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Estimation of the error distribution in nonparametric regression with cross-sectional data Heuchenne, Cédric ; Laurent, Géraldine E-print/Working paper (2014) In this article, we study the nonparametric regression model Y=m(X)+varepsilon where m(x)=E[Y|X=x] and sigma²(x)=Var[varepsilon|X=x] are unknown smooth functions, and the error varepsilon has zero mean ... [more ▼] In this article, we study the nonparametric regression model Y=m(X)+varepsilon where m(x)=E[Y|X=x] and sigma²(x)=Var[varepsilon|X=x] are unknown smooth functions, and the error varepsilon has zero mean and finite variance conditionally on X=x. The problem consists in estimating the cumulative distribution function of the error in a nonparametric way when the couple (X,Y) is obtained by cross-sectional sampling while the positive response Y can be right-censored. We propose a new estimator for the error distribution function based on the estimators of m(.) and sigma²(.) described in Heuchenne and Laurent 2014. A bootstrap procedure is developed to solve the critical problem of the smoothing parameter choice. We assess the performance of the proposed estimator through simulations. Finally, a data set based on the mortality of diabetics is analyzed. (Heuchenne Cédric and Laurent Géraldine, Nonparametric regression with cross-sectional data: an alternative to conditional product-limit estimators, 2014) [less ▲] Detailed reference viewed: 41 (10 ULg)Parametric conditional variance estimation in location-scale models with censored data Heuchenne, Cédric ; Laurent, Géraldine E-print/Working paper (2014) Suppose the random vector (X,Y) satisfies the regression model Y=m(X)+sigma(X)*varepsilon, where m(.)=E(Y|.), sigma²(.)=Var(Y|.) belongs to some parametric class {sigma _theta(.): theta in Theta} and ... [more ▼] Suppose the random vector (X,Y) satisfies the regression model Y=m(X)+sigma(X)*varepsilon, where m(.)=E(Y|.), sigma²(.)=Var(Y|.) belongs to some parametric class {sigma _theta(.): theta in Theta} and varepsilon is independent of X. The response Y is subject to random right censoring and the covariate X is completely observed. A new estimation procedure is proposed for sigma_theta(.) when m(.) is unknown. It is based on nonlinear least squares estimation extended to conditional variance in the censored case. The consistency and asymptotic normality of the proposed estimator are established. The estimator is studied via simulations and an important application is devoted to fatigue life data analysis. [less ▲] Detailed reference viewed: 44 (3 ULg)Nonparametric regression with cross-sectional data: an alternative to conditional product-limit estimators Heuchenne, Cédric ; Laurent, Géraldine E-print/Working paper (2014) Suppose the random vector (X,Y) satisfies the nonparametric regression model Y=m(X)+varepsilon, where m(x)=E[Y|X=x] and sigma²(x)=Var[varepsilon|X=x] are unknown smooth functions and the error varepsilon ... [more ▼] Suppose the random vector (X,Y) satisfies the nonparametric regression model Y=m(X)+varepsilon, where m(x)=E[Y|X=x] and sigma²(x)=Var[varepsilon|X=x] are unknown smooth functions and the error varepsilon has zero mean and finite variance conditionally on X=x. The pair (X,Y) is obtained by cross-sectional sampling involving left-truncated and right-censored responses. The considered model is completely nonparametric but the conditional truncation distribution is assumed to be known. The novelty of this work is twofold: first, it extends the results on cross-sectional data to the conditional case and second, it generalizes the length bias results in the conditional case to right censoring and to any truncation distribution. New estimators for m(.) and sigma²(.) are constructed and relevant tools are used to quickly provide the main asymptotic properties for this kind of estimators. Extensive simulations are carried out and show that the new estimators outperform classical nonparametric estimators for left-truncated and right-censored data (when the truncation model is known). Finally, a data set on the mortality of diabetics is analyzed. [less ▲] Detailed reference viewed: 43 (3 ULg)From censored to cross-sectional data: non and semiparametric new developments Laurent, Géraldine Doctoral thesis (2014) In many statistical studies, an observation is evident: the available data are regularly right-censored. A censorship arises when, for different reasons, the data time of interest can not be observed. A ... [more ▼] In many statistical studies, an observation is evident: the available data are regularly right-censored. A censorship arises when, for different reasons, the data time of interest can not be observed. A data is so right-censored if, instead of observing its time of interest, a lower bound of this time