Maximum-likelihood estimation for the removal method

We prove that the profile log-likelihood function for the removal method of estimating population size is unimodal. The result is obtained by a variation-diminishing property of the Laplace transform. An implication of this result is that the likelihood-ratio confidence region for the population size is always an interval. Necessary and sufficient conditions for the existence of a finite maximum-likelihood estimator are presented. We also present evidence that the likelihood-ratio confidence interval for the population size has acceptable small-sample coverage properties.     

Pitman closeness for hierarchical bayes predictors in mixed linear models

The present article considers the Pitman Closeness (PC) criterion of certain hierarchical Bayes (HB) predictors derived under a normal mixed linear models for known ratios of variance components using a uniform prior for the vector of fixed effects and some proper or improper prior on the error variance. For a generalized Euclidean error, simultaneous HB predictors of several linear combinations of vector of effects are shown to be the Pitman-closest in the frequentist sense in the class of equivariant predictors for location group of transformations. The normality assumption can be relaxed to show that these HB predictors are the Pitman-closest for location-scale group of transformations for a wider family of elliptically symmetric distributions. Also for this family, the HB predictors turn out to be Pitman-closest in the class of all linear unbiased predictors (LUPs). All these results are extended for the HB predictor of finite population mean vector in the context of finite population sampling.     

Robust Bayesian prediction under a general linear-exponential posterior risk function and its application in finite population

We consider robust Bayesian prediction of a function of unobserved data based on observed data under an asymmetric loss function. Under a general linear-exponential posterior risk function, the posterior regret gamma-minimax (PRGM), conditional gamma-minimax (CGM), and most stable (MS) predictors are obtained when the prior distribution belongs to a general class of prior distributions. We use this general form to find the PRGM, CGM, and MS predictors of a general linear combination of the finite population values under LINEX loss function on the basis of two classes of priors in a normal model. Also, under the general ε-contamination class of prior distributions, the PRGM predictor of a general linear combination of the finite population values is obtained. Finally, we provide a real-life example to predict a finite population mean and compare the estimated risk and risk bias of the obtained predictors under the LINEX loss function by a simulation study.     

Bayes Prediction for a Heteroscedastic Regression Superpopulation Model Using Balanced Loss Function

We consider Prais–Houthakker heteroscedastic normal regression model having variance of the dependent variable same as square of its expectation. Bayes predictors for the regression coefficient and the mean of a finite population are derived using Zellner's balanced loss function. Bayes predictive expected losses are obtained and compared with those of classical predictors and Bayes predictors under squared error loss function to examine their loss robustness.     

Profiled adaptive Elastic-Net procedure for partially linear models with high-dimensional covariates

We study variable selection for partially linear models when the dimension of covariates diverges with the sample size. We combine the ideas of profiling and adaptive Elastic-Net. The resulting procedure has oracle properties and can handle collinearity well. A by-product is the uniform bound for the absolute difference between the profiled and original predictors. We further examine finite sample performance of the proposed procedure by simulation studies and analysis of a labor-market dataset for an illustration.     

A note on finite population prediction under asymetric loss functions

The purpose of the present note is to derive optimal population total predictors relative to the Linex (Zellner, 1986) loss function under some well known superpopulation models. The risk function and Bayes risk are derived and compared with those of usual predictors. Minimax and and admissibility properties of some of the derived predictors are also investigated.     

An adaptive rule for stopping a discovery process under time censorship

Consider a finite population of large but unknown size of hidden objects. Consider searching for these objects for a period of time, at a certain cost, and receiving a reward depending on the sizes of the objects found. Suppose that the size and discovery time of the objects both have unknown distributions, but the conditional distribution of time given size is exponential with an unknown non-negative and non-decreasing function of the size as failure rate. The goal is to find an optimal way to stop the discovery process. Assuming that the above parameters are known, an optimal stopping time is derived and its asymptotic properties are studied. Then, an adaptive rule based on order restricted estimates of the distributions from truncated data is presented. This adaptive rule is shown to perform nearly as well as the optimal stopping time for large population size.     

Nonparametric Bayes Stochastically Ordered Latent Class Models

Latent class models (LCMs) are used increasingly for addressing a broad variety of problems, including sparse modeling of multivariate and longitudinal data, model-based clustering, and flexible inferences on predictor effects. Typical frequentist LCMs require estimation of a single finite number of classes, which does not increase with the sample size, and have a well-known sensitivity to parametric assumptions on the distributions within a class. Bayesian nonparametric methods have been developed to allow an infinite number of classes in the general population, with the number represented in a sample increasing with sample size. In this article, we propose a new nonparametric Bayes model that allows predictors to flexibly impact the allocation to latent classes, while limiting sensitivity to parametric assumptions by allowing class-specific distributions to be unknown subject to a stochastic ordering constraint. An efficient MCMC algorithm is developed for posterior computation. The methods are validated using simulation studies and applied to the problem of ranking medical procedures in terms of the distribution of patient morbidity.     

Equivariant prediction in finite populations

A theory of equivariant prediction is developed for predicting the population total in finite populations. Minimum risk equivariant predictors (MREP) are derived under the location, scale and locationscale superpopulation models. Under the general linear model, it is shown that the best(linear) unbiased predictor (B(L)UP) is an MREP.     

Partial functional linear regression

It is frequently the case that a response will be related to both a vector of finite length and a function-valued random variable as predictor variables. In this paper, we propose new estimators for the parameters of a partial functional linear model which explores the relationship between a scalar response variable and mixed-type predictors. Asymptotic properties of the proposed estimators are established and finite sample behavior is studied through a small simulation experiment.     

