Optimal designs for estimating the control values in multi-univariate regression models

This paper considers a linear regression model with a one-dimensional control variable x and an m-dimensional response vector . The components of are correlated with a known covariance matrix. Based on the assumed regression model, it is of interest to obtain a suitable estimation of the corresponding control value for a given target vector on the expected responses. Due to the fact that there is more than one target value to be achieved in the multiresponse case, the m expected responses may meet their target values at different respective control values. Consideration on the performance of an estimator for the control value includes the difference of the expected response E(y_i) from its corresponding target value T_i for each component and the optimal value of control point, say x₀, is defined to be the one which minimizes the weighted sum of squares of those standardized differences within the range of x. The objective of this study is to find a locally optimal design for estimating x₀, which minimizes the mean squared error of the estimator of x₀. It is shown that the optimality criterion is equivalent to a c-criterion under certain conditions and explicit solutions with dual response under linear and quadratic polynomial regressions are obtained.     

A two-step method for estimating high-dimensional Gaussian graphical models

The problem of estimating high-dimensional Gaussian graphical models has gained much attention in recent years. Most existing methods can be considered as one-step approaches, being either regression-based or likelihood-based. In this paper, we propose a two-step method for estimating the high-dimensional Gaussian graphical model. Specifically, the first step serves as a screening step, in which many entries of the concentration matrix are identified as zeros and thus removed from further consideration. Then in the second step, we focus on the remaining entries of the concentration matrix and perform selection and estimation for nonzero entries of the concentration matrix. Since the dimension of the parameter space is effectively reduced by the screening step,the estimation accuracy of the estimated concentration matrix can be potentially improved. We show that the proposed method enjoys desirable asymptotic properties. Numerical comparisons of the proposed method with several existing methods indicate that the proposed method works well. We also apply the proposed method to a breast cancer microarray data set and obtain some biologically meaningful results.     

Optimal designs for estimating the coefficients of the lower frequencies in trigonometric regression models

In the common Fourier regression model we determine the optimal designs for estimating the coefficients corresponding to the lower frequencies. An analytical solution is provided which is found by an alternative characterization of c-optimal designs. Several examples are provided and the performance of the D-optimal design with respect to the estimation of the lower order coefficients is investigated. The results give a complete answer to an open question which was recently raised in the literature.     

A heuristic method for estimating time-series models for forecasting. I

The aim is to develop a heuristic method for estimating time-series models for forecasting. The study consists of two parts. This one presents the analytical framework of the proposed procedure; the second will present the actual algorithm and numerical evaluations of the process. Our approach makes use of the frequency-domain theory of second-order stochastic processes to remedy several of the problems that we encounter in fitting ARIMA-type models for forecasting. Within this framework some of the problems that the present study addresses are: sample size of time series, initial estimates of the coefficients, convergence of difficult data to stable estimates, and computing time.     

非线性半参数统计模型估计函数的几何方法

将非线性半参数统计模型的概率密度函数族视为统计流形,利用微分几何方法,建立非线性半参数统计模型相对应的Hilbert空间,进而研究非线性半参数统计模型的估计函数问题.利用两类得分函数张成的子空间对Hilbert空间进行正交分解,进而讨论估计函数所在的集合,以及如何选取最优估计函数的问题.最后,通过实例分析来验证此方法的有效性.     

The Monte Carlo EM method for estimating multinomial probit latent variable models

We propose a multinomial probit (MNP) model that is defined by a factor analysis model with covariates for analyzing unordered categorical data, and discuss its identification. Some useful MNP models are special cases of the proposed model. To obtain maximum likelihood estimates, we use the EM algorithm with its M-step greatly simplified under Conditional Maximization and its E-step made feasible by Monte Carlo simulation. Standard errors are calculated by inverting a Monte Carlo approximation of the information matrix using Louis’s method. The methodology is illustrated with a simulated data.     

Heteroscedasticity checks for regression models

For checking on heteroscedasticity in regression models, a unified approach is proposed to constructing test statistics in parametric and nonparametric regression models. For nonparametric regression, the test is not affected sensitively by the choice of smoothing parameters which are involved in estimation of the nonparametric regression function. The limiting null distribution of the test statistic remains the same in a wide range of the smoothing parameters. When the covariate is one-dimensional, the tests are, under some conditions, asymptotically distribution-free. In the high-dimensional cases, the validity of bootstrap approximations is investigated. It is shown that a variant of the wild bootstrap is consistent while the classical bootstrap is not in the general case, but is applicable if some extra assumption on conditional variance of the squared error is imposed. A simulation study is performed to provide evidence of how the tests work and compare with tests that have appeared in the literature. The approach may readily be extended to handle partial linear, and linear autoregressive models.     

A variable selection method based on Tabu search for logistic regression models

A Tabu search method is proposed and analysed for selecting variables that are subsequently used in Logistic Regression Models. The aim is to find from among a set of m variables a smaller subset which enables the efficient classification of cases. Reducing dimensionality has some very well-known advantages that are summarized in literature. The specific problem consists in finding, for a small integer value of p, a subset of size p of the original set of variables that yields the greatest percentage of hits in Logistic Regression. The proposed Tabu search method performs a deep search in the solution space that alternates between a basic phase (that uses simple moves) and a diversification phase (to explore regions not previously visited). Testing shows that it obtains significantly better results than the Stepwise, Backward or Forward methods used by classic statistical packages. Some results of applying these methods are presented.     

