Pairwise Likelihood Research Articles

Spatio-temporal variability of dry and wet spells and their influence on crop yields

Spatial modelling of extreme dry (wet) spells is relevant to design agricultural insurances and establish risk management strategies. We characterised dry (wet) spells during the period 1971–2010 in 133 rainfall stations and investigated their impact on crop yields across Belgium. A day was considered as a dry (wet) day when the total amount of rainfall is below (above) 0.2 mm. A two-way ANOVA demonstrated significant differences between low and high yields across 12 agricultural regions in the number of consecutive dry (wet) days, rainfall amount during the wet days and their total during the crop growing season. Generalised Extreme Value distributions were fitted to the maxima series using maximum likelihood to estimate the distribution parameters. The number of consecutive dry (wet) days and rainfall amount was considered during the cropping season between March and October. We modelled maxima observed at rainfall stations by means of max-stable processes, an extension of multivariate extreme value theory. Spatial extreme modelling was performed through pairwise likelihood inference by a madogram and enabled 20-year return levels to be simulated across regions based on geographic coordinates and altitude as covariates. Spatial return levels confirmed the exceptionality of 2016 and 2018 as extreme wet and dry years with high return periods. The 2018 drought resulted in yield reductions of 74% for maize and 42% for potato, while the 2016 rainfall induced yield losses of up to 39% for winter cereals, 45% for maize and 53% for potato. The max-stable models provided a good fit to the joint behaviour of extremes in the number of consecutive dry (wet) days during the growing season as an indication for crop stress. The method allows for spatio-temporal characterisation of generalised extreme value distributions across regions, and provides a tool for weather based insurances and risk management.

Approximate pairwise likelihood inference in SGLM models with skew normal latent variables

Spatial generalized linear mixed models are commonly employed for modeling discrete spatial responses that are acquired on a continuous area. A standard assumption in these models is that the latent variables are normally distributed, however skewed residuals appear in some spatial generalized linear mixed models. In this study, we consider a closed skew Gaussian random field for the spatial latent variables in the spatial generalized linear mixed models and present a new approximate pairwise likelihood approach to estimate parameters. In order to introduce a new algorithm to obtain the pairwise maximum likelihood estimates for the parameters, we use a linearization method in the composite marginal likelihood and EM algorithm. Also, techniques to calculate parameter estimates and spatial prediction in this class of skew models are proposed. The performance of the proposed model and method are illustrated through a simulation study, and applied the Tehran air quality index data set.

Truncated pair-wise likelihood for the Brown-Resnick process with applications to maximum temperature data

Max-stable processes are a natural extension of multivariate extreme value theory important for modeling the spatial dependence of environmental extremes. Inference for max-stable processes observed at several spatial locations is challenging due to the intractability of the full likelihood function. Composite likelihood methods avoid these difficulties by combining a number of low-dimensional likelihood objects, typically defined on pairs or triplets of spatial locations. This work develops a new truncation procedure based on ℓ1-penalty to reduce the complexity and computational cost associated with the composite likelihood function. The new method is shown to achieve a favorable trade-off between computational burden and statistical efficiency by dropping a number of noisy sub-likelihoods. The properties of the new method are illustrated through numerical simulations and an application to real extreme temperature data.

Pairwise likelihood estimation for confirmatory factor analysis models with categorical variables and data that are missing at random.

Methods for the treatment of item non-response in attitudinal scales and in large-scale assessments under the pairwise likelihood (PL) estimation framework and under a missing at random (MAR) mechanism are proposed. Under a full information likelihood estimation framework and MAR, ignorability of the missing data mechanism does not lead to biased estimates. However, this is not the case for pseudo-likelihood approaches such as the PL. We develop and study the performance of three strategies for incorporating missing values into confirmatory factor analysis under the PL framework, the complete-pairs (CP), the available-cases (AC) and the doubly robust (DR) approaches. The CP and AC require only a model for the observed data and standard errors are easy to compute. Doubly-robust versions of the PL estimation require a predictive model for the missing responses given the observed ones and are computationally more demanding than the AC and CP. A simulation study is used to compare the proposed methods. The proposed methods are employed to analyze the UK data on numeracy and literacy collected as part of the OECD Survey of Adult Skills.

