This paper studies hypothesis testing and parameter estimation in the context of the divide-and-conquer algorithm. In a unified likelihood based framework, we propose new test statistics and point estimators obtained by aggregating various statistics from k subsamples of size n/k, where n is the sample size. In both low dimensional and sparse high dimensional settings, we address the important question of how large k can be, as n grows large, such that the loss of efficiency due to the divide-and-conquer algorithm is negligible. In other words, the resulting estimators have the same inferential efficiencies and estimation rates as an oracle with access to the full sample. Thorough numerical results are provided to back up the theory.

Financial econometrics is an interdisciplinary subject that uses statistical methods and economic theory to address a variety of quantitative problems in finance. This compact, master's-level textbook focuses on methodology and includes real financial data illustrations throughout. The mathematical level is purposely kept moderate, allowing the power of the quantitative methods to be understood without too much technical detail. Wherever possible, the authors indicate where to find the relevant R codes to implement the various methods. This book grew out of a course at Princeton University which is one of the world's flagship programs in computational finance and financial engineering. It will therefore be useful for those with an economics and finance background who are looking to sharpen their quantitative skills, and also for those with strong quantitative skills who want to learn how to apply them to finance.

Long term prediction such as multi-step time series prediction is a challenging prognostics problem. This paper proposes an improved AR time series model called ND-AR model (Nonlinear Degradation AutoRegression) for Remaining Useful Life (RUL) estimation of lithium-ion batteries. The nonlinear degradation feature of the lithiumion battery capacity degradation is analyzed and then the non-linear accelerated degradation factor is extracted to improve the linear AR model. In this model, the nonlinear degradation factor can be obtained with curve fitting, and then the ND-AR model can be applied as an adaptive datadriven prognostics method to monitor degradation time series data. Experimental results with CALCE battery data set show that the proposed nonlinear degradation AR model can realize satisfied prognostics for various lithium-ion batteries with low computing complexity.

We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations, a problem which is typically ill-posed. Regularization of these problems using L2 function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem—namely whether the subjective choice of regularization is compatible with prior knowledge. Using path-integral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative path-integral approaches, while offering alternatives to computational approaches like Markov-Chain-Monte-Carlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.

This article is concerned with screening features in ultrahigh-dimensional data analysis, which has become increasingly important in diverse scientific fields. We develop a sure independence screening procedure based on the distance correlation (DC-SIS). The DC-SIS can be implemented as easily as the sure independence screening (SIS) procedure based on the Pearson correlation proposed by Fan and Lv. However, the DC-SIS can significantly improve the SIS. Fan and Lv established the sure screening property for the SIS based on linear models, but the sure screening property is valid for the DC-SIS under more general settings, including linear models. Furthermore, the implementation of the DC-SIS does not require model specification (e.g., linear model or generalized linear model) for responses or predictors. This is a very appealing property in ultrahigh-dimensional data analysis. Moreover, the DC-SIS can be used directly to screen grouped predictor variables and multivariate response variables. We establish the sure screening property for the DC-SIS, and conduct simulations to examine its finite sample performance. A numerical comparison indicates that the DC-SIS performs much better than the SIS in various models. We also illustrate the DC-SIS through a real-data example.

Recently a computational-based experimental design strategy called rerandomization has been proposed as an alternative or complement to traditional blocked designs. The idea of rerandomization is to remove, from consideration, those allocations with large imbalances in observed covariates according to a balance criterion, and then randomize within the set of acceptable allocations. Based on the Mahalanobis distance criterion for balancing the covariates, we show that asymptotic inference to the population, from which the units in the sample are randomly drawn, is possible using only the set of best, or ‘optimal’, allocations. Finally, we show that for the optimal and near optimal designs, the quite complex asymptotic sampling distribution derived by Li et al. (2018), is well approximated by a normal distribution.

This article considers experimental design based on the strategy of rerandomization to increase the efficiency in experiments. Two aspects of rerandomization are addressed. First, we propose a two-stage allocation sample scheme for randomization inference to the units in experiments that guarantees that the difference-in-mean estimator is an unbiased estimator of the sample average treatment effect for any experiment, conserves the exactness of randomization inference, and halves the time consumption of the rerandomization design. Second, we propose a rank-based covariate-balance measure which can take into account the estimated relative weight of each covariate. Several strategies for estimating these weights using pre-experimental data are proposed. Using Monte Carlo simulations, the proposed strategies are compared to complete randomization and Mahalanobis-based rerandomization. An empirical example is given where the power of a mean difference test of electricity consumption of 54 households is increased by 99%, in comparison to complete randomization, using one of the proposed designs based on high frequency longitudinal electricity consumption data. Supplementary materials for this article are available online.

Many statistical models seek relationship between variables via subspaces of reduced dimensions. For instance, in factor models, variables are roughly distributed around a low dimensional subspace determined by the loading matrix; in mixed linear regression models, the coefficient vectors for different mixtures form a subspace that captures all regression functions; in multiple index models, the effect of covariates is summarized by the effective dimension reduction space.

Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve 1, 2. Understanding evolutionary game dynamics in structured populations remains difficult. Mathematical results are known for special structures in which all individuals have the same number of neighbours 3, 4, 5, 6, 7, 8. The general case, in which the number of neighbours can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class that suggests there is no efficient algorithm 9. Whether a simple solution for weak selection exists has remained unanswered. Here we provide a solution for weak selection that applies to any graph or network. Our method relies on calculating the coalescence times 10, 11 of random walks 12. We evaluate large numbers of diverse population structures for their propensity to favour cooperation. We study how small

Frequentists' inference often delivers point estimators associated with condence intervals or sets for parameters of interest. Constructing the condence intervals or sets requires understanding the sampling distributions of the point estimators, which, in many but not all cases, are related to asymptotic Normal distributions ensured by central limit theorems. Although previous literature has established various forms of central limit theorems for statistical inference in super population models, we still need general and convenient forms of central limit theorems for some randomization-based causal analysis of experimental data, where the parameters of interests are functions of a nite population and randomness comes solely from the treatment assignment. We use central limit theorems for sample surveys and rank statistics to establish general forms of the nite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under the sharp null hypothesis of zero individual causal eects, and for obtaining the asymptotic repeated sampling distributions of the causal eect estimators. The new central limit theorems hold for general experimental designs with multiple treatment levels, multiple treatment factors and vector outcomes, and are immediately applicable for studying the asymptotic properties of many methods in causal inference, including instrumental variable, regression adjustment, rerandomization, clustered randomized experiments, and so on. Previously, the asymptotic properties of these problems are often based on heuristic arguments, which in fact rely on general forms of nite population central limit theorems that have not been established before. Our new theorems ll in this gap by providing more solid theoretical foundation for asymptotic randomization-based causal inference.