James-stein estimation with constraints on the norm

Consider the problem of estimating a multivariate mean 0(pxl), p>3, based on a sample x^ ..., xn with quadratic loss function. We find an optimal decision rule within the class of James-Stein type decision rules when the underlying distribution is that of a variance mixture of normals and when the norm ||0|| is known. When the norm is restricted to a known interval, typically no optimal James-Stein type rule exists but we characterize a minimal complete class within the class of James-Stein type decision rules. We also characterize the subclass of James-Stein type decision rules that dominate the sample mean.     

Classification into two normal populations with a common mean and unequal variances

The problem of classification into two univariate normal populations with a common mean is considered. Several classification rules are proposed based on efficient estimators of the common mean. Detailed numerical comparisons of probabilities of misclassifications using these rules have been carried out. It is shown that the classification rule based on the Graybill-Deal estimator of the common mean performs the best. Classification rules are also proposed for the case when variances are assumed to be ordered. Comparison of these rules with the rule based on the Graybill-Deal estimator has been done with respect to individual probabilities of misclassification.     

A prior-valued estimator applied to multinomial classification

A multinomial classification rule is proposed based on a prior-valued smoothing for the state probabilities. Asymptotically, the proposed rule has an error rate that converges uniformly and strongly to that of the Bayes rule. For a fixed sample size the prior-valued smoothing is effective in obtaining reason¬able classifications to the situations such as missing data. Empirically, the proposed rule is compared favorably with other commonly used multinomial classification rules via Monte Carlo sampling experiments     

A response-adaptive design in crossover trial

In the present work, whenever the response variables are binary, we frame an adaptive allocation rule for a two-treatment two-period crossover design in the presence of possible carry-over effects. The proposed rule is a combination of the play-the-winner and randomized play-the-winner rules. We study various properties of the proposed rule through asymptotics and simulations. Some related inferential problems are also considered. The proposed procedure is compared with some possible competitor.     

Larger posterior mode wavelet thresholding and applications

This paper explores the thresholding rules induced by a variation of the Bayesian MAP principle. The MAP rules are Bayes actions that maximize the posterior. The proposed rule is thresholding and always picks the mode of the posterior larger in absolute value, thus the name LPM. We demonstrate that the introduced shrinkage performs comparably to several popular shrinkage techniques. The exact risk properties of the thresholding rule are explored, as well. We provide extensive simulational analysis and apply the proposed methodology to real-life experimental data coming from the field of atomic force microscopy.     

Isotonic rules for selecting good exponential populations

The problem of selecting exponential populations better than a control under a simple ordering prior is investigated. Based on some prior information, it is appropriate to set lower bounds for the concerned parameters. The information about the lower bounds of the concerned parameters is taken into account to derive isotonic selection rules for the control known case. An isotonic selection rule for the control unknown case is also proposed. A criterion is proposed to evaluate the performance of the selection rules. Simulation comparisons among the performances of several selection rules are carried out. The simulation results indicate that for the control known case, the new proposed selection rules perform better than some earlier existing selection rules.     

A comparative study of the K-means algorithm and the normal mixture model for clustering: Bivariate homoscedastic case

The K-means algorithm and the normal mixture model method are two common clustering methods. The K-means algorithm is a popular heuristic approach which gives reasonable clustering results if the component clusters are ball-shaped. Currently, there are no analytical results for this algorithm if the component distributions deviate from the ball-shape. This paper analytically studies how the K-means algorithm changes its classification rule as the normal component distributions become more elongated under the homoscedastic assumption and compares this rule with that of the Bayes rule from the mixture model method. We show that the classification rules of both methods are linear, but the slopes of the two classification lines change in the opposite direction as the component distributions become more elongated. The classification performance of the K-means algorithm is then compared to that of the mixture model method via simulation. The comparison, which is limited to two clusters, shows that the K-means algorithm provides poor classification performances consistently as the component distributions become more elongated while the mixture model method can potentially, but not necessarily, take advantage of this change and provide a much better classification performance.     