Modeling Nonstationary Extreme Dependence With Stationary Max-Stable Processes and Multidimensional Scaling

Modeling the joint distribution of extreme events at multiple locations is a challenging task with important applications. In this study, we use max-stable models to study extreme daily precipitation events in Switzerland. The nonstationarity of the spatial process at hand involves important challenges, which are often dealt with by using a stationary model in a so-called climate space, with well-chosen covariates. Here, we instead choose to warp the weather stations under study in a latent space of higher dimension using multidimensional scaling (MDS). Two methods are proposed to define target dissimilarity matrices, based respectively on extremal coefficients and on pairwise likelihoods. Results suggest that the proposed methods allow capturing complex spatial dependences of spatial extreme precipitations, enabling in turn to reliably extrapolate functionals such as extremal coefficients. Supplemental materials for this article are available online.

The Hyvärinen scoring rule in Gaussian linear time series models

In this work we study stationary linear time-series models, and construct and analyse “score-matching” estimators based on the Hyvärinen scoring rule. We consider two scenarios: a single series of increasing length, and an increasing number of independent series of fixed length. In the latter case there are two variants, one based on the full data, and another based on a sufficient statistic.We study the empirical performance of these estimators in three special cases, autoregressive (AR), moving average (MA) and fractionally differenced white noise (ARFIMA) models, and make comparisons with full and pairwise likelihood estimators. The results are somewhat model-dependent, with the new estimators doing well for MA and ARFIMA models, but less so for AR models.

Max‐infinitely divisible models and inference for spatial extremes

AbstractFor many environmental processes, recent studies have shown that the dependence strength is decreasing when quantile levels increase. This implies that the popular max‐stable models are inadequate to capture the rate of joint tail decay, and to estimate joint extremal probabilities beyond observed levels. We here develop a more flexible modeling framework based on the class of max‐infinitely divisible processes, which extend max‐stable processes while retaining dependence properties that are natural for maxima. We propose two parametric constructions for max‐infinitely divisible models, which relax the max‐stability property but remain close to some popular max‐stable models obtained as special cases. The first model considers maxima over a finite, random number of independent observations, while the second model generalizes the spectral representation of max‐stable processes. Inference is performed using a pairwise likelihood. We illustrate the benefits of our new modeling framework on Dutch wind gust maxima calculated over different time units. Results strongly suggest that our proposed models outperform other natural models, such as the Student‐t copula process and its max‐stable limit, even for large block sizes.

On modeling positive continuous data with spatiotemporal dependence

AbstractIn this article, we concentrate on an alternative modeling strategy for positive data that exhibit spatial or spatiotemporal dependence. Specifically, we propose to consider stochastic processes obtained through a monotone transformation of scaled version of χ2 random processes. The latter is well known in the specialized literature and originates by summing independent copies of a squared Gaussian process. However, their use as stochastic models and related inference has not been much considered. Motivated by a spatiotemporal analysis of wind speed data from a network of meteorological stations in the Netherlands, we exemplify our modeling strategy by means of a nonstationary process with Weibull marginal distributions. For the proposed Weibull process we study the second‐order and geometrical properties and we provide analytic expressions for the bivariate distribution. Since the likelihood is intractable, even for a relatively small data set, we suggest adopting the pairwise likelihood as a tool for inference. Moreover, we tackle the prediction problem and we propose to use a linear prediction. The effectiveness of our modeling strategy is illustrated by analyzing the aforementioned Netherland wind speed data that we integrate with a simulation study. The proposed method is implemented in the R package GeoModels.

Pairwise likelihood estimation of latent autoregressive count models.

Latent autoregressive models are useful time series models for the analysis of infectious disease data. Evaluation of the likelihood function of latent autoregressive models is intractable and its approximation through simulation-based methods appears as a standard practice. Although simulation methods may make the inferential problem feasible, they are often computationally intensive and the quality of the numerical approximation may be difficult to assess. We consider instead a weighted pairwise likelihood approach and explore several computational and methodological aspects including estimation of robust standard errors and the role of numerical integration. The suggested approach is illustrated using monthly data on invasive meningococcal disease infection in Greece and Italy.

