An Approach to Constructing Explicit Estimators in Nonlinear Regression

We address the problem of estimating and removing localized image blur, as it for example arises from moving objects in a scene, or when the depth of field is insufficient to sharply render all objects of interest. Unlike the case of camera shake, such blur changes abruptly at the object boundaries. To cope with this, we propose an automated sharp image recovery method that simultaneously determines blurred regions and estimates their responsible blur kernels. To address a wide range of different scenarios, our model is not restricted to a discrete set of candidate blurs, but allows for arbitrary, non-parametric blur kernels. Moreover, our approach does not require specialized hardware, an alpha matte, or user annotation of the blurred region. Unlike previous methods, we show that localized blur estimation can be accomplished by incorporating a pixel-wise latent variable to indicate the active blur kernel. Furthermore, we generalize the marginal likelihood technique of blind deblurring to the case of localized blur. Specifically, we integrate out the latent image derivatives to permit marginal density estimates of both blur kernels and their regions of influence. We obtain sharp images in applications to both object motion blur and defocus blur removal. Quantitative results on two novel datasets as well as qualitative results comparing to a range of specialized methods demonstrate the versatility and effectiveness of our non-parametric approach.

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

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

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.

Parametric factor copula models typically work well in modeling multivariate dependencies due to their flexibility and ability to capture complex dependency structures. However, accurately estimating the linking copulas within these models remains challenging, especially when working with high-dimensional data. This paper proposes a novel approach for estimating linking copulas based on a non-parametric kernel estimator. Unlike conventional parametric methods, our approach utilizes the flexibility of kernel density estimation to capture the underlying dependencies more accurately, particularly in scenarios where the underlying copula structure is complex or unknown. We show that the proposed estimator is consistent under mild conditions and demonstrate its effectiveness through extensive simulation studies. Our findings suggest that the proposed approach offers a promising avenue for modeling multivariate dependencies, particularly in applications requiring robust and efficient estimation of copula-based models.

Testing Over-Identifying Restrictions Without Consistent Estimation of the Asymptotic Covariance Matrix

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.

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.

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.

The role of price risk in sow farrowings is investigated by using bivariate ARCH-M and GARCH-M models and a nonparametric kernel estimator. To account for the relevant time horizon of irreversible supply decisions, predictions for mean price and conditional price variance are iterated forward. The empirical results vary markedly in terms of their implications for risk response in hog supply decisions, with the ARCH-M and GARCH-M models suggesting a small and negative risk effect. Estimates of the marginal risk premium also indicate moderate and variable departures from marginal cost pricing in sow farrowing supply decisions.

In this multicenter study, a new semiautomatic method for segmenting the cervical cord from C2 to C5 was used to investigate the correlation between cord atrophy and clinical disability in a large sample of patients with multiple sclerosis (MS). T2 and 3-dimensional T1-weighted cervical cord scans and dual-echo brain scans were acquired from 143 healthy controls, 22 patients with clinically isolated syndromes (CIS), 101 patients with relapsing-remitting MS (RRMS), 79 patients with secondary progressive MS (SPMS), 58 patients with benign MS (BMS), and 75 patients with primary progressive MS (PPMS) in 3 European centers. Normalized cervical cord cross-sectional area (CSAn) was measured by an active surface cord model. Between-group comparisons were performed using linear mixed-effect models. A nonparametric kernel estimator was used to obtain smoothed plots of CSA along the cervical cord. Cord CSAn was significantly lower in PPMS vs healthy controls, BMS vs RRMS, SPMS vs BMS, and RRMS. From C2 to C5, a net separation and definition of the plots of patients with BMS, PPMS, and SPMS was seen with respect to those of the other study groups. CSAn was correlated with Expanded Disability Status Scale (r = -0.49, p < 0.0001), with a differential effect among disease clinical phenotypes: no association in either CIS or in BMS; association in RRMS (r = -0.30, p = 0.001), SPMS (r = -0.34, p = 0.001), and PPMS (r = -0.27, p = 0.01). Cervical cord atrophy provides a relevant and useful marker for the characterization of clinical heterogeneity of patients with MS. The stability of this measure among different centers supports its use as potential outcome measure to monitor disease progression in multicenter trials.