Inference Function For Margins Research Articles

Parametric and semiparametric approaches for copula-based regression estimation

Based on the normality assumption on dependent variable, regression analysis is one of the most popular statistical techniques for studying the dependence between response and explanatory variables. However, violation of this assumption in the data makes regression analysis inappropriate in several real life situations. Copula is a powerful tool for modeling multivariate data and have recently been employed in regression analysis. The key concept behind copula-based regression approach is to formulate conditional expectation in terms of copula density and marginal distributions. In this paper, we explore parametric and semiparametric estimations of the copula-based regression function. The maximum likelihood (ML), inference functions for margins (IFM), and pseudo maximum likelihood (PML) techniques are adopted here for estimation purposes. Extensive numerical experiments are performed to illustrate the performance of the proposed copula-based regression estimators under specified and misspecified scenarios of copulas and marginals. Finally, two real data applications are also presented to demonstrate the performance of the considered estimators.

A Copula-Based Bivariate Composite Model for Modelling Claim Costs

This paper aims to develop a new family of bivariate distributions for modelling different types of claims and their associated costs jointly in a flexible manner. The proposed bivariate distributions can be viewed as a continuous copula distribution paired with two marginals based on composite distributions. For expository purposes, the details of one of the proposed bivarite composite distributions is provided. The dependence measures for the resulting bivariate copula-based composite distribution are studied, and its fitting is compared with other bivariate composite distributions and existing bivariate distributions. The parameters of the proposed bivariate composite model are estimated via the inference functions for margins (IFM) method. The suitability of the proposed bivariate distribution is examined using two real-world insurance datasets, namely the motor third-party liability (MTPL) insurance dataset and Danish fire insurance dataset.

Optimizing multilevel maintenance in multi-component systems under s-dependent competing risks

This paper presents a novel reliability-centered multilevel maintenance optimization model tailored for multicomponent systems subjected to stochastic dependent (s-dependent) competing risks (CRs). Four types of maintenance actions, including minimal repair (MR), medium preventive maintenance (MPM), overhaul preventive maintenance (OPM), and preventive replacement (PR), are incorporated into the development of the multilevel maintenance model. The Copula function is utilized to quantify the s-dependence among failure risks, and subsequently, all unknown parameters are estimated via the inference function for margins (IFM) method. Three types of decision variables are designed to minimize the long-run expected cost rate of the system. Finally, sequential decision process method is utilized to solve the model, and a numerical example of a metro door system illustrates the effectiveness of our proposed model.

Dependence Modelling of Lifetimes in Egyptian Families

In this study, we analyse a large sample of Egyptian social pension data which covers, by law, the policyholder’s spouse, children, parents and siblings. This data set uniquely enables the study and comparison of pairwise dependence between multiple familial relationships beyond the well-known husband and wife case. Applying Bayesian Markov Chain Monte Carlo (MCMC) estimation techniques with the two-step inference functions for margins (IFM) method, we model dependence between lifetimes in spousal, parent–child and child–parent relationships, using copulas to capture the strength of association. Dependence is observed to be strongest in child–parent relationships and, in comparison to the high-income countries of data sets previously studied, of lesser significance in the husband and wife case, often referred to as broken-heart syndrome. Given the traditional use of UK mortality tables in the modelling of mortality in Egypt, the findings of this paper will help to inform appropriate mortality assumptions specific to the unique structure of the Egyptian scheme.

Copulas and Histogram-Valued Data

Histogram-valued data are emerging increasingly often as a consequence of the aggregation of large datasets. One statistic that underpins many methodologies especially regression and principal component analyses is the covariance function. To date, no method exists for calculating these functions directly from the marginal histogram observations. This article develops techniques through copula functions to develop a parametric distribution for multivariate histogram-valued data. In particular, maximum likelihood, inference function for margins, and canonical maximum likelihood estimation methods are proposed. A numerical study helps to ascertain which copulas are best to use in various cases, and thence to calculate the covariances. The results are applied to a real dataset.

