Papers by Daniela Rodriguez
In partially linear models the dependence of the response y on (x^T,t) is modeled through the rel... more In partially linear models the dependence of the response y on (x^T,t) is modeled through the relationship y=\x^T \beta+g(t)+\epsilon where \epsilon is independent of (x^T,t). In this paper, estimators of \beta and g are constructed when the explanatory variables t take values on a Riemannian manifold. Our proposal combine the flexibility of these models with the complex structure of a set of explanatory variables. We prove that the resulting estimator of \beta is asymptotically normal under the suitable conditions. Through a simulation study, we explored the performance of the estimators. Finally, we applied the studied model to an example based on real dataset.
Journal of Mathematical Imaging and Vision, 2009
The paper concerns the strong uniform consistency and the asymptotic distribution of the kernel d... more The paper concerns the strong uniform consistency and the asymptotic distribution of the kernel density estimator of random objects on a Riemannian manifolds, proposed by Pelletier (Stat. Probab. Lett., 73(3):297–304, 2005). The estimator is illustrated via one example based on a real data.
In many situations, when dealing with several populations with different covariance operators, eq... more In many situations, when dealing with several populations with different covariance operators, equality of the operators is assumed. Usually, if this assumption does not hold, one estimates the covariance operator of each group separately, which leads to a large number of parameters. As in the multivariate setting, this is not satisfactory since the covariance operators may exhibit some common structure. In this paper, we discuss the extension to the functional setting of common principal component model that has been widely studied when dealing with multivariate observations. Moreover, we also consider a proportional model in which the covariance operators are assumed to be equal up to a multiplicative constant. For both models, we present estimators of the unknown parameters and we obtain their asymptotic distribution. A test for equality against proportionality is also considered.
Computational Statistics & Data Analysis, 2008
In this paper, under a semiparametric partly linear regression model with fixed design, we introd... more In this paper, under a semiparametric partly linear regression model with fixed design, we introduce a family of robust procedures to select the bandwidth parameter. The robust plug-in proposal is based on nonparametric robust estimates of the ν−th derivatives and under mild conditions, it converges to the optimal bandwidth. A robust cross-validation bandwidth is also considered and the performance of the different proposals is compared through a Monte Carlo study. We define an empirical influence measure for data-driven bandwidth selectors and, through it, we study the sensitivity of the data-driven bandwidth selectors. It appears that the robust selector compares favorably to its classical competitor, despite the need to select a pilot bandwidth when considering plug-in bandwidths. Moreover, the plug-in procedure seems to be less sensitive than the cross-validation in particular, when introducing several outliers. When combined with the three-step procedure proposed by , the robust selectors lead to robust data-driven estimates of both the regression function and the regression parameter. 1 2 ( β ls (h) − β), and n MSE(h) = EU 2 /σ 2 its standardized mean square error. Then, when the smoothing procedure corresponds to local means, under general conditions, that include that the design points are almost uniform design points, i.e., {t i } n i=1 are fixed design points in [0, 1], 0 ≤ t 1 ≤ . . . ≤ t n ≤ 1, such that t 0 = 0 and t n+1 = 1 and max 1≤i≤n+1 |(t i − t i−1 ) − 1/n| = O(n −δ ) for some δ > 1, we have that, for ν ≥ 2, MSE(h) = n −1 {1 + (nh) −1 A 2 + o(n −2µ ) + (n 1 2 h 2ν A 1 + o(n −µ )) 2 } ,
Test, 2012
A natural generalization of the well known generalized linear models is to allow only for some of... more A natural generalization of the well known generalized linear models is to allow only for some of the predictors to be modeled linearly while others are modeled nonparametrically. However, this model can face the so called “curse of dimensionality” problem that can be solved by imposing a nonparametric dependence on some unknown projection of the carriers. More precisely, we assume that the observations (y i ,x i ,t i ), 1≤i≤n, are such that t i ∈ℝq , x i ∈ℝp and y i |(x i ,t i )∼F(⋅,μ i ) with \(\mu_{i}=H (\eta(\boldsymbol{\alpha}^{\mathrm{T}}\mathbf{t}_{i})+\mathbf {x}_{i}^{\mathrm{T}}\boldsymbol{\beta} )\) , for some known distribution function F and link function H. The function η:ℝ→ℝ and the parameters α and β are unknown and to be estimated. This model is known as the generalized partly linear single-index model. In this paper, we introduce a family of robust estimates for the parametric and nonparametric components under a generalized partially linear single-index model. It is shown that the estimates of α and β are root-n consistent and asymptotically normally distributed. Through a Monte Carlo study, we compare the performance of the proposed estimators with that of the classical ones.
