Dverview and Main Advances in Permutation Tests for Linear Regression Models

When the population, from which the samples are extracted, is not normally distributed, or if the sample size is particularly reduced, become preferable the use of not parametric statistic test. An alternative to the normal model is the permutation or randomization model. The permutation model is nonparametric because no formal assumptions are made about the population parameters of the reference distribution, i.e., the distribution to which an obtained result is compared to determine its probability when the null hypothesis is true. Typically the reference distribution is a sampling distribution for parametric tests and a permutation distribution for many nonparametric tests. Within the regression models, it is possible to use the permutation tests, considering their ownerships of optimality, especially in the multivariate context and the normal distribution of the response variables is not guaranteed. In the literature there are numerous permutation tests applicable to the estimation of the regression models. The purpose of this study is to examine different kinds of permutation tests applied to linear models, focused our attention on the specific test statistic on which they are based. In this paper we focused our attention on permutation test of the independent variables, proposed by Oja, and other methods to effect the inference in non parametric way, in a regression model. Moreover, we show the recent advances in this context and try to compare them.

Bayesian Segmentation of Piecewise Linear Regression Models Using Reversible Jump MCMC Algorithm

Piecewise linear regression models are very flexible models for modeling the data. If the piecewise linear regression models are matched against the data, then the parameters are generally not known. This paper studies the problem of parameter estimation ofpiecewise linear regression models. The method used to estimate the parameters ofpicewise linear regression models is Bayesian method. But the Bayes estimator can not be found analytically. To overcome these problems, the reversible jump MCMC （Marcov Chain Monte Carlo） algorithm is proposed. Reversible jump MCMC algorithm generates the Markov chain converges to the limit distribution of the posterior distribution of the parameters ofpicewise linear regression models. The resulting Markov chain is used to calculate the Bayes estimator for the parameters of picewise linear regression models.

Robust Linear Regression Models:Use of a Stable Distribution for the Response Data

In this paper, we study some robustness aspects of linear regression models of the presence of outliers or discordant observations considering the use of stable distributions for the response in place of the usual normality assumption. It is well known that, in general, there is no closed form for the probability density function of stable distributions. However, under a Bayesian approach, the use of a latent or auxiliary random variable gives some simplification to obtain any posterior distribution when related to stable distributions. To show the usefulness of the computational aspects, the methodology is applied to two examples: one is related to a standard linear regression model with an explanatory variable and the other is related to a simulated data set assuming a 23 factorial experiment. Posterior summaries of interest are obtained using MCMC (Markov Chain Monte Carlo) methods and the OpenBugs software.

The Consistency of LSE Estimators in Partial Linear Regression Models under Mixing Random Errors

In this paper,we consider the partial linear regression model y_(i)=x_(i)β^(*)+g(ti)+ε_(i),i=1,2,...,n,where(x_(i),ti)are known fixed design points,g(·)is an unknown function,andβ^(*)is an unknown parameter to be estimated,random errorsε_(i)are(α,β)-mix_(i)ng random variables.The p-th(p>1)mean consistency,strong consistency and complete consistency for least squares estimators ofβ^(*)and g(·)are investigated under some mild conditions.In addition,a numerical simulation is carried out to study the finite sample performance of the theoretical results.Finally,a real data analysis is provided to further verify the effect of the model.

Integrating Multiple Linear Regression and Infectious Disease Models for Predicting Information Dissemination in Social Networks

Social network is the mainstream medium of current information dissemination,and it is particularly important to accurately predict its propagation law.In this paper,we introduce a social network propagation model integrating multiple linear regression and infectious disease model.Firstly,we proposed the features that affect social network communication from three dimensions.Then,we predicted the node influence via multiple linear regression.Lastly,we used the node influence as the state transition of the infectious disease model to predict the trend of information dissemination in social networks.The experimental results on a real social network dataset showed that the prediction results of the model are consistent with the actual information dissemination trends.

Asymptotic Properties in Semiparametric Partially Linear Regression Models for Functional Data

We consider the semiparametric partially linear regression models with mean function XTβ ＋ g（z）, where X and z are functional data. The new estimators of β and g（z） are presented and some asymptotic results are given. The strong convergence rates of the proposed estimators are obtained. In our estimation, the observation number of each subject will be completely flexible. Some simulation study is conducted to investigate the finite sample performance of the proposed estimators.

