arXiv subjects

Ezequiel Smucler

Publications and source records attributed to Ezequiel Smucler.

17 recordsLinked to original sources

arXiv2025

A note on the relation between one-step, outcome regression and IPW-type estimators of parameters with the mixed bias property

Bruns-Smith et al. (2025) established an algebraic identity between the one-step estimator and a specific outcome regression-type estimator for a class of parameters that forms a strict subset of the class introduced in Chernozhukov et al. (2022), assuming both nuisance functions are estimated as linear combinations of given features. They conjectured that this identity extends to the broader mixed bias class introduced in Rotnitzky et al. (2021). In this note, we prove their conjecture and further extend the result to allow one of the nuisance estimators to be non-linear. We also relate these findings to the work of Robins et al. (2007), who established other identities linking one-step estimators to outcome regression-type and IPW-type estimators.

math.ST↗

arXiv2025

On the asymptotic validity of confidence sets for linear functionals of solutions to integral equations

This paper examines the construction of confidence sets for parameters defined as linear functionals of a function of W and X whose conditional mean given Z and X equals the conditional mean of another variable Y given Z and X. Many estimands of interest in causal inference can be expressed in this form, including the average treatment effect in proximal causal inference and treatment effect contrasts in instrumental variable models. We derive a necessary condition for a confidence set to be uniformly valid over a model that allows for the dependence between W and Z given X to be arbitrarily weak. Specifically, we show that for any such confidence set, there must exist some laws in the model under which, with high probability, the confidence set has a diameter greater than or equal to the diameter of the parameter's range. In particular, consistent with the weak instruments literature, Wald confidence intervals are not uniformly valid over the aforementioned model when the parameter's range is infinite. Furthermore, we argue that inverting the score test, a successful approach in that literature, generally fails for the broader class of parameters considered here. We present a method for constructing uniformly valid confidence sets in the special case where all variables, but possibly Y, are binary and discuss its limitations. Finally, we emphasize that developing uniformly valid confidence sets for the class of parameters considered in this paper remains an open problem.

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arXiv2025

A note on the properties of the confidence set for the local average treatment effect obtained by inverting the score test

We study the properties of the score confidence set for the local average treatment effect in non and semiparametric instrumental variable models. This confidence set is constructed by inverting a score test based on an estimate of the nonparametric influence function for the estimand, and is known to be uniformly valid in models that allow for arbitrarily weak instruments; because of this, the confidence set can have infinite diameter at some laws. We characterize the six possible forms the score confidence set can take: a finite interval, an infinite interval (or a union of them), the whole real line, an empty set, or a single point. Moreover, we show that, at any fixed law, the score confidence set asymptotically coincides, up to a term of order 1/n, with the Wald confidence interval based on the doubly robust estimator which solves the estimating equation associated with the nonparametric influence function. This result implies that, in models where the efficient influence function coincides with the nonparametric influence function, the score confidence set is, in a sense, optimal in terms of its diameter. We also show that under weak instrument asymptotics, where the strength of the instrument is modelled as local to zero, the doubly robust estimator is asymptotically biased and does not follow a normal distribution. A simulation study confirms that, as expected, the doubly robust estimator performs poorly when instruments are weak, whereas the score confidence set retains good finite-sample properties in both strong and weak instrument settings. Finally, we provide an algorithm to compute the score confidence set, which is now available in the DoubleML package for double machine learning.

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arXiv2022

A note on efficient minimum cost adjustment sets in causal graphical models

We study the selection of adjustment sets for estimating the interventional mean under an individualized treatment rule. We assume a non-parametric causal graphical model with, possibly, hidden variables and at least one adjustment set comprised of observable variables. Moreover, we assume that observable variables have positive costs associated with them. We define the cost of an observable adjustment set as the sum of the costs of the variables that comprise it. We show that in this setting there exist adjustment sets that are minimum cost optimal, in the sense that they yield non-parametric estimators of the interventional mean with the smallest asymptotic variance among those that control for observable adjustment sets that have minimum cost. Our results are based on the construction of a special flow network associated with the original causal graph. We show that a minimum cost optimal adjustment set can be found by computing a maximum flow on the network, and then finding the set of vertices that are reachable from the source by augmenting paths. The optimaladj Python package implements the algorithms introduced in this paper.

