23 results on '"Irène Gijbels"'
Search Results
2. A new distance based measure of asymmetry
- Author
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Jonas Baillien, Irène Gijbels, and Anneleen Verhasselt
- Subjects
Statistics and Probability ,Numerical Analysis ,Statistics, Probability and Uncertainty - Published
- 2023
3. Omnibus test for covariate effects in conditional copula models
- Author
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Noël Veraverbeke, Marek Omelka, and Irène Gijbels
- Subjects
Statistics and Probability ,Numerical Analysis ,Conditional dependence ,Omnibus test ,Nonparametric statistics ,Asymptotic distribution ,Covariate ,Econometrics ,Test statistic ,Statistics::Methodology ,Statistics, Probability and Uncertainty ,Marginal distribution ,Null hypothesis ,Mathematics - Abstract
Conditional copulas describe the conditional dependence and the influence that covariates have on the dependence structure between two (or more) variables. Of interest is to test the null hypothesis that the covariates have a specific effect. This paper proposes an omnibus test for testing the null hypothesis of a specified effect of the covariates. The test statistic is designed for having power against many alternatives, and can be used to test for a variety of covariate effects (no effects, linear effects, partial effects, etc.). A special case is the testing problem that the covariates do not affect the dependence structure. In this semiparametric framework the marginal distribution functions are estimated using nonparametric kernel techniques and the parametric dependence model is estimated using maximum likelihood estimation. We establish the asymptotic distribution of the test statistic under the null hypothesis, and evaluate the finite-sample performance of the test via a simulation study, which also includes comparisons with alternative tests. A real data analysis illustrates the practical use of the test.
- Published
- 2021
4. Semiparametric quantile regression using family of quantile-based asymmetric densities
- Author
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Anneleen Verhasselt, Irène Gijbels, and Rezaul Karim
- Subjects
Statistics and Probability ,Statistics::Theory ,Applied Mathematics ,Nonparametric statistics ,Asymptotic distribution ,Estimator ,Statistics::Computation ,Quantile regression ,Nonparametric regression ,Computational Mathematics ,Computational Theory and Mathematics ,Linear regression ,Parametric model ,Econometrics ,Statistics::Methodology ,Mathematics ,Quantile - Abstract
Quantile regression is an important tool in data analysis. Linear regression, or more generally, parametric quantile regression imposes often too restrictive assumptions. Nonparametric regression avoids making distributional assumptions, but might have the disadvantage of not exploiting distributional modelling elements that might be brought in. A semiparametric approach towards estimating conditional quantile curves is proposed. It is based on a recently studied large family of asymmetric densities of which the location parameter is a quantile (and not a mean). Passing to conditional densities and exploiting local likelihood techniques in a multiparameter functional setting then leads to a semiparametric estimation procedure. For the local maximum likelihood estimators the asymptotic distributional properties are established, and it is discussed how to assess finite sample bias and variance. Due to the appealing semiparametric framework, one can discuss in detail the bandwidth selection issue, and provide several practical bandwidth selectors. The practical use of the semiparametric method is illustrated in the analysis of maximum winds speeds of hurricanes in the North Atlantic region, and of bone density data. A simulation study includes a comparison with nonparametric local linear quantile regression as well as an investigation of robustness against miss-specifying the parametric model part.
- Published
- 2021
5. On the specification of multivariate association measures and their behaviour with increasing dimension
- Author
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Irène Gijbels, Marek Omelka, and Vojtěch Kika
- Subjects
Statistics and Probability ,Numerical Analysis ,Multivariate statistics ,Multivariate random variable ,Generalization ,Nonparametric statistics ,020206 networking & telecommunications ,02 engineering and technology ,Bivariate analysis ,01 natural sciences ,010104 statistics & probability ,Random variate ,Dimension (vector space) ,0202 electrical engineering, electronic engineering, information engineering ,Statistics::Methodology ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
In this paper the interest is to elaborate on the generalization of bivariate association measures, namely Spearman’s rho, Kendall’s tau, Blomqvist’s beta and Gini’s gamma, for a general dimension d ≥ 2 . Desirable properties and axioms for such generalizations are discussed, where special attention is given to the impact of the addition of: (i) an independent random variable to a random vector; (ii) a conical combination of all components; (iii) a set of arbitrary random components. Existing generalizations are evaluated with respect to the axiom set. For a d -variate Gini’s gamma, a simplified formula is developed, making its analytical computation easier. Further, for Archimedean and meta-elliptical copulas the asymptotic behaviour when the dimension d increases is studied. Nonparametric estimation of the considered generalizations of multivariate association measures is reviewed and a nonparametric estimator of the multivariate Gini’s gamma is introduced. The practical use of multivariate association measures is illustrated on a real data example.