In this paper, we consider a k-nearest neighbor kernel type estimator when the random variables b... more In this paper, we consider a k-nearest neighbor kernel type estimator when the random variables belong in a Riemannian manifolds. We study asymptotic properties such as the consistency and the asymptotic distribution. A simulation study is also consider to evaluate the performance of the proposal. Finally, to illustrate the potential applications of the proposed estimator, we analyzed two real example where two different manifolds are considered.
Statistics & Probability Letters, 2006
In this paper, robust estimates for the derivatives of order ν of the regression function are con... more In this paper, robust estimates for the derivatives of order ν of the regression function are considered. This estimator extend the proposals given when ν = 1, 2. Uniform consistency, which allows to construct a robust data-driven bandwidth, is established. Besides, the robust estimates introduced are asymptotically normally distributed and their asymptotic efficiency is that of the related M −location estimators.
In many situations, when dealing with several populations, equality of the covariance operators i... more In many situations, when dealing with several populations, equality of the covariance operators is assumed. In this work, we will study a hypothesis test to validate this assumption.
Under a partially linear models we study a family of robust estimates for the regression paramete... more Under a partially linear models we study a family of robust estimates for the regression parameter and the regression function when some of the predictor variables take values on a Riemannian manifold. We obtain the consistency and the asymptotic normality of the proposed estimators. Also, we consider a robust cross validation procedure to select the smoothing parameter. Simulations and application to real data show the performance of our proposal under small samples and contamination.
Journal of Multivariate Analysis, 2010
In many situations, when dealing with several populations with different covariance operators, eq... more In many situations, when dealing with several populations with different covariance operators, equality of the operators is assumed. Usually, if this assumption does not hold, one estimates the covariance operator of each group separately, which leads to a large number of parameters. As in the multivariate setting, this is not satisfactory since the covariance operators may exhibit some common structure. In this paper, we discuss the extension to the functional setting of common principal component model that has been widely studied when dealing with multivariate observations. Moreover, we also consider a proportional model in which the covariance operators are assumed to be equal up to a multiplicative constant. For both models, we present estimators of the unknown parameters and we obtain their asymptotic distribution. A test for equality against proportionality is also considered.
Journal of Nonparametric Statistics, 2009
Robust inference in generalized partially linear models
Computational Statistics & Data Analysis, 2010
In many situations, data follow a generalized partly linear model in which the mean of the respon... more In many situations, data follow a generalized partly linear model in which the mean of the responses is modeled, through a link function, linearly on some covariates and nonparametrically on the remaining ones. A new class of robust estimates for the smooth function η, associated to the nonparametric component, and for the parameter β, related to the linear one, is
Partly linear models on Riemannian manifolds
Journal of Applied Statistics, 2012
In partly linear models, the dependence of the response y on (x T, t) is modeled through the rela... more In partly linear models, the dependence of the response y on (x T, t) is modeled through the relationship y=x Tβ+g(t)+ϵ, where ϵ is independent of (x T, t). We are interested in developing an estimation procedure that allows us to combine the flexibility of the partly linear models, studied by several authors, but including some variables that belong to a non-Euclidean space. The motivating application of this paper deals with the explanation of the atmospheric SO2 pollution incidents using these models when some of the predictive variables belong in a cylinder. In this paper, the estimators of β and g are constructed when the explanatory variables t take values on a Riemannian manifold and the asymptotic properties of the proposed estimators are obtained under suitable conditions. We illustrate the use of this estimation approach using an environmental data set and we explore the performance of the estimators through a simulation study.
In partially linear models the dependence of the response y on (x^T,t) is modeled through the rel... more In partially linear models the dependence of the response y on (x^T,t) is modeled through the relationship y=\x^T \beta+g(t)+\epsilon where \epsilon is independent of (x^T,t). In this paper, estimators of \beta and g are constructed when the explanatory variables t take values on a Riemannian manifold. Our proposal combine the flexibility of these models with the complex structure of a set of explanatory variables. We prove that the resulting estimator of \beta is asymptotically normal under the suitable conditions. Through a simulation study, we explored the performance of the estimators. Finally, we applied the studied model to an example based on real dataset.
Journal of Mathematical Imaging and Vision, 2009
The paper concerns the strong uniform consistency and the asymptotic distribution of the kernel d... more The paper concerns the strong uniform consistency and the asymptotic distribution of the kernel density estimator of random objects on a Riemannian manifolds, proposed by Pelletier (Stat. Probab. Lett., 73(3):297–304, 2005). The estimator is illustrated via one example based on a real data.