Non‐Gaussian geostatistical modeling using (skew) t processes

AbstractWe propose a new model for regression and dependence analysis when addressing spatial data with possibly heavy tails and an asymmetric marginal distribution. We first propose a stationary process with t marginals obtained through scale mixing of a Gaussian process with an inverse square root process with Gamma marginals. We then generalize this construction by considering a skew‐Gaussian process, thus obtaining a process with skew‐t marginal distributions. For the proposed (skew) t process, we study the second‐order and geometrical properties and in the t case, we provide analytic expressions for the bivariate distribution. In an extensive simulation study, we investigate the use of the weighted pairwise likelihood as a method of estimation for the t process. Moreover we compare the performance of the optimal linear predictor of the t process versus the optimal Gaussian predictor. Finally, the effectiveness of our methodology is illustrated by analyzing a georeferenced dataset on maximum temperatures in Australia.

A joint mean-correlation modeling approach for longitudinal zero-inflated count data

Longitudinal zero-inflated count data are widely encountered in many fields, while modeling the correlation between measurements for the same subject is more challenge due to the lack of suitable multivariate joint distributions. This paper studies a novel mean-correlation modeling approach for longitudinal zero-inflated regression model, solving both problems of specifying joint distribution and parsimoniously modeling correlations with no constraint. The joint distribution of zero-inflated discrete longitudinal responses is modeled by a copula model whose correlation parameters are innovatively represented in hyper-spherical coordinates. To overcome the computational intractability in maximizing the full likelihood function of the model, we further propose a computationally efficient pairwise likelihood approach. We then propose separated mean and correlation regression models to model these key quantities, such modeling approach can also handle irregularly and possibly subject-specific times points. The resulting estimators are shown to be consistent and asymptotically normal. Data example and simulations support the effectiveness of the proposed approach.

Asymptotic Theory for the Inference of the Latent Trawl Model for Extreme Values

This article develops statistical inference methods and their asymptotic theory for the so-called latent trawl model for extremes, which captures serial dependence in the time series of exceedances above a threshold. We review two inference methods based on the pairwise likelihood (PL) approach, a classical single-stage and a two-stage estimation; in the latter, we first estimate the marginal parameters, then the parameters describing the serial dependence. We derive the asymptotic theory in both cases and find that the asymptotic behaviour and goodness-of-fit are similar to the two-stage method being slightly preferred in our empirical work. However, we also show that PL-based approaches tend to underestimate the serial dependence in the extremes, and we propose a generalised method of moments procedure (both a single-stage and a two-stage procedure) based on autocovariance matching to overcome this shortcoming. We prove the strong mixing property of our model and provide central limit theorems for the four inference approaches. For single-stage approaches, the convergence is shown in the sense of weakly approaching sequences of distributions for strongly mixing sequences. For the two-stage strategies, we provide an asymptotic variance for the serial parameter which takes the estimation error arising from the first stage into account. In an empirical illustration using London air pollution data featuring six pollutants, we find that the two-stage autocovariance matching scheme seems to be performing best.

Simultaneous Optimization of Multiple Responses That Involve Correlated Continuous and Ordinal Responses According to the Gaussian Copula Models

This study investigates the simultaneous optimization of multiple correlated responses that involve mixed ordinal and continuous responses. The proposed approach is applicable for responses that have either an all ordinal categorical form are continuous but have different marginal distributions, or when standard multivariate distribution of responses is not applicable or does not exist. These multiple responses have rarely been the focus of studies despite their high occurrence during experiments. The copula functions have been used to construct a multivariate model for mixed responses. To resolve the computational problems of estimation under a high dimension of responses, we have estimated parameters of the model according to a pairwise likelihood estimation method. We adapted the generalized distance approach to determine settings of the factors that simultaneously optimized the mean of continuous responses and desired cumulative categories of the ordinal responses. A simulation study was used to evaluate the performance of the estimators from the pairwise likelihood approach. Finally, we presented an application of the proposed method in a real data example of a semiconductor manufacturing process.

