Boosting distributional copula regression for bivariate binary, discrete and mixed responses

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.

In this paper we discuss the problem on parametric and non parametric estimation of the distributions generated by the Marshall-Olkin copula. This copula comes from the Marshall-Olkin bivariate exponential distribution used in reliability analysis. We generalize this model by the copula and different marginal distributions to construct several bivariate survival functions. The cumulative distribution functions are not absolutely continuous and they unknown parameters are often not be obtained in explicit form. In order to estimate the parameters we propose an easy procedure based on the moments. This method consist in two steps: in the first step we estimate only the parameters of marginal distributions and in the second step we estimate only the copula parameter. This procedure can be used to estimate the parameters of complex survival functions in which it is difficult to find an explicit expression of the mixed moments. Moreover it is preferred to the maximum likelihood one for its simplex mathematic form; in particular for distributions whose maximum likelihood parameters estimators can not be obtained in explicit form.

In recent years, there has been an increasing interest in developing dose finding methods incorporating both efficacy and toxicity outcomes. It is reasonable to assume that efficacy and toxicity are associated; therefore, we need to model their stochastic dependence. Copula functions are very useful tools to model different kinds of dependence with arbitrary marginal distributions. We consider a binary efficacy-toxicity response with logit marginal distributions. Since the dose which maximizes the probability of efficacy without toxicity (P-optimal dose) changes depending on different copula functions, we propose a criterion which is useful for choosing between the rival copula models but also protects patients against doses that are far away from the P-optimal dose. The performance of this compromise criterion (called PKL) is illustrated for different choices of the parameter values.

This study investigates the sensitivity and uncertainty of hydrological droughts frequencies and severity in the Weihe Basin, China during 1960–2012, by using six commonly used univariate probability distributions and three Archimedean copulas to fit the marginal and joint distributions of drought characteristics. The Anderson-Darling method is used for testing the goodness-of-fit of the univariate model, and the Akaike information criterion (AIC) is applied to select the best distribution and copula functions. The results demonstrate that there is a very strong correlation between drought duration and drought severity in three stations. The drought return period varies depending on the selected marginal distributions and copula functions and, with an increase of the return period, the differences become larger. In addition, the estimated return periods (both co-occurrence and joint) from the best-fitted copulas are the closet to those from empirical distribution. Therefore, it is critical to select the appropriate marginal distribution and copula function to model the hydrological drought frequency and severity. The results of this study can not only help drought investigation to select a suitable probability distribution and copulas function, but are also useful for regional water resource management. However, a few limitations remain in this study, such as the assumption of stationary of runoff series.

A statistical analysis method for significant wave height and spectral peak frequency considering the random and time-varying effects based on copula function and Bayesian inference

Quantitative studies in many fields involve the analysis of multivariate data of diverse types, including measurements that we may consider binary, ordinal and continuous. One approach to the analysis of such mixed data is to use a copula model, in which the associations among the variables are parameterized separately from their univariate marginal distributions. The purpose of this article is to provide a simple, general method of semiparametric inference for copula models via a type of rank likelihood function for the association parameters. The proposed method of inference can be viewed as a generalization of marginal likelihood estimation, in which inference for a parameter of interest is based on a summary statistic whose sampling distribution is not a function of any nuisance parameters. In the context of copula estimation, the extended rank likelihood is a function of the association parameters only and its applicability does not depend on any assumptions about the marginal distributions of the data, thus making it appropriate for the analysis of mixed continuous and discrete data with arbitrary marginal distributions. Estimation and inference for parameters of the Gaussian copula are available via a straightforward Markov chain Monte Carlo algorithm based on Gibbs sampling. Specification of prior distributions or a parametric form for the univariate marginal distributions of the data is not necessary.