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arXiv2020

Uncovering differential equations from data with hidden variables

SINDy is a method for learning system of differential equations from data by solving a sparse linear regression optimization problem [Brunton et al., 2016]. In this article, we propose an extension of the SINDy method that learns systems of differential equations in cases where some of the variables are not observed. Our extension is based on regressing a higher order time derivative of a target variable onto a dictionary of functions that includes lower order time derivatives of the target variable. We evaluate our method by measuring the prediction accuracy of the learned dynamical systems on synthetic data and on a real data-set of temperature time series provided by the Réseau de Transport d'Électricité (RTE). Our method provides high quality short-term forecasts and it is orders of magnitude faster than competing methods for learning differential equations with latent variables.

stat.ML↗

arXiv2020

Clustering high dimensional meteorological scenarios: results and performance index

The Reseau de Transport d'Electricité (RTE) is the French main electricity network operational manager and dedicates large number of resources and efforts towards understanding climate time series data. We discuss here the problem and the methodology of grouping and selecting representatives of possible climate scenarios among a large number of climate simulations provided by RTE. The data used is composed of temperature times series for 200 different possible scenarios on a grid of geographical locations in France. These should be clustered in order to detect common patterns regarding temperatures curves and help to choose representative scenarios for network simulations, which in turn can be used for energy optimisation. We first show that the choice of the distance used for the clustering has a strong impact on the meaning of the results: depending on the type of distance used, either spatial or temporal patterns prevail. Then we discuss the difficulty of fine-tuning the distance choice (combined with a dimension reduction procedure) and we propose a methodology based on a carefully designed index.

stat.AP↗

arXiv2020

Efficient adjustment sets in causal graphical models with hidden variables

We study the selection of covariate adjustment sets for estimating the value of point exposure dynamic policies, also known as dynamic treatment regimes, assuming a non-parametric causal graphical model with hidden variables, in which at least one adjustment set is fully observable. We show that recently developed criteria, for graphs without hidden variables, to compare the asymptotic variance of non-parametric estimators of static policy values that control for certain adjustment sets, are also valid under dynamic policies and graphs with hidden variables. We show that there exist adjustment sets that are optimal minimal (minimum), in the sense of yielding estimators with the smallest variance among those that control for adjustment sets that are minimal (of minimum cardinality). Moreover, we show that if either no variables are hidden or if all the observable variables are ancestors of either treatment, outcome, or the variables that are used to decide treatment, a globally optimal adjustment set exists. We provide polynomial time algorithms to compute the globally optimal (when it exists), optimal minimal, and optimal minimum adjustment sets. Our results are based on the construction of an undirected graph in which vertex cuts between the treatment and outcome variables correspond to adjustment sets. In this undirected graph, a partial order between minimal vertex cuts can be defined that makes the set of minimal cuts a lattice. This partial order corresponds directly to the ordering of the asymptotic variances of the corresponding non-parametrically adjusted estimators.

math.ST↗

arXiv2020

Orthant probabilities and the attainment of maxima on a vertex of a simplex

We calculate bounds for orthant probabilities for the equicorrelated multivariate normal distribution and use these bounds to show the following: for degree $k>4$, the probability that a $k$-homogeneous polynomial in $n$ variables attains a relative maximum on a vertex of the $n$-dimensional simplex tends to one as the dimension $n$ grows. The bounds we obtain for the orthant probabilities are tight up to $\log(n)$ factors.

math.PR↗

arXiv2019

Efficient adjustment sets for population average treatment effect estimation in non-parametric causal graphical models