Structure of bivariate Rayleigh proportional hazard rate model with its associated copula applied on COVID-19 data.

Visualizing the fatality of coronavirus is a very tricky point through the world. In this paper, a new construction via the proportional hazard rate model with Rayleigh marginal is introduced and applied on COVID-19 data set. The statistical and reliability characteristics of bivariate Rayleigh proportional hazard (BRPH) distribution are derived. The copula dependence structure and its properties are studied. The point estimation of the marginal and dependence parameters is introduced via maximum likelihood, method of moments, and inference function for margins (IFM) method. A simulation study is carried out to examine the effectiveness and the performance of the parameter estimates. Finally, an application on COVID-19 data is used in a comparison study between BRPH model and other constructed bivariate models. This application concerned with modeling the fatality on COVID-19. Throughout the results of goodness-of-fit criteria, BRPH provides a better fit than different competitors constructed bivariate models which reflects its flexibility and applicability on modeling the fatality of COVID-19.

Estimation in copula models with two-piece skewed margins using the inference for margins method

Copulas provide a versatile tool in the modelling of multivariate distributions. With an increased awareness for possible asymmetry in data, skewed copulas in combination with classical margins have been employed to appropriately model these data. The reverse, skewed margins with a (classical) copula has also been considered, but mainly with classical skew-symmetrical margins. An alternative approach is to rely on a large family of asymmetric two-piece distributions for the univariate marginal distributions. Together with any copula this family of asymmetric univariate distributions provides a powerful tool for skewed multivariate distributions. Maximum likelihood estimation of all parameters involved is discussed. A key step in achieving statistical inference results is an extension of the theory available for generalized method of moments, under non-standard conditions. This together with the inference results for the family of univariate distributions, allows to establish consistency and asymptotic normality of the estimators obtained through the method of ‘inference functions for margins’. The theoretical results are complemented by a simulation study and the practical use of the method is demonstrated on real data examples.

An Application of the IFM Method for the Risk Assessment of Financial Instruments

External influences or behavioral biases can affect the way risk is perceived. This paper studies the prediction of VaR (Value at Risk) as a measure of the risk of loss for investments on financial products. Our aim is to predict the percentage of loss that a financial product would have in the future to assess the risks and determine the potential loss of a security in the stock market, thus reducing reasoning influenced by feelings for bank and financial firms seeking to deploy AI and advanced automation. We used the IFM (inference function for margins) method in different market scenarios, with particular emphasis on the strengths and weaknesses of it. The study is assessed on single product level with the skewed studen-t GARCH(1,1) model and portfolio level with t-copulas for the inter-dependencies. It has been shown that under normal market conditions the risk is predicted properly for both levels. However, when an unexpected market event occurs, the prediction fails. To address this limitation, a combined model with sentiment analysis and regression is proposed for further investigation as a future work.

Effective estimation algorithm for parameters of multivariate Farlie\u2013Gumbel\u2013Morgenstern copula

This paper focuses on the parameter estimation for the d-variate Farlie–Gumbel–Morgenstern (FGM) copula (dge 2), which has 2^d-d-1 dependence parameters to be estimated; therefore, maximum likelihood estimation is not practical for a large d from the viewpoint of computational complexity. Besides, the restriction for the FGM copula’s parameters becomes increasingly complex as d becomes large, which makes parameter estimation difficult. We propose an effective estimation algorithm for the d-variate FGM copula by using the method of inference functions for margins under the restriction of the parameters. We then discuss its asymptotic normality as well as its performance determined through simulation studies. The proposed method is also applied to real data analysis of bearing reliability.

