We propose, for multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize efficiency of the pseudo-likelihood estimator and adaptivity of the model with respect to the unknown marginal distributions. For correlation matrices structured according to a factor model, the pseudo-likelihood estimator turns out to be semiparametrically efficient. On the other hand, for Toeplitz correlation matrices, the asymptotic relative efficiency of the pseudo-likelihood estimator can be as low as 20%. These findings are confirmed by Monte Carlo simulations. We indicate how our results can be extended to joint regression models.
We propose a new class of multivariate survival models based on archimedean copulas with margins modeled by the Yang and Prentice (YP) model. The Ali-Mikhail-Haq (AMH), Clayton, Frank, Gumbel-Hougaard (GH), and Joe copulas are employed to accommodate the dependency among marginal distributions. Baseline distributions are modeled semiparametrically by the piecewise exponential (PE) distribution and the Bernstein polynomials. The new class of models possesses some attractive features: i) the ability to take into account survival data with crossing survival curves; ii) the inclusion of the well-k…
BMW Group（宝马集团）；Technical University of Dortmund（多特蒙德工业大学）
In this work we present a rigorous application of the Expectation Maximization algorithm to determine the marginal distributions and the dependence structure in a Gaussian copula model with missing data. We further show how to circumvent a priori assumptions on the marginals with semiparametric modelling. The joint distribution learned through this algorithm is considerably closer to the underlying distribution than existing methods.
Penalized EM algorithm and copula skeptic graphical models for inferring networks for mixed variables
本文研究了高斯copula图模型中稀疏精度矩阵的估计方法，对连续变量、二元变量、计数变量和离散序变量的网络进行重构。我们提出两种方法：$\ell_1$用Monte Carlo期望最大化算法（copula EM glasso）惩罚扩展秩似然，用copula Gaussian图模型的成对copula估计惩罚copula怀疑。所提出的方法有助于推断网络所产生的非正态和混合变量。我们通过对乳腺癌基因组和临床数据以及玉米遗传学数据的模拟研究和分析来证明我们的方法的性能。
Semiparametric Copula Quantile Regression for Complete or Censored Data
当面对多元协变量时，一般的半参数回归技术可以提出灵活的模型，这些模型不受维数灾难的影响。在这项工作中，半参数Copula为基础的估计条件分位数完全或右删失数据进行了研究。在精神上，该方法扩展了Noh et al.（2013）和Noh et al.（2015）的近期工作，因为主要思想在于根据多变量copula和边缘分布适当定义分位数回归。事先估计后者和简单的插件导致一个容易实现的估计表示，无论上下文中有或没有审查，作为一个加权分位数的观察到的响应变量。此外，与文献中的初始建议相反，研究了多变量Copula密度的半参数估计方案，其动机是纯参数方法的可能缺点，并由回归上下文驱动。由此得到的分位数回归估计具有在分位数水平上自动单调的有价值的性质，并且在经典的正则条件下得到了完全数据和删失数据的渐近正态性。最后，数值例子以及实际数据的应用，以说明所提出的方法的有效性和有限样本性能。
This research is motivated by discovering and underpinning genetic causes for the progression of a bilateral eye disease, Age-related Macular Degeneration (AMD), of which the primary outcomes, progression times to late-AMD, are bivariate and interval-censored due to intermittent assessment times. We propose a novel class of copula-based semiparametric transformation models for bivariate data under general interval censoring, which includes the case 1 interval censoring (current status data) and case 2 interval censoring. Specifically, the joint likelihood is modeled through a two-parameter Arc…
The purpose of this paper is to introduce two semiparametric methods for the estimation of copula parameter. These methods are based on minimum Alpha-Divergence between a non-parametric estimation of copula density using local likelihood probit transformation method and a true copula density function. A Monte Carlo study is performed to measure the performance of these methods based on Hellinger distance and Neyman divergence as special cases of Alpha-Divergence. Simulation results are compared to the Maximum Pseudo-Likelihood (MPL) estimation as a conventional estimation method in well-known …
In this paper, a Bayesian semiparametric copula approach is used to model the underlying multivariate distribution $F_{true}$. First, the Dirichlet process is constructed on the unknown marginal distributions of $F_{true}$. Then a Gaussian copula model is utilized to capture the dependence structure of $F_{true}$. As a result, a Bayesian multivariate normality test is developed by combining the relative belief ratio and the Energy distance. Several interesting theoretical results of the approach are derived. Finally, through several simulated examples and a real data set, the proposed approach…
A typical situation in competing risks analysis is that the researcher is only interested in a subset of risks. This paper considers a depending competing risks model with the distribution of one risk being a parametric or semi-parametric model, while the model for the other risks being unknown. Identifiability is shown for popular classes of parametric models and the semiparametric proportional hazards model. The identifiability of the parametric models does not require a covariate, while the semiparametric model requires at least one. Estimation approaches are suggested which are shown to be…
Semiparametric Copula Estimation for Spatially Correlated Multivariate Mixed Outcomes: Analyzing Visual Sightings of Fin Whales from a Line Transect Survey
Tokyo University of Science（东京理科大学）；Meisei University（明海大学）；Keio University（庆应义塾大学）；Institute of Marine Research（海洋研究所）；RIKEN Center for Brain Science（理化学研究所脑科学中心）
Extreme-value copulas arise as the limiting dependence structure of component-wise maxima. Defined in terms of a functional parameter, they are one of the most widespread copula families due to their flexibility and ability to capture asymmetry. Despite this, meeting the complex analytical properties of this parameter in an unconstrained setting remains a challenge, restricting most uses to models with very few parameters or nonparametric models. Focusing on the bivariate case, we propose a novel semiparametric approach. Our procedure relies on a series of transformations, including Williamson…