- Published
- 2021
6. Bias reduced tail estimation for censored Pareto type distributions
- Author
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Anastasios Bardoutsos, Irène Gijbels, I. de Wet, and Jan Beirlant
- Subjects
Statistics and Probability ,Estimation ,Pareto interpolation ,05 social sciences ,Pareto principle ,Estimator ,Type (model theory) ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Heavy-tailed distribution ,0502 economics and business ,Statistics ,symbols ,Lomax distribution ,Pareto distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,050205 econometrics ,Mathematics - Abstract
We consider bias reduced estimators for the tail index and tail probabilities under random right censoring in case of Pareto-type distributions. The solution is based on second-order refined peaks-over-threshold modelling as developed in Beirlant et al. (2009).
- Published
- 2016
7. Consistency of non-integrated depths for functional data
- Author
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Stanislav Nagy and Irène Gijbels
- Subjects
Statistics and Probability ,Numerical Analysis ,Class (set theory) ,Mathematical analysis ,Strong consistency ,Sample (statistics) ,Type (model theory) ,Glivenko–Cantelli theorem ,Consistency (statistics) ,Statistics, Probability and Uncertainty ,Special case ,Algorithm ,Counterexample ,Mathematics - Abstract
© 2015 Elsevier Inc. In the analysis of functional data, the concept of data depth is of importance. Strong consistency of a sample version of a data depth is among the basic statistical properties that need to hold. In this paper we discuss consistency properties of three popular types of functional depth: the band depth, the half-region depth and the infimal depth. The latter is a special case of the recently introduced general class of Φ-depths. All three considered depth functions are of a non-integrated type. Counterexamples illustrate some problems with consistency results for these data depths. The main contribution of this paper consists of providing sufficient conditions for consistency of these non-integrated data depths to hold. publisher: Elsevier articletitle: Consistency of non-integrated depths for functional data journaltitle: Journal of Multivariate Analysis articlelink: http://dx.doi.org/10.1016/j.jmva.2015.05.012 content_type: article copyright: Copyright © 2015 Elsevier Inc. All rights reserved. ispartof: Journal of Multivariate Analysis vol:140 pages:259-282 status: published
- Published
- 2015
8. Robust nonnegative garrote variable selection in linear regression
- Author
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Inge Vrinssen and Irène Gijbels
- Subjects
Statistics and Probability ,Computational Mathematics ,Mathematical optimization ,Computational Theory and Mathematics ,Applied Mathematics ,Linear regression ,Outlier ,Leverage (statistics) ,Regression analysis ,Feature selection ,Data application ,Mathematics - Abstract
© 2014 Elsevier B.V. All rights reserved. Robust selection of variables in a linear regression model is investigated. Many variable selection methods are available, but very few methods are designed to avoid sensitivity to vertical outliers as well as to leverage points. The nonnegative garrote method is a powerful variable selection method, developed originally for linear regression but recently successfully extended to more complex regression models. The method has good performances and its theoretical properties have been established. The aim is to robustify the nonnegative garrote method for linear regression as to make it robust to vertical outliers and leverage points. Several approaches are discussed, and recommendations towards a final good performing robust nonnegative garrote method are given. The proposed method is evaluated via a simulation study that also includes a comparison with existing methods. The method performs very well, and often outperforms existing methods. A real data application illustrates the use of the method in practice. publisher: Elsevier articletitle: Robust nonnegative garrote variable selection in linear regression journaltitle: Computational Statistics & Data Analysis articlelink: http://dx.doi.org/10.1016/j.csda.2014.11.009 content_type: article copyright: Copyright © 2014 Elsevier B.V. All rights reserved. ispartof: Computational Statistics & Data Analysis vol:85 pages:1-22 status: published
- Published
- 2015
9. On the distribution of sums of random variables with copula-induced dependence
- Author
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Irène Gijbels and Klaus Herrmann
- Subjects
Statistics and Probability ,Economics and Econometrics ,Multivariate random variable ,Quantile function ,Conditional expectation ,Copula (probability theory) ,Expected shortfall ,Joint probability distribution ,Econometrics ,Applied mathematics ,Statistics, Probability and Uncertainty ,Marginal distribution ,Random variable ,Mathematics - Abstract
We investigate distributional properties of the sum of d possibly unbounded random variables. The joint distribution of the random vector is formulated by means of an absolutely continuous copula, allowing for a variety of different dependence structures between the summands. The obtained expression for the distribution of the sum features a separation property into marginal and dependence structure contributions typical for copula approaches. Along the same lines we obtain the formulation of a conditional expectation closely related to the expected shortfall common in actuarial and financial literature. We further exploit the separation to introduce new numerical algorithms to compute the distribution and quantile function, as well as this conditional expectation. A comparison with the most common competitors shows that the discussed Path Integration algorithm is the most suitable method for computing these quantities. In our example, we apply the theory to compute Value-at-Risk forecasts for a trivariate portfolio of index returns.
