This paper presents a new method to estimate large-scale multivariate normal probabilities. The approach combines a hierarchical representation with processing of the covariance matrix that decomposes the n-dimensional problem into a sequence of smaller m-dimensional ones. It also includes a d-dimensional conditioning method that further decomposes the m-dimensional problems into smaller d-dimensional problems. The resulting two-level hierarchical-block conditioning method requires Monte Carlo simulations to be performed only in d dimensions, with d n, and allows the complexity of the algorithm's major cost to be O(n log n). The run-time cost of the method depends on two parameters, m and d, where m represents the diagonal block size and controls the sizes of the blocks of the covariance matrix that are replaced by low-rank approximations, and d allows a trade-off of accuracy for expensive computations in the evaluation of the probabilities of m-dimensional blocks. We also introduce an inexpensive block reordering strategy to provide improved accuracy in the overall probability computation. Numerical simulations on problems from 2D spatial statistics with dimensions up to 16,384 indicate that the algorithm achieves a 1% error level and improves the run time over a one-level hierarchical Quasi-Monte Carlo method by a factor between 5 and 10.
We present a preconditioned Monte Carlo method for computing high-dimensional multivariate normal and Student-t probabilities arising in spatial statistics. The approach combines a tile-low-rank representation of covariance matrices with a block-reordering scheme for efficient Quasi-Monte Carlo simulation. The tile-low-rank representation decomposes the highdimensional problem into many diagonal-block-size problems and low-rank connections. The block-reordering scheme reorders between and within the diagonal blocks to reduce the impact of integration variables from right to left, thus improving the Monte Carlo convergence rate. Simulations up to dimension 65,536 suggest that the new method can improve the run time by an order of magnitude compared with the non-reordered tile-low-rank Quasi-Monte Carlo method and two orders of magnitude compared with the dense Quasi-Monte Carlo method. Our method also forms a strong substitute for the approximate conditioning methods as a more robust estimation with error guarantees. An application study to wind stochastic generators is provided to illustrate that the new computational method makes the maximum likelihood estimation feasible for high-dimensional skew-normal random fields.
Initially proposed by Marcoulides and further expanded by Raykov and Marcoulides, a structural equation modeling approach can be used in generalizability theory estimation. This article examines the utility of incorporating auxiliary variables into the structural equation modeling approach when missing data is present. In particular, the authors assert that by adapting a saturated correlates model strategy to structural equation modeling generalizability theory models, one can reduce any biased effects caused by missingness. Traditional approaches such as an analysis of variance do not possess such a feature. This article provides detailed instructions for adding auxiliary variables into a structural equation modeling generalizability theory model, demonstrates the corresponding benefits of bias reduction in generalizability coefficient estimate via simulations, and discusses issues relevant to the proposed approach.
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