Electronic Journal of Statistics

Weak convergence of the least concave majorant of estimators for a concave distribution function

Brendan K. Beare and Zheng Fang

Full-text: Open access

Abstract

We study the asymptotic behavior of the least concave majorant of an estimator of a concave distribution function under general conditions. The true concave distribution function is permitted to violate strict concavity, so that the empirical distribution function and its least concave majorant are not asymptotically equivalent. Our results are proved by demonstrating the Hadamard directional differentiability of the least concave majorant operator. Standard approaches to bootstrapping fail to deliver valid inference when the true distribution function is not strictly concave. While the rescaled bootstrap of Dümbgen delivers asymptotically valid inference, its performance in small samples can be poor, and depends upon the selection of a tuning parameter. We show that two alternative bootstrap procedures—one obtained by approximating a conservative upper bound, the other by resampling from the Grenander estimator—can be used to construct reliable confidence bands for the true distribution. Some related results on isotonic regression are provided.

Article information

Source
Electron. J. Statist., Volume 11, Number 2 (2017), 3841-3870.

Dates
Received: October 2016
First available in Project Euclid: 18 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1508292528

Digital Object Identifier
doi:10.1214/17-EJS1349

Mathematical Reviews number (MathSciNet)
MR3714300

Zentralblatt MATH identifier
06796557

Subjects
Primary: 62G09: Resampling methods 62G20: Asymptotic properties

Keywords
Least concave majorant Hadamard directional derivative Grenander estimator rescaled bootstrap

Rights
Creative Commons Attribution 4.0 International License.

Citation

Beare, Brendan K.; Fang, Zheng. Weak convergence of the least concave majorant of estimators for a concave distribution function. Electron. J. Statist. 11 (2017), no. 2, 3841--3870. doi:10.1214/17-EJS1349. https://projecteuclid.org/euclid.ejs/1508292528


