Estimation of a Convex Function: Characterizations and Asymptotic Theory

Groeneboom, Piet; Jongbloed, Geurt; Wellner, Jon A.

doi:10.1214/aos/1015345958

Cited by 185 publications

(208 citation statements)

References 17 publications

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“…For instance, in the convex single index model where Ψ 0 is known to be convex (as in the counterexample of Figure 5), one has to consider C to be the set of all convex functions. In that case, the profile estimator that minimizes Ψ → M(α, Ψ) over Ψ ∈ C will take a form that is very different from the one we obtain in Section 2.2: instead of being connected to the theory of isotonic regression as is the case in the setting we consider, it will be connected to the theory of convex estimation as in Groeneboom et al (2001). We feel that our results from Section 2.3 could be extended to such cases, but the analysis will certainly involve new arguments.…”

Section: Discussionmentioning

confidence: 99%

On the population least‐squares criterion in the monotone single index model

Balabdaoui

Durot

Fragneau

2021

Statistica Neerlandica

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Monotone single index models have gained over the past decades increasing popularity due to their flexibility and versatile use in diverse areas. Semi-parametric estimators such as the least squares and maximum likelihood estimators of the unknown index and monotone ridge function were considered to make inference in such models without having to choose some tuning parameter. Description of the asymptotic behavior of those estimators crucially depends on acquiring a good understanding of the optimization problems associated with the corresponding population criteria. In this paper, we give several insights into these criteria by proving existence of minimizers thereof over general classes of parameters. In order to describe these minimizers, we prove different results which give the direction of variation of the population criteria in general and in the special case where the common distribution of the covariates is Gaussian. A complementary simulation study was performed and whose results give support to our main theorems.

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Section: Discussionmentioning

confidence: 99%

On the population least‐squares criterion in the monotone single index model

Balabdaoui

Durot

Fragneau

2021

Statistica Neerlandica

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“…If that is not the case, more flexible approaches perform better which use less assumption on distribution patterns. For this reason to track long term changes in phytoplankton spring blooms we propose to derive the cardinal dates using a non-parametric shape constrained method, namely logconcave regression (Groeneboom et al, 2001;Groeneboom and Jongbloed, 2014;Doss, 2019). Log-concave regression meets this flexibility requirement as it does not require any tuning parameters and can be directly applied on the annual bimodal time series without any pre-processing.…”

Section: Tracking Phytoplankton Spring Bloom Dynamicsmentioning

confidence: 99%

“…In order to track long term changes in phytoplankton spring blooms we propose to derive the cardinal dates using a non-parametric shape constrained method, namely concave regression (Groeneboom et al, 2001;Groeneboom and Jongbloed, 2014;Doss, 2019). The concave or convex regression setup for a data set of size {n :(x i , y i ) : i = 1, .…”

Section: Tracking Phytoplankton Spring Bloom Dynamicsmentioning

confidence: 99%

Climate Change Induced Trends and Uncertainties in Phytoplankton Spring Bloom Dynamics

et al. 2021

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Spring phytoplankton blooms in the southern North Sea substantially contribute to annual primary production and largely influence food web dynamics. Studying long-term changes in spring bloom dynamics is therefore crucial for understanding future climate responses and predicting implications on the marine ecosystem. This paper aims to study long term changes in spring bloom dynamics in the Dutch coastal waters, using historical coastal in-situ data and satellite observations as well as projected future solar radiation and air temperature trajectories from regional climate models as driving forces covering the twenty-first century. The main objective is to derive long-term trends and quantify climate induced uncertainties in future coastal phytoplankton phenology. The three main methodological steps to achieve this goal include (1) developing a data fusion model to interlace coastal in-situ measurements and satellite chlorophyll-a observations into a single multi-decadal signal; (2) applying a Bayesian structural time series model to produce long-term projections of chlorophyll-a concentrations over the twenty-first century; and (3) developing a feature extraction method to derive the cardinal dates (beginning, peak, end) of the spring bloom to track the historical and the projected changes in its dynamics. The data fusion model produced an enhanced chlorophyll-a time series with improved accuracy by correcting the satellite observed signal with in-situ observations. The applied structural time series model proved to have sufficient goodness-of-fit to produce long term chlorophyll-a projections, and the feature extraction method was found to be robust in detecting cardinal dates when spring blooms were present. The main research findings indicate that at the study site location the spring bloom characteristics are impacted by the changing climatic conditions. Our results suggest that toward the end of the twenty-first century spring blooms will steadily shift earlier, resulting in longer spring bloom duration. Spring bloom magnitudes are also projected to increase with a 0.4% year−1 trend. Based on the ensemble simulation the largest uncertainty lies in the timing of the spring bloom beginning and -end timing, while the peak timing has less variation. Further studies would be required to link the findings of this paper and ecosystem behavior to better understand possible consequences to the ecosystem.

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“…To this end, different shape constraints have been considered: monotonicity (studied in Grenander, 1956 for d =1 and extended in Polonik, 1998 to d >1), convexity (studied in Groeneboom, Jongbloed, & Wellner, 2001 for d =1 and extended in Seregin & Wellner, 2010 to d >1), and, most prominently, log‐concavity. The log‐concave MLE

{\hat{f}}_{n}

was first introduced by Walther (2002) and has been studied in great detail recently.…”

Section: Introductionmentioning

confidence: 99%

Maximum likelihood estimation for totally positive log‐concave densities

Robeva

Sturmfels

Tran

et al. 2020

Scandinavian J Statistics

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We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP2) distributions and log-L -concave (LLC ) distributions. In both cases we also assume logconcavity in order to ensure boundedness of the likelihood function. Given n independent and identically distributed random vectors in R d from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when n ≥ 3. This holds independently of the ambient dimension d. We conjecture that the MLE is always the exponential of a tent function. We prove this result for samples in {0, 1} d or in R 2 under MTP2, and for samples in Q d under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.MSC 2010 subject classifications: Primary 62G07, 62H12; secondary 52B55

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Estimation of a Convex Function: Characterizations and Asymptotic Theory

Cited by 185 publications

References 17 publications

On the population least‐squares criterion in the monotone single index model

On the population least‐squares criterion in the monotone single index model

Climate Change Induced Trends and Uncertainties in Phytoplankton Spring Bloom Dynamics

Maximum likelihood estimation for totally positive log‐concave densities

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