Shifeng Xiong

Sequential Design and Analysis of High-Accuracy and Low-Accuracy Computer Codes

A growing trend in engineering and science is to use multiple computer codes with different levels of accuracy to study the same complex system. We propose a framework for sequential design and analysis of a pair of high-accuracy and low-accuracy computer codes. It first runs the two codes with a pair of nested Latin hypercube designs (NLHDs). Data from the initial experiment are used to fit a prediction model. If the accuracy of the fitted model is less than a prespecified threshold, the two codes are evaluated again with input values chosen in an elaborate fashion so that their expanded scenario sets still form a pair of NLHDs. The nested relationship between the two scenario sets makes it easier to model and calibrate the difference between the two sources. If necessary, this augmentation process can be repeated a number of times until the prediction model based on all available data has reasonable accuracy. The effectiveness of the proposed method is illustrated with several examples. Matlab codes are provided in the online supplement to this article.

Some Notes on the Nonnegative Garrote

Some notes on the use of the nonnegative garrote (NG) are given in this article. The main result is that, compared with other penalized least-squares methods, the NG has a natural selection of penalty function according to an estimator of prediction risk. Furthermore, two natural estimators of the tuning parameter which only involve very simple computations are proposed corresponding to the Akaike information criterion (AIC) and Bayesian information criterion (BIC), respectively. This indicates that to select tuning parameters, it may be unnecessary to optimize a model selection criterion repeatedly. Several reasonable NG estimators with natural tuning parameters are proposed for multicollinearity problems and other problems. Simulation results and a real data analysis are reported for testing the results obtained previously.

GENERALIZED CONFIDENCE REGIONS OF FIXED EFFECTS IN THE TWO-WAY ANOVA

The authors discuss the unbalanced two-way ANOVA model under heteroscedasticity. By taking the generalized approach, the authors derive the generalized p-values for testing the equality of fixed effects and the generalized confidence regions for these effects. The authors also provide their frequentist properties in large-sample cases. Simulation studies show that the generalized confidence regions have good coverage probabilities.

Simultaneous confidence intervals for one-way layout based on generalized pivotal quantities

In this paper, we consider simultaneous confidence intervals for all-pairwise comparisons of treatment means in a one-way layout under heteroscedasticity. Two kinds of simultaneous intervals are provided based on the fiducial generalized pivotal quantities of the interest parameters. We prove that they both have asymptotically correct coverage. Simulation results and an example are also reported. It is concluded from calculational evidence that the second kind of simultaneous confidence intervals, which we provide, performs better than existing methods.

Generalized Inference for a Class of Linear Models Under Heteroscedasticity

In this article we study inferences for a class of linear models under heteroscedasticity. Using the generalized inference approach, we obtain the generalized p-values of two-sided hypotheses for the multi-dimensional location parameters and one-sided hypotheses for the scale parameters, respectively. Some frequentist properties in small-sample cases and large-sample cases are proven.

An asymptotics look at the generalized inference

This paper provides an asymptotics look at the generalized inference through showing connections between the generalized inference and two widely used asymptotic methods, the bootstrap and plug-in method. A generalized bootstrap method and a generalized plug-in method are introduced. The generalized bootstrap method can not only be used to prove asymptotic frequentist properties of existing generalized confidence regions through viewing fiducial generalized pivotal quantities as generalized bootstrap variables, but also yield new confidence regions for the situations where the generalized inference is unavailable. Some examples are presented to illustrate the method. In addition, the generalized F-test (Weerahandi, 1995 [26]) can be derived by the generalized plug-in method, then its asymptotic validity is obtained.

Nonparametric Interval Estimation in One-Way Random-Effects Models

The structural method provided by Hannig et al. (2006) has proved to be a useful tool for constructing confidence intervals. However, it is difficult to apply this method to nonparametric problems since the pivotal quantity required in using it exists only in some special parametric models. Based on an extended structural method, this article discusses nonparametric interval estimation for smooth functions of the variances in one-way random-effects models. We use the bootstrap distribution estimator of a statistic to construct an approximate pivotal equation, and prove that the confidence interval derived by the approximate pivotal equation has asymptotically correct coverage probability. Simulation results are presented and show that the normal fiducial interval is not robust against non normality and that the proposed confidence interval has better finite-sample behaviors than the naive interval based on normal approximation.

Orthogonalizing EM: A design-based least squares algorithm.

We introduce an efficient iterative algorithm, intended for various least squares problems, based on a design of experiments perspective. The algorithm, called orthogonalizing EM (OEM), works for ordinary least squares (OLS) and can be easily extended to penalized least squares. The main idea of the procedure is to orthogonalize a design matrix by adding new rows and then solve the original problem by embedding the augmented design in a missing data framework. We establish several attractive theoretical properties concerning OEM. For the OLS with a singular regression matrix, an OEM sequence converges to the Moore-Penrose generalized inverse-based least squares estimator. For ordinary and penalized least squares with various penalties, it converges to a point having grouping coherence for fully aliased regression matrices. Convergence and the convergence rate of the algorithm are examined. Finally, we demonstrate that OEM is highly efficient for large-scale least squares and penalized least squares problems, and is considerably faster than competing methods when n is much larger than p. Supplementary materials for this article are available online.

Local optimization-based statistical inference

This paper introduces a local optimization-based approach to test statistical hypotheses and to construct confidence intervals. This approach can be viewed as an extension of bootstrap, and yields asymptotically valid tests and confidence intervals as long as there exist consistent estimators of unknown parameters. We present simple algorithms including a neighborhood bootstrap method to implement the approach. Several examples in which theoretical analysis is not easy are presented to show the effectiveness of the proposed approach.

Inference on System Reliability for Independent Series Components

In this article, we study inferences for reliability functions of the system having two components connected in series. Suppose that the lifetime of one component has a lognormal distribution. Lognormal, exponential, and weibull distributions are considered for the lifetime of the other component. Using the generalized inference approach, we obtain confidence intervals of our interested parameters with good coverage. Some frequentist properties in small-sample cases and large-sample cases are proved.