Analyzing Binomial Data in a Split-Plot Design: Classical Approach or Modern Techniques?

Modeling data that are non-normally distributed with random effects is the major challenge in analyzing binomial data in split-plot designs. Seven methods for analyzing such data using mixed, generalized linear, or generalized linear mixed models are compared for the size and power of the tests. This study shows that analyzing random effects properly is more important than adjusting the analysis for non-normality. Methods based on mixed and generalized linear mixed models hold Type I error rates better than generalized linear models. Mixed model methods tend to have higher power than generalized linear mixed models when the sample size is small.     

Best linear unbiased estimators of parameters of a simple linear regression model based on ordered ranked set samples

As an alternative to an estimation based on a simple random sample (BLUE-SRS) for the simple linear regression model, Moussa-Hamouda and Leone [E. Moussa-Hamouda and F.C. Leone, The o-blue estimators for complete and censored samples in linear regression, Technometrics, 16 (3) (1974), pp. 441–446.] discussed the best linear unbiased estimators based on order statistics (BLUE-OS), and showed that BLUE-OS is more efficient than BLUE-SRS for normal data. Using the ranked set sampling, Barreto and Barnett [M.C.M. Barreto and V. Barnett, Best linear unbiased estimators for the simple linear regression model using ranked set sampling. Environ. Ecoll. Stat. 6 (1999), pp. 119–133.] derived the best linear unbiased estimators (BLUE-RSS) for simple linear regression model and showed that BLUE-RSS is more efficient for the estimation of the regression parameters (intercept and slope) than BLUE-SRS for normal data, but not so for the estimation of the residual standard deviation in the case of small sample size. As an alternative to RSS, this paper considers the best linear unbiased estimators based on order statistics from a ranked set sample (BLUE-ORSS) and shows that BLUE-ORSS is uniformly more efficient than BLUE-RSS and BLUE-OS for normal data.     

Extended linear empirical bayes estimation

In this paper, the linear empirical Bayes estimation method, which is based on approximation of the Bayes estimator by a linear function, is generalized to an extended linear empirical Bayes estimation technique which represents the Bayes estimator by a series of algebraic polynomials. The extended linear empirical Bayes estimators are elaborated in the case of a location or a scale parameter. The theory is illustrated by examples of its application to the normal distribution with a location parameter and the gamma distribution with a scale parameter. The linear and the extended linear empirical Bayes estimators are constructed in these two cases and, then, studied numerically via Monte Carlo simulations. The simulations show that the extended linear empirical Bayes estimators have better convergence rates than the traditional linear empirical Bayes estimators.     

Confidence Intervals Based on Local Linear Smoother

Point-wise confidence intervals for a non-parametric regression function in conjunction with the popular local linear smoother are considered. The confidence intervals are based on the asymptotic normal distribution of the local linear smoother. Their coverage accuracy is evaluated by developing Edgeworth expansion for the coverage probability. It is found that the coverage error near the boundary of the support of the regression function is of a larger order than that in the interior, which implies that the local linear smoother is not adaptive to the boundary in terms of coverage. This is quite unexpected as the local linear smoother is adaptive to the boundary in terms of the mean squared error.     

Comparison of Two Estimation Methods of Missing Values Using Pitman-Closeness Criterion

In this article, a proof is given that the linear interpolator is Pitman-closer than the linear predictor with respect to a missing value of a stationary first-order autoregressive process.     

Recovery of inter-block information: extensions in a two variance component model

For a two variance component mixed linear model, it is shown that under suitable conditions there exists a nonlinear unbiased estimator that is better than a best linear unbiased estimator defined with respect to a given singular covariance matrix. It is also shown how this result applies to improving on intra-block estimators and on estimators like the unweighted means estimator in a random one-way model.     

Component reduction in linear discriminant analysis

The linear discriminant function is transformed into a linear combination of independent random variables. It is shown that reducing dimensionality using the smallest distance criterion results in smaller increase in the error rate than using the smallest variance criterion. Three error rates are used to prove this.     