- Published
- 2014
10. Testing tail monotonicity by constrained copula estimation
- Author
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Dominik Sznajder and Irène Gijbels
- Subjects
Statistics and Probability ,Economics and Econometrics ,Mathematical optimization ,Resampling ,Nonparametric statistics ,Test statistic ,Estimator ,Monotonic function ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics ,Copula (probability theory) ,Parametric statistics - Abstract
In this paper the interest is in testing for tail monotonicity dependence structures between two random variables. The main focus in the presentation of the statistical methodology is on left tail decreasingness, but the developed procedures can also be used for testing for other specific tail monotonicity dependence structures. In order to assess the p -values of the test statistic, we resample from a constrained copula estimator. This can be done in a nonparametric or in a parametric way. The main difficulty is the construction of a constrained estimator and the development of a resampling technique. The finite-sample performances of the proposed testing procedures are investigated in a simulation study and illustrations on real data examples are provided.
- Published
- 2013
11. Bootstrapping the conditional copula
- Author
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Noël Veraverbeke, Marek Omelka, and Irène Gijbels
- Subjects
Statistics and Probability ,Statistics::Theory ,Conditional dependence ,Applied Mathematics ,Nonparametric statistics ,Conditional probability distribution ,Copula (probability theory) ,Regular conditional probability ,Statistics ,Covariate ,Econometrics ,Statistics::Methodology ,Statistics, Probability and Uncertainty ,Marginal distribution ,Conditional variance ,Mathematics - Abstract
This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.
- Published
- 2013
12. Semiparametric estimation of conditional copulas
- Author
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Noël Veraverbeke, Irène Gijbels, and Fentaw Abegaz
- Subjects
Statistics and Probability ,Numerical Analysis ,Bandwidth (signal processing) ,Estimator ,Asymptotic distribution ,Local polynomial fitting ,Semiparametric estimation ,Covariate ,Econometrics ,Asymptotic normality ,Statistics::Methodology ,Applied mathematics ,Conditional copula ,Consistency ,Statistics, Probability and Uncertainty ,Random variable ,Parametric statistics ,Mathematics - Abstract
The manner in which two random variables influence one another often depends on covariates. A way to model this dependence is via a conditional copula function. This paper contributes to the study of semiparametric estimation of conditional copulas by starting from a parametric copula function in which the parameter varies with a covariate, and leaving the marginals unspecified. Consequently, the unknown parts in the model are the parameter function and the unknown marginals. The authors use a local pseudo-likelihood with nonparametrically estimated marginals approximating the unknown parameter function locally by a polynomial. Under this general setting, they prove the consistency of the estimators of the parameter function as well as its derivatives; they also establish asymptotic normality. Furthermore, they derive an expression for the theoretical optimal bandwidth and discuss practical bandwidth selection. They illustrate the performance of the estimation procedure with data-driven bandwidth selection via a simulation study and a real-data case.
