Self-exciting threshold models for time series of counts with a finite range

In this article, an integer-valued self-exciting threshold model with a finite range based on the binomial INARCH(1) model is proposed. Important stochastic properties are derived, and approaches for parameter estimation are discussed. A real-data example about the regional spread of public drunkenness in Pittsburgh demonstrates the applicability of the new model in comparison to existing models. Feasible modifications of the model are presented, which are designed to handle special features such as zero-inflation.     

New innovational outlier unit root test with a break at an unknown time

The Perron test which is based on a Dickey–Fuller test regression is a commonly employed approach to test for a unit root in the presence of a structural break of unknown timing. In the case of an innovational outlier (IO), the Perron test tends to exhibit spurious rejections in finite samples when the break occurs under the null hypothesis. In the present paper, a new Perron-type IO unit root test is developed. It is shown in Monte Carlo experiments that the new test does not over-reject the null hypothesis. Even for the case of a level and slope break for trending data, the empirical size is near its nominal level. The test distribution equals the case of a known break date. Furthermore, the test is able to identify the true break date very accurately even for small breaks. As an application serves the Nelson–Plosser data set.     

Another look at the robustness of colton's model for comparing two medical treatments

The robustness of an extended version of Colton's decision theoretic model is considered. The extended version includes the losses due to the patients who are not entered in the experiment, but require treatment while the experiment is in progress. Among the topics considered are the effects of risk of using a sample size considerably less than the optimum, use of an incorrect patient horizon, application of a modified loss function, and use of a two point prior distribution. It is shown that the investigated model is robust with respect to all these changes with the exception of the use of the modified prior density.     

Extremum sieve estimation in k-out-of-n systems

This article considers the non parametric estimation of absolutely continuous distribution functions of independent lifetimes of non identical components in k-out-of-n systems, 2 ? k ? n, from the observed “autopsy” data. In economics, ascending “button” or “clock” auctions with n heterogeneous bidders with independent private values present 2-out-of-n systems. Classical competing risks models are examples of n-out-of-n systems. Under weak conditions on the underlying distributions, the estimation problem is shown to be well-posed and the suggested extremum sieve estimator is proven to be consistent. This article considers the sieve spaces of Bernstein polynomials which allow to easily implement constraints on the monotonicity of estimated distribution functions.     

Modelling of connected processes

The problem of comparing, contrasting and combining information from different sets of data is an enduring one in many practical applications of statistics. A specific problem of combining information from different sources arose in integrating information from three different sets of data generated by three different sampling campaigns at the input stage as well as at the output stage of a grey-water treatment process. For each stage, a common process trend function needs to be estimated to describe the input and output material process behaviours. Once the common input and output process models are established, it is required to estimate the efficiency of the grey-water treatment method. A synthesized tool for modelling different sets of process data is created by assembling and organizing a number of existing techniques: (i) a mixed model of fixed and random effects, extended to allow for a nonlinear fixed effect, (ii) variogram modelling, a geostatistical technique, (iii) a weighted least squares regression embedded in an iterative maximum-likelihood technique to handle linear/nonlinear fixed and random effects and (iv) a formulation of a transfer-function model for the input and output processes together with a corresponding nonlinear maximum-likelihood method for estimation of a transfer function. The synthesized tool is demonstrated, in a new case study, to contrast and combine information from connected process models and to determine the change in one quality characteristic, namely pH, of the input and output materials of a grey-water filtering process.     

Numerical evaluation of tests based on different heteroskedasticity-consistent covariance matrix estimators

This article considers the issue of performing tests in linear heteroskedastic models when the test statistic employs a consistent variance estimator. Several different estimators are considered, namely: HC0, HC1, HC2, HC3, and their bias-adjusted versions. The numerical evaluation is performed using numerical integration methods; the Imhof algorithm is used to that end. The results show that bias-adjustment of variance estimators used to construct test statistics delivers more reliable tests when they are performed for the HC0 and HC1 estimators, but the same does not hold for the HC3 estimator. Overall, the most reliable test is the HC3-based one.     

Bayesian penalized B-spline estimation approach for epidemic models

Ordinary differential equations (ODEs) are normally used to model dynamic processes in applied sciences such as biology, engineering, physics, and many other areas. In these models, the parameters are usually unknown, and thus they are often specified artificially or empirically. Alternatively, a feasible method is to estimate the parameters based on observed data. In this study, we propose a Bayesian penalized B-spline approach to estimate the parameters and initial values for ODEs used in epidemiology. We evaluated the efficiency of the proposed method based on simulations using the Markov chain Monte Carlo algorithm for the Kermack–McKendrick model. The proposed approach is also illustrated based on a real application to the transmission dynamics of hepatitis C virus in mainland China.     

Bayesian optimal sequential design for nonparametric regression via inhomogeneous evolutionary MCMC

We develop a novel computational methodology for Bayesian optimal sequential design for nonparametric regression. This computational methodology, that we call inhomogeneous evolutionary Markov chain Monte Carlo, combines ideas of simulated annealing, genetic or evolutionary algorithms, and Markov chain Monte Carlo. Our framework allows optimality criteria with general utility functions and general classes of priors for the underlying regression function. We illustrate the usefulness of our novel methodology with applications to experimental design for nonparametric function estimation using Gaussian process priors and free-knot cubic splines priors.     

Maximum likelihood estimation for generalized conditionally autoregressive models of spatial data

Conditionally autoregressive (CAR) models are often used to analyze a spatial process observed over a lattice or a set of irregular regions. The neighborhoods within a CAR model are generally formed deterministically using the inter-distances or boundaries between the regions. To accommodate directional and inherent anisotropy variation, a new class of spatial models is proposed that adaptively determines neighbors based on a bivariate kernel using the distances and angles between the centroid of the regions. The newly proposed model generalizes the usual CAR model in a sense of accounting for adaptively determined weights. Maximum likelihood estimators are derived and simulation studies are presented for the sampling properties of the estimates on the new model, which is compared to the CAR model. Finally the method is illustrated using a data set on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

Composite Estimation for Single‐Index Models with Responses Subject to Detection Limits

We propose a semiparametric estimator for single‐index models with censored responses due to detection limits. In the presence of left censoring, the mean function cannot be identified without any parametric distributional assumptions, but the quantile function is still identifiable at upper quantile levels. To avoid parametric distributional assumption, we propose to fit censored quantile regression and combine information across quantile levels to estimate the unknown smooth link function and the index parameter. Under some regularity conditions, we show that the estimated link function achieves the non‐parametric optimal convergence rate, and the estimated index parameter is asymptotically normal. The simulation study shows that the proposed estimator is competitive with the omniscient least squares estimator based on the latent uncensored responses for data with normal errors but much more efficient for heavy‐tailed data under light and moderate censoring. The practical value of the proposed method is demonstrated through the analysis of a human immunodeficiency virus antibody data set.