Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity
TL;DR: In this article, the authors consider confidence intervals for high-dimensional linear regression with random design and establish the convergence rates of the minimax expected length for confidence intervals in the oracle setting where the sparsity parameter is given.
Abstract: Confidence sets play a fundamental role in statistical inference. In this paper, we consider confidence intervals for high-dimensional linear regression with random design. We first establish the convergence rates of the minimax expected length for confidence intervals in the oracle setting where the sparsity parameter is given. The focus is then on the problem of adaptation to sparsity for the construction of confidence intervals. Ideally, an adaptive confidence interval should have its length automatically adjusted to the sparsity of the unknown regression vector, while maintaining a pre-specified coverage probability. It is shown that such a goal is in general not attainable, except when the sparsity parameter is restricted to a small region over which the confidence intervals have the optimal length of the usual parametric rate. It is further demonstrated that the lack of adaptivity is not due to the conservativeness of the minimax framework, but is fundamentally caused by the difficulty of learning the bias accurately.
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TL;DR: A method for debiasing penalized regression adjustments to allow sparse regression methods like the lasso to be used for √n‐consistent inference of average treatment effects in high dimensional linear models.
Abstract: There are many settings where researchers are interested in estimating average treatment effects and are willing to rely on the unconfoundedness assumption, which requires that the treatment assignment be as good as random conditional on pretreatment variables. The unconfoundedness assumption is often more plausible if a large number of pretreatment variables are included in the analysis, but this can worsen the performance of standard approaches to treatment effect estimation. We develop a method for debiasing penalized regression adjustments to allow sparse regression methods like the lasso to be used for √n‐consistent inference of average treatment effects in high dimensional linear models. Given linearity, we do not need to assume that the treatment propensities are estimable, or that the average treatment effect is a sparse contrast of the outcome model parameters. Rather, in addition to standard assumptions used to make lasso regression on the outcome model consistent under 1‐norm error, we require only overlap, i.e. that the propensity score be uniformly bounded away from 0 and 1. Procedurally, our method combines balancing weights with a regularized regression adjustment.
326 citations
TL;DR: This work studies the problem of treatment effect estimation in randomized experiments with high-dimensional covariate information and shows that essentially any risk-consistent regression adjustment can be used to obtain efficient estimates of the average treatment effect.
Abstract: We study the problem of treatment effect estimation in randomized experiments with high-dimensional covariate information and show that essentially any risk-consistent regression adjustment can be used to obtain efficient estimates of the average treatment effect. Our results considerably extend the range of settings where high-dimensional regression adjustments are guaranteed to provide valid inference about the population average treatment effect. We then propose cross-estimation, a simple method for obtaining finite-sample-unbiased treatment effect estimates that leverages high-dimensional regression adjustments. Our method can be used when the regression model is estimated using the lasso, the elastic net, subset selection, etc. Finally, we extend our analysis to allow for adaptive specification search via cross-validation and flexible nonparametric regression adjustments with machine-learning methods such as random forests or neural networks.
97 citations
TL;DR: In this article, the authors consider the problem of constructing confidence intervals (CIs) for a linear functional of a regression function, such as its value at a point, the regression discontinuity parameter, or a regression coefficient in a linear or partly linear regression.
Abstract: We consider the problem of constructing confidence intervals (CIs) for a linear functional of a regression function, such as its value at a point, the regression discontinuity parameter, or a regression coefficient in a linear or partly linear regression. Our main assumption is that the regression function is known to lie in a convex function class, which covers most smoothness and/or shape assumptions used in econometrics. We derive finite-sample optimal CIs and sharp efficiency bounds under normal errors with known variance. We show that these results translate to uniform (over the function class) asymptotic results when the error distribution is not known. When the function class is centrosymmetric, these efficiency bounds imply thatminimax CIs are close to efficient at smooth regression functions. This implies, in particular, that it is impossible to form CIs that are tighter using data-dependent tuning parameters, and maintain coverage over the whole function class. We specialize our results to inference in a linear regression, and inference on the regression discontinuity parameter, and illustrate them in an empirical application.
89 citations
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TL;DR: It is proved that the maximum-likelihood estimate (MLE) is biased, the variability of the MLE is far greater than classically estimated, and the likelihood-ratio test (LRT) is not distributed as a χ2.
Abstract: Every student in statistics or data science learns early on that when the sample size largely exceeds the number of variables, fitting a logistic model produces estimates that are approximately unbiased. Every student also learns that there are formulas to predict the variability of these estimates which are used for the purpose of statistical inference; for instance, to produce p-values for testing the significance of regression coefficients. Although these formulas come from large sample asymptotics, we are often told that we are on reasonably safe grounds when $n$ is large in such a way that $n \ge 5p$ or $n \ge 10p$. This paper shows that this is far from the case, and consequently, inferences routinely produced by common software packages are often unreliable.
