Nonparametric hazard rate estimation by orthogonal wavelet methods

The ability to accurately predict natural backgrounds over a wide range of potential image acquisition conditions is key to wide-area change detection. The lack of robustness in pure image-based techniques has led to the development of site modeling and model-supported exploitation approaches. Instead of using multiple images to build a site model, we consider the possibility of using them directly to estimate the background of a new image for change detection. Our method uses a nonlinear mean-squared estimation technique to compute image backgrounds from multiple reference images. In comparing linear and nonlinear estimators, we find that although the performance of both improves as the number of reference images increases, the nonlinear estimator yields significantly better background estimates and is much less sensitive to registration errors between images than the linear estimator. We show that as the number of references increases, the pixels within a region become physically significant; i.e., they correspond to parts of the scene with similar physical properties. Synthetic aperture radar (SAR) background modeling and change detection results including a crosssensor change detection example in which SAR reference images are used to predict the background of an electro-optical (EO) image are presented. Significant performance gains over linear change detection (up to two orders of magnitude reduction in the false alarm rate) are demonstrated in each case over a 60-deg range of SAR sensor aspect angles.

Nonparametric conditional hazard rate estimation: A local linear approach

Abstract. Hazard rate estimation is an alternative to density estimation for positive variables that is of interest when variables are times to event. In particular, it is here shown that hazard rate estimation is useful for seismic hazard assessment. This paper suggests a simple, but flexible, Bayesian method for non‐parametric hazard rate estimation, based on building the prior hazard rate as the convolution mixture of a Gaussian kernel with an exponential jump‐size compound Poisson process. Conditions are given for a compound Poisson process prior to be well‐defined and to select smooth hazard rates, an elicitation procedure is devised to assign a constant prior expected hazard rate while controlling prior variability, and a Markov chain Monte Carlo approximation of the posterior distribution is obtained. Finally, the suggested method is validated in a simulation study, and some Italian seismic event data are analysed.

Nonlinear wavelet-based estimation to spectral density for stationary non-Gaussian linear processes

We consider the nonparametric regression model with long memory data that are not necessarily Gaussian and provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet-based mean regression function estimators. We show this MISE expansion, when the underlying mean regression function is only piecewise smooth, is the same as analogous expansion for the kernel estimators. However, for the kernel estimators, this MISE expansion generally fails if an additional smoothness assumption is absent.

In this paper, we provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet estimator of survival density for a censorship model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary and α‐mixing sequence. This asymptotic MISE expansion, when the density is only piecewise smooth, is same. However, for the kernel estimators, the MISE expansion fails if the additional smoothness assumption is absent. Also, we establish the asymptotic normality of the nonlinear wavelet estimator. Copyright © 2011 John Wiley & Sons, Ltd.

In a distributed estimation system, the local sensor estimates are transferred to fusion center and fused to be an optimal estimation according to some criterion. The well-known best linear unbiased estimation (BLUE) fusion optimally combines the local estimates by minimising the covariance matrix of estimation errors. However, the BLUE fusion need know the cross-covariances of sensor estimation errors exactly, which are not attainable in most of practical applications. In this article, the random matrix is employed to describe the uncertainty of the estimation error covariance among the sensors. By minimising the estimation error covariance only for the most favorable realisations of the random matrix, we model it as an optimisation problem with chance constraint. With appropriate relaxation method, a robust linear unbiased estimation fusion is proposed analytically. Furthermore, an upper bound on the relaxation gap is proposed. Finally, we demonstrate through some numerical simulations that the presented estimation fusion always has smaller absolute estimation error than that of the covariance intersection filter.

This paper considers nonparametric estimations of a density function in a two-class mixture model. A linear wavelet estimator and an adaptive wavelet estimator are constructed. Upper bound estimations over $ L^{p}\; (1\leq p < +\infty) $ risk of those wavelet estimators are proved in Besov spaces. When $ \tilde{p}\geq p\geq1 $, the convergence rate of adaptive wavelet estimator is the same as the linear estimator up to a $ \ln n $ factor. The adaptive wavelet estimator can get better than the linear estimator in the case of $ 1\leq \tilde{p} < p $. Finally, some numerical experiments are presented to validate the theoretical results.

This paper proposes the nonparametric kernel method for the hazard rate (HR) function estimation in the context of positively skewed data. The class of generalized Birnbaum-Saunders (GBS) kernels is considered because of its several interesting properties and flexibility. Some asymptotic properties, such as bias, variance and mean integrated squared error (MISE) are established for the proposed estimator. In addition, we demonstrate that, the GBS-HR estimator is strongly consistent and asymptotically normal. The choice of bandwidth is also investigated by rule of thumb and unbiased cross validation approaches. Finally, performances of the HR estimator based on GBS kernels and comparison of the two bandwidth selection methods are illustrated by a simulation study and real applications.

