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Sequential Monte Carlo (SMC) methods are a class of Monte Carlo methods that are used to obtain random samples of a high dimensional random variable in a sequential fashion. Many problems encountered in applications often involve different types of constraints. These constraints can make the problem much more challenging. In this paper, we formulate a general framework of using SMC for constrained sampling problems based on forward and backward pilot resampling strategies. We review some existing methods under the framework and develop several new algorithms. It is noted that all information observed or imposed on the underlying system can be viewed as constraints. Hence the approach outlined in this paper can be useful in many applications.

Estimation of the high-dimensional banded covariance matrix is widely used in multivariate statistical analysis. To ensure the validity of estimation, we aim to test the hypothesis that the covariance matrix is banded with a certain bandwidth under the high-dimensional framework. Though several testing methods have been proposed in the literature, the existing tests are only powerful for some alternatives with certain sparsity levels, whereas they may not be powerful for alternatives with other sparsity structures. The goal of this paper is to propose a new test for the bandedness of high-dimensional covariance matrix, which is powerful for alternatives with various sparsity levels. The proposed new test also be used for testing the banded structure of covariance matrices of error vectors in high-dimensional factor models. Based on these statistics, a consistent bandwidth estimator is also introduced for a banded high dimensional covariance matrix. Extensive simulation studies and an application to a prostate cancer dataset from protein mass spectroscopy are conducted for evaluating the effectiveness of the proposed adaptive tests blue and bandwidth estimator for the banded covariance matrix.

This paper aims to test the number of spikes in a generalized spiked covariance matrix, the spiked eigenvalues of which may be extremely larger or smaller than the non-spiked ones. For a high-dimensional problem, we first propose a general test statistic and derive its central limit theorem by random matrix theory without a Gaussian population constraint. We then apply the result to estimate the noise variance and test the equality of the smallest roots in generalized spiked models. Simulation studies showed that the proposed test method was correctly sized, and the power outcomes showed the robustness of our statistic to deviations from a Gaussian population. Moreover, our estimator of the noise variance resulted in much smaller mean absolute errors and mean squared errors than existing methods. In contrast to previously developed methods, we eliminated the strict conditions of diagonal or block-wise diagonal form of the population covariance matrix and extend the work to a wider range without the assumption of normality. Thus, the proposed method is more suitable for real problems., 23 pages,3 figures

A composite likelihood is a combination of low-dimensional likelihood objects useful in applications where the data have complex structure. Although composite likelihood construction is a crucial aspect influencing both computing and statistical properties of the resulting estimator, currently there does not seem to exist a universal rule to combine low-dimensional likelihood objects that is statistically justified and fast in execution. This paper develops a methodology to select and combine the most informative low-dimensional likelihoods from a large set of candidates while carrying out parameter estimation. The new procedure minimizes the distance between composite likelihood and full likelihood scores subject to a constraint representing the afforded computing cost. The selected composite likelihood is sparse in the sense that it contains a relatively small number of informative sub-likelihoods while the noisy terms are dropped. The resulting estimator is found to have asymptotic variance close to that of the minimum-variance estimator constructed using all the low-dimensional likelihoods.

In this paper, we propose a new spectral-based approach to hypothesis testing for populations of networks. The primary goal is to develop a test to determine whether two given samples of networks come from the same random model or distribution. Our test statistic is based on the trace of the third order for a centered and scaled adjacency matrix, which we prove converges to the standard normal distribution as the number of nodes tends to infinity. The asymptotic power guarantee of the test is also provided. The proper interplay between the number of networks and the number of nodes for each network is explored in characterizing the theoretical properties of the proposed testing statistics. Our tests are applicable to both binary and weighted networks, operate under a very general framework where the networks are allowed to be large and sparse, and can be extended to multiple-sample testing. We provide an extensive simulation study to demonstrate the superior performance of our test over existing methods and apply our test to three real datasets.

Selecting hyperparameters for unsupervised learning problems is challenging in general due to the lack of ground truth for validation. Despite the prevalence of this issue in statistics and machine learning, especially in clustering problems, there are not many methods for tuning these hyperparameters with theoretical guarantees. In this paper, we provide a framework with provable guarantees for selecting hyperparameters in a number of distinct models. We consider both the subgaussian mixture model and network models to serve as examples of i.i.d. and non-i.i.d. data. We demonstrate that the same framework can be used to choose the Lagrange multipliers of penalty terms in semi-definite programming (SDP) relaxations for community detection, and the bandwidth parameter for constructing kernel similarity matrices for spectral clustering. By incorporating a cross-validation procedure, we show the framework can also do consistent model selection for network models. Using a variety of simulated and real data examples, we show that our framework outperforms other widely used tuning procedures in a broad range of parameter settings.

High-dimensional self-exciting point processes have been widely used in many application areas to model discrete event data in which past and current events affect the likelihood of future events. In this paper, we are concerned with detecting abrupt changes of the coefficient matrices in discrete-time high-dimensional self-exciting Poisson processes, which have yet to be studied in the existing literature due to both theoretical and computational challenges rooted in the non-stationary and high-dimensional nature of the underlying process. We propose a penalized dynamic programming approach which is supported by a theoretical rate analysis and numerical evidence.