Univariate log-concave density estimation with symmetry or modal constraints

Doss, Charles R.; Wellner, Jon A.

doi:10.1214/19-ejs1574

Cited by 9 publications

(76 citation statements)

References 49 publications

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“…For the remainder of the proof of Theorem 3, one considers cases where on either the left side, the right side, or both sides, there is no sequence of shared touch points converging to infinity, and deriving a contradiction. The argument follows as in the proof of Theorem 5.2 of [14]. This completes the proof of Theorem 3.…”

Section: Now Letmentioning

confidence: 53%

“…Then the remainder of the proof follows as in the proof of Theorem 6.3 of [24] (see also [34]) and of Theorem 5.8 of [14]. By Lemma 4 below, n 1/5 τ n,R = O p (1), and this allows us to also conclude thatH n,R and its first, second, and third derivatives are all tight in appropriate metric spaces.…”

Section: Define Alsõmentioning

confidence: 78%

“…Proof The result follows from Lemma 1, via the analysis used in the the proof of Lemma 8.10 of [14] (see also Lemma 2.7 of [23]).…”

Section: Now Letmentioning

confidence: 99%

“…Remark 1 If X is replaced by X a,σ (t) := σW (t) − 4at 3 for constants a, σ > 0, then the conclusion of Theorem 3 still holds; in this case, we denote the process r 0 of the theorem by r 0 a,σ . Remark 2 The knot definitions in Theorem 3 differ from those in Theorem 5.2 of [14], in the context of a mode constraint. Condition (iii) of Theorem 5.2 of [14] (which is analogous to Condition 3 of Theorem 2) is based on knots τ 0 + and τ 0 − , one (but almost surely not both) of which may be 0.…”

Section: Now Letmentioning

confidence: 99%

“…Remark 2 The knot definitions in Theorem 3 differ from those in Theorem 5.2 of [14], in the context of a mode constraint. Condition (iii) of Theorem 5.2 of [14] (which is analogous to Condition 3 of Theorem 2) is based on knots τ 0 + and τ 0 − , one (but almost surely not both) of which may be 0. These knots are potentially distinct from the knots τ L and τ R in that setup, where τ L , τ R can never be 0.…”

Section: Now Letmentioning

confidence: 99%

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Concave regression: value-constrained estimation and likelihood ratio-based inference

Doss

2018

Math. Program.

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We propose a likelihood ratio statistic for forming hypothesis tests and confidence intervals for a nonparametrically estimated univariate regression function, based on the shape restriction of concavity (alternatively, convexity). Dealing with the likelihood ratio statistic requires studying an estimator satisfying a null hypothesis, that is, studying a concave least-squares estimator satisfying a further equality constraint. We study this null hypothesis least-squares estimator (NLSE) here, and use it to study our likelihood ratio statistic. The NLSE is the solution to a convex program, and we find a set of inequality and equality constraints that characterize the solution. We also study a corresponding limiting version of the convex program based on observing a Brownian motion with drift. The solution to the limit problem is a stochastic process. We study the optimality conditions for the solution to the limit problem and find that they match those we derived for the solution to the finite sample problem. This allows us to show the limit stochastic process yields the limit distribution of the (finite sample) NLSE. We conjecture that the likelihood ratio statistic is asymptotically pivotal, meaning that it has a limit distribution with no nuisance parameters to be estimated, which makes it a very effective tool for this difficult inference problem. We provide a partial proof of this conjecture, and we also provide simulation evidence strongly supporting this conjecture.

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Section: Now Letmentioning

confidence: 53%

Section: Define Alsõmentioning

confidence: 78%

“…Proof The result follows from Lemma 1, via the analysis used in the the proof of Lemma 8.10 of [14] (see also Lemma 2.7 of [23]).…”

Section: Now Letmentioning

confidence: 99%

Section: Now Letmentioning

confidence: 99%

Section: Now Letmentioning

confidence: 99%

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Concave regression: value-constrained estimation and likelihood ratio-based inference

Doss

2018

Math. Program.

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Modal linear regression using log-concave distributions

Kim

Seo

2020

J. Korean Stat. Soc.

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The modal linear regression suggested by Yao and Li (Scand J Stat 41(3):656-671, 2014) models the conditional mode of a response Y given a vector of covariates as a linear function of . To identify the conditional mode of Y given , existing methods utilize a kernel density estimator to obtain the distribution of Y given . Like other kernel-based methods, these require a suitable choice of tuning parameters, and no unified objective function exists for estimating regression parameters. In this paper, we propose a model-based modal linear regression using a family of log-concave distributions. The proposed method does not require tuning parameters and enables us to construct an explicit likelihood function. To estimate the regression parameters with an estimated log-concave density, we turned the log-likelihood into a sum of affine functions using a dual representation of piecewise linear concave functions so that well-known linear programming techniques can be adopted. Simulation studies reveal that the proposed method produces more efficient estimators than kernel-based methods. A real data example is also presented to illustrate the applicability of the proposed method.

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Inference for the mode of a log-concave density

Doss

Wellner²

2019

Ann. Statist.

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We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihood estimator of a logconcave density and the constrained maximum likelihood estimator where the constraint is that the mode of the density is fixed, say at m. The constrained estimation problem is studied in detail in Doss and Wellner [2018].Here the results of that paper are used to show that, under the null hypothesis (and strict curvature of − log f at the mode), the likelihood ratio statistic is asymptotically pivotal: that is, it converges in distribution to a limiting distribution which is free of nuisance parameters, thus playing the role of the χ 2 1 distribution in classical parametric statistical problems. By inverting this family of tests we obtain new (likelihood ratio based) confidence intervals for the mode of a log-concave density f . These new intervals do not depend on any smoothing parameters. We study the new confidence intervals via Monte Carlo methods and illustrate them with two real data sets. The new intervals seem to have several advantages over existing procedures. Software implementing the test and confidence intervals is available in the R package logcondens.mode.

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Univariate log-concave density estimation with symmetry or modal constraints

Cited by 9 publications

References 49 publications

Concave regression: value-constrained estimation and likelihood ratio-based inference

Concave regression: value-constrained estimation and likelihood ratio-based inference

Modal linear regression using log-concave distributions

Inference for the mode of a log-concave density

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