arXiv subjects

Hendrik P. Lopuhaä

Publications and source records attributed to Hendrik P. Lopuhaä.

16 recordsLinked to original sources

arXiv2018

Central limit theorems for the $L_p$-error of smooth isotonic estimators

We investigate the asymptotic behavior of the $L_p$-distance between a monotone function on a compact interval and a smooth estimator of this function. Our main result is a central limit theorem for the $L_p$-error of smooth isotonic estimators obtained by smoothing a Grenander-type estimator or isotonizing the ordinary kernel estimator. As a preliminary result we establish a similar result for ordinary kernel estimators. Our results are obtained in a general setting, which includes estimation of a monotone density, regression function and hazard rate. We also perform a simulation study for testing monotonicity on the basis of the $L_2$-distance between the kernel estimator and the smoothed Grenander-type estimator.

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arXiv2018

Smooth estimation of a monotone hazard and a monotone density under random censoring

We consider kernel smoothed Grenander-type estimators for a monotone hazard rate and a monotone density in the presence of randomly right censored data. We show that they converge at rate $n^{2/5}$ and that the limit distribution at a fixed point is Gaussian with explicitly given mean and variance. It is well-known that standard kernel smoothing leads to inconsistency problems at the boundary points. It turns out that, also by using a boundary correction, we can only establish uniform consistency on intervals that stay away from the end point of the support (though we can go arbitrarily close to the right boundary).

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arXiv2018

Smoothed isotonic estimators of a monotone baseline hazard in the Cox model

We consider the smoothed maximum likelihood estimator and the smoothed Grenander-type estimator for a monotone baseline hazard rate $λ_0$ in the Cox model. We analyze their asymptotic behavior and show that they are asymptotically normal at rate $n^{m/(2m+1)}$, when~$λ_0$ is $m\geq 2$ times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.

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arXiv2018

Isotonized smooth estimators of a monotone baseline hazard in the Cox model

We consider two isotonic smooth estimators for a monotone baseline hazard in the Cox model, a maximum smooth likelihood estimator and a Grenander-type estimator based on the smoothed Breslow estimator for the cumulative baseline hazard. We show that they are both asymptotically normal at rate $n^{m/(2m+1)}$, where $m\geq 2$ denotes the level of smoothness considered, and we relate their limit behavior to kernel smoothed isotonic estimators studied in Lopuhaä and Musta (2016). It turns out that the Grenander-type estimator is asymptotically equivalent to the kernel smoothed isotonic estimators, while the maximum smoothed likelihood estimator exhibits the same asymptotic variance but a different bias. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.

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arXiv2018

The distance between a naive cumulative estimator and its least concave majorant

We consider the process $\widehatΛ_n-Λ_n$, where $Λ_n$ is a cadlag step estimator for the primitive $Λ$ of a nonincreasing function $λ$ on $[0,1]$, and $\widehatΛ_n$ is the least concave majorant of $Λ_n$. We extend the results in Kulikov and Lopuhaä (2006, 2008) to the general setting considered in Durot (2007). Under this setting we prove that a suitably scaled version of $\widehatΛ_n-Λ_n$ converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the $L_p$-distance between $\widehatΛ_n$ and $Λ_n$.

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arXiv2016

A central limit theorem for the Hellinger loss of Grenander type estimators

We consider Grenander type estimators for a monotone function $λ:[0,1]\to\mathbb{R}$, obtained as the slope of a concave (convex) estimate of the primitive of $λ$. Our main result is a central limit theorem for the Hellinger loss, which applies to statistical models that satisfy the setup in Durot (2007). This includes estimation of a monotone density, for which the limiting variance of the Hellinger loss turns out to be independent of $λ$.

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arXiv2016

Functional central limit theorems for single-stage samplings designs

For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz-Thompson empirical process and the Hájek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation.

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arXiv2014

A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation

We consider Grenander type estimators for monotone functions $f$ in a very general setting, which includes estimation of monotone regression curves, monotone densities, and monotone failure rates. These estimators are defined as the left-hand slope of the least concave majorant $\hat{F}_n$ of a naive estimator $F_n$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\hat{F}_n$ and $F_n$ is of the order $O_p(n^{-1}\log n)^{2/(4-τ)}$, for some $τ\in[0,4)$ that characterizes the tail probabilities of an approximating process for $F_n$. In typical examples, the approximating process is Gaussian and $τ=1$, in which case the convergence rate is $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the one obtained by Kiefer and Wolfowitz (1976) for the special case of estimating a decreasing density. We also obtain a similar result for the primitive of $F_n$, in which case $τ=2$, leading to a faster rate $n^{-1}\log n$, also found by Wang and Woodfroofe (2007). As an application in our general setup, we show that a smoothed Grenander type estimator and its derivative are asymptotically equivalent to the ordinary kernel estimator and its derivative in first order.