- Published
- 2012
13. Conditional copulas, association measures and their applications
- Author
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Irène Gijbels, Marel Omelka, and Noël Veraverbeke
- Subjects
Statistics and Probability ,Statistics::Theory ,Applied Mathematics ,Nonparametric statistics ,Estimator ,Conditional probability distribution ,Spearman's rank correlation coefficient ,Copula (probability theory) ,Computational Mathematics ,Computational Theory and Mathematics ,Joint probability distribution ,Covariate ,Statistics ,Econometrics ,Statistics::Methodology ,Conditional variance ,Mathematics - Abstract
One way to model a dependence structure is through the copula function which is a mean to capture the dependence structure in the joint distribution of variables. Association measures such as Kendall's tau or Spearman's rho can be expressed as functionals of the copula. The dependence structure between two variables can be highly influenced by a covariate, and it is of real interest to know how this dependence structure changes with the value taken by the covariate. This motivates the need for introducing conditional copulas, and the associated conditional Kendall's tau and Spearman's rho association measures. After the introduction and motivation of these concepts, two nonparametric estimators for a conditional copula are proposed and discussed. Then nonparametric estimates for the conditional association measures are derived. A key issue is that these measures are now looked at as functions in the covariate. The performances of all estimators are investigated via a simulation study which also includes a data-driven algorithm for choosing the smoothing parameters. The usefulness of the methods is illustrated on two real data examples.
- Published
- 2011
14. Penalized wavelet monotone regression
- Author
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Anestis Antoniadis, Jérémie Bigot, Irène Gijbels, Statistique et Modélisation Stochatisque (SMS), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Leuven Statistics Research Centre (LStat), Catholic University of Leuven - Katholieke Universiteit Leuven (KU Leuven), Centre National de la Recherche Scientifique - CNRS (FRANCE), Institut National des Sciences Appliquées de Toulouse - INSA (FRANCE), Université Toulouse III - Paul Sabatier - UT3 (FRANCE), Université Toulouse - Jean Jaurès - UT2J (FRANCE), Université Toulouse 1 Capitole - UT1 (FRANCE), Université Joseph Fourier Grenoble 1 - UJF (FRANCE), Katholieke Universiteit Leuven - KU LEUVEN (BELGIUM), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Splines ,Statistics and Probability ,Monotonicity ,Statistics::Theory ,Mathematical optimization ,Optimization problem ,Wavelets ,01 natural sciences ,Statistics::Machine Learning ,010104 statistics & probability ,Minimum-variance unbiased estimator ,Wavelet ,Constrained curve fitting ,[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,0502 economics and business ,Statistics::Methodology ,0101 mathematics ,Wavelet nonparametric regression ,050205 econometrics ,Mathematics ,Wavelet thresholding ,05 social sciences ,Constrained optimization ,Estimator ,[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH] ,MSC: 62G07, 65Dxx ,Nonparametric regression ,Besov spaces ,Convex optimization ,Statistiques ,Statistics, Probability and Uncertainty ,Minimax estimator - Abstract
International audience; In this paper we focus on nonparametric estimation of a constrained regression function using penalized wavelet regression techniques. This results into a convex optimization problem under linear constraints. Necessary and sufficient conditions for existence of a unique solution are discussed. The estimator is easily obtained via the dual formulation of the optimization problem. In particular we investigate a penalized wavelet monotone regression estimator. We establish the rate of convergence of this estimator, and illustrate its finite sample performance via a simulation study. We also compare its performance with that of a recently proposed constrained estimator. An illustration to some real data is given.
