Penalized inverse probability weighted estimators for weighted rank regression with missing covariates

Abstract

In this article, we study the variable selection and estimation for linear regression models with missing covariates. The proposed estimation method is almost as efficient as the popular least-squares-based estimation method for normal random errors and empirically shown to be much more efficient and robust with respect to heavy tailed errors or outliers in the responses and covariates. To achieve sparsity, a variable selection procedure based on SCAD is proposed to conduct estimation and variable selection simultaneously. The procedure is shown to possess the oracle property. To deal with the covariates missing, we consider the inverse probability weighted estimators for the linear model when the selection probability is known or unknown. It is shown that the estimator by using estimated selection probability has a smaller asymptotic variance than that with true selection probability, thus is more efficient. Therefore, the important Horvitz-Thompson property is verified for penalized rank estimator with the covariates missing in the linear model. Some numerical examples are provided to demonstrate the performance of the estimators.     

Zone control charts with estimated parameters for detecting prespecified changes in linear profiles

ABSTRACT

In profile monitoring, control charts are proposed to detect unanticipated changes, and it is usually assumed that the in-control parameters are known. However, due to the characteristics of a system or process, the prespecified changes would appear in the process. Moreover, in most applications, the in-control parameters are usually unknown. To overcome these issues, we develop the zone control charts with estimated parameters to detect small shifts of these prespecified changes. The effects of estimation error have been investigated on the performance of the proposed charts. To account for the practitioner-to-practitioner variability, the expected average run length (ARL) and the standard deviation of the average run length (SDARL) is used as the performance metrics. Our results show that the estimation error results in the significant variation in the ARL distribution. Furthermore, in order to adequately reduce the variability, more phase I samples are required in terms of the SDARL metric than that in terms of the expected ARL metric. In addition, more observations on each sampled profile are suggested to improve the charts' performance, especially for small phase I sample sizes. Finally, an illustrative example is given to show the performance of the proposed zone control charts.     

Finite nonparametric grach model for foreign exchange volatility

GARCH model has been commonly used to describe the volatility of foreign exchange returns, which typically depends on returns many lags before, While the GARCH model provides a simple geometric decaying structure for persistence in time, it restricts tiie impact of variables to Quadratic functions. A finite nonparametric GARCH model is proposed that allows the variables' impact to be a smooth function of any form. A direct local polynomial estimation method for this finite GARCH model is proposed based on results on proportional additive model, and is applied to the German Mark (DEM)/US Dollar (USD) daily returns data. Estimators uf both the decaying rate and the impact function are obtained. Diagnostics show satisfactory out-of-sampie prediction based on the proposed model, which helps to better understand the dynamics of foreign exchange volatility.     

The MLE of Geometric Parameter for a Geometric Process

In this article, we provide the MLE of the ratio parameter of a geometric process and discuss its consistency and asymptotic normality.     

Ridge Estimation to the Restricted Linear Model

This article is concerned with the problem of multicollinearity in a linear model with linear restrictions. After introducing a spheral restricted condition, a new restricted ridge estimation method is proposed by minimizing the sum of squared residuals. The property of the new estimator in its superiority over the ordinary restricted least squares estimation is then theoretically analyzed. Furthermore, a sufficient and necessary condition for selecting the ridge parameter k is obtained. To simplify the selection of the ridge parameter, a sufficient condition is also given. Finally, a numerical example demonstrates the merit of the new method in the aspect of solving the multicollinearity over the ordinary restricted least squares estimation.     

The Research on Two Kinds of Restricted Biased Estimators Based on Mean Squared Error Matrix

Sakall?oglu et al. (2001 Sakall?oglu , Kaç?ranlar , Akdeniz ( 2001 ). Mean squared error comparisons of some biased estimators . Commun. Statist. Theor. Meth. 30 : 347 – 361 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) dealt with the comparisons among the ridge estimator, Liu estimator, and iteration estimator. Akdeniz and Erol (2003 Akdeniz , F. , Erol , H. ( 2003 ). Mean squared error matrix comparisons of some biased estimators in linear regression . Commun. Statist. Theor. Meth. 32 : 2389 – 2413 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) have compared the (almost unbiased) generalized ridge regression estimator with the (almost unbiased) generalized Liu estimator in the matrix mean squared error sense. In this article, we study the ridge estimator and Liu estimator with respect to linear equality restriction, and establish some sufficient conditions for the superiority of the restricted ridge estimator over the restricted Liu estimator and the superiority of the restricted Liu estimator over the restricted ridge estimator under mean squared error matrix, respectively. Furthermore, we give a numerical example.     

Ridge Regression Estimation for Survey Samples

This paper describes procedure for constructing a vector of regression weights. Under the regression superpopulation model, the ridge regression estimator that has minimum model mean squared error is derived. Through a simulation study, we compare the ridge regression weights, regression weights, quadratic programming weights, and raking ratio weights. The ridge regression procedure with weights bounded by zero performed very well.     

A Note on the Lower Bounds on Maximum Number of Clear Two-Factor Interactions for (2 m−p III) and (2 m−p IV) Designs

ABSTRACT

In the reliability analysis of mechanical repairable equipment subjected to reliability deterioration with operating time, two forms of the non-homogeneous Poisson processes, namely the Power-Law (PL) and the Log-Linear (LL) model, have found general acceptance in the literature. Inferential procedures, conditioned on the assumption of the PL or LL model, underestimate the overall uncertainty about a quantity of interest because the PL and LL models can provide different estimates of the quantity of interest, even when both of them adequately fit the observed data. In this paper, a composite estimation procedure, which uses the PL and LL models as competing models, is proposed in the framework of Bayesian statistics, thus allowing the uncertainty involved in model selection to be considered. A model-free approach is then proposed for incorporating technical information on the failure mechanism into the inferential procedure. Such an approach, which is based on two model-free quantities defined irrespectively of the functional form of the failure model, prevents that the prior information on the failure mechanism can improperly introduce prior probabilities on the adequacy of each model to fit the observed data. Finally, numerical applications are provided to illustrate the proposed procedures.     

Concurrent Design for Process Quality,Statistical Tolerance,and SPC

The rapid response to the requirements of customers and markets promotes the concurrent engineering (CE) technique in product and process design. The decision making for process quality target, SPC method, sampling plan, and control chart parameter design can be done at the stage of process quality plan based on historical data and process knowledge database. Therefore, it is a reasonable trend to introduce the concepts and achievements on process quality evaluation and process capability analysis, CE, and SPC techniques into process plan and tolerance design. A new systematic method for concurrent design of process quality, statistical tolerance (ST), and control chart is presented based on a NSFC research program. A set of standardized process quality indices (PQIs) for variables is introduced for meeting the measurement and evaluation to process yield, process centering, and quality loss. This index system that has relatively strong compatibility and adaptability is based on raisonne grading by using the series of preferred numbers and arithmetical progression. The expected process quality based on this system can be assured by a standardized interface between PQIs and SPC, that is, quality-oriented statistical tolerance zone. A quality-oriented ST and SPC approach that quantitatively specifies what a desired process is and how to assure it will realize the optimal control for a process toward a predetermined quality target.     

A brief proof on the generalized variance bound of the relative efficiency in statistics

In this paper, we obtain a brief proof on the generalized varance bound of the relative efficiency that was simultaneously given by Bloomfield and Watson(1975) and by Knott (1975).