Empirical Uncertain Bayes Methods in Area‐level Models

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area‐level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus, it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one‐point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area‐level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson‐gamma and the binomial‐beta, and the normal‐normal (Fay–Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well‐known expectation‐maximization algorithm and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second‐order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.     

Tests for Structural Changes in Time Series of Counts

We propose methods for detecting structural changes in time series with discrete‐valued observations. The detector statistics come in familiar L2‐type formulations incorporating the empirical probability generating function. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. For both models, we study mainly structural changes due to a change in distribution, but we also comment for the classical problem of parameter change. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is also included along with a real data example.     

Monotonic change point estimation of generalized linear model-based regression profiles

In this article, a maximum likelihood estimator is derived in the generalized linear model-based regression profiles under monotonic change in Phase II. The performance of the proposed estimator is comprehensively investigated through some special cases, and compared with estimators under step change and drift. The results show that the proposed estimator has better performance in small and medium shifts under different increasing changes. Finally, the applicability of the proposed estimator is illustrated using a real case.     

Robust modal estimation and variable selection for single-index varying-coefficient models

In this article, we present a new efficient iteration estimation approach based on local modal regression for single-index varying-coefficient models. The resulted estimators are shown to be robust with regardless of outliers and error distributions. The asymptotic properties of the estimators are established under some regularity conditions and a practical modified EM algorithm is proposed for the new method. Moreover, to achieve sparse estimator when there exists irrelevant variables in the index parameters, a variable selection procedure based on SCAD penalty is developed to select significant parametric covariates and the well-known oracle properties are also derived. Finally, some numerical examples with various distributed errors and a real data analysis are conducted to illustrate the validity and feasibility of our proposed method.     

A note on efficient simulation of multidimensional spatial autoregressive processes

In applications of spatial statistics, it is necessary to compute the product of some matrix W of spatial weights and a vector y of observations. The weighting matrix often needs to be adapted to the specific problems, such that the computation of Wy cannot necessarily be done with available R-packages. Hence, this article suggests one possibility treating such issues. The proposed technique avoids the computation of the matrix product by calculating each entry of Wy separately. Initially, a specific spatial autoregressive process is introduced. The performance of the proposed program is briefly compared to a basic program using the matrix multiplication.     

Variable selection for semiparametric proportional hazards model under progressive Type-II censoring

Variable selection is an effective methodology for dealing with models with numerous covariates. We consider the methods of variable selection for semiparametric Cox proportional hazards model under the progressive Type-II censoring scheme. The Cox proportional hazards model is used to model the influence coefficients of the environmental covariates. By applying Breslow’s “least information” idea, we obtain a profile likelihood function to estimate the coefficients. Lasso-type penalized profile likelihood estimation as well as stepwise variable selection method are explored as means to find the important covariates. Numerical simulations are conducted and Veteran’s Administration Lung Cancer data are exploited to evaluate the performance of the proposed method.     

Variable selection in partially linear additive models for modal regression

Based on B-spline basis functions and smoothly clipped absolute deviation (SCAD) penalty, we present a new estimation and variable selection procedure based on modal regression for partially linear additive models. The outstanding merit of the new method is that it is robust against outliers or heavy-tail error distributions and performs no worse than the least-square-based estimation for normal error case. The main difference is that the standard quadratic loss is replaced by a kernel function depending on a bandwidth that can be automatically selected based on the observed data. With appropriate selection of the regularization parameters, the new method possesses the consistency in variable selection and oracle property in estimation. Finally, both simulation study and real data analysis are performed to examine the performance of our approach.     

Bayesian parameter inference for stochastic SIR epidemic model with hyperbolic diffusion

This paper is concerned with the Bayesian estimation parameters of the stochastic SIR (Susceptible-Infective-Removed) epidemic model from the trajectory data. Specifically, the data from the count of both infectives and susceptibles is assumed to be available on some time grid as the epidemic progresses. The diffusion approximation of the appropriate jump process is then used to estimate missing data between every pair of observation times. If the time step of imputations is small enough, we derive the posterior distributions of the infection and recovery rates using the Milstein scheme. The paper also presents Markov-chain Monte Carlo (MCMC) simulation that demonstrates that the method provides accurate estimates, as illustrated by the synthetic data from SIR epidemic model and the real data.     

Convergence rate of wavelet density estimator with data missing randomly when covariables are present

In this article, we study global L₂ error of non linear wavelet estimator of density in the Besov space B^s_pq for missing data model when covariables are present and prove that the estimator can achieve the optimal rate of convergence, which is similar to the result studied by Donoho et al. (1996) Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Stat. 24:508–539.[Crossref], [Web of Science ®] , [Google Scholar] in complete independent data case with term-by-term thresholding of the empirical wavelet coefficients. Finite-sample behavior of the proposed estimator is explored via simulations.     

t-Type corrected-loss estimation for error-in-variable model

In this article, we consider a linear model in which the covariates are measured with errors. We propose a t-type corrected-loss estimation of the covariate effect, when the measurement error follows the Laplace distribution. The proposed estimator is asymptotically normal. In practical studies, some outliers that diminish the robustness of the estimation occur. Simulation studies show that the estimators are resistant to vertical outliers and an application of 6-minute walk test is presented to show that the proposed method performs well.