Random Weighting T-Statistics in Linear Regression Models

In this paper, we have constructed a random weighting statistic to approximate the distribution of studentized least square estimator in a linear regression model with ideal accuracy o(n-1/2). Thus, we have provided a more practical distribution approximating method.

Multiple linear regression models of urban runoff pollutant load and event mean concentration considering rainfall variables

Rainfall is an important factor in estimating the event mean concentration （EMC） which is used to quantify the washed-off pollutant concentrations from non-point sources （NPSs）. Pollutant loads could also be calculated using rainfall, catchment area and runoff coefficient. In this study, runoff quantity and quality data gathered from a 28-month monitoring conducted on the road and parking lot sites in Korea were evaluated using multiple linear regression （MLR） to develop equations for estimating pollutant loads and EMCs as a function of rainfall variables. The results revealed that total event rainfall and average rainfall intensity are possible predictors of pollutant loads. Overall, the models are indicators of the high uncertainties of NPSs; perhaps estimation of EMCs and loads could be accurately obtained by means of water quality sampling or a long term monitoring is needed to gather more data that can be used for the development of estimation models.

Delete-group Jackknife Estimate in Partially Linear Regression Models with Heteroscedasticity

Consider a partially linear regression model with an unknown vector parameter , an unknown function g(·), and unknown heteroscedastic error variances. Chen, You[23] proposed a semiparametric generalized least squares estimator (SGLSE) for , which takes the heteroscedasticity into account to increase efficiency. For inference based on this SGLSE, it is necessary to construct a consistent estimator for its asymptotic covariance matrix. However, when there exists within-group correlation, the traditional delta method and the delete-1 jackknife estimation fail to offer such a consistent estimator. In this paper, by deleting grouped partial residuals a delete-group jackknife method is examined. It is shown that the delete-group jackknife method indeed can provide a consistent estimator for the asymptotic covariance matrix in the presence of within-group correlations. This result is an extension of that in [21].

Evaluation of accuracy of linear regression models in predicting urban stormwater discharge characteristics

Stormwater runoff has been identified as a source of pollution for the environment, especially for receiving waters. In order to quantify and manage the impacts of stormwater runoff on the environment, predictive models and mathematical models have been developed. Predictive tools such as regression models have been widely used to predict stormwater discharge characteristics. Storm event characteristics, such as antecedent dry days （ADD）, have been related to response variables, such as pollutant loads and concentrations. However it has been a controversial issue among many studies to consider ADD as an important variable in predicting stormwater discharge characteristics. In this study, we examined the accuracy of general linear regression models in predicting discharge characteristics of roadway runoff. A total of 17 storm events were monitored in two highway segments, located in Gwangju, Korea. Data from the monitoring were used to calibrate United States Environmental Protection Agency＇s Storm Water Management Model （SWMM）. The calibrated SWMM was simulated for 55 storm events, and the results of total suspended solid （TSS） discharge loads and event mean concentrations （EMC） were extracted. From these data, linear regression models were developed. R2 and p-values of the regression of ADD for both TSS loads and EMCs were investigated. Results showed that pollutant loads were better predicted than pollutant EMC in the multiple regression models. Regression may not provide the true effect of site-specific characteristics, due to uncertainty in the data.

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Consider the partly linear regression model , where y i ’s are responses, are known and nonrandom design points, is a compact set in the real line , β = (β 1, ··· , β p )' is an unknown parameter vector, g(·) is an unknown function and {ε i } is a linear process, i.e., , where e j are i.i.d. random variables with zero mean and variance . Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {ε i }. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {ε i } are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coeffcients of the process. Moreover, our result can be used to construct the asymptotically effcient estimators for parameters in the ARMA error process.

Jackknifed Liu Estimator in Linear Regression Models

In this paper, we introduce a generalized Liu estimator and jackknifed Liu estimator in a linear regression model with correlated or heteroscedastic errors. Therefore, we extend the Liu estimator. Under the mean square error（MSE）, the jackknifed estimator is superior to the Liu estimator and the jackknifed ridge estimator. We also give a method to select the biasing parameter for d. Furthermore, a numerical example is given to illustvate these theoretical results.