On the classical estimation of bivariate copula-based Seemingly unrelated tobit models through the proposed inference function for augmented margins method

This paper extends the analysis of the bivariate Seemingly Unrelated (SUR) Tobit by modeling its nonlinear dependence structure through copula and assuming non-normal marginal error distributions. For model estimation, the use of copula methods enables the use of the (classical) Inference Function for Margins (IFM) method by Joe and Xu (1996), which is more computationally attractive (feasible) than the full maximum likelihood approach. However, our simulation study shows that the IFM method provides a biased estimate of the copula parameter in the presence of censored observations in both margins. In order to obtain an unbiased estimate of the copula association parameter, we propose/develop a modified version of the IFM method, which we refer to as Inference Function for Augmented Margins (IFAM). Since the usual asymptotic approach, that is the computation of the asymptotic covariance matrix of the parameter estimates, is troublesome, we propose the use of resampling procedures (bootstrap methods) to obtain confidence intervals for the copula-based SUR Tobit model parameters. The satisfactory results from the simulation and empirical studies indicate the adequate performance of our proposed model and methods. We illustrate our procedure using bivariate data on consumption of salad dressings and lettuce by U.S. individuals.

A New Dynamic Mixture Copula Mechanism to Examine the Nonlinear and Asymmetric Tail Dependence Between Stock and Exchange Rate Returns

This paper develops a new time-varying mixture copula, in which the dynamic weights of four distinct copulas are determined by a two-stratum process, to investigate the magnitude of tail dependence in four independent quadrants. In the two-stratum process, the weight of each copula is determined firstly by the relative importance of positive and negative dependence structures, and then by its own past values and adjustment processes. The weighting mechanism is time-varying in each stratum. This new specification is applied to analyze the asymmetric tail dependencies between the stock and exchange rate markets. Empirical results show four interesting findings. First, the quasi-maximum likelihood estimation (QMLE) has a better fitting ability than does the inference function for margins. The relative efficiency of the QMLE is irrespective of marginal specifications. Second, the goodness-of-fit tests of the new time-varying mixture copula are crucially affected by the marginal specifications. Third, estimation methods impact mixture weights. Four distinct tail dependencies are observed, revealing the importance of considering all four tails concurrently, and not just parts of the four tails. Fourth, the asymmetric positive and negative dependencies are significant. Each country shows a similar pattern of asymmetric negative dependence, but a different pattern of asymmetric positive dependence. These empirical findings provide important portfolio allocation implications.

Study on Chip Reliability Modeling Based on Mutually Dependent Competing Failure of Solder Joints in Different Failure Modes

The chip is a core functional component. Its reliability plays a vital role in electronic equipment normal operation. As the typical cause for chip malfunction, the solder joint degradation is selected to study chip reliability. The degradation models of solder joint in different failure modes are established through data-driven and failure physical model, and chip reliability model is constructed based on mutually competing failures of multiple solder joints. First, the chip reliability degradation test and finite element modeling(FEM) are carried out under coupled environment stress. The solder joint failure modes and degradation processes are studied through the analysis of test data, microstructure and mechanical simulation. Then, solder joint degradation models are established based on Coffin-Manson and Paris functions that have been modified by a data-driven method. Taking the solder joint failure time of degradation model as the characteristic parameter of Weibull distribution, the solder joint reliability function is obtained. Finally, mutually dependent competing failure theory is cited to describe the correlation about solder joint reliability of different failure modes, then the chip reliability model is established. The parameter estimation is realized by the inference function for margins (IFM) method. From verification tests, results show models are highly consistent with the actual reliability, indicting our reliability modeling method achieves the transition from underlying solder joint level to integrated component level. The joint application of different models can make up for the deficiencies of the single model and obtain more accurate results. In addition, we determined that intermetallic compounds (IMC) are main source of model error.

