A note on uniform consistency of monotone function estimators

Neumeyer, Natalie

doi:10.1016/j.spl.2006.11.004

Cited by 12 publications

(11 citation statements)

References 34 publications

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“…That means,f I is pointwise consistent iff is pointwise consistent and both have the same asymptotic normal distribution. The consistency off I follows from results stated in Neumeyer (2007). For the sake of simplicity the asymptotic considerations are stated for A = [0, 1] but they hold true for any compact interval A Theorem 1 Let the density f ∈ C 2 ([0, 1]) be strictly increasing and K be a kernel of order 2.…”

Section: Monotone Density Estimatorsmentioning

confidence: 93%

Shape constrained kernel density estimation

Birke

2009

Journal of Statistical Planning and Inference

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In this paper, a method for estimating monotone, convex and log-concave densities is proposed. The estimation procedure consists of an unconstrained kernel estimator which is modified in a second step with respect to the desired shape constraint by using monotone rearrangements. It is shown that the resulting estimate is a density itself and shares the asymptotic properties of the unconstrained estimate. A short simulation study shows the finite sample behavior.

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Section: Monotone Density Estimatorsmentioning

confidence: 93%

Shape constrained kernel density estimation

Birke

2009

Journal of Statistical Planning and Inference

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“…See for example the relatively recent contributions by Groeneboom (2001), Hall and Huang (2001, 2002), Hall and Kang (2005), Dette et al (2006), Antoniadis et al (2007), Neumeyer (2007), Pal and Woodroofe (2007), Birke and Dette (2007), Birke (2009), Dümbgen and Rufibach (2009), Cule et al (2010) and the references therein. Earlier work includes that of Friedman and Tibshirani (1984), Mukerjee (1988), Kelly and Rice (1990), Mammen (1991, 1995) and Sun and Woodroofe (1996).…”

Section: Introductionmentioning

confidence: 99%

Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error

Carroll

Delaigle

Hall

2011

Journal of the American Statistical Association

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In many applications we can expect that, or are interested to know if, a density function or a regression curve satisfies some specific shape constraints. For example, when the explanatory variable, X, represents the value taken by a treatment or dosage, the conditional mean of the response, Y , is often anticipated to be a monotone function of X. Indeed, if this regression mean is not monotone (in the appropriate direction) then the medical or commercial value of the treatment is likely to be significantly curtailed, at least for values of X that lie beyond the point at which monotonicity fails. In the case of a density, common shape constraints include log-concavity and unimodality. If we can correctly guess the shape of a curve, then nonparametric estimators can be improved by taking this information into account. Addressing such problems requires a method for testing the hypothesis that the curve of interest satisfies a shape constraint, and, if the conclusion of the test is positive, a technique for estimating the curve subject to the constraint. Nonparametric methodology for solving these problems already exists, but only in cases where the covariates are observed precisely. However in many problems, data can only be observed with measurement errors, and the methods employed in the error-free case typically do not carry over to this error context. In this paper we develop a novel approach to hypothesis testing and function estimation under shape constraints, which is valid in the context of measurement errors. Our method is based on tilting an estimator of the density or the regression mean until it satisfies the shape constraint, and we take as our test statistic the distance through which it is tilted. Bootstrap methods are used to calibrate the test. The constrained curve estimators that we develop are also based on tilting, and in that context our work has points of contact with methodology in the errorfree case.

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“…Theorem 3.1 can now be proved by similar arguments as in [25], which give Proof of Theorem 3.2 The assertion follows by combining several lemmas which we prove in the following. We begin the asymptotic analysis with a Taylor expansion of the differencem −1…”

Section: Appendixmentioning

confidence: 67%

A note on estimating a smooth monotone regression by combining kernel and density estimates

Birke

Dette

2008

Journal of Nonparametric Statistics

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In a recent paper Dette, Neumeyer and Pilz (H. Dette, N. Neumeyer, and K.F. Pilz, A simple nonparametric estimator of a strictly monotone regression function, Bernoulli 12 (2006), pp. 469-490) proposed a new nonparametric estimate of a smooth monotone regression function. This method is based on a nondecreasing rearrangement of an arbitrary unconstrained nonparametric estimator. Under the assumption of a twice continuously differentiable regression function, the estimate is first-order asymptotic equivalent to the unconstrained estimate and other types of smooth monotone estimates. In this note, we provide a more refined asymptotic analysis of the monotone regression estimate. In the case where the regression function is increasing but only once continuously differentiable, we prove the asymptotic normality of an appropriately standardised version of the estimate, where the asymptotic variance is of order n −2/3−ε , the bias is of order n −1/3+ε and ε > 0 is small. Therefore, the rate of convergence of the new estimate is worse than the rate n −1/3 of the (unsmoothed) monotone least squares estimate. On the other hand, if the derivative of the regression function is Hölder continuous, the rate of convergence of the new estimate is faster than n −1/3 .

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A note on uniform consistency of monotone function estimators

Cited by 12 publications

References 34 publications

Shape constrained kernel density estimation

Shape constrained kernel density estimation

Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error

A note on estimating a smooth monotone regression by combining kernel and density estimates

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