is considered for this data. For example, the study duration can be shorter than the time of interest leading then to a correspondence between the observed times and the study end time. Moreover, these data can be obtained from cross-sectional process. Cross-sectional process selects only data in progress at a fixed time to constitute the studied sample, determining the data followed for the study. Therefore, cross-sectional process introduces left truncation. A data is described as left-truncated if its time of interest is larger or equal to a fixed time. It is in this context this thesis has been elaborated. The considered estimation problems for such data will be studied with a nonparametric or semiparametric approach. An approach is nonparametric or semiparametric if none assumption is supposed about the belonging to parametric family for the time of interest distribution function, solely based on qualitative hypotheses. These estimation methods have thus the advantage to be based on weaker assumptions in comparison with the parametric approaches. The aim of the different researches developed in this thesis is to improve the current estimation techniques. This thesis is organised in four parts. The first part (first chapter) determines the context of our researches through practical examples and a significant but not exhaustive literature overview as well as our motivation about the different researches presented in this thesis. To conclude this first part, our contributions in these researches are briefly explained. The second part (second chapter) presents a new estimation procedure for the parameters of the parametric conditional variance in the heteroscedastic regression situation applied to right-censored data. This procedure constructs artificial data to replace censored data exploiting a heteroscedastic regression model and then defines the optimal parameters from the least squares method. The interest of this research is to fill a gap in the current literature. The third part (third and fourth chapters) studies, in a regression context, the cross-sectional data, i.e. left-truncated and right-censored data, where the conditional truncation distribution function is supposed to be known. The innovation of the method proposed here consists in the use of information contained in the conditional truncation distribution function for the nonparametric estimation methods. Finally, the fourth part (fifth chapter) is devoted to the cross-sectional data examination but this time for nonparametric estimation of the time of interest distribution function. In this chapter, the truncation distribution function is supposed to belong to a parametric family and not known anymore. The relevance of this approach is due to this weaker assumption than one in the above part. This information about the truncation distribution function is also introduced in the nonparametric estimation. This thesis concludes with a set of suggestions related to possible future researches in these statistical fields. [less ▲] Detailed reference viewed: 41 (11 ULg)Estimation from cross-sectional data under a semiparametric truncation model ; Heuchenne, Cédric ; Laurent, Géraldine E-print/Working paper (2013) Cross-sectional sampling is often used when investigating inter-event times, resulting in left-truncated and right-censored data. In this paper we consider a semiparametric truncation model in which the ... [more ▼] Cross-sectional sampling is often used when investigating inter-event times, resulting in left-truncated and right-censored data. In this paper we consider a semiparametric truncation model in which the truncating variable is assumed to belong to a certain parametric family, while nothing is assumed on lifetime and censoring distributions. The novelty of this work is in the fact that it introduces estimators of this semiparametric model in the presence of censoring. Two alternative methods are considered, based on conditional and full likelihood considerations. Asymptotic representations of the estimators for the lifetime distribution are obtained, and their weak convergence is established. The finite sample performance of the new estimators is explored through simulations, and two real data illustrations are provided. One of the conclusions of our research is that both estimators perform better than Wang's NPMLE when the parametric family for the truncation variable is valid) in the sense of the integrated mean squared error, and that the full likelihood approach is preferable to the conditional likelihood approach for the estimation of the lifetime distribution but not necessarily for the estimation of the truncation distribution. [less ▲] Detailed reference viewed: 28 (3 ULg)Nonparametric regression with right-censored and generalized selection biased data