We consider the problem of comparing probability densities between two groups. To model the complex pattern of the underlying densities, we formulate the problem as a nonparametric density hypothesis testing problem. The major difficulty is that conventional tests may fail to distinguish the alternative from the null hypothesis under the controlled type I error. In this paper, we model log-transformed densities in a tensor product reproducing kernel Hilbert space (RKHS) and propose a probabilistic decomposition of this space. Under such a decomposition, we quantify the difference of the densities between two groups by the component norm in the probabilistic decomposition. Based on the Bernstein width, a sharp minimax lower bound of the distinguishable rate is established for the nonparametric two-sample test. We then propose a penalized likelihood ratio (PLR) test possessing the Wilks' phenomenon with an asymptotically Chi-square distributed test statistic and achieving the established minimax testing rate. Simulations and real applications demonstrate that the proposed test outperforms the conventional approaches under various scenarios.
Testing the hypothesis of parallelism is a fundamental statistical problem arising from many applied sciences. In this paper, we develop a nonparametric parallelism test for inferring whether the trends are parallel in treatment and control groups. In particular, the proposed nonparametric parallelism test is a Wald type test based on a smoothing spline ANOVA (SSANOVA) model which can characterize the complex patterns of the data. We derive that the asymptotic null distribution of the test statistic is a Chi-square distribution, unveiling a new version of Wilks phenomenon. Notably, we establish the minimax sharp lower bound of the distinguishable rate for the nonparametric parallelism test by using the information theory, and further prove that the proposed test is minimax optimal. Simulation studies are conducted to investigate the empirical performance of the proposed test. DNA methylation and neuroimaging studies are presented to illustrate potential applications of the test. The software is available at \url{https://github.com/BioAlgs/Parallelism} .
For massive data, the family of subsampling algorithms is popular to downsize the data volume and reduce computational burden. Existing studies focus on approximating the ordinary least squares estimate in linear regression, where statistical leverage scores are often used to define subsampling probabilities. In this paper, we propose fast subsampling algorithms to efficiently approximate the maximum likelihood estimate in logistic regression. We first establish consistency and asymptotic normality of the estimator from a general subsampling algorithm, and then derive optimal subsampling probabilities that minimize the asymptotic mean squared error of the resultant estimator. An alternative minimization criterion is also proposed to further reduce the computational cost. The optimal subsampling probabilities depend on the full data estimate, so we develop a two-step algorithm to approximate the optimal subsampling procedure. This algorithm is computationally efficient and has a significant reduction in computing time compared to the full data approach. Consistency and asymptotic normality of the estimator from a two-step algorithm are also established. Synthetic and real data sets are used to evaluate the practical performance of the proposed method.
A significant hurdle for analyzing large sample data is the lack of effective statistical computing and inference methods. An emerging powerful approach for analyzing large sample data is subsampling, by which one takes a random subsample from the original full sample and uses it as a surrogate for subsequent computation and estimation. In this paper, we study subsampling methods under two scenarios: approximating the full sample ordinary least-square (OLS) estimator and estimating the coefficients in linear regression. We present two algorithms, weighted estimation algorithm and unweighted estimation algorithm, and analyze asymptotic behaviors of their resulting subsample estimators under general conditions. For the weighted estimation algorithm, we propose a criterion for selecting the optimal sampling probability by making use of the asymptotic results. On the basis of the criterion, we provide two novel subsampling methods, the optimal subsampling and the predictor- length subsampling methods. The predictor-length subsampling method is based on the L2 norm of predictors rather than leverage scores. Its computational cost is scalable. For unweighted estimation algorithm, we show that its resulting subsample estimator is not consistent to the full sample OLS estimator. However, it has better performance than the weighted estimation algorithm for estimating the coefficients. Simulation studies and a real data example are used to demonstrate the effectiveness of our proposed subsampling methods.
One popular method for dealing with large-scale data sets is sampling. For example, by using the empirical statistical leverage scores as an importance sampling distribution, the method of algorithmic leveraging samples and rescales rows/columns of data matrices to reduce the data size before performing computations on the subproblem. This method has been successful in improving computational efficiency of algorithms for matrix problems such as least-squares approximation, least absolute deviations approximation, and low-rank matrix approximation. Existing work has focused on algorithmic issues such as worst-case running times and numerical issues associated with providing high-quality implementations, but none of it addresses statistical aspects of this method. In this paper, we provide a simple yet effective framework to evaluate the statistical properties of algorithmic leveraging in the context of estimating parameters in a linear regression model with a fixed number of predictors. We show that from the statistical perspective of bias and variance, neither leverage-based sampling nor uniform sampling dominates the other. This result is particularly striking, given the well-known result that, from the algorithmic perspective of worst-case analysis, leverage-based sampling provides uniformly superior worst-case algorithmic results, when compared with uniform sampling. Based on these theoretical results, we propose and analyze two new leveraging algorithms. A detailed empirical evaluation of existing leverage-based methods as well as these two new methods is carried out on both synthetic and real data sets. The empirical results indicate that our theory is a good predictor of practical performance of existing and new leverage-based algorithms and that the new algorithms achieve improved performance.