Joint probability distribution of winds and waves from wave simulation of 20 years (1989-2008) in Bohai Bay

Thanks to their ability to capture complex dependence structures, copulas are frequently used to glue random variables into a joint model with arbitrary marginal distributions. More recently, they have been applied to solve statistical learning problems such as regression or classification. Framing such approaches as solutions of estimating equations, we generalize them in a unified framework. We can then obtain simultaneous, coherent inferences across multiple regression-like problems. We derive consistency, asymptotic normality, and validity of the bootstrap for corresponding estimators. The conditions allow for both continuous and discrete data as well as parametric, nonparametric, and semiparametric estimators of the copula and marginal distributions. The versatility of this methodology is illustrated by several theoretical examples, a simulation study, and an application to financial portfolio allocation. Supplementary materials for this article are available online.

Groundwater is an essential water resource in the Yarkant River Basin Irrigation District, which is the largest oasis in Xinjiang, China. This study used a novel approach to analyze the relationship between groundwater depth and three driving factors by developing Copula Functions from monthly time series data collected from 16 monitoring wells. More specifically, marginal distribution and joint distribution functions were established, and the conditional probabilities for three data ranges were calculated using both two- and three-dimensional Copula Functions. The developed statistical models showed that groundwater exploitation, runoff, and surface water irrigation significantly affected groundwater depth. The most significant effect on water table declines was associated with groundwater exploitation and lagged 1-month behind the groundwater withdrawals. The amount of runoff and irrigation were both inversely related to water table depth due to groundwater recharge. Frank Copula Functions were found to produce the best fit to the joint distribution of the variables and were used herein, allowing for the establishment of a detailed probability distribution of the change in groundwater depth under the combined effect of multiple controlling factors. The results provided key insights into how irrigation and groundwater withdrawals can be adjusted to effectively manage groundwater resources.

Contemporary financial risk management is significantly based on the analysis of time series of returns. One of the most significant errors frequently committed by analysts is the predominant use of normal distributions when it is clear that the returns are not normal. Copula models and models for non-normal multivariate distributions provide new tools to solve the problem because the obtained results are immediately applicable in portfolio management, option pricing and measuring risk without assuming normality. Therefore, both a theoretician and a practitioner are interested in multivariate models for returns and copula functions. The copula function models provide an effective and interesting technique of constructing multivariate distribution starting from marginal ones. Due to Sklar\'s result established in 1959, we can present any multivariate distribution with a help of corresponding marginal distributions and a selected copula function. In this work we present an application of copula function to construct multivariate conditional distributions of times series. In the last part of this paper dynamic models such as DCC-MVGARCH and conditional copula are analyzed. Moreover, we also present an application of bootstrap in the context of copula function. This work is appended by examples showing practical application of our work.

This study seeks to apply Khoudraji asymmetric copula in a trivariate analysis of drought, focusing on the uncertainty assessment of the severity–duration–magnitude–frequency (SDMF) curve. The analysis, influenced by input data and copula parameters, was conducted in the Mina Basin, Algeria. The results showed: (1) the dependence structure of SDMF of drought characteristics can be well modelled by Khoudraji copula; (2) Gumbel and Frank copulas depart more than other copula functions in estimating SDMF curves; (3) the uncertainty of SDMF curves caused by input data is much bigger than the uncertainty caused by copula parameters; (4) an increase (decrease) in the conditional probability causes a decrease (increase) in the uncertainty band, which applies to both sources of uncertainty in this law. Conversely, increasing the threshold M also increases the uncertainty. These results are very useful for reducing uncertainty caused by estimating SDMF curves and preparing strategies for different levels of drought.