- Published
- 2007
15. Copula Directed Acyclic Graphs
- Author
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Irène Gijbels, Eugen Pircalabelu, and Gerda Claeskens
- Subjects
Statistics and Probability ,Theoretical computer science ,Selection (relational algebra) ,Computer science ,Copula (linguistics) ,Multivariate normal distribution ,Statistics::Other Statistics ,Bivariate analysis ,010501 environmental sciences ,Model selection ,01 natural sciences ,Directed acyclic graph ,D-vine ,Theoretical Computer Science ,010104 statistics & probability ,Joint probability distribution ,Statistics::Methodology ,0101 mathematics ,0105 earth and related environmental sciences ,Mathematics ,C-vine ,business.industry ,Bayesian network ,Pattern recognition ,Continuous data ,Statistics::Computation ,Computational Theory and Mathematics ,Copula ,Artificial intelligence ,Statistics, Probability and Uncertainty ,business - Abstract
A new methodology for selecting a Bayesian network for continuous data outside the widely used class of multivariate normal distributions is developed. The ‘copula DAGs’ combine directed acyclic graphs and their associated probability models with copula C/D-vines. Bivariate copula densities introduce flexibility in the joint distributions of pairs of nodes in the network. An information criterion is studied for graph selection tailored to the joint modeling of data based on graphs and copulas. Examples and simulation studies show the flexibility and properties of the method. ispartof: Statistics and Computing vol:27 issue:1 pages:55-78 status: published
- Published
- 2015
16. Data-driven boundary estimation in deconvolution problems
- Author
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Aurore Delaigle and Irène Gijbels
- Subjects
Statistics and Probability ,Applied Mathematics ,Kernel density estimation ,Estimator ,Probability density function ,Density estimation ,Classification of discontinuities ,Data-driven ,Computational Mathematics ,Kernel method ,Computational Theory and Mathematics ,Statistics ,Deconvolution ,Algorithm ,Mathematics - Abstract
Estimation of the support of a density function is considered, when only a contaminated sample from the density is available. A kernel-based method has been proposed in the literature, where the authors study theoretical bias and variance of the estimator. Practical implementation issues of this method are considered here, which are a necessary supplement to the theoretical results to get to a data-driven method that is widely applicable. Two such practical data-driven procedures are proposed. Simulation results show that they perform well for a wide variety of densities (including quite difficult cases). The methods can also be applied for error-free data and as such also present data-driven procedures for estimation of boundaries in the case of non-contaminated data. Moreover they can be applied for estimating discontinuities of a density, as is shown. The proposed data-driven boundary estimation procedures are illustrated in frontier estimation.
- Published
- 2006
17. Practical bandwidth selection in deconvolution kernel density estimation
- Author
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Aurore Delaigle and Irène Gijbels
- Subjects
Statistics and Probability ,Statistics::Theory ,Mathematical optimization ,Applied Mathematics ,Kernel density estimation ,Bandwidth (signal processing) ,Density estimation ,Cross-validation ,Statistics::Computation ,Statistics::Machine Learning ,Computational Mathematics ,Density based ,ComputingMethodologies_PATTERNRECOGNITION ,Computational Theory and Mathematics ,TheoryofComputation_LOGICSANDMEANINGSOFPROGRAMS ,Statistics::Methodology ,Errors-in-variables models ,Deconvolution ,Software_PROGRAMMINGLANGUAGES ,Algorithm ,Normal density ,Mathematics - Abstract
Kernel estimation of a density based on contaminated data is considered and the important issue of how to choose the bandwidth parameter in practice is discussed. Some plug-in (PI) type of bandwidth selectors, which are based on non-parametric estimation of an approximation of the mean integrated squared error, are proposed. The selectors are a refinement of the simple normal reference bandwidth selector, which is obtained by parametrically estimating the approximated mean integrated squared error by referring to a normal density. A simulation study compares these PI bandwidth selectors with a bootstrap (BT) and a cross-validated (CV) bandwidth selector. It is concluded that in finite samples, an appropriately chosen PI bandwidth selector and the BT bandwidth selector perform comparably and both outperform the CV bandwidth. The use of the various practical bandwidth selectors is illustrated on a real data example.
- Published
- 2004
18. A nonparametric least-squares test for checking a polynomial relationship
- Author
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Irène Gijbels and Valentin Rousson
- Subjects
Statistics and Probability ,Polynomial regression ,Polynomial ,Goodness of fit ,Homogeneous polynomial ,Linear model ,Nonparametric statistics ,Calculus ,Applied mathematics ,Statistics, Probability and Uncertainty ,Pseudo-polynomial time ,Mathematics ,Square-free polynomial - Abstract
In this paper the interest is in testing whether a regression function is a polynomial of a certain degree. One possible approach to this testing problem is to do a parametric polynomial fit and a nonparametric fit and to reject the null hypothesis of a polynomial function if the distance between the two fits is too large. Another approach consists of looking at the residuals from the parametric fit. In this paper we propose an entirely new approach to deal with the testing problem. When testing whether a regression function is a polynomial of degree smaller than or equal to p, the key idea is to first obtain a nonparametric focal polynomial estimate of the pth derivative of the unknown regression function, and then to proceed with a classical least-squares test for a general linear model for testing whether this derivative is constant. This is a quite appealing approach since it just relies on ordinary least-squares tests, and hence is simple to use. The performance of the method is illustrated via a simulation study. (C) 2001 Elsevier Science B.V. All rights reserved.
