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CONSISTENCY OF L(P)-BEST MONOTONE APPROXIMATIONS

JOURNAL OF STATISTICAL PLANNING AND INFERENCE 1995年第3期47卷 295-318页

作者： CUESTAALBERTOS, JA DOMINGUEZMENCHERO, JS MATRAN, C UNIV CANTABRIA DEPT MATEMATE-39005 SANTANDERSPAIN UNIV VALLADOLID DEPT ESTADIST & INVEST OPERATVALLADOLIDSPAIN UNIV OVIEDO DEPT MATEMATOVIEDOSPAIN

The paper studies consistency properties of the empirical L(p)-best monotone approximations of estimates of an unknown function. Under the hypothesis of the estimates being 'well-behaved approximations', we prove the almost sure and sample-L(p)-consistency of the procedure. As a main consequence we obtain the consistency of the L(p)-best monotone approximation of the kernel regression estimate. The obtention of this result involves as a previous fact of independent interest the empirical L(p)-consistency of the kernel estimate of the regression function. The paper includes some simulations that illustrate the performance of the suggested method.

关键词： L(P)-BEST MONOTONE APPROXIMATION L(P)-DIP L(P)-CONSISTENCY regression function KERNEL regression ESTIMATE

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RANDOM APPROXIMATIONS TO SOME MEASURES OF ACCURACY IN NONPARAMETRIC CURVE ESTIMATION

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JOURNAL OF MULTIVARIATE ANALYSIS 1986年第1期20卷 91-113页

作者： MARRON, JS HARDLE, W UNIV HEIDELBERG SONDERFORSCH BEREICH 123D-6900 HEIDELBERGFED REP GER

This paper deals with a quite general nonparametric statistical curve estimation setting. Special cases include estimation or probability density functions, regression functions, and hazard functions. The class of “fractional delta sequence estimators” is defined and treated here. This class includes the familiar kernel, orthogonal series, and histogram methods. It is seen that, under some mild assumptions, both the average square error and integrated square error provide reasonable (random) approximations to the mean integrated square error . This is important for two reasons. First, it provides theoretical backing to a practice that has been employed in several simulation studies. Second, it provides a vital tool for proving theorems about selecting smoothing parameters for several different nonparametric curve estimators.

关键词： 62H12 62G05 Hazard functions mean integrated square error nonparametric estimation regression function

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On characterizing discrete distributions via conditional expectations of weak record values

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METRIKA 2007年第2期66卷 129-138页

作者： Danielak, Katarzyna Dembinska, Anna Polish Acad Sci Inst Math PL-00956 Warsaw Poland Warsaw Univ Technol Fac Math & Informat Sci PL-00661 Warsaw Poland

Let (W (n) ,n >= 0) denote the sequence of weak records from a distribution with support S = {alpha(0),alpha(1),...,alpha (N) }. In this paper, we consider regression functions of the form psi (n) (x) = E(h(W (n) ) |W (n+1) = x), where h(.) is some strictly increasing function. We show that a single function psi (n) (.) determines F uniquely up to F(alpha(0)). Then we derive an inversion formula which enables us to obtain F from knowledge of psi (n) (.), psi (n-1)(.), h(.) and F(alpha(0)).

关键词： characterization weak records regression function

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Nonparametric regression with parametric help

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ELECTRONIC JOURNAL OF STATISTICS 2020年第2期14卷 3845-3868页

作者： Lee, Young K. Mammen, Enno Nielsen, Jens P. Park, Byeong U. Kangwon Natl Univ Chunchon South Korea Heidelberg Univ Heidelberg Germany City Univ London London England Seoul Natl Univ Seoul South Korea

In this paper we propose a new nonparametric regression technique. Our proposal has common ground with existing two-step procedures in that it starts with a parametric model. However, our approach differs from others in the choice of parametric start within the parametric family. Our proposal chooses a function that is the projection of the unknown regression function onto the parametric family in a certain metric, while the existing methods select the best approximation in the usual L-2 metric. We find that the difference leads to substantial improvement in the performance of regression estimators in comparison with direct one-step estimation, irrespective of the choice of a parametric model. This is in contrast with the existing two-step methods, which fail if the chosen parametric model is largely misspecified. We demonstrate this with sound theory and numerical experiment.