基于聚类关联规则的缺失数据处理研究

 本文提出了基于聚类和关联规则的缺失数据处理新方法,通过聚类方法将含有缺失数据的数据集相近的记录归到一类,然后利用改进后的关联规则方法对各子数据集挖掘变量间的关联性,并利用这种关联性来填补缺失数据。通过实例分析,发现该方法对缺失数据处理,尤其是海量数据集具有较好的效果。     

Testing multiple primary endpoints in clinical trials with sample size adaptation

In this paper, we propose a design that uses a short‐term endpoint for accelerated approval at interim analysis and a long‐term endpoint for full approval at final analysis with sample size adaptation based on the long‐term endpoint. Two sample size adaptation rules are compared: an adaptation rule to maintain the conditional power at a prespecified level and a step function type adaptation rule to better address the bias issue. Three testing procedures are proposed: alpha splitting between the two endpoints; alpha exhaustive between the endpoints; and alpha exhaustive with improved critical value based on correlation. Family‐wise error rate is proved to be strongly controlled for the two endpoints, sample size adaptation, and two analysis time points with the proposed designs. We show that using alpha exhaustive designs greatly improve the power when both endpoints are effective, and the power difference between the two adaptation rules is minimal. The proposed design can be extended to more general settings. Copyright © 2015 John Wiley & Sons, Ltd.     

On a threshold representation for complex load-sharing systems

Complex load-sharing systems are studied to incorporate dependencies among components through a load-sharing rule. As the load on the system increases, a series of cycles of Phase I/II failures occur where Phase I failure is a single component failure, which then causes a cascade of component failures (Phase II) due to the load transfer as these components fail. A threshold representation for the process of system failure is given. This representation is a gamma-type mixture representation when the component strengths are independent exponentials. In this case, for a given breaking pattern the mixture is over the gamma scale parameter and is based on a convolution of uniforms defined by the load-sharing parameters. Such convolutions can be approximated by normal densities which reduces the dimension of the parameter space. This representation can be generalized to independent component strengths with arbitrary distributions by transforming the strength and load-sharing to pseudo-strength and pseudo-load-sharing rules.     

Bootstrap adjustments of signed scoring rule root statistics

Scoring rules give rise to methods for statistical inference and are useful tools to achieve robustness or reduce computations. Scoring rule inference is generally performed through first-order approximations to the distribution of the scoring rule estimator or of the ratio-type statistic. In order to improve the accuracy of first-order methods even in simple models, we propose bootstrap adjustments of signed scoring rule root statistics for a scalar parameter of interest in presence of nuisance parameters. The method relies on the parametric bootstrap approach that avoids onerous calculations specific of analytical adjustments. Numerical examples illustrate the accuracy of the proposed method.     

Label switching and its solutions for frequentist mixture models

In this article, the label switching problem and the importance of solving it are discussed for frequentist mixture models if a simulation study is used to evaluate the performance of mixture model estimators. Two effective labelling methods are proposed by using true label for each observation. The empirical studies demonstrate that the new proposed methods work well and provide better results than the rule of thumb method of order constraint labelling. In addition, a Monte Carlo study also demonstrates that simple order constraint labelling can sometimes produce severely biased, and possibly meaningless, estimated bias and standard errors.     

A new per-field classification method using mixture discriminant analysis

In this study, a new per-field classification method is proposed for supervised classification of remotely sensed multispectral image data of an agricultural area using Gaussian mixture discriminant analysis (MDA). For the proposed per-field classification method, multivariate Gaussian mixture models constructed for control and test fields can have fixed or different number of components and each component can have different or common covariance matrix structure. The discrimination function and the decision rule of this method are established according to the average Bhattacharyya distance and the minimum values of the average Bhattacharyya distances, respectively. The proposed per-field classification method is analyzed for different structures of a covariance matrix with fixed and different number of components. Also, we classify the remotely sensed multispectral image data using the per-pixel classification method based on Gaussian MDA.     

Prbposterior expected loss as a scoring rule for prior distributions

A scoring rule for evaluating the usefulness of an assessed prior distribution should reflect the purpose for which the distribution is to be used. In this paper we suppose that sample data is to become available and that the posterior distribution will be used to estimate some quantity under a quadratic loss function. The utility of a prior distribution is consequently determined by its preposterior expected quadratic loss. It is shown that this loss function has properties desirable in a scoring rule and formulae are derived for calculating the scores it gives in some common problems. Many scoring rules give a very poor score to any improper prior distribution but, in contrast, the scoring rule proposed here provides a meaningful measure for comparing the usefulness of assessed prior distributions and non-informative (improper) prior distributions. Results for making this comparison in various situations are also given.     