A bayesian wls approach to generalized linear models

We use a Bayesian approach to fitting a linear regression model to transformations of the natural parameter for the exponential class of distributions. The usual Bayesian approach is to assume that a linear model exactly describes the relationship among the natural parameters. We assume only that a linear model is approximately in force. We approximate the theta-links by using a linear model obtained by minimizing the posterior expectation of a loss function.While some posterior results can be obtained analytically considerable generality follows from an exact Monte Carlo method for obtaining random samples of parameter values or functions of parameter values from their respective posterior distributions. The approach that is presented is justified for small samples, requires only one-dimensional numerical integrations, and allows for the use of regression matrices with less than full column rank. Two numerical examples are provided.     

Bayesian regression with multivariate linear splines

We present a Bayesian analysis of a piecewise linear model constructed by using basis functions which generalizes the univariate linear spline to higher dimensions. Prior distributions are adopted on both the number and the locations of the splines, which leads to a model averaging approach to prediction with predictive distributions that take into account model uncertainty. Conditioning on the data produces a Bayes local linear model with distributions on both predictions and local linear parameters. The method is spatially adaptive and covariate selection is achieved by using splines of lower dimension than the data.     

Data Driven Rank Test for Two-Sample Problem

Traditional linear rank tests are known to possess low power for large spectrum of alternatives. In this paper we introduce a new rank test possessing a considerably larger range of sensitivity than linear rank tests. The new test statistic is a sum of squares of some linear rank statistics while the number of summands is chosen via a data-based selection rule. Simulations show that the new test possesses high and stable power in situations when linear rank tests completely break down, while simultaneously it has almost the same power under alternatives which can be detected by standard linear rank tests. Our approach is illustrated by some practical examples. Theoretical support is given by deriving asymptotic null distribution of the test statistic and proving consistency of the new test under essentially any alternative.     

A New Regression Model: Modal Linear Regression

The mode of a distribution provides an important summary of data and is often estimated on the basis of some non‐parametric kernel density estimator. This article develops a new data analysis tool called modal linear regression in order to explore high‐dimensional data. Modal linear regression models the conditional mode of a response Y given a set of predictors x as a linear function of x . Modal linear regression differs from standard linear regression in that standard linear regression models the conditional mean (as opposed to mode) of Y as a linear function of x . We propose an expectation–maximization algorithm in order to estimate the regression coefficients of modal linear regression. We also provide asymptotic properties for the proposed estimator without the symmetric assumption of the error density. Our empirical studies with simulated data and real data demonstrate that the proposed modal regression gives shorter predictive intervals than mean linear regression, median linear regression and MM‐estimators.     

Outlier Resistant Estimation in Difference-Based Semiparametric Partially Linear Models

Partially linear models are extensions of linear models that include a nonparametric function of some covariate allowing an adequate and more flexible handling of explanatory variables than in linear models. The difference-based estimation in partially linear models is an approach designed to estimate parametric component by using the ordinary least squares estimator after removing the nonparametric component from the model by differencing. However, it is known that least squares estimates do not provide useful information for the majority of data when the error distribution is not normal, particularly when the errors are heavy-tailed and when outliers are present in the dataset. This paper aims to find an outlier-resistant fit that represents the information in the majority of the data by robustly estimating the parametric and the nonparametric components of the partially linear model. Simulations and a real data example are used to illustrate the feasibility of the proposed methodology and to compare it with the classical difference-based estimator when outliers exist.     

Characterization of Admissible Linear Estimators in Multivariate Linear Model with Respect to Inequality Constraints under Matrix Loss Function

In this article, we study the characterization of admissible linear estimators in a multivariate linear model with inequality constraint, under a matrix loss function. In the homogeneous class, we present several equivalent, necessary and sufficient conditions for a linear estimator of estimable functions to be admissible. In the inhomogeneous class, we find that the necessary and sufficient conditions depend on the rank of the matrix in the constraint. When the rank is greater than one, the necessary and sufficient conditions are obtained. When the rank is equal to one, we have necessary conditions and sufficient conditions separately. We also obtain the necessary and sufficient conditions for a linear estimator of inestimable function to be admissible in both classes.     

A comparison of estimators for the mean in the inverse gaussian distribution with a known coefficient of variation

In this paper, we discuss an estimation problem of the mean in the inverse Gaussian distribution with a known coefficient of variation. Two types of linear estimators for the mean, the linear minimum variance unbiased estimator and the linear minimum mean squared error estimator, are constructed by using the squared error loss function and their properties are examined. It is observed that, for small samples the performance of the proposed estimators is better than that of the maximum likelihood estimator, when the coefficient of variation is large.     