Iterative Weighted Semiparametric Least Squares Estimation in Repeated Measurement Partially Linear Regression Models

Consider a repeated measurement partially linear regression model with anunknown vector parameter β_1, an unknown function g(·), and unknown heteroscedastic errorvariances. In order to improve the semiparametric generalized least squares estimator (SGLSE) of ,we propose an iterative weighted semiparametric least squares estimator (IWSLSE) and show that itimproves upon the SGLSE in terms of asymptotic covariance matrix. An adaptive procedure is given todetermine the number of iterations. We also show that when the number of replicates is less than orequal to two, the IWSLSE can not improve upon the SGLSE. These results are generalizations of thosein [2] to the case of semiparametric regressions.

Selection of the Linear Regression Model According to the Parameter Estimation

In this paper, based on the theory of parameter estimation, we give a selection method and, in a sense of a good character of the parameter estimation, we think that it is very reasonable. Moreover, we offer a calculation method of selection statistic and an applied example.

STRONG CONVERGENCE RATES OF SEVERAL ESTIMATORS IN SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS

This article is concerned with the estimating problem of semiparametric varyingcoefficient partially linear regression models. By combining the local polynomial and least squares procedures Fan and Huang （2005） proposed a profile least squares estimator for the parametric component and established its asymptotic normality. We further show that the profile least squares estimator can achieve the law of iterated logarithm. Moreover, we study the estimators of the functions characterizing the non-linear part as well as the error variance. The strong convergence rate and the law of iterated logarithm are derived for them, respectively.

EMPIRICAL BAYES ESTIMATION FOR ESTIMABLE FUNCTION OF REGRESSION COEFFICIENT IN A MULTIPLE LINEAR REGRESSION MODEL

In this paper we consider the empirical Bayes (EB) estimation problem for estimable function of regression coefficient in a multiple linear regression model Y=Xβ+e. where e with given β has a multivariate standard normal distribution. We get the EB estimators by using kernel estimation of multivariate density function and its first order partial derivatives. It is shown that the convergence rates of the EB estimators are under the condition where an integer k > 1 . is an arbitrary small number and m is the dimension of the vector Y.

A LARGE SAMPLE ESTIMATE IN MEDIAN LINEAR REGRESSION MODEL Ⅰ: NONTRUNCATED CASE

This paper uses a grouping-adjusting procedure to the data from a median linear regression model, and estimtes the regression coefficients by the method of weighted least squares. This method simplifies computation and in the meantime, preserves the same asymptotic normal distribution for the estimator, as in the ordinary minimum L_1-norm estimates.

Statistical Analysis of Fuzzy Linear Regression Model Based on Centroid Method

This paper transforms fuzzy number into clear number using the centroid method, thus we can research the traditional linear regression model which is transformed from the fuzzy linear regression model. The model’s input and output are fuzzy numbers, and the regression coefficients are clear numbers. This paper considers the parameter estimation and impact analysis based on data deletion. Through the study of example and comparison with other models, it can be concluded that the model in this paper is applied easily and better.

Application of Grey System GM (1,1) model and unary linear regression model in coal consumption of Jilin Province

The data on the coal production and consumption in Jilin Province for the last ten years were collected,and the Grey System GM( 1,1) model and unary linear regression model were applied to predict the coal consumption of Jilin Production in 2014 and 2015. Through calculation,the predictive value on the coal consumption of Jilin Province was attained,namely consumption of 2014 is 114. 84 × 106 t and of 2015 is 117. 98 ×106t,respectively. Analysis of error data indicated that the predicted accuracy of Grey System GM( 1,1) model on the coal consumption in Jilin Province improved 0. 21% in comparison to unary linear regression model.

Parametric estimation for the simple linear regression model under moving extremes ranked set sampling design

Cost effective sampling design is a major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming.Ranked set sampling(RSS)was first proposed by McIntyre[1952.A method for unbiased selective sampling,using ranked sets.Australian Journal of Agricultural Research 3,385-390]as an effective way to estimate the pasture mean.In the current paper,a modification of ranked set sampling called moving extremes ranked set sampling(MERSS)is considered for the best linear unbiased estimators(BLUEs)for the simple linear regression model.The BLUEs for this model under MERSS are derived.The BLUEs under MERSS are shown to be markedly more efficient for normal data when compared with the BLUEs under simple random sampling.