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Binary spatial observations arise in environmental and ecological studies, where Markov random field (MRF) models are often applied. Despite the prevalence and the long history of MRF models for spatial binary data, appropriate model diagnostics have remained an unresolved issue in practice. A complicating factor is that such models involve neighborhood specifications, which are difficult to assess for binary data. To address this, we propose a formal goodness-of-fit (GOF) test for diagnosing an MRF model for spatial binary values. The test statistic involves a type of conditional Moran's I based on the fitted conditional probabilities, which can detect departures in model form, including neighborhood structure. Numerical studies show that the GOF test can perform well in detecting deviations from a null model, with a focus on neighborhoods as a difficult issue. We illustrate the spatial test with an application to Besag's historical endive data as well as the breeding pattern of grasshopper sparrows across Iowa. [ABSTRACT FROM AUTHOR]

Nonparametric cointegrating regression models have been extensively used in financial markets, stock prices, heavy traffic, climate datasets, and energy markets. Models with parametric regression functions can be more appealing in practice compared to nonparametric forms, but do result in potential functional misspecification. Thus, there exists a vast literature on developing a model specification test for parametric forms of regression functions. In this article, we develop two test statistics which are applicable for the endogenous regressors driven by long memory and semi-long memory input shocks in the regression model. The limit distributions of the test statistics under these two scenarios are complicated and cannot be effectively used in practice. To overcome this difficulty, we use the subsampling method and compute the test statistics on smaller blocks of the data to construct their empirical distributions. Throughout, Monte Carlo simulation studies are used to illustrate the properties of test statistics. We also provide an empirical example of relating gross domestic product to total output of carbon dioxide in two European countries. for this article are available online. [ABSTRACT FROM AUTHOR]

Bootstrapping spectral mean statistics has been a notoriously difficult problem over the past 25 years. Many frequency domain bootstraps are valid only for certain time series structures, e.g. linear processes, or for special types of statistics, i.e. ratio statistics, because such bootstraps fail to capture the limiting variance of spectral statistics in general settings. We address this issue with a different form of resampling, namely, subsampling. While not considered previously, subsampling provides consistent variance estimation under much weaker conditions than any existing bootstrap in the frequency domain. Mixing is not used, as is often standard with subsampling. Rather, subsampling can be generally justified under the same conditions needed for original spectral mean statistics to have distributional limits in the first place. This result has impacts for other bootstrap methods. Subsampling then applies to extending the validity of recent state-of-the-art bootstraps in the frequency domain. We nontrivially link subsampling to such bootstraps, which broadens their range, as moment and block assumptions needed for these are cut by more than half. Essentially, state-of-the-art bootstraps then require no more stringent assumptions than those needed for a target limit distribution to exist, which is unusual in the bootstrap world. We also close a gap in the theory of subsampling for time series with distributional approximations, in addition to variance estimation, for frequency domain statistics. [ABSTRACT FROM AUTHOR]

This article describes prediction methods for the number of future events from a population of units associated with an on-going time-to-event process. Examples include the prediction of warranty returns and the prediction of the number of future product failures that could cause serious threats to property or life. Important decisions such as whether a product recall should be mandated are often based on such predictions. Data, generally right-censored (and sometimes left truncated and right-censored), are used to estimate the parameters of a time-to-event distribution. This distribution can then be used to predict the number of events over future periods of time. Such predictions are sometimes called within-sample predictions and differ from other prediction problems considered in most of the prediction literature. This article shows that the plug-in (also known as estimative or naive) prediction method is not asymptotically correct (i.e., for large amounts of data, the coverage probability always fails to converge to the nominal confidence level). However, a commonly used prediction calibration method is shown to be asymptotically correct for within-sample predictions, and two alternative predictive-distribution-based methods that perform better than the calibration method are presented and justified. for this article are available online. [ABSTRACT FROM AUTHOR]

Random forests are among the most popular machine learning techniques for prediction problems. When using random forests to predict a quantitative response, an important but often overlooked challenge is the determination of prediction intervals that will contain an unobserved response value with a specified probability. We propose new random forest prediction intervals that are based on the empirical distribution of out-of-bag prediction errors. These intervals can be obtained as a by-product of a single random forest. Under regularity conditions, we prove that the proposed intervals have asymptotically correct coverage rates. Simulation studies and analysis of 60 real datasets are used to compare the finite-sample properties of the proposed intervals with quantile regression forests and recently proposed split conformal intervals. The results indicate that intervals constructed with our proposed method tend to be narrower than those of competing methods while still maintaining marginal coverage rates approximately equal to nominal levels. [ABSTRACT FROM AUTHOR]

A probability model exhibits instability if small changes in a data outcome result in large and, often unanticipated, changes in probability. This instability is a property of the probability model, given by a distributional form and a given configuration of parameters. For correlated data structures found in several application areas, there is increasing interest in identifying such sensitivity in model probability structure. We consider the problem of quantifying instability for general probability models defined on sequences of observations, where each sequence of length |$N$| has a finite number of possible values that can be taken at each point. A sequence of probability models, indexed by |$N$|⁠ , and an associated parameter sequence result to accommodate data of expanding dimension. Model instability is formally shown to occur when a certain log probability ratio under such models grows faster than |$N$|⁠. In this case, a one component change in the data sequence can shift probability by orders of magnitude. Also, as instability becomes more extreme, the resulting probability models are shown to tend to degeneracy, placing all their probability on potentially small portions of the sample space. These results on instability apply to large classes of models commonly used in random graphs, network analysis and machine learning contexts. [ABSTRACT FROM AUTHOR]

