Non-parametric estimation techniques of factor copula model using proxies

Factor copula models are useful for modeling high-dimensional data sets that typically exhibit complex dependence structures. However, the practical merits of factor copula models have not been fully explored from an estimation point of view. This thesis proposes a simple, yet useful framework for estimating factor copula models, which allows for a richer and more precise understanding of the underlying dependence in the data sets.

Between the initial colonization of North America and the European settlement period, Indigenous American land use practices shaped North American landscapes and ecosystems, but a critical question is the extent of these impacts on the land, and how these influenced the distributions of the flora and fauna. The present study addresses this question by estimating the spatial correlation between continental-scale records of fossil pollen and archaeological radiocarbon data, and provides a detailed analysis of the spatiotemporal relationship between palaeo-populations and ten important North American pollen taxa. Maps of Indigenous American population density, based on the Canadian Archaeological Radiocarbon Database, are compared to maps of plant abundance as estimated by pollen records from the Neotoma Paleoecology Database, using nonparametric kernel estimators and cross-correlation techniques. Periods of high spatial cross-correlation (either positive or negative) between population density and plant abundance were identified, but these associations were intermittent and did not increase towards the present. In many cases, high values of population density corresponded with high values of a particular taxon in one region, but simultaneously corresponded with low values in other regions, lessening the overall correlation between the two fields. This analysis suggests that human impacts were not significant enough to be identified at a continental scale, either due to low population numbers or land use, implying significant impacts of ancient human activities on the vegetation were regional rather than continental.

We propose a new monitoring procedure based on moving sums (MOSUM) for detecting single or multiple structural breaks in factor copula models. The test compares parameter estimates from a rolling window to those from a historical data set and analyzes the behavior under the null hypothesis of no parameter change. The case of multiple breaks is also treated. In the model, the joint copula is given by the copula of random variables which arise from a factor model. This is particularly useful for analyzing high dimensional data. Parameters are estimated with the simulated method of moments (SMM). We analyze the behavior of the monitoring procedure in Monte Carlo simulations and a real data application. We consider an online procedure for predicting the day-ahead Value-at-risk based on the suggested monitoring procedure.

Dependent Defaults and Losses with Factor Copula Models

Computing marginal probabilities is an important and fundamental issue in Bayesian inference. We present a simple method which arises from a likelihood identity for computation. The likelihood identity, called Candidate's formula, sets the marginal probability as a ratio of the prior likelihood to the posterior density. Based on Markov chain Monte Carlo output simulated from the posterior distribution, a nonparametric kernel estimate is used to estimate the posterior density contained in that ratio. This derived nonparametric Candidate's estimate requires only one evaluation of the posterior density estimate at a point. The optimal point for such evaluation can be chosen to minimize the expected mean square relative error. The results show that the best point is not necessarily the posterior mode, but rather a point compromising between high density and low Hessian. For high dimensional problems, we introduce a variance reduction approach to ease the tension caused by data sparseness. A simulation study is presented.

In this article, a semiparametric time‐varying nonlinear vector autoregressive (NVAR) model is proposed to model nonlinear vector time series data. We consider a combination of parametric and nonparametric estimation approaches to estimate the NVAR function for both independent and dependent errors. We use the multivariate Taylor series expansion of the link function up to the second order which has a parametric framework as a representation of the nonlinear vector regression function. After the unknown parameters are estimated by the maximum likelihood estimation procedure, the obtained NVAR function is adjusted by a nonparametric diagonal matrix, where the proposed adjusted matrix is estimated by the nonparametric kernel estimator. The asymptotic consistency properties of the proposed estimators are established. Simulation studies are conducted to evaluate the performance of the proposed semiparametric method. A real data example on short‐run interest rates and long‐run interest rates of United States Treasury securities is analyzed to demonstrate the application of the proposed approach.The Canadian Journal of Statistics47: 668–687; 2019 © 2019 Statistical Society of Canada

This article develops a simultaneous model and moment selection procedure for factor copula models. Since the density of the factor copula is generally not known in closed form, widely used likelihood or moment-based model selection criteria cannot be directly applied on factor copulas. The new approach is inspired by the methods for generalized methods of moments proposed by Andrews (1999) and Andrews and Lu (2001). The consistency of the procedure is proved and Monte Carlo simulations show its good performance in finite samples in different scenarios of sample sizes and dimensions. The impact of the choice of moments in selected regions of the support on model selection and value-at-risk prediction is further examined by simulation and an application to a portfolio consisting of ten stocks in the Deutscher Aktienindex (DAX30) index.

Parametric and nonparametric income distribution estimators in CGE micro-simulation modeling

Adaptive background updating is an important step in motion segmentations of video sequences. However, the irregular distributions of background pixel values make the background modeling complicated. In this work, a method for background pixel classification based on the mean shift algorithm is proposed, which can classify the background pixels as single mode or multiple mode pixels so that different updating methods can be applied, for example, the single-mode pixel values can be updated with a simple and fast method such as IIR filter, while the multi-mode pixel values are modeled by a more complex updating algorithm such as a mixture of K Gaussian distributions or the non-parametric kernel estimation, which is robust to small motions (noisy motions, repetitive motions). Since in most scenes, the number of static background pixels (single-mode pixels) is far more than the number of dynamic background pixels (multi-mode pixels), thus the presented method can help improve the speed of background reconstruction without reducing its precision.

We propose an easy to use derivative-based two-step estimation procedure for semiparametric index models, where the number of indexes is not known a priori. In the first step various functionals involving the derivatives of the unknown function are estimated using nonparametric kernel estimators, in particular the average outer product of the gradient (AOPG). By testing the rank of the AOPG we determine the required number of indexes. Subsequently, we estimate the index parameters in a method of moments framework, with moment conditions constructed using the estimated average derivative functionals. The estimator readily extends to multiple equation models and is shown to be root-N-consistent and asymptotically normal.

The aim of this paper is to present goodness-of-fit tests of a specified density function, based on a quadratic measure of deviation of a nonparametric kernel estimator. The results presented here, considering degenerate U-statistics technics, are generalisations, to the multidimensional case in a context of asymptotic independence, of the corresponding ones obtained by Bickel and Rosenblatt [1973]. A simulation is also proposed.

We construct pointwise confidence intervals for regression functions. The method uses nonparametric kernel estimates and the moment-oriented bootstrap method of Bunke which is a wild bootstrap based on smoothed local estimators of higher order error moments. We show that our bootstrap consistently estimates the distribution of mh(x0) - m(xo). In the present paper we focus on fully data-driven procedures and prove that the confidence intervals give asymptotically correct coverage probabilities.

The structure of forest stands in virgin and managed peatlands: a comparison between Finnish and Russian Karelia

Conditional mode function is often used in various fields such as statistics, data analysis, and machine learning to understand the distribution of data and make decisions based on specific conditions or criteria. In this paper, we propose a non-parametric kernel estimator of the conditional mode function, when the variable of interest is subject to random doubly truncation. We prove the suggested estimator's strong consistency with a rate and state its asymptotic normality under some regularity assumptions. Our results are based on special mathematical techniques such as the iterative non-parametric maximum likelihood estimators (NPMLE) of the distribution function and the Vapnik-Cervonenkis classes for which uniform exponential inequalities are available. This study will be a valuable resource for scholars and practitioners interested in non-parametric kernel estimation methods for doubly truncated data.

We consider the problem of constructing explicit consistent estimators of finite-dimensionalparameters of nonlinear regression models using various nonparametric kernel estimators.