Canonical correlation analysis investigates linear relationships between two sets of variables, but often works poorly on modern data sets due to high-dimensionality and mixed data types such as continuous, binary and zero-inflated. To overcome these challenges, we propose a semiparametric approach for sparse canonical correlation analysis based on Gaussian copula. Our main contribution is a truncated latent Gaussian copula model for data with excess zeros, which allows us to derive a rank-based estimator of the latent correlation matrix for mixed variable types without the estimation of margi…
Area Under the Curve (AUC) is arguably the most popular measure of classification accuracy. We use a semiparametric framework to introduce a latent scale-invariant $R^2$, a novel measure of variation explained for an observed binary outcome and an observed continuous predictor, and then directly link the latent $R^2$ to AUC. This enables a mutually consistent simultaneous use of AUC as a measure of classification accuracy and the latent $R^2$ as a scale-invariant measure of explained variation. Specifically, we employ Semiparametric Gaussian Copula (SGC) to model a joint dependence between obs…
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…
We propose a new semiparametric approach for modelling nonlinear univariate diffusions, where the observed process is a nonparametric transformation of an underlying parametric diffusion (UPD). This modelling strategy yields a general class of semiparametric Markov diffusion models with parametric dynamic copulas and nonparametric marginal distributions. We provide primitive conditions for the identification of the UPD parameters together with the unknown transformations from discrete samples. Likelihood-based estimators of both parametric and nonparametric components are developed and we anal…
In this work we propose a semiparametric bivariate copula whose density is defined by a piecewise constant function on disjoint squares. We obtain the maximum likelihood estimators of model parameters and prove that they reduce to the sample copula under specific conditions. We further propose to carry out a full Bayesian analysis of the model and introduce a spatial dependent prior distribution for the model parameters. This prior allows the parameters to borrow strength across neighbouring regions to produce smooth posterior estimates. To characterise the posterior distribution, via the full…
Sequencing-based technologies provide an abundance of high-dimensional biological datasets with skewed and zero-inflated measurements. Classification of such data with linear discriminant analysis leads to poor performance due to the violation of the Gaussian distribution assumption. To address this limitation, we propose a new semiparametric discriminant analysis framework based on the truncated latent Gaussian copula model that accommodates both skewness and zero inflation. By applying sparsity regularization, we demonstrate that the proposed method leads to the consistent estimation of clas…
近年来，大量的工作致力于推广线性判别分析，以克服其在高维分类方面的不足（Witten & Tibshirani 2011，Cai & Liu 2011，Mai et al. 2012，Fan et al. 2012）。在本文中，我们开发了高维半参数稀疏判别分析（HD-SeSDA），一般正常理论判别分析在两个方面：它放松了高斯假设，可以处理非多项式（NP）维分类问题。如果底层贝叶斯规则是稀疏的，HD-SeSDA可以估计贝叶斯规则和选择的真实特征，同时以压倒性的概率，只要对数的维数增长慢于样本量的立方根。仿真和实际例子被用来证明有限样本性能的HD-SeSDA。在理论的核心是一个新的指数浓度界的半参数高斯copula，这是独立的兴趣。
Missing data imputation forms the first critical step of many data analysis pipelines. The challenge is greatest for mixed data sets, including real, Boolean, and ordinal data, where standard techniques for imputation fail basic sanity checks: for example, the imputed values may not follow the same distributions as the data. This paper proposes a new semiparametric algorithm to impute missing values, with no tuning parameters. The algorithm models mixed data as a Gaussian copula. This model can fit arbitrary marginals for continuous variables and can handle ordinal variables with many levels, …
Maximum pseudo-likelihood (MPL) is a semiparametric estimation method often used to obtain the dependence parameters in copula models from data. It has been shown that despite being consistent, and in some cases efficient, MPL estimation can overestimate the level of dependence especially for small weakly dependent samples. We show that the MPL method uses the expected value of order statistics and we propose to use instead the median or the mode of the same order statistics. In a simulation study we compare the finite-sample performance of the proposed estimators with that of the original MPL…
Conditional copula graphic estimator for semi-competing risks data
Children’s Hospital Research Institute of Manitoba, University of Manitoba（曼尼托巴儿童医院研究所，曼尼托巴大学）；Department of Mathematics and Statistics, University of Guelph（圭尔夫大学数学与统计系）；Hospital for Sick Children（多伦多多伦多儿童医院）；Department of Statistics, University of Manitoba（曼尼托巴大学统计系）
Causal mediation in cluster-randomized trials with multiple mediators: spillover-aware decomposition, identification, and semiparametric efficient inference
This paper is concerned with modeling the dependence structure of two (or more) time-series in the presence of a (possible multivariate) covariate which may include past values of the time series. We assume that the covariate influences only the conditional mean and the conditional variance of each of the time series but the distribution of the standardized innovations is not influenced by the covariate and is stable in time. The joint distribution of the time series is then determined by the conditional means, the conditional variances and the marginal distributions of the innovations, which …