For spatial and network data, we consider models formed from a Markov random field (MRF) structure and the specification of a conditional distribution for each observation. Fast simulation from such MRF models is often an important consideration, particularly when repeated generation of large numbers of datasets is required. However, a standard Gibbs strategy for simulating from MRF models involves single-site updates, performed with the conditional univariate distribution of each observation in a sequential manner, whereby a complete Gibbs iteration may become computationally involved even for moderate samples. As an alternative, we describe a general way to simulate from MRF models using Gibbs sampling with "concliques" (i.e., groups of nonneighboring observations). Compared to standard Gibbs sampling, this simulation scheme can be much faster by reducing Gibbs steps and independently updating all observations per conclique at once. The speed improvement depends on the number of concliques relative to the sample size for simulation, and order-of-magnitude speed increases are possible with many MRF models (e.g., having appropriately bounded neighborhoods). We detail the simulation method, establish its validity, and assess its computational performance through numerical studies, where speed advantages are shown for several spatial and network examples. for this article are available online. [ABSTRACT FROM AUTHOR]

This article develops empirical likelihood methodology for a class of long range dependent processes driven by a stationary Gaussian process. We consider population parameters that are defined by estimating equations in the time domain. It is shown that the standard block empirical likelihood (BEL) method, with a suitable scaling, has a non‐standard limit distribution based on a multiple Wiener–Itô integral. Unlike the short memory time series case, the scaling constant involves unknown population quantities that may be difficult to estimate. Alternative versions of the empirical likelihood method, involving the expansive BEL (EBEL) methods are considered. It is shown that the EBEL renditions do not require an explicit scaling and, therefore, remove this undesirable feature of the standard BEL. However, the limit law involves the long memory parameter, which may be estimated from the data. Results from a moderately large simulation study on finite sample properties of tests and confidence intervals based on different empirical likelihood methods are also reported. [ABSTRACT FROM AUTHOR]

One major aim of time series analysis, particularly in the physical and geo‐sciences, is the estimation of the spectral density function. With weakly dependent time processes, non‐parametric, kernel‐based methods are available for spectral density estimation, which involves smoothing the periodogram by a kernel function. However, a similar non‐parametric approach is presently unavailable for strongly, or long‐range, dependent processes. In particular, as the spectral density function under long‐range dependence commonly has a pole at the origin, kernel‐based methods developed for weakly dependent processes (i.e., with bounded spectral densities) do not apply readily for long‐range dependence without suitable modification. To address this, we propose a non‐parametric kernel‐based method for spectral density estimation, which is valid under both weak and strong dependence. Based on the initial or pilot estimator of the long‐memory parameter, the method involves a frequency domain transformation to dampen the dependence in periodogram ordinates and mimic kernel‐based estimation under weak dependence. Under mild assumptions, the proposed non‐parametric spectral density estimator is shown to be uniformly consistent, and general expressions are provided for rates of estimation error and optimal kernel bandwidths. The method is investigated through simulation and illustrated through data examples, which also consider bandwidth selection. [ABSTRACT FROM AUTHOR]

We present the R package fourierin (Basulto-Elias, 2017) for evaluating functions defined as Fourier-type integrals over a collection of argument values. The integrals are finitely supported with integrands involving continuous functions of one or two variables. As an important application, such Fourier integrals arise in so-called "inversion formulas", where one seeks to evaluate a probability density at a series of points from a given characteristic function (or vice versa) through Fourier transforms. This paper intends to fill a gap in current R software, where tools for repeated evaluation of functions as Fourier integrals are not directly available. We implement two approaches for such computations with numerical integration. In particular, if the argument collection for evaluation corresponds to a regular grid, then an algorithm from Inverarity (2002) may be employed based on a fast Fourier transform, which creates significant improvements in the speed over a second approach to numerical Fourier integration (where the latter also applies to cases where the points for evaluation are not on a grid). We illustrate the package with the computation of probability densities and characteristic functions through Fourier integrals/transforms, for both univariate and bivariate examples. [ABSTRACT FROM AUTHOR]

Unlike with independent data, smoothed bootstraps have received little consideration for time series, although data smoothing within resampling can improve bootstrap approximations, especially when target distributions depend on smooth population quantities (e.g., marginal densities). For approximating a broad class statistics formulated through statistical functionals (e.g., LL-estimators, and sample quantiles), we propose a smooth bootstrap by modifying a state-of-the-art (extended) tapered block bootstrap (TBB). Our treatment shows that the smooth TBB applies to time series inference cases not formally established with other TBB versions. Simulations also indicate that smoothing enhances the block bootstrap. [ABSTRACT FROM AUTHOR]

We summarize advances in empirical likelihood (EL) for time series data. The EL formulation for independent data is briefly presented, which can apply for inference in special time series problems, reproducing the Wilks phenomenon of chi-square limits for log-ratio statistics. For more general inference with time series, versions of time domain block-based EL, and its generalizations based on divergence measures, are described along with their distributional properties; some approaches are intended for mixing time processes and others are tailored to time series with a Markovian structure. We also present frequency domain EL methods based on the periodogram. Finally, EL for long-range dependent processes is reviewed as well as recent advantages in EL for high dimensional problems. Some illustrative numerical examples are given along with a summary of open research issues for EL with dependent data. [ABSTRACT FROM AUTHOR]

This paper considers blockwise empirical likelihood for real-valued linear time processes which may exhibit either short- or long-range dependence. Empirical likelihood approaches intended for weakly dependent time series can fail in the presence of strong dependence. However, a modified blockwise method is proposed for confidence interval estimation of the process mean, which is valid for various dependence structures including long-range dependence. The finite-sample performance of the method is evaluated through a simulation study and compared with other confidence interval procedures involving subsampling or normal approximations. [ABSTRACT FROM AUTHOR]