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During the last decades, there has been an explosion in computation and information technology. This development comes with an expansion of complex observational studies and clinical trials in a variety of fields such as medicine, biology, epidemiology, sociology, and economics among many others, which involve collection of large amounts of data on subjects or organisms over time. The goal of such studies can be formulated as estimation of a finite dimensional parameter of the population distribution corresponding to the observed time-dependent process. Such estimation problems arise in survival analysis, causal inference and regression analysis. This book provides a fundamental statistical framework for the analysis of complex longitudinal data. It provides the first comprehensive description of optimal estimation techniques based on time-dependent data structures subject to informative censoring and treatment assignment in so called semiparametric models. Semiparametric models are particularly attractive since they allow the presence of large unmodeled nuisance parameters. These techniques include estimation of regression parameters in the familiar (multivariate) generalized linear regression and multiplicative intensity models. They go beyond standard statistical approaches by incorporating all the observed data to allow for informative censoring, to obtain maximal efficiency, and by developing estimators of causal effects. It can be used to teach masters and Ph.D. students in biostatistics and statistics and is suitable for researchers in statistics with a strong interest in the analysis of complex longitudinal data.

Nonparametric statistics --- Longitudinal method --- Estimation theory --- Statistique non-paramétrique --- Méthode longitudinale --- Théorie de l'estimation --- Censored observations (Statistics) --- Estimation theory. --- Longitudinal method. --- Nonparametric statistics. --- Censored observations (Statistics).

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We revisit the classic semiparametric problem of inference on a low dimensional parameter θ_0 in the presence of high-dimensional nuisance parameters η_0. We depart from the classical setting by allowing for η_0 to be so high-dimensional that the traditional assumptions, such as Donsker properties, that limit complexity of the parameter space for this object break down. To estimate η_0, we consider the use of statistical or machine learning (ML) methods which are particularly well-suited to estimation in modern, very high-dimensional cases. ML methods perform well by employing regularization to reduce variance and trading off regularization bias with overfitting in practice. However, both regularization bias and overfitting in estimating η_0 cause a heavy bias in estimators of θ_0 that are obtained by naively plugging ML estimators of η_0 into estimating equations for θ_0. This bias results in the naive estimator failing to be N^(-1/2) consistent, where N is the sample size. We show that the impact of regularization bias and overfitting on estimation of the parameter of interest θ_0 can be removed by using two simple, yet critical, ingredients: (1) using Neyman-orthogonal moments/scores that have reduced sensitivity with respect to nuisance parameters to estimate θ_0, and (2) making use of cross-fitting which provides an efficient form of data-splitting. We call the resulting set of methods double or debiased ML (DML). We verify that DML delivers point estimators that concentrate in a N^(-1/2)-neighborhood of the true parameter values and are approximately unbiased and normally distributed, which allows construction of valid confidence statements. The generic statistical theory of DML is elementary and simultaneously relies on only weak theoretical requirements which will admit the use of a broad array of modern ML methods for estimating the nuisance parameters such as random forests, lasso, ridge, deep neural nets, boosted trees, and various hybrids and ensembles of these methods. We illustrate the general theory by applying it to provide theoretical properties of DML applied to learn the main regression parameter in a partially linear regression model, DML applied to learn the coefficient on an endogenous variable in a partially linear instrumental variables model, DML applied to learn the average treatment effect and the average treatment effect on the treated under unconfoundedness, and DML applied to learn the local average treatment effect in an instrumental variables setting. In addition to these theoretical applications, we also illustrate the use of DML in three empirical examples.