Bayes Predictor of One-Parameter Exponential Family Type Population Mean Under Balanced Loss Function

We obtain a Bayes predictor and a Bayes prediction risk of the mean of a finite population relative to the balanced loss function. The predictive expected losses associated with classical and standard Bayes predictors are derived and compared with that of a Bayes predictor under a balanced loss function. Specific expressions for a regular exponential family distributed superpopulation are presented and illustrated for some well-known superpopulations.     

Finite-population prediction under error-in-variables superpopulation models

The prediction problem in finite populations is considered under error-in-variables super population models. The models considered are the usual regression models involving at most two variables, x and y, where both may be measured with error. Properties of some classical predictors are investigated. A Bayesian approach is proposed.     

Mean estimate in ranked set sampling using a length-biased concomitant variable

In this paper, a ranked set sampling procedure with ranking based on a length-biased concomitant variable is proposed. The estimate for population mean based on this sample is given. It is proved that the estimate based on ranked set samples is asymptotically more efficient than the estimate based on simple random samples. Simulation studies are conducted to present the properties of the proposed estimate for finite sample size. Moreover, the consequence of ignoring length bias is also addressed by simulation studies and the real data analysis.     

Surface Estimation, Variable Selection, and the Nonparametric Oracle Property

Variable selection for multivariate nonparametric regression is an important, yet challenging, problem due, in part, to the infinite dimensionality of the function space. An ideal selection procedure should be automatic, stable, easy to use, and have desirable asymptotic properties. In particular, we define a selection procedure to be nonparametric oracle (np-oracle) if it consistently selects the correct subset of predictors and at the same time estimates the smooth surface at the optimal nonparametric rate, as the sample size goes to infinity. In this paper, we propose a model selection procedure for nonparametric models, and explore the conditions under which the new method enjoys the aforementioned properties. Developed in the framework of smoothing spline ANOVA, our estimator is obtained via solving a regularization problem with a novel adaptive penalty on the sum of functional component norms. Theoretical properties of the new estimator are established. Additionally, numerous simulated and real examples further demonstrate that the new approach substantially outperforms other existing methods in the finite sample setting.     

A note on the determination of sample sizes for hypergeometric distributions

Sample size determination for testing the hypothesis of equality of two proportions against an alternative with specified type I and type II error probabilities is considered for two finite populations. When two finite populations involved are quite different in sizes, the equal size assumption may not be appropriate. In this paper, we impose a balanced sampling condition to determine the necessary samples taken without replacement from the finite populations. It is found that our solution requires smaller samples as compared to those using binomial distributions. Furthermore, our solution is consistent with the sampling with replacement or when population size is large. Finally, three examples are given to show the application of the derived sample size formula.     

Predicting random effects with an expanded finite population mixed model

Prediction of random effects is an important problem with expanding applications. In the simplest context, the problem corresponds to prediction of the latent value (the mean) of a realized cluster selected via two-stage sampling. Recently, Stanek and Singer [Predicting random effects from finite population clustered samples with response error. J. Amer. Statist. Assoc. 99, 119–130] developed best linear unbiased predictors (BLUP) under a finite population mixed model that outperform BLUPs from mixed models and superpopulation models. Their setup, however, does not allow for unequally sized clusters. To overcome this drawback, we consider an expanded finite population mixed model based on a larger set of random variables that span a higher dimensional space than those typically applied to such problems. We show that BLUPs for linear combinations of the realized cluster means derived under such a model have considerably smaller mean squared error (MSE) than those obtained from mixed models, superpopulation models, and finite population mixed models. We motivate our general approach by an example developed for two-stage cluster sampling and show that it faithfully captures the stochastic aspects of sampling in the problem. We also consider simulation studies to illustrate the increased accuracy of the BLUP obtained under the expanded finite population mixed model.     

PREDICTION OF THE FINITE POPULATION DISTRIBUTION FUNCTION UNDER GAUSSIAN SUPERPOPULATION MODELS

This article considers optimal prediction of the finite population distribution function under Gaussian superpopulation models, which allows auxiliary prior information to be incorporated into the estimation process. Large sample approximations for the variance of the optimal predictors are derived in some special important cases. A small scale Monte Carlo study illustrates comparisons between the optimal predictor and some others which are proposed in the literature. The conclusion is that the optimal predictor can be considerably more efficient in situations where the normal superpopulation model is adequate.     

Nonlinear regression analysis for complex surveys 1

In a clustered finite population, it is assumed that a given function depending on an unknown parameter may be adopted to reveal the relationship among the variables of interest. The finite population parameter corresponding to this unknown parameter is defined as a solution of an estimating equation defined by a properly chosen population loss function. An estimation procedure that takes sample weights into account is considered. Use of this function in estimating the population mean per cluster is discussed. Large sample properties of estimators are investigated.     

Asymptotic properties of a generalized regression-type predictor of a finite population variance in probability sampling

A system of predictors for estimating a finite population variance is defined and shown to be asymptotically design-unbiased (ADU) and asymptotically design-consistent (ADC) under probability sampling. An asymptotic mean squared error (MSE) of a generalized regression-type predictor, generated from the system, is obtained. The suggested predictor attains the minimum expected variance of any design-unbiased estimator when the superpopulation model is correct. The generalized regression-type predictor and the predictor suggested by Mukhopadhyay (1990) are compared.