关键词： regression function bias profiling technique local linear estimation cross-validatory bandwidth selectors

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Developing the volume models for 5 major species of street trees in Gwangju metropolitan city of Korea

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URBAN FORESTRY & URBAN GREENING 2016年 18卷 53-58页

作者： Kim, Hyun-Jun Lee, Sang-Hyun Korea Univ Plus Ecoleader Educ Ctr BK21 Anam Ro 145 Seoul 02841 South Korea Korea Univ Dept Environm Sci & Ecol Engn Anam Ro 145 Seoul 02841 South Korea Chonbuk Natl Univ Dept Forest Environm Sci Baekjedae Ro 567 Jeonju 54896 South Korea

Street trees are one of the most important parts of an urban forest system because they have various functions including environmental conservation, harmony of scenery, traffic safety, and prevention. Moreover, of the analysis factors considered for street trees, volume is mainly used not only for further research on health assessment, environment analysis, carbon storage estimation, and economic analysis, but also for local government management of street trees. For this reason, this study was performed to develop volume models for the 5 major species of street tree of Gwangju metropolitan city in Korea. After selecting the 5 major species-Ginkgo biloba, Zelkova serrata, Metasequoia glytroboides, Prunus serrulata, and Platanus occidentalis-one hundred sample trees of each species were randomly chosen considering diameter at the breast height (DBH), and then diameter at every 2 m height, height, and crown radius were measured. Volume was calculated using the Huber equation. regression analysis, variance analysis, and descriptive statistical analysis were conducted using the 4 regression equations, and the best volume model was chosen by comparing precision, accuracy, credibility, and normality. From the results, all species showed the same patterns according to model type. That is, models that included diameter and height out-performed models with crown basal area had the highest precision, while model type IV using crown basal area (CBA) as a parameter showed the lowest goodness of fit because CBA can be influenced by volume, planting distance and space, or pollarding. Thus, using DBH and H together should be suitable for designing the volume models for urban-grown trees. However, properly trained experts are very important in order to avoid measurement error. Moreover, a large amount of data on Platanus ocidentalis is required to avoid large error in volume due to pollarding treatment. To sum up, model type III was chosen as the best fit models for all species wit

关键词： Gwangju metropolitan city regression function Street tree Urban forestry Volume model

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Estimating covariate functions associated to multivariate risks: a level set approach (vol 78, pg 497, 2015)

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METRIKA 2015年第5期78卷 527-527页

作者： Di Bernardino, Elena Laloe, Thomas Servien, Remi IMATH Conservatoire Natl Arts & Metiers EA4629 F-75003 Paris France Univ Nice Sophia Antipolis CNRS LJAD UMR 7351 F-06100 Nice France Univ Toulouse Res Ctr Food Toxicol INRA Toxalim UMR1331 F-31027 Toulouse France

The aim of this paper is to study the behavior of a covariate function in a multivariate risks scenario. The first part of this paper deals with the problem of estimating the $$c$$ -upper level sets $${L(c)= \{F(x) \ge c \}}$$ , with $$c \in (0,1)$$ , of an unknown distribution function $$F$$ on $$\mathbb {R}^d_+$$ . A plug-in approach is followed. We state consistency results with respect to the volume of the symmetric difference. In the second part, we obtain the $$L_p$$ -consistency, with a convergence rate, for the regression function estimate on these level sets $$L(c)$$ . We also consider a new multivariate risk measure: the Covariate-Conditional-Tail-Expectation. We provide a consistent estimator for this measure with a convergence rate. We propose a consistent estimate when the regression cannot be estimated on the whole data set. Then, we investigate the effects of scaling data on our consistency results. All these results are proven in a non-compact setting. A complete simulation study is detailed and a comparison with parametric and semi-parametric approaches is provided. Finally, a real environmental application of our risk measure is provided.

关键词： Multidimensional distribution function Plug-in estimation regression function

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ON THE ALMOST EVERYWHERE PROPERTIES OF THE KERNEL regression ESTIMATE

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ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 1991年第2期43卷 311-326页

作者： PAWLAK, M UNIV MANITOBA DEPT ELECT & COMP ENGNWINNIPEG R3T 2N2MANITOBACANADA

The regression m(x) = E{Y \ X = x} is estimated by the kernel regression estimate m triple-overdot (x) calculated from a sequence (X1, Y1),..., (X(n), Y(n)) of independent identically distributed random vectors from R(d) x R. The second order asymptotic expansions for Em triple-overdot (x) and var m triple-overdot (x) are derived. The expansions hold for almost all (mu) x is-an-element-of R(d), mu is the probability measure of X. No smoothing conditions on mu and m are imposed. As a result, the asymptotic distribution-free normality for a stochastic component of m triple-overdot (x) is established. Also some bandwidth-selection rule is suggested and bias adjustment is proposed.