Dynamic dependence modelling of wind power uncertainty considering heteroscedastic effect

The inherent stochastic nature of wind power poses significant challenges to the safe and economic operation of power systems, and an accurate model of wind power forecast error can give operators great help in dealing with the negative impact of wind power uncertainty on the power system. To obtain accurate forecast error intervals, a wind power forecast error (WPFE) model based on dynamic Copula theory is proposed in this paper. By combining Autoregressive integrated moving average (ARIMA) method and generalized autoregressive conditional heteroscedasticity (GARCH) model, the marginal distribution of WPFE model with the ability to describe its time-varying feature is achieved. In this proposed model, 4 dynamic Copula functions are estimated and evaluated by the combination method of Inference Functions for Margins (IFM) and Akaike Information Criterion (AIC), and the best-fitted one is selected to be applied in a set of synchronous data of wind power and its forecast. The result shows that compared with the conventional method, the proposed method can provide more accurate forecast error intervals. This advantage can benefit the scheduling of the power system with high wind power penetration rate, which is also verified by a stochastic unit commitment case in modified IEEE 118-Bus system.

Copula-based Bivariate Cointegration Model

This article focuses on a bivariate cointegrating model with non-normal errors. In particular, we propose a bivariate error distribution constructed using two non-identical marginals through a copula. The model parameters are estimated using the method of inference functions for margins and maximum likelihood. Applications of the proposed model are illustrated through real life examples. AMS classification codes: 62M10

Copula information criterion for model selection with two-stage maximum likelihood estimation

In parametric copula setups, where both the margins and copula have parametric forms, two-stage maximum likelihood estimation, often referred to as inference functions for margins, is used as an attractive alternative to the full maximum likelihood estimation strategy. Exploiting the existing model robust inference of two-stage maximum likelihood estimation, a copula information criterion (CIC) for model selection is developed. In a nutshell, CIC aims for the model that minimizes the Kullback–Leibler divergence from the real data generating mechanism. CIC does not assume that the chosen parametric model captures this true model, unlike what is assumed for AIC. In this sense, CIC is analogous to the Takeuchi Information Criterion (TIC), which is defined for the full maximum likelihood. If the additional assumption that a candidate model is correctly specified is made, then CIC for that model simplifies to AIC. Additionally, CIC can easily be extended to the conditional copula setup where covariates are parametrically linked to the copula model. As a numerical illustration, simulation studies were performed to show that the better model according to CIC also has better prediction performance in general. The result also shows that the bias correction term of CIC penalizes the misspecified model more heavily. This bias correction term has a strong positive relationship with the prediction performance of the model. So, a model with bad prediction performance is being penalized more by CIC. Although this behavior of the bias correction part is an important conceptual advance of CIC, this is not sufficient to make CIC outperform AIC in practice. This is because each misspecified model has the bias correction term and they grow at different speeds, depending on the model. The difference between CIC and AIC becomes minimal as sample size grows because the log-likelihood part outgrows the bias correction part.

A generalized bivariate copula for flood analysis in Peninsular Malaysia

This study generalized the best copula to characterize the joint probability distribution between rainfall severity and duration in Peninsular Malaysia using two dimensional copulas. Specifically, to construct copulas, Inference Function for Margins (IFM) and Canonical Maximum Likelihood (CML) methods were specially exploited. For the purpose of achieving copula fitting, the derived rainfall variables by making use of the Standardized Precipitation Index (SPI) were fitted into several distributions. Five copulas, namely Gaussian, Clayton, Frank, Joe and Gumbel were put to the tests to establish the best data fitted copula. The tests produced acknowledged and satisfactory results of copula fitting for rainfall severity and duration. Surveying the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), only three copulas produced a better fit for parametric and semi parametric approaches. Finally, two consistency tests were conducted and the results shown that Frank Copula produced consistent results.