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arXiv2013

Testing equality of functions under monotonicity constraints

We consider the problem of testing equality of functions $f_j:[0,1]\to \mathbb{R}$ for $j=1,2,...,J$ the basis of $J$ independent samples from possibly different distributions under the assumption that the functions are monotone. We provide a uniform approach that covers testing equality of monotone regression curves, equality of monotone densities and equality of monotone hazards in the random censorship model. Two test statistics are proposed based on $L_1$-distances. We show that both statistics are asymptotically normal and we provide bootstrap implementations, which are shown to have critical regions with asymptotic level $α$.

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arXiv2013

Shape constrained nonparametric estimators of the baseline distribution in Cox proportional hazards model

We investigate nonparametric estimation of a monotone baseline hazard and a decreasing baseline density within the Cox model. Two estimators of a nondecreasing baseline hazard function are proposed. We derive the nonparametric maximum likelihood estimator and consider a Grenander type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. We demonstrate that the two estimators are strong consistent and asymptotically equivalent and derive their common limit distribution at a fixed point. Both estimators of a nonincreasing baseline hazard and their asymptotic properties are acquired in a similar manner. Furthermore, we introduce a Grenander type estimator for a nonincreasing baseline density, defined as the left-hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function, derived from the Breslow estimator. We show that this estimator is strong consistent and derive its asymptotic distribution at a fixed point.

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arXiv2012

The limit distribution of the $L_{\infty}$-error of Grenander-type estimators

Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n/\log n)^{-1/3}$ and the limiting distribution to be Gumbel.

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arXiv2012

Approximation of rejective sampling inclusion probabilities and application to high order correlations

This paper is devoted to rejective sampling. We provide an expansion of joint inclusion probabilities of any order in terms of the inclusion probabilities of order one, extending previous results by Hájek (1964) and Hájek (1981) and making the remainder term more precise. Following Hájek (1981), the proof is based on Edgeworth expansions. The main result is applied to derive bounds on higher order correlations, which are needed for the consistency and asymptotic normality of several complex estimators.

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arXiv2012

Central limit theorem and influence function for the MCD estimators at general multivariate distributions

We define the minimum covariance determinant functionals for multivariate location and scatter through trimming functions and establish their existence at any multivariate distribution. We provide a precise characterization including a separating ellipsoid property and prove that the functionals are continuous. Moreover, we establish asymptotic normality for both the location and covariance estimator and derive the influence function. These results are obtained in a very general multivariate setting.

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arXiv2007

Asymptotics: Particles, Processes and Inverse Problems. Festschrift for Piet Groeneboom

In September 2006, Piet Groeneboom officially retired as professor of statistics at Delft University of Technology and the Vrije Universiteit in Amsterdam. He did so by delivering his farewell lecture `Summa Cogitatio' ([42] in Piet's publication list) in the Aula of the university in Delft. To celebrate Piet's impressive contributions to statistics and probability, the workshop `Asymptotics: particles, processes and inverse problems' was held from July 10 until July 14, 2006, at the Lorentz Center in Leiden. Many leading researchers in the fields of probability and statistics gave talks at this workshop, and it became a memorable event for all who attended, including the organizers and Piet himself. This volume serves as a Festschrift for Piet Groeneboom. It contains papers that were presented at the workshop as well as some other contributions, and it represents the state of the art in the areas in statistics and probability where Piet has been (and still is) most active. Furthermore, a short CV of Piet Groeneboom and a list of his publications are included.

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arXiv2006

The behavior of the NPMLE of a decreasing density near the boundaries of the support

We investigate the behavior of the nonparametric maximum likelihood estimator $\hat{f}_n$ for a decreasing density $f$ near the boundaries of the support of $f$. We establish the limiting distribution of $\hat{f}_n(n^{-α})$, where we need to distinguish between different values of $0<α<1$. Similar results are obtained for the upper endpoint of the support, in the case it is finite. This yields consistent estimators for the values of $f$ at the boundaries of the support. The limit distribution of these estimators is established and their performance is compared with the penalized maximum likelihood estimator.

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arXiv2006

Asymptotic normality of the $L_k$-error of the Grenander estimator

We investigate the limit behavior of the $L_k$-distance between a decreasing density $f$ and its nonparametric maximum likelihood estimator $\hat{f}_n$ for $k\geq1$. Due to the inconsistency of $\hat{f}_n$ at zero, the case $k=2.5$ turns out to be a kind of transition point. We extend asymptotic normality of the $L_1$-distance to the $L_k$-distance for $1\leq k<2.5$, and obtain the analogous limiting result for a modification of the $L_k$-distance for $k\geq2.5$. Since the $L_1$-distance is the area between $f$ and $\hat{f}_n$, which is also the area between the inverse $g$ of $f$ and the more tractable inverse $U_n$ of $\hat{f}_n$, the problem can be reduced immediately to deriving asymptotic normality of the $L_1$-distance between $U_n$ and $g$. Although we lose this easy correspondence for $k>1$, we show that the $L_k$-distance between $f$ and $\hat{f}_n$ is asymptotically equivalent to the $L_k$-distance between $U_n$ and $g$.

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