Statistics and Computing最新文献

We discuss, and give examples of, methods for randomly implementing some minimax robust designs from the literature. These have the advantage, over their deterministic counterparts, of having bounded maximum loss in large and very rich neighbourhoods of the, almost certainly inexact, response model fitted by the experimenter. Their maximum loss rivals that of the theoretically best possible, but not implementable, minimax designs. The procedures are then extended to more general robust designs. For two-dimensional designs we sample from contractions of Voronoi tessellations, generated by selected basis points, which partition the design space. These ideas are then extended to k-dimensional designs for general k.

Outlier-prone data sets are of immense interest in diverse areas including economics, finance, statistical physics, signal processing, telecommunications and so on. Stable laws (also known as (alpha )- stable laws) are often found to be useful in modeling outlier-prone data containing important information and exhibiting heavy tailed phenomenon. In this article, an asymptotic distribution of a unbiased and consistent estimator of the stability index (alpha ) is proposed based on jackknife empirical likelihood (JEL) and adjusted JEL method. Next, using the sum-preserving property of stable random variables and exploiting U-statistic theory, we have developed a goodness-of-fit test procedure for (alpha )-stable distributions where the stability index (alpha ) is specified. Extensive simulation studies are performed in order to assess the finite sample performance of the proposed test. Finally, two appealing real life data examples related to the daily closing price of German Stock Index and Bitcoin cryptocurrency are analysed in detail for illustration purposes.

In some applied scenarios, the availability of complete data is restricted, often due to privacy concerns; only aggregated, robust and inefficient statistics derived from the data are made accessible. These robust statistics are not sufficient, but they demonstrate reduced sensitivity to outliers and offer enhanced data protection due to their higher breakdown point. We consider a parametric framework and propose a method to sample from the posterior distribution of parameters conditioned on various robust and inefficient statistics: specifically, the pairs (median, MAD) or (median, IQR), or a collection of quantiles. Our approach leverages a Gibbs sampler and simulates latent augmented data, which facilitates simulation from the posterior distribution of parameters belonging to specific families of distributions. A by-product of these samples from the joint posterior distribution of parameters and data given the observed statistics is that we can estimate Bayes factors based on observed statistics via bridge sampling. We validate and outline the limitations of the proposed methods through toy examples and an application to real-world income data.

This paper investigates the performance of routinely used optimization algorithms in application to the Generalized Covariance estimator (GCov) for univariate and multivariate mixed causal and noncausal models. The GCov is a semi-parametric estimator with an objective function based on nonlinear autocovariances to identify causal and noncausal orders. When the number and type of nonlinear autocovariances included in the objective function are insufficient/inadequate, or the error density is too close to the Gaussian, identification issues can arise. These issues result in local minima in the objective function, which correspond to parameter values associated with incorrect causal and noncausal orders. Then, depending on the starting point and the optimization algorithm employed, the algorithm can converge to a local minimum. The paper proposes the Simulated Annealing (SA) optimization algorithm as an alternative to conventional numerical optimization methods. The results demonstrate that SA performs well in its application to mixed causal and noncausal models, successfully eliminating the effects of local minima. The proposed approach is illustrated by an empirical study of a bivariate series of commodity prices.

Semi-parametric Gaussian mixtures of non-parametric regressions (SPGMNRs) are a flexible extension of Gaussian mixtures of linear regressions (GMLRs). The model assumes that the component regression functions (CRFs) are non-parametric functions of the covariate(s) whereas the component mixing proportions and variances are constants. Unfortunately, the model cannot be reliably estimated using traditional methods. A local-likelihood approach for estimating the CRFs requires that we maximize a set of local-likelihood functions. Using the Expectation-Maximization (EM) algorithm to separately maximize each local-likelihood function may lead to label-switching. This is because the posterior probabilities calculated at the local E-step are not guaranteed to be aligned. The consequence of this label-switching is wiggly and non-smooth estimates of the CRFs. In this paper, we propose a unified approach to address label-switching and obtain sensible estimates. The proposed approach has two stages. In the first stage, we propose a model-based approach to address the label-switching problem. We first note that each local-likelihood function is a likelihood function of a Gaussian mixture model (GMM). Next, we reformulate the SPGMNRs model as a mixture of these GMMs. Lastly, using a modified version of the Expectation Conditional Maximization (ECM) algorithm, we estimate the mixture of GMMs. In addition, using the mixing weights of the local GMMs, we can automatically choose the local points where local-likelihood estimation takes place. In the second stage, we propose one-step backfitting estimates of the parametric and non-parametric terms. The effectiveness of the proposed approach is demonstrated on simulated data and real data analysis.