The method of covariate adjustment is often used for estimation of population average treatment effects in observational studies. Graphical rules for determining all valid covariate adjustment sets from an assumed causal graphical model are well known. Restricting attention to causal linear models, a recent article derived two novel graphical criteria: one to compare the asymptotic variance of linear regression treatment effect estimators that control for certain distinct adjustment sets and another to identify the optimal adjustment set that yields the least squares treatment effect estimator with the smallest asymptotic variance among consistent adjusted least squares estimators. In this paper we show that the same graphical criteria can be used in non-parametric causal graphical models when treatment effects are estimated by contrasts involving non-parametrically adjusted estimators of the interventional means. We also provide a graphical criterion for determining the optimal adjustment set among the minimal adjustment sets, which is valid for both linear and non-parametric estimators. We provide a new graphical criterion for comparing time dependent adjustment sets, that is, sets comprised by covariates that adjust for future treatments and that are themselves affected by earlier treatments. We show by example that uniformly optimal time dependent adjustment sets do not always exist. In addition, for point interventions, we provide a sound and complete graphical criterion for determining when a non-parametric optimally adjusted estimator of an interventional mean, or of a contrast of interventional means, is as efficient as an efficient estimator of the same parameter that exploits the information in the conditional independencies encoded in the non-parametric causal graphical model.

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arXiv2019

Split Regularized Regression

We propose an approach for fitting linear regression models that splits the set of covariates into groups. The optimal split of the variables into groups and the regularized estimation of the regression coefficients are performed by minimizing an objective function that encourages sparsity within each group and diversity among them. The estimated coefficients are then pooled together to form the final fit. Our procedure works on top of a given penalized linear regression estimator (e.g., Lasso, elastic net) by fitting it to possibly overlapping groups of features, encouraging diversity among these groups to reduce the correlation of the corresponding predictions. For the case of two groups, elastic net penalty and orthogonal predictors, we give a closed form solution for the regression coefficients in each group. We establish the consistency of our method with the number of predictors possibly increasing with the sample size. An extensive simulation study and real-data applications show that in general the proposed method improves the prediction accuracy of the base estimator used in the procedure. Possible extensions to GLMs and other models are discussed. The supplemental material for this article, available online, contains the proofs of our theoretical results and the full results of our simulation study.

stat.ME↗

arXiv2019

A unifying approach for doubly-robust $\ell_1$ regularized estimation of causal contrasts

We consider inference about a scalar parameter under a non-parametric model based on a one-step estimator computed as a plug in estimator plus the empirical mean of an estimator of the parameter's influence function. We focus on a class of parameters that have influence function which depends on two infinite dimensional nuisance functions and such that the bias of the one-step estimator of the parameter of interest is the expectation of the product of the estimation errors of the two nuisance functions. Our class includes many important treatment effect contrasts of interest in causal inference and econometrics, such as ATE, ATT, an integrated causal contrast with a continuous treatment, and the mean of an outcome missing not at random. We propose estimators of the target parameter that entertain approximately sparse regression models for the nuisance functions allowing for the number of potential confounders to be even larger than the sample size. By employing sample splitting, cross-fitting and $\ell_1$-regularized regression estimators of the nuisance functions based on objective functions whose directional derivatives agree with those of the parameter's influence function, we obtain estimators of the target parameter with two desirable robustness properties: (1) they are rate doubly-robust in that they are root-n consistent and asymptotically normal when both nuisance functions follow approximately sparse models, even if one function has a very non-sparse regression coefficient, so long as the other has a sufficiently sparse regression coefficient, and (2) they are model doubly-robust in that they are root-n consistent and asymptotically normal even if one of the nuisance functions does not follow an approximately sparse model so long as the other nuisance function follows an approximately sparse model with a sufficiently sparse regression coefficient.

math.ST↗

arXiv2019

Characterization of parameters with a mixed bias property

In this article we study a class of parameters with the so-called `mixed bias property'. For parameters with this property, the bias of the semiparametric efficient one step estimator is equal to the mean of the product of the estimation errors of two nuisance functions. In non-parametric models, parameters with the mixed bias property admit so-called rate doubly robust estimators, i.e. estimators that are consistent and asymptotically normal when one succeeds in estimating both nuisance functions at sufficiently fast rates, with the possibility of trading off slower rates of convergence for the estimator of one of the nuisance functions with faster rates for the estimator of the other nuisance. We show that the class of parameters with the mixed bias property strictly includes two recently studied classes of parameters which, in turn, include many parameters of interest in causal inference. We characterize the form of parameters with the mixed bias property and of their influence functions. Furthermore, we derive two functional moment equations, each being solved at one of the two nuisance functions, as well as, two functional loss functions, each being minimized at one of the two nuisance functions. These loss functions can be used to derive loss based penalized estimators of the nuisance functions.

