Deconvolution methods for non-parametric inference in two-level mixed models

Summary.  We develop a general non-parametric approach to the analysis of clustered data via random effects. Assuming only that the link function is known, the regression functions and the distributions of both cluster means and observation errors are treated non-parametrically. Our argument proceeds by viewing the observation error at the cluster mean level as though it were a measurement error in an errors-in-variables problem, and using a deconvolution argument to access the distribution of the cluster mean. A Fourier deconvolution approach could be used if the distribution of the error-in-variables were known. In practice it is unknown, of course, but it can be estimated from repeated measurements, and in this way deconvolution can be achieved in an approximate sense. This argument might be interpreted as implying that large numbers of replicates are necessary for each cluster mean distribution, but that is not so; we avoid this requirement by incorporating statistical smoothing over values of nearby explanatory variables. Empirical rules are developed for the choice of smoothing parameter. Numerical simulations, and an application to real data, demonstrate small sample performance for this package of methodology. We also develop theory establishing statistical consistency.     

Hierarchical bayes quality measurement plan

Quality Measurement Plan (QMP) as developed by Hoadley (1981) is a statistical method for analyzing discrete quality audit data which consist of the expected number of defects given the standard quality. The QMP is based on an empirical Bayes (EB) model of the audit sampling process. Despite its wide publicity, Hoadley's method has often been described as heuristic. In this paper we offer an hierarchical Bayes (HB) alternative to Hoadley's EB model, and overcome much of the criticism against this model. Gibbs sampling is used to implement the HB model proposed in this paper. Also, the convergence of the Gibbs sampler is monitored via the algorithm of Gelman and Rubin (1992).     

Variable selection and estimation for high-dimensional spatial autoregressive models

Spatial regression models are important tools for many scientific disciplines including economics, business, and social science. In this article, we investigate postmodel selection estimators that apply least squares estimation to the model selected by penalized estimation in high-dimensional regression models with spatial autoregressive errors. We show that by separating the model selection and estimation process, the postmodel selection estimator performs at least as well as the simultaneous variable selection and estimation method in terms of the rate of convergence. Moreover, under perfect model selection, the ℓ₂ rate of convergence is the oracle rate of $\sqrt{s/n}$, compared with the convergence rate of $\text{◂√▸}\sqrt{s\log p/n}$ in the general case. Here, n is the sample size and p, s are the model dimension and number of significant covariates, respectively. We further provide the convergence rate of the estimation error in the form of $\sup$ norm, and ideally the rate can reach as fast as $\text{◂√▸}\sqrt{\log s/n}$.     

Asymptotic design-cum-model based estimation of variances of estimated linear regression coefficients in survey sampling with unequal probabilities

Postulating a super-population linear regression model for a variable of interest on an auxiliary variable we consider design-based estimation of regression coefficients on drawing a sample with unequal probabilities from a survey population. Asymptotic design-cum-model based variance estimation procedures are proposed.     

The use of the weighted likelihood in the natural exponential families with quadratic variance

The authors propose a weighted likelihood concept for the estimation of parameters in natural exponential families with quadratic variance. They apply the results to both simulated and real data.     

Accurate Confidence Interval Estimation of Small Area Parameters Under the Fay–Herriot Model

Small area estimation has long been a popular and important research topic due to its growing demand in public and private sectors. We consider here the basic area level model, popularly known as the Fay–Herriot model. Although much of current research is predominantly focused on second order unbiased estimation of mean squared prediction errors, we concentrate on developing confidence intervals (CIs) for the small area means that are second order correct. The corrected CI can be readily implemented, because it only requires quantities that are already estimated as part of the mean squared error estimation. We extend the approach to a CI for the difference of two small area means. The findings are illustrated with a simulation study.     

Variance estimation in model assisted survey sampling

Two versions of Yates-Grundy type variance estimators are usually employed for large samples when estimating a survey population total by a generalized regression (Greg, in brief) predictor motivated by consideration of a linear regression model. Their two alternative modifications are developed so that the limiting values of the design expectations of the model expectations of variance estimators 'match' respectively the (I) model expectations of the Taylor approximation of the design variance of the Greg predictor and the (II) limiting value of the design expectation of the model expectation of the squared difference between the Greg predictor and the population total. The exercise is extended to yield modifications needed when randomized response (RR) is only available rather than direct response (DR) when one encounters sensitive issues demanding protection of privacy. A comparative study based on simulation is presented for illustration..     

Functional Mixed Effects Model for Small Area Estimation

Functional data analysis has become an important area of research because of its ability of handling high‐dimensional and complex data structures. However, the development is limited in the context of linear mixed effect models and, in particular, for small area estimation. The linear mixed effect models are the backbone of small area estimation. In this article, we consider area‐level data and fit a varying coefficient linear mixed effect model where the varying coefficients are semiparametrically modelled via B‐splines. We propose a method of estimating the fixed effect parameters and consider prediction of random effects that can be implemented using a standard software. For measuring prediction uncertainties, we derive an analytical expression for the mean squared errors and propose a method of estimating the mean squared errors. The procedure is illustrated via a real data example, and operating characteristics of the method are judged using finite sample simulation studies.     

Bias correction in logistic regression with missing categorical covariates

Logistic regression plays an important role in many fields. In practice, we often encounter missing covariates in different applied sectors, particularly in biomedical sciences. Ibrahim (1990) proposed a method to handle missing covariates in generalized linear model (GLM) setup. It is well known that logistic regression estimates using small or medium sized missing data are biased. Considering the missing data that are missing at random, in this paper we have reduced the bias by two methods; first we have derived a closed form bias expression using Cox and Snell (1968), and second we have used likelihood based modification similar to Firth (1993). Here we have analytically shown that the Firth type likelihood modification in Ibrahim led to the second order bias reduction. The proposed methods are simple to apply on an existing method, need no analytical work, with the exception of a little change in the optimization function. We have carried out extensive simulation studies comparing the methods, and our simulation results are also supported by a real world data.     

Small-area estimation based on survey data from a left-censored Fay-Herriot model

This paper develops methodology for survey estimation and small-area prediction using Fay-Herriot (1979) models in which the responses are left-censored. Parameter and small-area estimators are derived both by censored-data likelihoods and by an estimating-equation approach which adjusts a Fay-Herriot analysis restricted to the uncensored observations. Formulas for variances of estimators and mean-squared errors of small-area predictions are provided and supported by a simulation study. The methodology is applied to provide diagnostics for the left-censored Fay-Herriot model which are illustrated in the context of the Census Bureau's ongoing Small-Area Income and Poverty Estimation (SAIPE) project.