BackgroundThe Resource Utilization in Dementia‐Lite (RUD‐Lite) scale measures both formal healthcare utilization and informal caregiver time required by individuals with cognitive impairment. The RUD‐Lite has been incorporated into multiple observational and interventional studies of participants with Alzheimer’s disease (AD) dementia of varying severity. However, RUD‐Lite data for earlier stages of AD such as mild cognitive impairment (MCI) is limited and is primarily from observational studies. As healthcare utilization may differ between participants in observational versus interventional studies, we analyzed data from the Tauriel study of semorinemab in MCI and mild dementia due to AD (NCT03289143).MethodsWe performed cross‐sectional and longitudinal RUD‐Lite comparisons between MCI (n = 160) and mild dementia (n = 288) cohorts of the Tauriel study, exploring formal healthcare usage (hospital admissions, emergency department visits, outpatient visits, and ancillary services) and informal caregiver time for assistance with basic (BADLs) and instrumental (IADLs) activities of daily living and participant supervision. Both unadjusted and adjusted (for baseline usage, age, sex) analyses were performed on longitudinal data.ResultsBaseline cross‐sectional analyses (Table 1) indicated that mild dementia participants required significantly greater caregiver time for assistance with IADLs than MCI participants. Other formal and informal healthcare utilization indices did not differ between groups. Unadjusted longitudinal analyses across 18 months (Figure 1) revealed significant increases in informal caregiver assistance with BADLs (MCI:+40%, mild dementia:+231%), IADLs (MCI:+31%, mild dementia:+34%), and supervision (mild dementia:+219%) and in ancillary service usage (mild dementia:+345%). Adjusted longitudinal analyses (Figure 2) revealed significantly greater increases in mild dementia versus MCI for assistance time with BADLs, IADLs, and supervision.ConclusionInformal caregiver time spent providing assistance in early AD increases with disease severity and longitudinal changes are detectable in MCI over 18 months. Formal healthcare utilization indices were less sensitive to disease progression. While the RUD‐Lite results from this interventional study are consistent with observational data from analogous populations, less caregiver assistance time was reported in our cohort. When using RUD‐Lite outcomes in future early AD studies, investigators may need to account for their participant population (e.g., observational versus interventional) in their study design.

We propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions. A joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions. This class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable, and its circumvention of the “curse of dimensionality” associated with purely nonparametric multivariate distributions. We show that the plug-in sieve maximum likelihood estimators (MLEs) of all smooth functionals, including the finite-dimensional copula parameters and the unknown marginal distributions, are semiparametrically efficient, and that their asymptotic variances can be estimated consistently. Moreover, prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve further efficiency gains. Two such cases are studied: (a) the marginal distributions are equal but otherwise unspecified, and (b) some but not all marginal distributions are parametric. Monte Carlo studies indicate that the sieve MLEs perform well in finite samples, especially when prior information on the marginal distributions is incorporated.

This paper investigates the impacts of macroeconomic variables, namely consumer price index, exchange rate, minimum loan rate and oil price movement, on the financials and the property & construction stock index return in the Stock Exchange of Thailand (SET). The monthly data is collected from January 2004 to November 2016, covering 155 observations. We employ a copula based SUR regression as a tool for this study. Ten copula functions are considered in this regression and the best copula function is selected based on Akaike’s Information Criterion (AIC) and Bayesian Information Criterion (BIC). The estimated results show that Gumbel 270 copula is the most appropriate function for being the linkage between the marginal distributions of residuals of financials sector and property & construction sector equations. In addition, the marginal distribution is also tested, and the result shows that normal distribution is the best fit for the marginal distribution for both financials and property & construction equations. Our results suggest that the exchange rate can exert significant impact on both sectors. The dependency parameter also suggests that dependency between financials sector and property & construction sector is negative, and very low dependency, meaning when the impact of macroeconomic variables in one of these two sectors, it just has a little effect to another one sector.

For smart electricity meter with multiple performance characteristics (PCs) with coupling relationships because of amounts of components experiencing multiple deteriorating processes, we develop a multivariate degradation modeling method via vine copulas to estimate the reliability of products with multiple PCs reflecting degradation states. In the multivariate model, pair‐copula construction and vine graphical representation are used to describe the mutual relationship among PCs to overcome the lack of multivariate copula in high‐dimensional cases. Each PC model of smart electricity meter is built by using drift Brownian motion to describe degradation processes of each PC on the basis of degradation mechanism analysis, and parameters are estimated by using likelihood estimation method. The Pearson correlation coefficient, Kendall's τ and product information are used to analyze correlation among those PCs. Furthermore, on the basis of conditional probability theory, the vine graph is used to construct a multivariate copula which can be decomposed into pair copulas. Akaike information criterion principle is utilized to choose the forms of pair‐copula functions in the correlation model. Finally, the reliability joint distribution of all PCs of smart electricity meter is obtained with combining all PCs' marginal distributions and copula functions. Copyright © 2016 John Wiley & Sons, Ltd.