Modified maximum likelihood estimation for rayleigh distribution

For defining a Modified Maximum Likelihood Estimate of the scale parameter of Rayleigh distribution, a hyperbolic approximation is used instead of linear approximation for a function which appears in the Maximum Likelihood equation. This estimate is shown to perform better, in the sense of accuracy and simplicity of calculation, than the one based on linear approximation for the same function. Also the estimate of the scale parameter obtained is shown to be asymptotically unbiased. Numerical computation for random samples of different sizes from Rayleigh distribution, using type I1 censoring is done and is shown to be better than that obtained by Lee et al. (1980)     

The Effect of Leap Years and Seasonal Trends on the Birthday Problem

Most data have a space and time label associated with them; data that are close together are usually more correlated than those that are far apart. Prediction (or forecasting) of a process at a particular label where there is no datum, from observed nearby data, is the subject of this article. One approach, known as geostatistics, is featured, from which linear methods of spatial prediction (kriging) will be considered. Brief reference is made to other linear/nonlinear, stochastic/deterministic predictors. The (linear) geostatistical method is applied to piezometric-head data around a potential nuclear-waste repository site.     

Partially Collapsed Gibbs Sampling for Linear Mixed-effects Models

This article presents a novel Bayesian analysis for linear mixed-effects models. The analysis is based on the method of partial collapsing that allows some components to be partially collapsed out of a model. The resulting partially collapsed Gibbs (PCG) sampler constructed to fit linear mixed-effects models is expected to exhibit much better convergence properties than the corresponding Gibbs sampler. In order to construct the PCG sampler without complicating component updates, we consider the reparameterization of model components by expressing a between-group variance in terms of a within-group variance in a linear mixed-effects model. The proposed method of partial collapsing with reparameterization is applied to the Merton’s jump diffusion model as well as general linear mixed-effects models with proper prior distributions and illustrated using simulated data and longitudinal data on sleep deprivation.     

Testing random effects in linear mixed models: another look at the F‐test (with discussion)

This article re‐examines the F‐test based on linear combinations of the responses, or FLC test, for testing random effects in linear mixed models. In current statistical practice, the FLC test is underused and we argue that it should be reconsidered as a valuable method for use with linear mixed models. We present a new, more general derivation of the FLC test which applies to a broad class of linear mixed models where the random effects can be correlated. We highlight three advantages of the FLC test that are often overlooked in modern applications of linear mixed models, namely its computation speed, its generality, and its exactness as a test. Empirical studies provide new insight into the finite sample performance of the FLC test, identifying cases where it is competitive or even outperforms modern methods in terms of power, as well as settings in which it performs worse than simulation‐based methods for testing random effects. In all circumstances, the FLC test is faster to compute.     

Maximum Likelihood Estimation for Multinomial‐Poisson Models: A Generalization of Birch's Numerical Invariance Results

Abstract. This study gives a generalization of Birch's log‐linear model numerical invariance result. The generalization is given in the form of a sufficient condition for numerical invariance that is simple to verify in practice and is applicable for a much broader class of models than log‐linear models. Unlike Birch's log‐linear result, the generalization herein does not rely on any relationship between sufficient statistics and maximum likelihood estimates. Indeed the generalization does not rely on the existence of a reduced set of sufficient statistics. Instead, the concept of homogeneity takes centre stage. Several examples illustrate the utility of non‐log‐linear models, the invariance (and non‐invariance) of fitted values, and the invariance (and non‐invariance) of certain approximating distributions.     

Bias Corrected Maximum Likelihood Estimator Under the Generalized Linear Model for a Binary Variable

Under the generalized linear models for a binary variable, an approximate bias of the maximum likelihood estimator of the coefficient, that is a special case of linear parameter in Cordeiro and McCullagh (1991), is derived without a calculation of the third-order derivative of the log likelihood function. Using the obtained approximate bias of the maximum likelihood estimator, a bias-corrected maximum likelihood estimator is defined. Through a simulation study, we show that the bias-corrected maximum likelihood estimator and its variance estimator have a better performance than the maximum likelihood estimator and its variance estimator.