Syntactic approach to constructions of generic models

A syntactic approach is described to constructing generic models which generalizes the known semantic one. A sufficient condition of a generic model being homogeneous is specified. It is shown that, within the syntactic approach, any countable homogeneous model is generic. Criteria and a sufficient condition are given for the generic models created in syntactic constructions to be saturated. Supported by RFBR grant No. 05-01-00411, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools via project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 2, pp. 244–268, March–April, 2007.     

Some new constructions of bivariate Weibull models

In this article, several approaches are advanced towards the construction of bivariate Weibull models from the consideration of failure behaviors of the components of a two-component system. First, a general method of construction of bivariate life models is developed in the setting of random environmental effects. Some new bivariate Weibull models are derived as special cases and added insights are provided for some of the existing ones. In the course of model formulation in terms of the dependence structure, a new bivariate family of life distributions is constructed so as to incorporate both positive and negative quadrant dependence in the same parametric setting, and a bivariate Weibull model is obtained as a special case. Finally, some distributional properties are presented for a bivariate Weibull model derived from the consideration of random hazards.     

Difference equation models for estimating athletic records

This article presents an analysis of the models that are frequently used for estimating the future records for atheletic events. It considers an innovative approach for modeling based on difference equations whose complementary solution yields one of these models. It does this by establishing a relationship between the models and certain difference equations. Moreover, it shows that each of those models can be converted to the exponential model by applying an appropriate substitution. Based on the derived relations it proposes a numerical procedure for estimating the model parameters and discusses its advantages over an existing procedure. This procedure reduces the computational cost and avoids the classical difficulties of non-linear optimization procedures such as the Newton's method. Finally, it tests the proposed numerical procedure using simulated data and applies the procedure to the fastest times for 400 m race.     

Improved estimation method for high dimension semimartingale regression models based on discrete data

Statistical Inference for Stochastic Processes - In this paper we study a high dimension (Big Data) regression model in continuous time observed in the discrete time moments with dependent noises...     

Diagnostic checking for multivariate regression models

Diagnostic checking for multivariate parametric models is investigated in this article. A nonparametric Monte Carlo Test (NMCT) procedure is proposed. This Monte Carlo approximation is easy to implement and can automatically make any test procedure scale-invariant even when the test statistic is not scale-invariant. With it we do not need plug-in estimation of the asymptotic covariance matrix that is used to normalize test statistic and then the power performance can be enhanced. The consistency of NMCT approximation is proved. For comparison, we also extend the score type test to one-dimensional cases. NMCT can also be applied to diverse problems such as a classical problem for which we test whether or not certain covariables in linear model has significant impact for response. Although the Wilks lambda, a likelihood ratio test, is a proven powerful test, NMCT outperforms it especially in non-normal cases. Simulations are carried out and an application to a real data set is illustrated.     

Robust algorithms for multiphase regression models

This paper proposes a robust procedure for solving multiphase regression problems that is efficient enough to deal with data contaminated by atypical observations due to measurement errors or those drawn from heavy-tailed distributions. Incorporating the expectation and maximization algorithm with the M-estimation technique, we simultaneously derive robust estimates of the change-points and regression parameters, yet as the proposed method is still not resistant to high leverage outliers we further suggest a modified version by first moderately trimming those outliers and then implementing the new procedure for the trimmed data. This study sets up two robust algorithms using the Huber loss function and Tukey's biweight function to respectively replace the least squares criterion in the normality-based expectation and maximization algorithm, illustrating the effectiveness and superiority of the proposed algorithms through extensive simulations and sensitivity analyses. Experimental results show the ability of the proposed method to withstand outliers and heavy-tailed distributions. Moreover, as resistance to high leverage outliers is particularly important due to their devastating effect on fitting a regression model to data, various real-world applications show the practicability of this approach.     

On statistical models for regression diagnostics

In regression diagnostics, the case deletion model (CDM) and the mean shift outlier model (MSOM) are commonly used in practice. In this paper we show that the estimates of CDM and MSOM are equal in a wide class of statistical models, which include LSE, MLE, Bayesian estimate andM-estimate in linear and nonlinear regression models; MLE in generalized linear models and exponential family nonlinear models; MLEs of transformation parameters of explanatory variables in a Box-Cox regression models and so on. Furthermore, we study some models, in which, the estimates are not exactly equal but are approximately equal for CDM and MSOM.     

Flexible regression models for carcinogenesis studies

Dynamic or flexible regression models are used more and more often in carcinogenesis studies to relate lifetime distribution to time-depending explanatory variables. In addition to the classical regression models, such as the Cox model, AFT model, Linear transformation model, Frailty model, etc., we expose so-called flexible regression models, which are well adapted to study cross-effects of survival functions. Such effects are sometimes observed in clinical trials. Classical examples are well-known data concerning effects of chemotherapy (CH) and chemotherapy plus radiotherapy (CH + R) on survival times of gastric cancers patients. In this paper, we give examples which illustrate possible applications of the Hsieh model (2001) and the SCE model proposed by Bagdonavicius and Nikulin and adapted to treat survival data with one crossing point. We compare both models. Biblipgraphy: 40 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 78–101.     

On resampling techniques for regression models

Resampling techniques like the bootstrap are examined for functions of the parameters of a linear model. A weighted resampling method analogous to the weighted jackknife developed by Hinkley (1977) is proposed for regression models.     

Influence diagnostic for censored regression models

This paper proposes a diagnostic measure to detect outliers either in the response variable or in the design space both for Schmee-Hahn estimator and Buckley-James estimator in censored regressions. The diagnostics consists of the leverage point and standardized predicted residual.