An evaluation of ridge estimator in linear mixed models: an example from kidney failure data

This paper is concerned with the ridge estimation of fixed and random effects in the context of Henderson's mixed model equations in the linear mixed model. For this purpose, a penalized likelihood method is proposed. A linear combination of ridge estimator for fixed and random effects is compared to a linear combination of best linear unbiased estimator for fixed and random effects under the mean-square error (MSE) matrix criterion. Additionally, for choosing the biasing parameter, a method of MSE under the ridge estimator is given. A real data analysis is provided to illustrate the theoretical results and a simulation study is conducted to characterize the performance of ridge and best linear unbiased estimators approach in the linear mixed model.     

The distribution of a linear predictor after model selection: conditional finite-sample distributions and asymptotic approximations

This paper studies the distribution of a linear predictor that is constructed after a data-driven model selection step in a linear regression model. The finite-sample cumulative distribution function (cdf) of the linear predictor is derived and a detailed analysis of the effects of the model selection step is given. Moreover, a simple approximation to the (complicated) finite-sample cdf is proposed. This approximation facilitates the study of the large-sample limit behavior of the linear predictor and its cdf, in the fixed-parameter case and under local alternatives. The focus of this paper is on the conditional distribution of a linear predictor, conditional on the event that a fixed (possibly incorrect) model has been selected. The unconditional distribution of a linear predictor is studied in the companion paper Leeb (The distribution of a linear predictor after model selection: unconditional finite-sample distributions and asymptotic approximations, Technical Report, Department of Statistics, University of Vienna, 2002).     

Asymptotic normality of the lengths of a class of nonparametric confidence intervals for a regression parameter

In the linear regression model, the asymptotic distributions of certain functions of confidence bounds of a class of confidence intervals for the regression parameter arc investigated. The class of confidence intervals we consider in this paper are based on the usual linear rank statistics (signed as well as unsigned). Under suitable assumptions, if the confidence intervals are based on the signed linear rank statistics, it is established that the lengths, properly normalized, of the confidence intervals converge in law to the standard normal distributions; if the confidence intervals arc based on the unsigned linear rank statistics, it is then proved that a linear function of the confidence bounds converges in law to a normal distribution.     

Linearly admissible estimators of stochastic regression coefficient under balanced loss function

The admissibility of linear estimators in a linear model with stochastic regression coefficient is investigated under a balanced loss function. The sufficient and necessary conditions for linear estimators to be admissible in classes of homogeneous and non-homogeneous linear estimators are obtained, respectively.     

Phase I Analysis of Multiple Linear Regression Profiles

This article considers the Phase I analysis of data when the quality of a process or product is characterized by a multiple linear regression model. This is usually referred to as the analysis of linear profiles in the statistical quality control literature. The literature includes several approaches for the analysis of simple linear regression profiles. Little work, however, has been done in the analysis of multiple linear regression profiles. This article proposes a new approach for the analysis of Phase I multiple linear regression profiles. Using this approach, regardless of the number of explanatory variables used to describe it, the profile response is monitored using only three parameters, an intercept, a slope, and a variance. Using simulation, the performance of the proposed method is compared to that of the existing methods for monitoring multiple linear profiles data in terms of the probability of a signal. The advantage of the proposed method over the existing methods is greatly improved detection of changes in the process parameters of linear profiles with high-dimensional space. The article also proposes useful diagnostic aids based on F-statistics to help in identifying the source of profile variation and the locations of out-of-control samples. Finally, the use of multiple linear profile methods is illustrated by a data set from a calibration application at National Aeronautics and Space Administration (NASA) Langley Research Center.     

Admissibility of Linear Predictors in the Superpopulation Model with Respect to Inequality Constraints under Matrix Loss Function

Admissibility of linear predictors for the linear quantity Qy is investigated in a superpopulation model with respect to some inequality constraints. Necessary and sufficient conditions for a linear predictor to be admissible in the class of homogeneous linear predictors and the class of inhomogeneous linear predictors are obtained, respectively, under matrix loss function.     

A class of partially linear transformation models for recurrent gap times

In this article, we propose a general class of partially linear transformation models for recurrent gap time data, which extends the linear transformation models by incorporating non linear covariate effects and includes the partially linear proportional hazards and the partially linear proportional odds models as special cases. Both global and local estimating equations are developed to estimate the parametric and non parametric covariate effects, and the asymptotic properties of the resulting estimators are established. The finite-sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a clinic study on chronic granulomatous disease is provided.     

Linearly admissible estimators of the common mean parameter under balanced loss function

Admissibility of linear estimators of the common mean parameter is investigated in the context of a linear model under balanced loss function. Sufficient and necessary conditions for linear estimators to be admissible in classes of homogeneous and non homogeneous linear estimators are obtained, respectively.     

Admissibilities of matrix linear estimators multivariate linear models

This article respectively provides sufficient conditions and necessary conditions of matrix linear estimators of an estimable parameter matrix linear function in multivariate linear models with and without the assumption that the underlying distribution is a normal one with completely unknown covariance matrix. In the latter model, a necessary and sufficient condition is given for matrix linear estimators to be admissible in the space of all matrix linear estimators under each of three different kinds of quadratic matrix loss functions, respectively. In the former model, a sufficient condition is first provided for matrix linear estimators to be admissible in the space of all matrix estimators having finite risks under each of the same loss functions, respectively. Furthermore in the former model, one of these sufficient conditions, correspondingly under one of the loss functions, is also proved to be necessary, if additional conditions are assumed.     