Screening to prevent fragility fractures among adults 40 years and older in primary care: protocol for a systematic review

PurposeTo inform recommendations by the Canadian Task Force on Preventive Health Care by systematically reviewing direct evidence on the effectiveness and acceptability of screening adults 40 years and older in primary care to reduce fragility fractures and related mortality and morbidity, and indirect evidence on the accuracy of fracture risk prediction tools. Evidence on the benefits and harms of pharmacological treatment will be reviewed, if needed to meaningfully influence the Task Force’s decision-making.MethodsA modified update of an existing systematic review will evaluate screening effectiveness, the accuracy of screening tools, and treatment benefits. For treatment harms, we will integrate studies from existing systematic reviews. A de novo review on acceptability will be conducted. Peer-reviewed searches (Medline, Embase, Cochrane Library, PsycINFO [acceptability only]), grey literature, and hand searches of reviews and included studies will update the literature. Based on pre-specified criteria, we will screen studies for inclusion following a liberal-accelerated approach. Final inclusion will be based on consensus. Data extraction for study results will be performed independently by two reviewers while other data will be verified by a second reviewer; there may be some reliance on extracted data from the existing reviews. The risk of bias assessments reported in the existing reviews will be verified and for new studies will be performed independently. When appropriate, results will be pooled using either pairwise random effects meta-analysis (screening and treatment) or restricted maximum likelihood estimation with Hartun-Knapp-Sidnick-Jonkman correction (risk prediction model calibration). Subgroups of interest to explain heterogeneity are age, sex, and menopausal status. Two independent reviewers will rate the certainty of evidence using the GRADE approach, with consensus reached for each outcome rated as critical or important by the Task Force.DiscussionSince the publication of other guidance in Canada, new trials have been published that are likely to improve understanding of screening in primary care settings to prevent fragility fractures. A systematic review is required to inform updated recommendations that align with the current evidence base.

Composite likelihood estimation for a Gaussian process under fixed domain asymptotics

We study the problem of estimating the covariance parameters of a one-dimensional Gaussian process with exponential covariance function under fixed-domain asymptotics. We show that the weighted pairwise maximum likelihood estimator of the microergodic parameter can be consistent or inconsistent. This depends on the range of admissible parameter values in the likelihood optimization. On the other hand, the weighted pairwise conditional maximum likelihood estimator is always consistent. Both estimators are also asymptotically Gaussian when they are consistent. Their asymptotic variances are larger or strictly larger than that of the maximum likelihood estimator. A simulation study is presented in order to compare the finite sample behavior of the pairwise likelihood estimators with their asymptotic distributions. For more general covariance functions, an additional inconsistency result is provided, for the weighted pairwise maximum likelihood estimator of a variance parameter.

Hierarchical Space-Time Modeling of Asymptotically Independent Exceedances With an Application to Precipitation Data

The statistical modeling of space-time extremes in environmental applications is key to understanding complex dependence structures in original event data and to generating realistic scenarios for impact models. In this context of high-dimensional data, we propose a novel hierarchical model for high threshold exceedances defined over continuous space and time by embedding a space-time Gamma process convolution for the rate of an exponential variable, leading to asymptotic independence in space and time. Its physically motivated anisotropic dependence structure is based on geometric objects moving through space-time according to a velocity vector. We demonstrate that inference based on weighted pairwise likelihood is fast and accurate. The usefulness of our model is illustrated by an application to hourly precipitation data from a study region in Southern France, where it clearly improves on an alternative censored Gaussian space-time random field model. While classical limit models based on threshold-stability fail to appropriately capture relatively fast joint tail decay rates between asymptotic dependence and classical independence, strong empirical evidence from our application and other recent case studies motivates the use of more realistic asymptotic independence models such as ours. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Discrete Longitudinal Data Modeling with a Mean-Correlation Regression Approach