Model robust inference with two-stage maximum likelihood estimation for copulas

This article is concerned with inference in parametric copula setups, where both the marginals and the copula have parametric forms. For such models, two-stage maximum likelihood estimation, often referred to as inference function for margins, is used as an attractive alternative to the full maximum likelihood estimation strategy. Previous studies of the two-stage maximum likelihood estimator have largely been based on the assumption that the chosen parametric model captures the true model that generated data. We study the impact of dropping this true model assumption, both theoretically and numerically. We first show that the two-stage maximum likelihood estimator is consistent for a well-defined least false parameter value, different from the analogous least false parameter associated with the full maximum likelihood procedure. Then we demonstrate limiting normality of the full vector of estimators, with concise matrix notation for the variance matrices involved. Along with consistent estimators for these, we have built a model-robust machinery for inference in parametric copula models. The special case where the parametric model is assumed to hold corresponds to situations studied earlier in the literature, with simpler formulas for variance matrices. As a numerical illustration, we perform a set of simulations. We also analyze five-dimensional Norwegian precipitation data. We find that the variance of the copula parameter estimate can both increase and decrease, by dropping the true model assumption. In addition, we observe that the two-stage maximum likelihood estimator is still highly efficient when the true model assumption is dropped and thus the model robust asymptotic variance formulas are used. Additionally, we discover that using highly misspecified models can lead to situations where the asymptotic variance of the two-stage maximum likelihood estimator is lower than that of full maximum likelihood estimator. Our results are also used to analyze the mean squared error properties for both the full and the two-stage maximum likelihood estimators of any focus parameter.

A meta-heuristic approach for multivariate design flood quantile estimation incorporating historical information

Abstract Design flood quantiles are crucial for hydraulic structures design, water resources planning and management, whereas previous multivariate hydrological quantile estimation methods usually do not consider historical flood information. To overcome such limitations, a meta-heuristic inference function for margins (MHIFM) approach, coupling meta-heuristic algorithm with a modified inference function for margins (IFM) method, is developed for modeling the joint distributions of flood peak and volumes with incorporation of historical flood information. Then, the most likely realization (MLR) and equivalent frequency combination (EFC) methods are employed for selecting multivariate design floods on a quantile iso-surface. The Danjiangkou reservoir located in Hanjiang River basin, the first pilot basin of most regulated water resources management policy in China, is selected as a case study. Application results indicate that the MHIFM approach shows good performance for estimating the parameters of marginal and joint distributions; moreover, the MLR method yields safer design flood quantiles than the EFC method in terms of highest routed reservoir water levels. The proposed MHIFM approach associated with the MLR method is safer and more rational for reservoir design, which would provide rich information as the reference for flood risk assessment, reservoir operation and management.

Semiparametric Estimation and Panel Data Clustering Analysis Based on D-Vine and C-Vine

This paper proposed a panel data clustering model based on D-vine and C-vine and supported a semiparametric estimation for parameters. These models include a two-step inference function for margins, two-step semiparameter estimation, and stepwise semiparametric estimation. In similarity measurement, similarity coefficients are constructed by a multivariate Hierarchical Nested Archimedean Copula (HNAC) model and compound PCC models, which are HNAC and D-vine compound model and HNAC and C-vine compound model. Estimation solutions and models evaluation are given for these models. In the case study, the clustering results of HNAC and D-vine compound model and HNAC and C-vine compound model are given, and the effect of different copula families on clustering results is also discussed. The result shows the models are effective and useful.

Probabilistic spatial prediction of categorical data using elliptical copulas

This study uses elliptical copulas and transition probabilities for uncertainty modeling of categorical spatial data. It begins by discussing the expressions of the cumulative distribution function and probability density function of two major elliptical copulas: Gaussian copula and t copula. The basic form of spatial copula discriminant function is then derived based on Bayes’ theorem, which consists of three parts: the prior probability, the conditional marginal densities, and the conditional copula density. Finally, three kinds of parameter estimation methods are discussed, including maximum likelihood estimation, inference functions for margins and canonical maximum likelihood (CML). To avoid making assumptions on the form of marginal distributions, the CML approach is adopted in the real-world case study. Results show that the occurrence probability maps generated by these two elliptical copulas are similar to each other. However, the prediction map interpolated by Gaussian copula has a relatively higher classification accuracy than t copula.