Generalized fused Lasso (GFL) is a powerful method based on adjacent relationships or the network structure of data. It is used in a number of research areas, including clustering, discrete smoothing, and spatio-temporal analysis. When applying GFL, the specific optimization method used is an important issue. In generalized linear models, efficient algorithms based on the coordinate descent method have been developed for trend filtering under the binomial and Poisson distributions. However, to apply GFL to other distributions, such as the negative binomial distribution, which is used to deal with overdispersion in the Poisson distribution, or the gamma and inverse Gaussian distributions, which are used for positive continuous data, an algorithm for each individual distribution must be developed. To unify GFL for distributions in the exponential family, this paper proposes a coordinate descent algorithm for generalized linear models. To illustrate the method, a real data example of spatio-temporal analysis is provided.

Censoring occurs when an outcome is unobserved beyond some threshold value. Methods that do not account for censoring produce biased predictions of the unobserved outcome. This paper introduces Type I Tobit Bayesian Additive Regression Tree (TOBART-1) models for censored outcomes. Simulation results and real data applications demonstrate that TOBART-1 produces accurate predictions of censored outcomes. TOBART-1 provides posterior intervals for the conditional expectation and other quantities of interest. The error term distribution can have a large impact on the expectation of the censored outcome. Therefore, the error is flexibly modeled as a Dirichlet process mixture of normal distributions. An R package is available at https://github.com/EoghanONeill/TobitBART.

In this paper, we introduce and study fused lasso nearly-isotonic signal approximation, which is a combination of fused lasso and generalized nearly-isotonic regression. We show how these three estimators relate to each other and derive solution to a general problem. Our estimator is computationally feasible and provides a trade-off between monotonicity, block sparsity, and goodness-of-fit. Next, we prove that fusion and near-isotonisation in a one-dimensional case can be applied interchangably, and this step-wise procedure gives the solution to the original optimization problem. This property of the estimator is very important, because it provides a direct way to construct a path solution when one of the penalization parameters is fixed. Also, we derive an unbiased estimator of degrees of freedom of the estimator.

Brute force cross-validation (CV) is a method for predictive assessment and model selection that is general and applicable to a wide range of Bayesian models. Naive or ‘brute force’ CV approaches are often too computationally costly for interactive modeling workflows, especially when inference relies on Markov chain Monte Carlo (MCMC). We propose overcoming this limitation using massively parallel MCMC. Using accelerator hardware such as graphics processor units, our approach can be about as fast (in wall clock time) as a single full-data model fit. Parallel CV is flexible because it can easily exploit a wide range data partitioning schemes, such as those designed for non-exchangeable data. It can also accommodate a range of scoring rules. We propose MCMC diagnostics, including a summary of MCMC mixing based on the popular potential scale reduction factor ((widehat{textrm{R}})) and MCMC effective sample size ((widehat{textrm{ESS}})) measures. We also describe a method for determining whether an (widehat{textrm{R}}) diagnostic indicates approximate stationarity of the chains, that may be of more general interest for applications beyond parallel CV. Finally, we show that parallel CV and its diagnostics can be implemented with online algorithms, allowing parallel CV to scale up to very large blocking designs on memory-constrained computing accelerators.

Sparse principal component analysis (SPCA) is a popular tool for dimensionality reduction in high-dimensional data. However, there is still a lack of theoretically justified Bayesian SPCA methods that can scale well computationally. One of the major challenges in Bayesian SPCA is selecting an appropriate prior for the loadings matrix, considering that principal components are mutually orthogonal. We propose a novel parameter-expanded coordinate ascent variational inference (PX-CAVI) algorithm. This algorithm utilizes a spike and slab prior, which incorporates parameter expansion to cope with the orthogonality constraint. Besides comparing to two popular SPCA approaches, we introduce the PX-EM algorithm as an EM analogue to the PX-CAVI algorithm for comparison. Through extensive numerical simulations, we demonstrate that the PX-CAVI algorithm outperforms these SPCA approaches, showcasing its superiority in terms of performance. We study the posterior contraction rate of the variational posterior, providing a novel contribution to the existing literature. The PX-CAVI algorithm is then applied to study a lung cancer gene expression dataset. The (textsf{R}) package (textsf{VBsparsePCA}) with an implementation of the algorithm is available on the Comprehensive R Archive Network (CRAN).