stat. Probab. Lett. 76(16), 1781–1786 (2006)
Robertson, T., Wright, F.T., Dykstra, R.L.: Order Restricted Inference. Wiley, New York (1988)
Wang, J.L.: Asymptotically minimax estimators for distributions with increasing failure rate. Ann. Stat. 14(3), 1113–1131 (1986)
Wang, J.L.: Estimators of a distribution function with increasing failure rate average. J. Stat. Plann. Inference 16(3), 415–427 (1987)
Wang, Y.: The limit distribution of the concave majorant of an empirical distribution function. Stat. Probab. Lett. 20, 81–84 (1994)
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Kulikov, V.N., Lopuhaä, H.P. Distribution of Global Measures of Deviation Between the Empirical Distribution Function and Its Concave Majorant. J Theor Probab 21, 356–377 (2008). https://doi.org/10.1007/s10959-007-0103-0
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DOI: https://doi.org/10.1007/s10959-007-0103-0