In many situations, when dealing with several populations with different covariance operators, eq... more In many situations, when dealing with several populations with different covariance operators, equality of the operators is assumed. Usually, if this assumption does not hold, one estimates the covariance operator of each group separately, which leads to a large number of parameters. As in the multivariate setting, this is not satisfactory since the covariance operators may exhibit some common structure. In this paper, we discuss the extension to the functional setting of common principal component model that has been widely studied when dealing with multivariate observations. Moreover, we also consider a proportional model in which the covariance operators are assumed to be equal up to a multiplicative constant. For both models, we present estimators of the unknown parameters and we obtain their asymptotic distribution. A test for equality against proportionality is also considered.
Computational Statistics & Data Analysis, 2008
In this paper, under a semiparametric partly linear regression model with fixed design, we introd... more In this paper, under a semiparametric partly linear regression model with fixed design, we introduce a family of robust procedures to select the bandwidth parameter. The robust plug-in proposal is based on nonparametric robust estimates of the ν−th derivatives and under mild conditions, it converges to the optimal bandwidth. A robust cross-validation bandwidth is also considered and the performance of the different proposals is compared through a Monte Carlo study. We define an empirical influence measure for data-driven bandwidth selectors and, through it, we study the sensitivity of the data-driven bandwidth selectors. It appears that the robust selector compares favorably to its classical competitor, despite the need to select a pilot bandwidth when considering plug-in bandwidths. Moreover, the plug-in procedure seems to be less sensitive than the cross-validation in particular, when introducing several outliers. When combined with the three-step procedure proposed by , the robust selectors lead to robust data-driven estimates of both the regression function and the regression parameter. 1 2 ( β ls (h) − β), and n MSE(h) = EU 2 /σ 2 its standardized mean square error. Then, when the smoothing procedure corresponds to local means, under general conditions, that include that the design points are almost uniform design points, i.e., {t i } n i=1 are fixed design points in [0, 1], 0 ≤ t 1 ≤ . . . ≤ t n ≤ 1, such that t 0 = 0 and t n+1 = 1 and max 1≤i≤n+1 |(t i − t i−1 ) − 1/n| = O(n −δ ) for some δ > 1, we have that, for ν ≥ 2, MSE(h) = n −1 {1 + (nh) −1 A 2 + o(n −2µ ) + (n 1 2 h 2ν A 1 + o(n −µ )) 2 } ,
Test, 2012
A natural generalization of the well known generalized linear models is to allow only for some of... more A natural generalization of the well known generalized linear models is to allow only for some of the predictors to be modeled linearly while others are modeled nonparametrically. However, this model can face the so called “curse of dimensionality” problem that can be solved by imposing a nonparametric dependence on some unknown projection of the carriers. More precisely, we assume that the observations (y i ,x i ,t i ), 1≤i≤n, are such that t i ∈ℝq , x i ∈ℝp and y i |(x i ,t i )∼F(⋅,μ i ) with \(\mu_{i}=H (\eta(\boldsymbol{\alpha}^{\mathrm{T}}\mathbf{t}_{i})+\mathbf {x}_{i}^{\mathrm{T}}\boldsymbol{\beta} )\) , for some known distribution function F and link function H. The function η:ℝ→ℝ and the parameters α and β are unknown and to be estimated. This model is known as the generalized partly linear single-index model. In this paper, we introduce a family of robust estimates for the parametric and nonparametric components under a generalized partially linear single-index model. It is shown that the estimates of α and β are root-n consistent and asymptotically normally distributed. Through a Monte Carlo study, we compare the performance of the proposed estimators with that of the classical ones.
In this paper, we consider a k-nearest neighbor kernel type estimator when the random variables b... more In this paper, we consider a k-nearest neighbor kernel type estimator when the random variables belong in a Riemannian manifolds. We study asymptotic properties such as the consistency and the asymptotic distribution. A simulation study is also consider to evaluate the performance of the proposal. Finally, to illustrate the potential applications of the proposed estimator, we analyzed two real example where two different manifolds are considered.
Statistics & Probability Letters, 2006
In this paper, robust estimates for the derivatives of order ν of the regression function are con... more In this paper, robust estimates for the derivatives of order ν of the regression function are considered. This estimator extend the proposals given when ν = 1, 2. Uniform consistency, which allows to construct a robust data-driven bandwidth, is established. Besides, the robust estimates introduced are asymptotically normally distributed and their asymptotic efficiency is that of the related M −location estimators.
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Papers by Daniela Rodriguez