math.ST↗

arXiv2017

Consistency of Generalized Dynamic Principal Components in Dynamic Factor Models

We study the theoretical properties of the generalized dynamic principal components introduced in Peña and Yohai (2016). In particular, we prove that when the data follows a dynamic factor model, the reconstruction provided by the procedure converges in mean square to the common part of the model as the number of series and periods diverge to infinity. The results of a simulation study support our findings.

math.ST↗

arXiv2017

Asymptotic theory for maximum likelihood estimates in reduced-rank multivariate generalised linear models

Reduced-rank regression is a dimensionality reduction method with many applications. The asymptotic theory for reduced rank estimators of parameter matrices in multivariate linear models has been studied extensively. In contrast, few theoretical results are available for reduced-rank multivariate generalised linear models. We develop M-estimation theory for concave criterion functions that are maximised over parameters spaces that are neither convex nor closed. These results are used to derive the consistency and asymptotic distribution of maximum likelihood estimators in reduced-rank multivariate generalised linear models, when the response and predictor vectors have a joint distribution. We illustrate our results in a real data classification problem with binary covariates.

math.ST↗

arXiv2017

Forecasting Multiple Time Series with One-Sided Dynamic Principal Components

We define one-sided dynamic principal components (ODPC) for time series as linear combinations of the present and past values of the series that minimize the reconstruction mean squared error. Previous definitions of dynamic principal components depend on past and future values of the series. For this reason, they are not appropriate for forecasting purposes. On the contrary, it is shown that the ODPC introduced in this paper can be successfully used for forecasting high-dimensional multiple time series. An alternating least squares algorithm to compute the proposed ODPC is presented. We prove that for stationary and ergodic time series the estimated values converge to their population analogues. We also prove that asymptotically, when both the number of series and the sample size go to infinity, if the data follows a dynamic factor model, the reconstruction obtained with ODPC converges, in mean squared error, to the common part of the factor model. Monte Carlo results shows that forecasts obtained by the ODPC compare favourably with other forecasting methods based on dynamic factor models.

stat.ME↗

arXiv2016

Asymptotic Statistical Properties of Redescending M-estimators in Linear Models with Increasing Dimension

This paper deals with the asymptotic statistical properties of a class of redescending M-estimators in linear models with increasing dimension. This class is wide enough to include popular high breakdown point estimators such as S-estimators and MM-estimators, which were not covered by existing results in the literature. We prove consistency assuming only that $p/n \rightarrow 0$ and asymptotic normality essentially if $p^{3}/n \rightarrow 0$, where $p$ is the number of covariates and $n$ is the sample size.

math.ST↗

arXiv2015

Robust and sparse estimators for linear regression models

Penalized regression estimators are a popular tool for the analysis of sparse and high-dimensional data sets. However, penalized regression estimators defined using an unbounded loss function can be very sensitive to the presence of outlying observations, especially high leverage outliers. Moreover, it can be particularly challenging to detect outliers in high-dimensional data sets. Thus, robust estimators for sparse and high-dimensional linear regression models are in need. In this paper, we study the robust and asymptotic properties of MM-Bridge and adaptive MM-Bridge estimators: $\ell_q$-penalized MM-estimators of regression and MM-estimators with an adaptive $\ell_t$ penalty. For the case of a fixed number of covariates, we derive the asymptotic distribution of MM-Bridge estimators for all $q>0$. We prove that for $q<1$ MM-Bridge estimators can have the oracle property defined in Fan and Li (2001). We prove that for all $t\leq 1$ adaptive MM-Bridge estimators can have the oracle property. The advantages of our proposed estimators are demonstrated through an extensive simulation study and the analysis of a real high-dimensional data set.

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