Heuchenne, Cédric ; Laurent, Géraldine Conference (2012, December 02) Detailed reference viewed: 33 (6 ULg)Error distribution estimation in nonparametric regression with right censored selection biased data Laurent, Géraldine ; Heuchenne, Cédric Conference (2012, October 25) In this presentation, we study the nonparametric regression model Y = m(X) +sigma(X) * epsilon where the error epsilon, with unknown distribution, is independent of the covariate X, and m(X) = E[Y|X] and ... [more ▼] In this presentation, we study the nonparametric regression model Y = m(X) +sigma(X) * epsilon where the error epsilon, with unknown distribution, is independent of the covariate X, and m(X) = E[Y|X] and sigma²(X) =Var[Y|X] are unknown smooth functions. The problem is to estimate the cumulative distribution function of the error in a nonparametric way when the couple (X;Y) is subject to generalized bias selection while the positive response Y can be right-censored. We propose a new estimator for the error distribution function. Asymptotic properties of the proposed estimator are established, namely the rate of convergence and the limiting distribution. A bootstrap procedure is developed to solve the critical problem of the smoothing parameter choice. The performance of the proposed estimator is investigated through simulations. Finally, a data set based on the mortality of diabetics is analyzed. [less ▲] Detailed reference viewed: 16 (1 ULg)Error distribution function for parametrically truncated and censored data Laurent, Géraldine ; Heuchenne, Cédric Conference (2012, September 14) Detailed reference viewed: 19 (5 ULg)Asymptotic properties of the error distribution estimation in right censored and selection biased regression models Laurent, Géraldine ; Heuchenne, Cédric Conference (2012, June 16) Suppose the random vector (X,Y) satisfies the nonparametric regression model Y=m(X)+sigma(X)*epsilon where m(X) =E [Y|X] and sigma^2(X) = Var [Y|X] are unknown smooth functions and the error epsilon, with ... [more ▼] Suppose the random vector (X,Y) satisfies the nonparametric regression model Y=m(X)+sigma(X)*epsilon where m(X) =E [Y|X] and sigma^2(X) = Var [Y|X] are unknown smooth functions and the error epsilon, with unknown distribution, is independent of the covariate X. The pair (X,Y) is subject to generalized bias selection and the response Y to right censoring. We define a new estimator for the cumulative distribution function of the error epsilon, where the estimators of m(.) and sigma^2(.) are obtained by extending the conditional estimation methods introduced in de Uña-Alvarez and Iglesias-Perez (2010). The asymptotic properties of the proposed estimator are established. A bootstrap technique is proposed to select the smoothing parameter involved in the procedure. Finally, this method is studied via extended simulations and applied to real data. [less ▲] Detailed reference viewed: 27 (7 ULg)Estimation under left parametric truncation and right censoring Heuchenne, Cédric ; Laurent, Géraldine Scientific conference (2012) Detailed reference viewed: 22 (7 ULg)Estimation under left parametric truncation and right censoring Heuchenne, Cédric ; Laurent, Géraldine Scientific conference (2012) Detailed reference viewed: 27 (6 ULg)Error distribution estimation in right censored and selection biased location-scale models Laurent, Géraldine ; Heuchenne, Cédric Poster (2011, June 23) Suppose the random vector (X;Y) satis es the regression model Y = m(X)+sigma(X)*epsilon where m(X) = E[Y|X] and sigma²(X) = Var[Y|X] are unknown smooth functions and the error epsilon, with unknown ... [more ▼] Suppose the random vector (X;Y) satis es the regression model Y = m(X)+sigma(X)*epsilon where m(X) = E[Y|X] and sigma²(X) = Var[Y|X] are unknown smooth functions and the error epsilon, with unknown distribution, is independent of the covariate X. The pair (X;Y) is subject to generalized selection biased and the response to right censoring. We construct a new estimator for the cumulative distribution function of the error epsilon, where the estimators of m(.) and sigma²(.) are obtained by extending the conditional estimation methods introduced in de Uña-Alvarez and Iglesias-Perez (2010). The asymptotic properties of the proposed estimator are established. A bootstrap technique is proposed to select the smoothing parameter involved in the procedure. This method is studied via extended simulations and applied to real unemployment data. Reference de UNA-ALVAREZ, J., IGLESIAS-PEREZ, M.C. (2010): Nonparametric estimation of a conditional distribution from length-biased data. Annals of the Institute of Statistical Mathematics, Vol. 62, 323-341. [less ▲] Detailed reference viewed: 30 (5 ULg)Nonparametric regression with parametric selection bias Heuchenne, Cédric ; Laurent, Géraldine in Proceedings of