A new search procedure of steepest ascent in response surface exploration

The method of steepest ascent direction has been widely accepted for process optimization in the applications of response surface methodology (RSM). The procedure of steepest ascent direction is performed on experiments run along the gradient of a fitted linear model. Therefore, the RSM practitioner needs to decide a suitable stopping rule such that the optimum point estimate in the search direction can be determined. However, the details of how to deflect and then halt a search in the steepest ascent direction are not thoroughly described in the literature. In common practice, it is convenient to use the simple stopping rules after one to three response deteriorations in a row after a series of fitted linear models used for exploration. In the literature, there are two formal stopping rules proposed, that is, Myers and Khuri's [A new procedure for steepest ascent, Comm. Statist. Theory Methods A 8(14) (1979), pp. 1359–1376] stopping rule and del Castillo's [Stopping rules for steepest ascent in experimental optimization, Comm. Statist. Simul. Comput. 26(4) (1997), pp. 1599–1615] stopping rule. This paper develops a new procedure for determining how to adjust and then when to stop a steepest ascent search in response surface exploration. This proposal wishes to provide the RSM practitioner with a clear-cut and easy-to-implement procedure that can attain the optimum mean response more accurately than the existing procedures. Through the study of simulation optimization, it shows that the average optimum point and response returned by using the new search procedure are considerably improved when compared with two existing stopping rules. The number of experimental trials required for convergence is greatly reduced as well.     

基于大数据规则挖掘的交通拥堵治理研究

随着中国城市机动车保有量的急剧增多,交通拥堵已经成为现代城市病。交通拥堵在道路网络中呈现向四周放射的传导特性,拥堵路段倾向于将拥堵扩散传导到其他相邻路段,该特性此前未被系统研究过,综合比较各种方法的适用性,从时间和大数据规则挖掘角度对拥堵建模;使用时间序列规则挖掘算法建立交通拥堵传导规律模型,并基于传导规则预测未来交通流状况;更重要的是,挖掘出来的拥堵传导规则直观可用,能够用于建立拥堵预警防治机制,完善道路路网建设规划中不合理的部分,从而达到提升交通效率的目的。研究结果证明本模型能够较好达到研究目的,挖掘出的拥堵传导规则可以精确分析交通拥堵状况并预测未来交通流状况,因此可以为交通拥堵治理决策提供重要参考。     

Two-sample post-stratified or subgroup analysis with restricted randomization rules

In a clinical trial to compare two treatments, subjects may be allocated sequentially to treatment groups by a restricted randomization rule. Suppose that at the end of the trial, the investigator is interested in a post-stratified or subgroup analysis with respect to a particular demographic or clinical factor which was not selected prior to the trial for stratified randomization. Under a randomization model, large sample theory of two-sample post-stratified permutational tests is developed with a broad class of restricted randomization treatment allocation rules. The test procedures proposed here are illustrated with a real-life example. The results of this example indicate that it is not always possible to ignore the treatment rule used in the trial in the design-based analysis.     

On the risk performance of extended sequence compound rules for classification between N(–1, 1) and N(1, 1) †

Since Robbins (1951) first introduced the compound decision problem, there has evolved a large literature on the subject for the most part dealing with the construction of compound rules whose excess risk over the simple envelope is no greater than zero in the limit as the number N of component problems goes to infinity. Such rules have compound risk which is asymptotically subminimax. Johns (1967) has introduced more stringent (extended) envelopes and has proposed extended compound rules whose risks achieve these envelopes in the limit. This paper reports some Monte Carlo results on the compound risk behavior of selected unextended and extended rules for moderate N values and certain parameter sequences for Robbins original example. The results show that the extended rules compare favorably with the minimax rule and the unextended rules for moderate N and parameter sequences exhibiting higher order empirical dependencies, for example, those generated by a Markov process.     

The table look-up rule

The table look-up rule problem can be described by the question: what is a good way for the table to represent the decision regions in the N-dimensional measurement space. This paper describes a quickly implementable table look-up rule based on Ashby’s representation of sets in his constraint analysis. A decision region for category c in the N-dimensional measurement space is considered to be the intersection of the inverse projections of the decision regions determined for category c by Bayes rules in smaller dimensional projection spaces. Error bounds for this composite decision rule are derived: any entry in the confusion matrix for the composite decision rule is bounded above by the minimum of that entry taken over all the confusion matrices of the Bayes decision rules in the smaller dimensional projection spaces.

On simulated Gaussian Data, probability of error with the table look-up rule is comparable to the optimum Bayes rule.