- Published
- 2001
19. Strong Representations of the Survival Function Estimator for Truncated and Censored Data with Applications
- Author
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Jane-Ling Wang and Irène Gijbels
- Subjects
Statistics and Probability ,Numerical Analysis ,Kernel density estimation ,Bernstein inequalities ,Estimator ,Asymptotic distribution ,Kernel method ,Compact space ,Survival function ,Statistics ,Applied mathematics ,Statistics, Probability and Uncertainty ,Kaplan–Meier estimator ,Mathematics - Abstract
A strong i.i.d. representation is obtained for the product-limit estimator of the survival function based on left truncated and right censored data. This extends the result of Chao and Lo (1988, Ann. Statist.16, 661-668) for truncated data. An improved rate of the approximation is also obtained on compact sets. Applications include density and hazard rate estimation. The advantage of the improved rate of the approximation is illustrated via kernel density estimation.
- Published
- 1993
20. Minimax estimation of a bounded squared mean
- Author
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Irène Gijbels and Jianqing Fan
- Subjects
Statistics and Probability ,Statistics::Theory ,Nonparametric statistics ,White noise ,Minimax ,Minimax approximation algorithm ,Quadratic equation ,Bounded function ,Statistics ,Applied mathematics ,Statistics, Probability and Uncertainty ,Minimax estimator ,Constant (mathematics) ,Mathematics - Abstract
Consider a normal model with unknown mean bounded by a known constant. This paper deals with minimax estimation of the squared mean. We establish an expression for the asymptotic minimax risk. This result is applied in nonparametric estimation of quadratic functionals.
- Published
- 1992
21. Unstable Volatility Functions: The Break Preserving Local Linear Estimator
- Author
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Isabel Casas and Irène Gijbels
- Subjects
Heteroscedasticity ,Stochastic volatility ,Markov chain ,Econometrics ,Estimator ,Applied mathematics ,Local regression ,Threshold model ,Volatility (finance) ,Conditional expectation ,Mathematics - Abstract
The objective of this paper is to introduce the break preserving local linear (BPLL) estimator for the estimation of unstable volatility functions. Breaks in the structure of the conditional mean and/or the volatility functions are common in Finance. Markov switching models (Hamilton, 1989) and threshold models (Lin and Terasvirta, 1994) are amongst the most popular models to describe the behaviour of data with structural breaks. The local linear (LL) estimator is not consistent at points where the volatility function has a break and it may even report negative values for finite samples. The estimator presented in this paper generalises the classical LL. The BPLL maintains the desirable properties of the LL with regard to the bias and the boundary estimation, it estimates the breaks consistently and it ensures that the volatility estimates are always positive.
- Published
- 2009
22. Sequential fixed-width confidence intervals for quantiles in the presence of censoring
- Author
-
Irène Gijbels and Noël Veraverbeke
- Subjects
Statistics and Probability ,Statistics::Theory ,Applied Mathematics ,Estimator ,Survival distribution ,Confidence interval ,Statistics::Computation ,Sequential method ,Censoring (clinical trials) ,Statistics ,Econometrics ,Statistics::Methodology ,Statistics, Probability and Uncertainty ,Fixed width ,Quantile ,Mathematics - Abstract
This paper studies the asymptotic properties of sequential fixed-width confidence intervals for quantiles of the survival distribution function in the random censorship model. The interval is formed by a pair of quantiles of the product-limit estimator. The set up requires new results on the almost sure behaviour of such empirical quantiles.
- Published
- 1989
23. Weak asymptotic representations for quantiles of the product-limit estimator
- Author
-
Irène Gijbels and Noël Veraverbeke
- Subjects
Statistics and Probability ,Statistics::Theory ,Weak convergence ,Representation theorem ,Estimation theory ,Applied Mathematics ,Mathematical statistics ,Estimator ,Random sequence ,Statistics ,Applied mathematics ,Statistics, Probability and Uncertainty ,Remainder ,Mathematics ,Quantile - Abstract
Sufficient conditions are given under which quantiles &'(p,,) of the product-limit estimator allow a Bahadur-type representation with remainder term o,D(~-"~). Here {p,} is either a deterministic or random sequence. This weak representation theorem and a uniform ver- sion of it lead to first-order asymptotic results in the estimation theory for quantiles of the lifetime distribution and of the residual lifetime distribution. AMS Subject ClassSficafion: Primary 62605; Secondary 60F05.
- Published
- 1988
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