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Isotonic estimators of monotone densities and distribution functions: basic facts

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  • P. Groeneboom
  • H. P. Lopuhaa

Abstract

We present short proofs of some basic results from isotonic regression theory. A straightforward argument is given to show that the left continuous version of the concave majorant of the empirical distribution function maximizes the likelihood function f↦f(X,)…f(Xn) within the class of non‐increasing densities. Similarly, it is shown that the nonparametric maximum likelihood estimator (NPMLE) of the distribution function of interval censored data has an interpretation in terms of the left derivative of a convex minor ant. Finally, a short proof is given to show that the number of vertices of the concave major ant of the uniform empirical distribution function is asymptotically normal with asymptotic mean and variance both equal to log n.

Suggested Citation

  • P. Groeneboom & H. P. Lopuhaa, 1993. "Isotonic estimators of monotone densities and distribution functions: basic facts," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 47(3), pages 175-183, September.
    • Handle: RePEc:bla:stanee:v:47:y:1993:i:3:p:175-183
    DOI: 10.1111/j.1467-9574.1993.tb01415.x
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    Cited by:

    1. Piet Groeneboom, 2021. "Grenander functionals and Cauchy's formula," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 275-294, March.
    2. Neumeyer, Natalie, 2005. "A note on uniform consistency of monotone function estimators," Technical Reports 2005,35, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.

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