Automatic structure discovery for varying-coefficient partially linear models

Varying-coefficient partially linear models provide a useful tools for modeling of covariate effects on the response variable in regression. One key question in varying-coefficient partially linear models is the choice of model structure, that is, how to decide which covariates have linear effect and which have non linear effect. In this article, we propose a profile method for identifying the covariates with linear effect or non linear effect. Our proposed method is a penalized regression approach based on group minimax concave penalty. Under suitable conditions, we show that the proposed method can correctly determine which covariates have a linear effect and which do not with high probability. The convergence rate of the linear estimator is established as well as the asymptotical normality. The performance of the proposed method is evaluated through a simulation study which supports our theoretical results.     

Entropic representation and estimation of diversity indices

This paper serves a twofold purpose. First, a unified perspective on diversity indices is introduced based on an entropic basis. It is shown that the class of all linear combinations of the entropic basis, referred to as the class of linear diversity indices, covers a wide range of diversity indices used in the literature. Second, a class of estimators for linear diversity indices is proposed and it is shown that these estimators have rapidly decaying biases and asymptotic normality.     

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.     

A test of the mean square error criterion for linear admissible estimators

A test for choosing between a linear admissible estimator and the least squares estimator (LSE) is developed. A characterization of linear admissible estimators useful for comparing estimators is presented and necessary and sufficient conditions for superiority of a linear admissible estimator over the LS estimetor is derived for the test. The test is based on the MSE matrix superiority, but also new resl?!ts concerning covariance matrix comparisons of linear estimators are derived. Further,shown that the test of Toro - Vizcarrondo and Wailace applies iioi only the restricted least squares estimators but also to certain estimators outside this class.     

Admissible Linear Predictors in the Superpopulation Model with Respect to Inequality Constraints

Admissibility of linear predictors of the linear quantity Qy is investigated under a general linear regression superpopulation model with some inequality constraints. The relation between admissible homogeneous and inhomogeneous linear predictors is characterized. Further, necessary and sufficient conditions for a linear predictor to be admissible in two cases of inequality constraints in the classes of homogeneous and inhomogeneous linear predictors are given, respectively.     

The geometry of case deletion and the assessment of influence in nonlinear regression

Numerous influence measures are available for use in linear regression. By contrast, very little has been done for nonlinear models. A notable exception is Chapter 4 of Cook and Weisberg (1982). The extension of measures based on case deletion from the linear to the nonlinear model usually involve linear approximation. In this paper, the geometry of case deletion is studied with a view to assessing the adequacy of linear approximation in the construction of influence measures for nonlinear regression. Of particular interest is the adequacy of the one-step estimator for the jack-knifed pseudoestimates of the unknown parameter vector.     

Comparison of estimators of the location parameter using pitman's closeness criterion

Necessary and sufficient conditions for a linear estimator to dominate another linear estimator of a location parameter under the Pitman's criterion of comparison are discussed. Consequently it is demonstrated that a linear biased estimator can not dominate a linear unbiased estimator under Pitman's criterion and that the sample mean is the Closest Linear Unbiased Estimator (CLUE). It is also shown that the ridge regression estimator with a known biasing constant can not dominate the ordinary least squares estimator. If an estimator δdominates an estimator δin the average loss sense then sufficient conditions are obtained under which δis also preferred over δunder Pitman's criterion. Further we obtain sufficient conditions under which preference under the Pitman's criterion will lead to preference under the mean squared error sense.     

A Simulation Study of the Performance of Five Discriminant Analysis Methods for Mixtures of Continuous and Binary Variables

The present study investigates the performance of fice discrimination methods for data consisting of a mixture of continuous and binary variables. The methods are Fisher’s linear discrimination, logistic discrimination, quadratic discrimination, a kernal model and an independence model. Six-dimensional data, consisting of three binary and three continuous variables, are simulated according to a location model. The results show an almost identical performance for Fisher’s linear discrimination and logistic discrimination. Only in situations with independently distributed variables the independence model does have a reasonable discriminatory ability for the dimensionality considered. If the log likelihood ratio is non-linear ratio is non-linear with respect to its continuous and binary part, the quadratic discrimination method is substantial better than linear and logistic discrimination, followed by the kernel method. A very good performance is obtained when in every situation the better one of linear and quardratic discrimination is used.     

The compound class of linear failure rate-power series distributions: Model,properties, and applications

In this article, a new class of distributions is introduced, which generalizes the linear failure rate distribution and is obtained by compounding this distribution and power series class of distributions. This new class of distributions is called the linear failure rate-power series distributions and contains some new distributions such as linear failure rate-geometric, linear failure rate-Poisson, linear failure rate-logarithmic, linear failure rate-binomial distributions, and Rayleigh-power series class of distributions. Some former works such as exponential-power series class of distributions, exponential-geometric, exponential-Poisson, and exponential-logarithmic distributions are special cases of the new proposed model. The ability of the linear failure rate-power series class of distributions is in covering five possible hazard rate function, that is, increasing, decreasing, upside-down bathtub (unimodal), bathtub and increasing-decreasing-increasing shaped. Several properties of this class of distributions such as moments, maximum likelihood estimation procedure via an EM-algorithm and inference for a large sample, are discussed in this article. In order to show the flexibility and potentiality, the fitted results of the new class of distributions and some of its submodels are compared using two real datasets.     

Numerical method for means of linear Hawkes processes

Abstract

Linear Hawkes processes are widely used in many fields and means are the basic and critical information of them. However, there is little research on linear Hawkes processes’ means. In this paper, we present a numerical method based on the Laplace transform and inverse Laplace transform for means of linear Hawkes processes. The advantage of this method is that whatever the kernel function is, we can always obtain the numerical solutions of means for a linear Hawkes process. In addition, this numerical method provides the basic information of linear Hawkes processes by means. As an application, the numerical method is applied in a WeChat network model.