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References

  • Balabdaoui, F., Jankowski, H., Pavlides, M., Seregin, A. and Wellner, J. A. (2011). On the Grenander estimator at zero., Statistica Sinica, 21(2):873–899.
  • Beare, B. K. (2010). Copulas and temporal dependence., Econometrica, 78(1):395–410.
  • Beare, B. K. (2012). Archimedean copulas and temporal dependence., Econometric Theory, 28(6):1165–1185.
  • Beare, B. K. and Moon, J. -M. (2015). Nonparametric tests of density ratio ordering., Econometric Theory, 31(3):471–492.
  • Beare, B. K. and Schmidt, L. D. W. (2016). An empirical test of pricing kernel monotonicity., Journal of Applied Econometrics, 31(2):338–356.
  • Beare, B. K. and Seo, J. (2014). Time irreversible copula-based Markov models., Econometric Theory, 30(5):923–960.
  • Beran, R. and Srivastava, M. S. (1985). Bootstrap tests and confidence regions for functions of a covariance matrix., Annals of Statistics, 13(1):95–115.
  • Beran, R. and Srivastava, M. S. (1987). Correction: Bootstrap tests and confidence regions for functions of a covariance matrix., Annals of Statistics, 15(1):470–471.
  • Bickel, P. J., Götze, F. and van Zwet, W. R. (1997). Resampling fewer than $n$ observations: gains, losses, and remedies for losses., Statistica Sinica, 7(1):1–31.
  • Bretagnolle, J. (1983). Lois limites du bootstrap de certaines fonctionelles., Annales de l’Institut Henri Poincaré (B), 19(3):281–296.
  • Brunk, H. D. (1958). On the estimation of parameters restricted by inequalities., Annals of Mathematical Statistics, 29:437–454.
  • Bücher, A. and Ruppert, M. (2013). Consistent testing for a constant copula under strong mixing based on the tapered block multiplier technique., Journal of Multivariate Analysis, 116:208–229.
  • Carolan, C. A. (2002). The least concave majorant of the empirical distribution function., Canadian Journal of Statistics, 30(2):317–328.
  • Carolan, C. A. and Dykstra, R. (1999). Asymptotic behavior of the Grenander estimator at density flat regions., Canadian Journal of Statistics, 27(3):557–566.
  • Carolan, C. A. and Dykstra, R. (2001). Marginal densities of the least concave majorant of Brownian motion., Annals of Statistics, 29(6):1732–1750.
  • Carolan, C. A. and Tebbs, J. M. (2005). Nonparametric tests for and against likelihood ratio ordering in the two sample problem., Biometrika, 92(1):159–171.
  • Dehling, H. and Philipp, W. (2002). Empirical process techniques for dependent data. In Dehling, H., Mikosch, T. and Sørensen, M. (Eds.), Empirical Process Techniques for Dependent Data, 3–113. Birkhäuser.
  • Delgado, M. A. and Escanciano, J. C. (2012). Distribution-free tests of stochastic monotonicity., Journal of Econometrics, 170(1):68–75.
  • Delgado, M. A. and Escanciano, J. C. (2013). Conditional stochastic dominance testing., Journal of Business and Economic Statistics, 31(1):16–28.
  • Delgado, M. A. and Escanciano, J. C. (2016). Distribution-free tests of conditional moment inequalities., Journal of Statistical Planning and Inference, 173:99–108.
  • Dümbgen, L. (1993). On nondifferentiable functions and the bootstrap., Probability Theory and Related Fields, 95(1):125–140.
  • Durot, C. (2007). On the $\mathbbL_p$-error of monotonicity constrained estimators., Annals of Statistics, 35(3):1080–1104.
  • Durot, C., Kulikov, V. N. and Lopuhaä, H. P. (2012). The limit distribution of the $L_\infty$-error of Grenander-type estimators., Annals of Statistics, 40(3):1578–1608.
  • Durot, C. and Tocquet, A. -S. (2003). On the distance between the empirical process and its concave majorant in a monotone regression framework., Annales de l’Institut Henri Poincaré (B), 39(2):217–240.
  • Eggermont, P. P. B. and LaRiccia, V. N. (2000). Maximum likelihood estimation of smooth monotone and unimodal densities., Annals of Statistics, 28(3):922–947.
  • Eggermont, P. P. B. and LaRiccia, V. N. (2001)., Maximum Penalized Likelihood Estimation, Volume I: Density Estimation. Springer Series in Statistics, Springer, New York.
  • Fang, Z. and Santos, A. (2015). Inference on directionally differentiable functions., arXiv:1404.3763v2
  • Giné, E. and Zinn, J. (1991). Gaussian characterization of uniform Donsker classes of functions., Annals of Probability, 19(2):758–782.
  • Grenander, U. (1956). On the theory of mortality measurement. Part II., Scandinavian Actuarial Journal, 39:125–153.
  • Groeneboom, P. (1985). Estimating a monotone density. In Le Cam, L. M. and Olshen, R. A. (Eds.), Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. 2, 539–555. Wadsworth Statistics/Probability Series, Wadsworth, Belmont.
  • Groeneboom, P., Hooghiemstra, G. and Lopuhaä, H. P. (1999). Asymptotic normality of the $L_1$ error of the Grenander estimator., Annals of Statistics, 27(4):1316–1347.
  • Groeneboom, P. and Jongbloed, G. (2014)., Nonparametric Estimation Under Shape Constraints:Estimators, Algorithms and Asymptotics. Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, New York.
  • Jankowski, H. (2014). Convergence of linear functionals of the Grenander estimator under misspecification., Annals of Statistics, 42(2):625–653.
  • Kiefer, J. and Wolfowitz, J. (1976). Asymptotically minimax estimation of concave and convex distribution functions., Probability Theory and Related Fields, 34(1):73–85.
  • Kosorok, M. R. (2008a). Bootstrapping the Grenander estimator. In Balakrishnan, N., Peña, E. A. and Silvapulle, M. J. (Eds.), Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, 282–292. Institute of Mathematical Statistics, Beachwood.
  • Kosorok, M. R. (2008b)., Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics, Springer, New York.
  • Kulikov, V. N. and Lopuhaä, H. P. (2005). Asymptotic normality of the $L_k$-error of the Grenander estimator., Annals of Statistics, 33(5):2228–2255.
  • Kulikov, V. N. and Lopuhaä, H. P. (2006a). The behavior of the NPMLE of a decreasing density near the boundaries of the support., Annals of Statistics, 34(2):742–768.
  • Kulikov, V. N. and Lopuhaä, H. P. (2006b). The limit process of the difference between the empirical distribution function and its concave majorant., Statistics and Probability Letters, 76(16):1781–1786.
  • Kulikov, V. N. and Lopuhaä, H. P. (2008). Distribution of global measures of deviation between the empirical distribution function and its concave majorant., Journal of Theoretical Probability, 21(2):356–377.
  • Mukerjee, H. (1988). Monotone nonparametric regression., Annals of Statistics, 16(2):741–750.
  • Paparoditis, E. and Politis, D. N. (2002). The local bootstrap for Markov processes., Journal of Statistical Planning and Inference, 108(1):301–328.
  • Pollard, D. (1984)., Convergence of Stochastic Processes. Springer Series in Statistics, Springer, New York.
  • Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density., Sankhyā Series A, 31:23–36.
  • Radulović, D. (2002). On the bootstrap and empirical processes for dependent sequences. In Dehling, H., Mikosch, T. and Sørensen, M. (Eds.), Empirical Process Techniques for Dependent Data, 345–364. Birkhäuser, Boston.
  • Sen, B., Banerjee, M. and Woodroofe, M. (2010). Inconsistency of bootstrap: the Grenander estimator., Annals of Statistics, 38(4):1953–1977.
  • Shapiro, A. (1990). On concepts of directional differentiability., Journal of Optimization Theory and Applications, 66(3):477–487.
  • Shapiro, A. (1991). Asymptotic analysis of stochastic programs., Annals of Operations Research, 30(1):169–186.
  • Shorack, G. R. and Wellner, J. A. (1986)., Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics, Wiley, New York.
  • van der Vaart, A. (1994). Weak convergence of smoothed empirical processes., Scandinavian Journal of Statistics, 21(4):501–504.
  • van der Vaart, A. (1998)., Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge.
  • van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes. Springer Series in Statistics, Springer, New York.
  • Volgushev, S. and Shao, X. (2014). A general approach to the joint asymptotic analysis of statistics from subsamples., Electronic Journal of Statistics, 8(1):390–431.
  • Walther, G. (2009) Inference and modeling with log-concave distributions., Statistical Science, 24(3):319–327.
  • Wang, Y. (1994). The limit distribution of the concave majorant of an empirical distribution function., Statistics and Probability Letters, 20(1):81–84.
  • Woodroofe, M. and Sun, J. (1993). A penalized maximum likelihood estimate of (f(0+)) when (f) is nonincreasing., Statistica Sinica, 3(2):501–515.