Joint mean-covariance regression modeling with unconstrained parametrization for continuous longitudinal data has provided statisticians and practitionersJoint mean-covariance regression modeling with unconstrained parametrization for continuous longitudinal data has provided statisticians and practitioners with a powerful analytical device. How to develop a delineation of such a regression framework amongst discrete longitudinal responses remains an open and more challenging problem. This paper studies a novel mean-correlation regression for a family of generic discrete responses. Targeting the joint distributions of the discrete longitudinal responses, our regression approach is constructed by using a copula model whose correlation parameters are represented in hyperspherical coordinates with no constraint on their support. To overcome computational intractability in maximizing the full likelihood function of the model, we propose a computationally efficient pairwise likelihood approach. A pairwise likelihood ratio test is then constructed and validated for statistical inferences. We show that the resulting estimators of our approaches are consistent and asymptotically normal. We demonstrate the effectiveness, parsimoniousness and desirable performance of the proposed approach by analyzing three data sets and conducting extensive simulations with a powerful analytical device. How to develop a delineation of such a regression framework amongst discrete longitudinal responses remains an open and more challenging problem. This paper studies a novel mean-correlation regression for a family of generic discrete responses. Targeting the joint distributions of the discrete longitudinal responses, our regression approach is constructed by using a copula model whose correlation parameters are represented in hyperspherical coordinates with no constraint on their support. To overcome computational intractability in maximizing the full likelihood function of the model, we propose a computationally efficient pairwise likelihood approach. A pairwise likelihood ratio test is then constructed and validated for statistical inferences. We show that the resulting estimators of our approaches are consistent and asymptotically normal. We demonstrate the effectiveness, parsimoniousness and desirable performance of the proposed approach by analyzing three data sets and conducting extensive simulations

Missing covariate data in generalized linear mixed models with distribution-free random effects

We consider generalized linear mixed models in which random effects are free of parametric distributions and missing at random data are present in some covariates. To overcome the problem of missing data, we propose two novel methods relying on auxiliary variables: a penalized conditional likelihood method when covariates are independent of random effects, and a two-step procedure consisting of a pairwise likelihood for estimating fixed effects in the first step and a penalized conditional likelihood for estimating random effects in the second step while covariates can be related to random effects. Our methods allow a nonparametric structure for the missing covariate data and do not rely on distribution assumptions for random effects, which are not observed in the data, thus providing great flexibility in capturing a board range of the missingness mechanism and behaviors of random effects. We show that the proposed estimators enjoy desirable theoretical properties by relaxing the conditions for a finite number of clusters or finite cluster size imposed in the literature. The finite sample performance of the estimators is assessed through extensive simulations. We illustrate the application of the methods using a longitudinal data set on forest health monitoring.

Multivariate ordinal regression models: an analysis of corporate credit ratings

Correlated ordinal data typically arises from multiple measurements on a collection of subjects. Motivated by an application in credit risk, where multiple credit rating agencies assess the creditworthiness of a firm on an ordinal scale, we consider multivariate ordinal regression models with a latent variable specification and correlated error terms. Two different link functions are employed, by assuming a multivariate normal and a multivariate logistic distribution for the latent variables underlying the ordinal outcomes. Composite likelihood methods, more specifically the pairwise and tripletwise likelihood approach, are applied for estimating the model parameters. Using simulated data sets with varying number of subjects, we investigate the performance of the pairwise likelihood estimates and find them to be robust for both link functions and reasonable sample size. The empirical application consists of an analysis of corporate credit ratings from the big three credit rating agencies (Standard & Poor’s, Moody’s and Fitch). Firm-level and stock price data for publicly traded US firms as well as an unbalanced panel of issuer credit ratings are collected and analyzed to illustrate the proposed framework.

Approximate composite marginal likelihood inference in spatial generalized linear mixed models

ABSTRACTNon-Gaussian spatial responses are usually modeled using spatial generalized linear mixed model with spatial random effects. The likelihood function of this model cannot usually be given in a closed form, thus the maximum likelihood approach is very challenging. There are numerical ways to maximize the likelihood function, such as Monte Carlo Expectation Maximization and Quadrature Pairwise Expectation Maximization algorithms. They can be applied but may in such cases be computationally very slow or even prohibitive. Gauss–Hermite quadrature approximation only suitable for low-dimensional latent variables and its accuracy depends on the number of quadrature points. Here, we propose a new approximate pairwise maximum likelihood method to the inference of the spatial generalized linear mixed model. This approximate method is fast and deterministic, using no sampling-based strategies. The performance of the proposed method is illustrated through two simulation examples and practical aspects are investigated through a case study on a rainfall data set.