关键词： regression function KERNEL ESTIMATE ASYMPTOTIC EXPANSIONS DISTRIBUTION-FREE PROPERTIES ASYMPTOTIC NORMALITY BANDWIDTH-SELECTION BIAS ADJUSTMENT

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AN EQUIVALENCE THEOREM FOR L1 CONVERGENCE OF THE KERNEL regression ESTIMATE

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 1989年第1期23卷 71-82页

作者： DEVROYE, L KRZYZAK, A MCGILL UNIV SCH COMP SCIMONTREAL H3A 2A7QUEBECCANADA MCGILL UNIV DEPT MATH & STATMONTREAL H3A 2A7QUEBECCANADA CONCORDIA UNIV DEPT COMP SCIMONTREAL H3G 1M8QUEBECCANADA

We show that all modes of convergence in L 1 (in probability, almost surely, complete) for the standard kernel regression estimate are equivalent.

关键词： Primary 62G05 regression function nonparametric estimation consistency strong convergence convergence kernel estimate

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Generalized kernel regression estimator for dependent size-biased data

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2012年第3期142卷 708-727页

作者： Chaubey, Yogendra P. Laib, Naamane Li, Jun Univ Paris 06 LSTA F-75252 Paris 05 France Concordia Univ Dept Math & Stat Montreal PQ H3G 1M8 Canada

This paper considers nonparametric regression estimation in the context of dependent biased nonnegative data using a generalized asymmetric kernel. It may be applied to a wider variety of practical situations, such as the length and size biased data. We derive theoretical results using a deep asymptotic analysis of the behavior of the estimator that provides consistency and asymptotic normality in addition to the evaluation of the asymptotic bias term. The asymptotic mean squared error is also derived in order to obtain the optimal value of smoothing parameters required in the proposed estimator. The results are stated under a stationary ergodic assumption, without assuming any traditional mixing conditions. A simulation study is carried out to compare the proposed estimator with the local linear regression estimate. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Ergodic process Gamma density function Length biased data Martingale difference Mixing MSE Normality regression function

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Reference values of ductus venosus flow velocities and calculated waveform indices

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PRENATAL DIAGNOSIS 2000年第8期20卷 623-634页

作者： Bahlmann, F Wellek, S Reinhardt, I Merz, E Steiner, E Welter, C Univ Mainz Dept Obstet & Gynecol D-55101 Mainz Germany Dept Biostat Mannheim Germany

In recent years. investigations of the venous vascular system have become increasingly important in the assessment of fetal myocardial function. The aim of the present Doppler ultrasound study was to establish both new reference ranges for blood flow velocity during the different phases of the cardiac cycle (S, SD, D, a) and various calculated indices (S - a)/S, (S - a)V-mean, (S - a)D, S/D, a/S, S/a) for the ductus venosus. Pulsed-wave colour Doppler was used in this prospective cross-sectional study to examine 696 women with low-risk pregnancies during the period from 14 to 41 weeks' gestation. Reference curves were constructed for the individual measuring parameters based on a growth function from a four-parameter class of monotonic continuous functions according to the smallest square principle. A significant increase in blood flow velocity from 48 cm/s to 65.8 cm/s was observed during ventricular systole (= S) from 14 to 41 week's gestation. Similarly, increases in blood flow velocity were recorded during the endsystolic phase (= SD) (35.5 cm/s to 50.7 cm/s during early ventricular diastole (= D ) (41.7 cm/s to 58 cm/s, p = 0.0001) and atrial contraction (= a) (11.2 cm/s to 35 cm/s, p = 0.0001), as well as for intensity-weighted mean velocity (30 cm/s to 48.3 cm/s). The venous indices were associated with significant decreases in the individual parameters with increasing gestational age: (S-a)/S from 0.77 to 0.47, (S-a)V-mean from 1.21 to 0.67, (S-a)/D from 0.89 to 0.53, S/a from 4.5 to 1.99. A significant increase from 0.23 to 0.53 was observed only for the quotient a/S. There were no changes in the S/D quotient (from 1.15 to 1.13). Regarding intra-observer reliability, more favourable results wen obtained for calculated indices than for measurements of absolute blood flow velocities. At constant measuring conditions, the reference ranges established by this study for blood flow velocities and calculated indices in the ductus venosus may serve as the basis for

关键词： Doppler reference ranges regression function ductus venosus fetus

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