Consider a logistic model with independent features in which $n$ and $p$ become increasingly large in a fixed ratio. Then we show that (1) the MLE is biased, (2) the variability of the MLE is far greater than classically predicted, and (3) the commonly used likelihood-ratio test (LRT) is not distributed as a chi-square. The bias of the MLE is extremely problematic as it yields completely wrong predictions for the probability of a case based on observed values of the covariates. We develop a new theory, which asymptotically predicts (1) the bias of the MLE, (2) the variability of the MLE, and (3) the distribution of the LRT. We empirically also demonstrate that these predictions are extremely accurate in finite samples. Further, an appealing feature is that these novel predictions depend on the unknown sequence of regression coefficients only through a single scalar, the overall strength of the signal. This suggests very concrete procedures to adjust inference; we describe one such procedure learning a single parameter from data and producing accurate inference
86 citations
TL;DR: In this article, the authors propose a methodology for testing linear hypothesis in high-dimensional linear models, which does not impose any restriction on the size of the model, i.e. model sparsity or the loading vector representing the hypothesis.
Abstract: We propose a methodology for testing linear hypothesis in high-dimensional linear models. The proposed test does not impose any restriction on the size of the model, i.e. model sparsity or the loading vector representing the hypothesis. Providing asymptotically valid methods for testing general linear functions of the regression parameters in high-dimensions is extremely challenging - especially without making restrictive or unverifiable assumptions on the number of non-zero elements. We propose to test the moment conditions related to the newly designed restructured regression, where the inputs are transformed and augmented features. These new features incorporate the structure of the null hypothesis directly. The test statistics are constructed in such a way that lack of sparsity in the original model parameter does not present a problem for the theoretical justification of our procedures. We establish asymptotically exact control on Type I error without imposing any sparsity assumptions on mode...
83 citations
References
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TL;DR: A new method for estimation in linear models called the lasso, which minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant, is proposed.
Abstract: SUMMARY We propose a new method for estimation in linear models. The 'lasso' minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models. Our simulation studies suggest that the lasso enjoys some of the favourable properties of both subset selection and ridge regression. It produces interpretable models like subset selection and exhibits the stability of ridge regression. There is also an interesting relationship with recent work in adaptive function estimation by Donoho and Johnstone. The lasso idea is quite general and can be applied in a variety of statistical models: extensions to generalized regression models and tree-based models are briefly described.
40,785 citations
TL;DR: In many important statistical applications, the number of variables or parameters p is much larger than the total number of observations n as discussed by the authors, and it is possible to estimate β reliably based on the noisy data y.
Abstract: In many important statistical applications, the number of variables or parameters p is much larger than the number of observations n. Suppose then that we have observations y=Xβ+z, where β∈Rp is a parameter vector of interest, X is a data matrix with possibly far fewer rows than columns, n≪p, and the zi’s are i.i.d. N(0, σ^2). Is it possible to estimate β reliably based on the noisy data y?
3,539 citations
TL;DR: In this article, the authors give a brief review of the basic idea and some history and then discuss some developments since the original paper on regression shrinkage and selection via the lasso.
Abstract: Summary. In the paper I give a brief review of the basic idea and some history and then discuss some developments since the original paper on regression shrinkage and selection via the lasso.
3,054 citations
TL;DR: In this article, the Lasso estimator and the Dantzig selector exhibit similar behavior under a sparsity scenario, and they derive, in parallel, oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the l p estimation loss for 1 ≤ p ≤ 2.
Abstract: We show that, under a sparsity scenario, the Lasso estimator and the Dantzig selector exhibit similar behavior. For both methods, we derive, in parallel, oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the l p estimation loss for 1 ≤ p ≤ 2 in the linear model when the number of variables can be much larger than the sample size.
2,524 citations
TL;DR: A constrained ℓ1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p-variate random variables and is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods.
Abstract: This article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p-variate random variables. The resulting estimator is shown to have a number of desirable properties. In particular, the rate of convergence between the estimator and the true s-sparse precision matrix under the spectral norm is when the population distribution has either exponential-type tails or polynomial-type tails. We present convergence rates under the elementwise l∞ norm and Frobenius norm. In addition, we consider graphical model selection. The procedure is easily implemented by linear programming. Numerical performance of the estimator is investigated using both simulated and real data. In particular, the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods.
947 citations