Consider the following three important problems in statistical inference, namely, constructing confidence intervals for (1) the error of a high-dimensional ( @math ) regression estimator, (2) the linear regression noise level, and (3) the genetic signal-to-noise ratio of a continuous-valued trait (related to the heritability). All three problems turn out to be closely related to the little-studied problem of performing inference on the @math -norm of the signal in high-dimensional linear regression. We derive a novel procedure for this, which is asymptotically correct when the covariates are multivariate Gaussian and produces valid confidence intervals in finite samples as well. The procedure, called EigenPrism, is computationally fast and makes no assumptions on coefficient sparsity or knowledge of the noise level. We investigate the width of the EigenPrism confidence intervals, including a comparison with a Bayesian setting in which our interval is just 5 wider than the Bayes credible interval. We are then able to unify the three aforementioned problems by showing that the EigenPrism procedure with only minor modifications is able to make important contributions to all three. We also investigate the robustness of coverage and find that the method applies in practice and in finite samples much more widely than just the case of multivariate Gaussian covariates. Finally, we apply EigenPrism to a genetic dataset to estimate the genetic signal-to-noise ratio for a number of continuous phenotypes.
When @math , ordinary least squares (OLS) theory gives us inference for @math and thus also for @math . When @math , the problem of estimating @math has been studied in @cite_5 . @cite_5 uses the method of moments on two statistics to estimate @math and @math without assumptions on @math , and with the same multivariate Gaussian random design assumptions used here . @cite_5 also derives asymptotic distributional results, but does not explore the estimation of the parameters of the asymptotic distributions, nor the coverage of any CI derived from it. The main contribution of our work is to provide tight, CIs which achieve nominal coverage even in finite samples.
@cite_5
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Let @math be an i.i.d. sequence of random variables in @math , @math . We show that, for any function @math , under regularity conditions, [n^ 1 2 (n^ -1 i=1 ^n (X_i) f ^(X_i) - (x) ,dx ) P 0, ] where @math is the classical kernel estimator of the density of @math . This result is striking because it speeds up traditional rates, in root @math , derived from the central limit theorem when @math . Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions @math and @math , the dimension @math and the bandwidth of the kernel estimator @math . Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of @math . As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations. We address the problem of fast estimation of ordinary least squares (OLS) from large amounts of data (n ≫ p). We propose three methods which solve the big data problem by subsampling the covariance matrix using either a single or two stage estimation. All three run in the order of size of input i.e. O(np) and our best method, Uluru, gives an error bound of O(√p n) which is independent of the amount of subsampling as long as it is above a threshold. We provide theoretical bounds for our algorithms in the fixed design (with Randomized Hadamard preconditioning) as well as sub-Gaussian random design setting. We also compare the performance of our methods on synthetic and real-world datasets and show that if observations are i.i.d., sub-Gaussian then one can directly subsample without the expensive Randomized Hadamard preconditioning without loss of accuracy. We consider the problem of estimating an unknown signal @math from noisy linear observations @math . In many practical instances, @math has a certain structure that can be captured by a structure inducing convex function @math . For example, @math norm can be used to encourage a sparse solution. To estimate @math with the aid of @math , we consider the well-known LASSO method and provide sharp characterization of its performance. We assume the entries of the measurement matrix @math and the noise vector @math have zero-mean normal distributions with variances @math and @math respectively. For the LASSO estimator @math , we attempt to calculate the Normalized Square Error (NSE) defined as @math as a function of the noise level @math , the number of observations @math and the structure of the signal. We show that, the structure of the signal @math and choice of the function @math enter the error formulae through the summary parameters @math and @math , which are defined as the Gaussian squared-distances to the subdifferential cone and to the @math -scaled subdifferential, respectively. The first LASSO estimator assumes a-priori knowledge of @math and is given by @math . We prove that its worst case NSE is achieved when @math and concentrates around @math . Secondly, we consider @math , for some @math . This time the NSE formula depends on the choice of @math and is given by @math . We then establish a mapping between this and the third estimator @math . Finally, for a number of important structured signal classes, we translate our abstract formulae to closed-form upper bounds on the NSE. We prove that the ordinary least-squares (OLS) estimator attains nearly minimax optimal performance for the identification of linear dynamical systems from a single observed trajectory. Our upper bound analysis relies on a generalization of Mendelson’s small-ball method to dependent data, eschewing the use of standard mixing-time arguments. Our lower bounds reveal that these upper bounds match up to logarithmic factors. In particular, we capture the correct signal-to-noise behavior of the problem, showing that more unstable linear systems are easier to estimate. This behavior is qualitatively different from arguments which rely on mixing-time calculations that suggest that unstable systems are more difficult to estimate. We generalize our technique to provide bounds for a more general class of linear response time-series.