the 14th Conference of the ASMDA International Society (ASMDA2011) (2011, June) Detailed reference viewed: 40 (10 ULg)Introduction of the asymptotic study of the estimation of the error distribution in right censored and selection biased regression models Laurent, Géraldine ; Heuchenne, Cédric Poster (2010, October) Consider the regression model Y = m(X) + σ(X) Ɛ where m(X) = E[Y|X] and σ²(X)=Var[Y|X] are unknown smooth functions and the error Ɛ , with unknown distribution, is independent of the covariate X. The pair ... [more ▼] Consider the regression model Y = m(X) + σ(X) Ɛ where m(X) = E[Y|X] and σ²(X)=Var[Y|X] are unknown smooth functions and the error Ɛ , with unknown distribution, is independent of the covariate X. The pair (X;Y) is subject to generalized bias selection and the response to right censoring. We construct a new estimator for the cumulative distribution function of the error Ɛ , where the estimators of m(.) and σ²(.) are obtained by extending the conditional estimation methods introduced in de Uña-Alvarez and Iglesias-Perez (2008). The asymptotic properties of the functions m(.) and σ(.) are obtained. A bootstrap technique is proposed to select the smoothing parameter involved in the procedure. This method is studied via extended simulations and applied to real unemployment data. [less ▲] Detailed reference viewed: 22 (3 ULg)Computational treatment of the error distribution in nonparametric regression with right-censored and selection-biased data Laurent, Géraldine ; Heuchenne, Cédric Conference (2010, August 24) Consider the regression model Y = m(X) + σ(X) Ɛ , where m(X) = E[Y|X] and σ²(X) = Var[Y|X] are unknown smooth functions and the error Ɛ (with unknown distribution) is independent of X. The pair (X;Y) is ... [more ▼] Consider the regression model Y = m(X) + σ(X) Ɛ , where m(X) = E[Y|X] and σ²(X) = Var[Y|X] are unknown smooth functions and the error Ɛ (with unknown distribution) is independent of X. The pair (X;Y) is subject to generalized selection bias and the response to right censoring. We construct a new estimator for the cumulative distribution function of the error Ɛ , and develop a bootstrap technique to select the smoothing parameter involved in the procedure. The estimator is studied via extended simulations and applied to real unemployment data. [less ▲] Detailed reference viewed: 12 (3 ULg)Computational study of the error distribution in right-censored and selection-biased regression models Laurent, Géraldine ; Heuchenne, Cédric Conference (2010, May 18) Consider the regression model Y = m(X) + σ(X) Ɛ where m(X) =E [Y|X] and σ²(X) = Var [Y|X] are unknown smooth functions and the error Ɛ, with unknown distribution, is independent of X. The pair (X,Y) is ... [more ▼] Consider the regression model Y = m(X) + σ(X) Ɛ where m(X) =E [Y|X] and σ²(X) = Var [Y|X] are unknown smooth functions and the error Ɛ, with unknown distribution, is independent of X. The pair (X,Y) is subject to generalized selection bias and the response to right censoring. We construct a new estimator for the cumulative distribution function of the error Ɛ, and develop a bootstrap technique to select the smoothing parameter involved in the procedure. The estimator is studied via extended simulations and applied to real unemployment data. [less ▲] Detailed reference viewed: 23 (5 ULg)Computational treatment of the error distribution in nonparametric regression with right-censored and selection-biased data Heuchenne, Cédric ; Laurent, Géraldine in Proceedings of the 19th International Conference on Computational Statistics (2010) Detailed reference viewed: 42 (12 ULg)Estimation of the error distribution in right censored and selection biased regression models Laurent, Géraldine ; Heuchenne, Cédric Conference (2009, October 15) Consider the location-scale regression model Y=m(X) + σ(X) Ɛ where the error Ɛ is independent of the covariate X and where m and σ are unknown smooth functions. The pair (X; Y ) is subject to generalized ... [more ▼] Consider the location-scale regression model Y=m(X) + σ(X) Ɛ where the error Ɛ is independent of the covariate X and where m and σ are unknown smooth functions. The pair (X; Y ) is subject to generalized bias selection and the response to right censoring. We construct an estimator for the cumulative distribution function of the error Ɛ, and develop a bootstrap procedure to select the smoothing parameter involved in the procedure. This method is studied via extension simulations and applied to real unemployment data. [less ▲] Detailed reference viewed: 22 (11 ULg)Estimation of the error distribution in nonparametric regression with right censored selection biased data Laurent, Géraldine ; Heuchenne, Cédric Poster (2009, May) Detailed reference viewed: 51 (21 ULg) |
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