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.     

The Kumaraswamy skew-t distribution and its related properties

Skew normal distribution is an alternative distribution to the normal distribution to accommodate asymmetry. Since then extensive studies have been done on applying Azzalini’s skewness mechanism to other well-known distributions, such as skew-t distribution, which is more flexible and can better accommodate long tailed data than the skew normal one. The Kumaraswamy generalized distribution (Kw ? F) is another new class of distribution which is capable of fitting skewed data that can not be fitted well by existing distributions. Such a distribution has been widely studied and various versions of generalization of this distribution family have been introduced. In this article, we introduce a new generalization of the skew-t distribution based on the Kumaraswamy generalized distribution. The new class of distribution, which we call the Kumaraswamy skew-t (KwST) has the ability of fitting skewed, long, and heavy-tailed data and is more flexible than the skew-t distribution as it contains the skew-t distribution as a special case. Related properties of this distribution family such as mathematical properties, moments, and order statistics are discussed. The proposed distribution is applied to a real dataset to illustrate the estimation procedure.     

Bayesian estimation and prediction for Weibull model with progressive censoring

This article presents the statistical inferences on Weibull parameters with the data that are progressively type II censored. The maximum likelihood estimators are derived. For incorporation of previous information with current data, the Bayesian approach is considered. We obtain the Bayes estimators under squared error loss with a bivariate prior distribution, and derive the credible intervals for the parameters of Weibull distribution. Also, the Bayes prediction intervals for future observations are obtained in the one- and two-sample cases. The method is shown to be practical, although a computer program is required for its implementation. A numerical example is presented for illustration and some simulation study are performed.     

Spectral Estimation of Covolatility from Noisy Observations Using Local Weights

We propose localized spectral estimators for the quadratic covariation and the spot covolatility of diffusion processes, which are observed discretely with additive observation noise. The appropriate estimation for time‐varying volatilities is based on an asymptotic equivalence of the underlying statistical model to a white‐noise model with correlation and volatility processes being constant over small time intervals. The asymptotic equivalence of the continuous‐time and discrete‐time experiments is proved by a construction with linear interpolation in one direction and local means for the other. The new estimator outperforms earlier non‐parametric methods in the literature for the considered model. We investigate its finite sample size characteristics in simulations and draw a comparison between various proposed methods.     

Shared frailty model based on reversed hazard rate for left censored data

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in the individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin or family data), the shared frailty models were suggested. In this article, we introduce the shared gamma frailty models with the reversed hazard rate. We develop the Bayesian estimation procedure using the Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. We apply the model to a real life bivariate survival dataset.