We study kernel density estimator from the ranked set samples (RSS). In the kernel density estimator, the selection of the bandwidth gives strong influence on the resulting estimate. In this article, we consider several different choices of the bandwidth and compare their asymptotic mean integrated square errors (MISE). We also propose a plug-in estimator of the bandwidth to minimize the asymptotic MISE. We numerically compare the MISE of the proposed kernel estimator (having the plug-in bandwidth estimator) to its simple random sampling counterpart. We further propose two estimators for a symmetric distribution, and show that they outperform in MISE all other estimators not considering symmetry. We finally apply the methods in this article to analyzing the tree height data from Platt et al. (1988) and Chen et al. (2003).

Token-Based Adaptive Time-Series Prediction by Ensembling Linear and Non-linear Estimators: A Machine Learning Approach for Predictive Analytics on big Stock Data

This paper proposes a novel feature ranking method, DensityRank, based on kernel estimation on the feature spaces to improve the classification performance. As the availability of raw data in many of today's applications continues to grow at an explosive rate, it is critical to assess the learning capabilities of different features and select the important subset of features to improve learning accuracy as well as reduce computational cost. In our approach, kernel methods are used to estimate the probability density function for each feature across different class labels. Discrepancy analysis based on the mean integrated square error (MISE) between pairs of such density estimations is used to provide the ranking values. Then, the ranked subspace method is adopted to select subsets of important features that are used to develop the learning models. Comparative study of this method with those of traditional ranking methods related to Fisher's discrimination ratio and information gain theory, as well as the random subspace algorithm and the bootstrap aggregating (bagging), are presented in this paper. Simulation results on various real-world data sets illustrate the effectiveness of the proposed method.

Accurate estimates of mortality at advanced ages are essential to improving forecasts of mortality and the population size of the oldest old age group. However, estimation of hazard rates at extremely old ages poses serious challenges to researchers: (1) The observed mortality deceleration may be at least partially an artifact of mixing different birth cohorts with different mortality (heterogeneity effect); (2) standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high at old ages and (3) ages of very old people may be exaggerated. One way of obtaining estimates of mortality at extreme ages is to pool together international records of persons surviving to extreme ages with subsequent efforts of strict age validation. This approach helps researchers to resolve the third of the above-mentioned problems but does not resolve the first two problems because of inevitable data heterogeneity when data for people belonging to different birth cohorts and countries are pooled together. In this paper we propose an alternative approach, which gives an opportunity to resolve the first two problems by compiling data for more homogeneous single-year birth cohorts with hazard rates measured at narrow (monthly) age intervals. Possible ways of resolving the third problem of hazard rate estimation are elaborated. This approach is based on data from the Social Security Administration Death Master File (DMF). Some birth cohorts covered by DMF could be studied by the method of extinct generations. Availability of month of birth and month of death information provides a unique opportunity to obtain hazard rate estimates for every month of age. Study of several single-year extinct birth cohorts shows that mortality trajectory at advanced ages follows the Gompertz law up to the ages 102–105 years without a noticeable deceleration. Earlier reports of mortality deceleration (deviation of mortality from the Gompertz law) at ages below 100 appear to be artifacts of mixing together several birth cohorts with different mortality levels and using cross-sectional instead of cohort data. Age exaggeration and crude assumptions applied to mortality estimates at advanced ages may also contribute to mortality underestimation at very advanced ages.

Repeated time-to-event (RTTE) models are the preferred method to characterize the repeated occurrence of clinical events. Commonly used diagnostics for parametric RTTE models require representative simulations, which may be difficult to generate in situations with dose titration or informative dropout. Here, we present a novel simulation-free diagnostic tool for parametric RTTE models; the kernel-based visual hazard comparison (kbVHC). The kbVHC aims to evaluate whether the mean predicted hazard rate of a parametric RTTE model is an adequate approximation of the true hazard rate. Because the true hazard rate cannot be directly observed, the predicted hazard is compared to a non-parametric kernel estimator of the hazard rate. With the degree of smoothing of the kernel estimator being determined by its bandwidth, the local kernel bandwidth is set to the lowest value that results in a bootstrap coefficient of variation (CV) of the hazard rate that is equal to or lower than a user-defined target value (CVtarget). The kbVHC was evaluated in simulated scenarios with different number of subjects, hazard rates, CVtarget values, and hazard models (Weibull, Gompertz, and circadian-varying hazard). The kbVHC was able to distinguish between Weibull and Gompertz hazard models, even when the hazard rate was relatively low (< 2 events per subject). Additionally, it was more sensitive than the Kaplan-Meier VPC to detect circadian variation of the hazard rate. An additional useful feature of the kernel estimator is that it can be generated prior to model development to explore the shape of the hazard rate function.

This study explores the Average Treatment Effects (ATE) estimator proposed by Terza (2009)’s Nonlinear Full Endogenous Treatment (NFES) model, where count dependent and binary treatment variables are present. Asymptotic distribution of ATE estimators based on NFES model is provided to show that nonlinear estimators have additional terms in asymptotic variance of which magnitudes depend on population coefficient. Due to their presence, the asymptotic variance of nonlinear estimators can be either larger or smaller than the linear counterparts depending on the values of coefficients. It turns out that the nonlinear ATE estimators are more efficient than linear estimators when the ATE conditional on covariates has small variance. An application to Botswana fertility is given.