Bootstrap confidence sets under model misspecification

A multiplier bootstrap procedure for construction of likelihood-based confidence sets is considered for finite samples and a possible model misspecification. Theoretical results justify the bootstrap validity for a small or moderate sample size and allow to control the impact of the parameter dimension $p$: the bootstrap approximation works if $p^3/n$ is small. The main result about bootstrap validity continues to apply even if the underlying parametric model is misspecified under the so-called small modelling bias condition. In the case when the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed confidence sets is increased by the modelling bias. We illustrate the results with numerical examples for misspecified linear and logistic regressions.Comment: Published at http://dx.doi.org/10.1214/15-AOS1355 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Non-Gaussianity in the Weak Lensing Correlation Function Likelihood -- Implications for Cosmological Parameter Biases

We study the significance of non-Gaussianity in the likelihood of weak lensing shear two-point correlation functions, detecting significantly non-zero skewness and kurtosis in one-dimensional marginal distributions of shear two-point correlation functions in simulated weak lensing data. We examine the implications in the context of future surveys, in particular LSST, with derivations of how the non-Gaussianity scales with survey area. We show that there is no significant bias in one-dimensional posteriors of $\Omega_{\rm m}$ and $\sigma_{\rm 8}$ due to the non-Gaussian likelihood distributions of shear correlations functions using the mock data ($100$ deg$^{2}$). We also present a systematic approach to constructing approximate multivariate likelihoods with one-dimensional parametric functions by assuming independence or more flexible non-parametric multivariate methods after decorrelating the data points using principal component analysis (PCA). While the use of PCA does not modify the non-Gaussianity of the multivariate likelihood, we find empirically that the one-dimensional marginal sampling distributions of the PCA components exhibit less skewness and kurtosis than the original shear correlation functions.Modeling the likelihood with marginal parametric functions based on the assumption of independence between PCA components thus gives a lower limit for the biases. We further demonstrate that the difference in cosmological parameter constraints between the multivariate Gaussian likelihood model and more complex non-Gaussian likelihood models would be even smaller for an LSST-like survey. In addition, the PCA approach automatically serves as a data compression method, enabling the retention of the majority of the cosmological information while reducing the dimensionality of the data vector by a factor of $\sim$5.Comment: 16 pages, 10 figures, published MNRA

U-Statistic Reduction: Higher-Order Accurate Risk Control and Statistical-Computational Trade-Off, with Application to Network Method-of-Moments

U-statistics play central roles in many statistical learning tools but face the haunting issue of scalability. Significant efforts have been devoted into accelerating computation by U-statistic reduction. However, existing results almost exclusively focus on power analysis, while little work addresses risk control accuracy -- comparatively, the latter requires distinct and much more challenging techniques. In this paper, we establish the first statistical inference procedure with provably higher-order accurate risk control for incomplete U-statistics. The sharpness of our new result enables us to reveal how risk control accuracy also trades off with speed for the first time in literature, which complements the well-known variance-speed trade-off. Our proposed general framework converts the long-standing challenge of formulating accurate statistical inference procedures for many different designs into a surprisingly routine task. This paper covers non-degenerate and degenerate U-statistics, and network moments. We conducted comprehensive numerical studies and observed results that validate our theory's sharpness. Our method also demonstrates effectiveness on real-world data applications

Conditional limit laws for goodness-of-fit tests

We study the conditional distribution of goodness of fit statistics of the Cram\'{e}r--von Mises type given the complete sufficient statistics in testing for exponential family models. We show that this distribution is close, in large samples, to that given by parametric bootstrapping, namely, the unconditional distribution of the statistic under the value of the parameter given by the maximum likelihood estimate. As part of the proof, we give uniform Edgeworth expansions of Rao--Blackwell estimates in these models.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ366 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Cram\'{e}r-type moderate deviations for Studentized two-sample UU-statistics with applications

Two-sample $U$-statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cram\'{e}r-type moderate deviation theorems for Studentized two-sample $U$-statistics in a general framework, including the two-sample $t$-statistic and Studentized Mann-Whitney test statistic as prototypical examples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample $t$-statistic. These results extend the applicability of the existing statistical methodologies from the one-sample $t$-statistic to more general nonlinear statistics. Applications to two-sample large-scale multiple testing problems with false discovery rate control and the regularized bootstrap method are also discussed.Comment: Published at http://dx.doi.org/10.1214/15-AOS1375 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Higher-Order Correct Fast Moving-Average Bootstrap for Dependent Data

We develop and implement a novel fast bootstrap for dependent data. Our scheme is based on the i.i.d. resampling of the smoothed moment indicators. We characterize the class of parametric and semi-parametric estimation problems for which the method is valid. We show the asymptotic refinements of the proposed procedure, proving that it is higher-order correct under mild assumptions on the time series, the estimating functions, and the smoothing kernel. We illustrate the applicability and the advantages of our procedure for Generalized Empirical Likelihood estimation. As a by-product, our fast bootstrap provides higher-order correct asymptotic confidence distributions. Monte Carlo simulations on an autoregressive conditional duration model provide numerical evidence that the novel bootstrap yields higher-order accurate confidence intervals. A real-data application on dynamics of trading volume of stocks illustrates the advantage of our method over the routinely-applied first-order asymptotic theory, when the underlying distribution of the test statistic is skewed or fat-tailed