As far as the nonlinear regression method is concerned, the condition when both independent and dependent variable take the Fuzzy value, while the parameter, θ∈ΘR m the real value, have been discussed in . But for...As far as the nonlinear regression method is concerned, the condition when both independent and dependent variable take the Fuzzy value, while the parameter, θ∈ΘR m the real value, have been discussed in . But for most of actual conditions, the independent variable generally takes the real value, while both parameter and dependent variable take the Fuzzy value. This paper propounded a method for the latter and its relevant Fuzzy regreession model. In addition the Fuzzy observation, matrix distribution and the rational estimation of modeling parameter have also been discussed. Furthermore, the Max min estimation of modeling parameter and its corresponding calculating sequence have also been offered to and the calculating example shows the method is feasible.展开更多
In order to study the work-ability and establish the optimum hot formation processing parameters for industrial 1060 pure aluminum, the compressive deformation behavior of pure aluminum was investigated at temperature...In order to study the work-ability and establish the optimum hot formation processing parameters for industrial 1060 pure aluminum, the compressive deformation behavior of pure aluminum was investigated at temperatures of 523?823 K and strain rates of 0.005?10 s?1 on a Gleeble?1500 thermo-simulation machine. The influence rule of processing parameters (strain, strain rate and temperature) on flow stress of pure aluminum was investigated. Nine analysis factors consisting of material parameters and according weights were optimized. Then, the constitutive equations of multilevel series rules, multilevel parallel rules and multilevel series ¶llel rules were established. The correlation coefficients (R) are 0.992, 0.988 and 0.990, respectively, and the average absolute relative errors (AAREs) are 6.77%, 8.70% and 7.63%, respectively, which proves that the constitutive equations of multilevel series rules can predict the flow stress of pure aluminum with good correlation and precision.展开更多
A geometric framework is proposed for semiparametric nonlinear regression models based on the concept of least favorable curve, introduced by Severini and Wong (1992). The authors use this framework to drive three kin...A geometric framework is proposed for semiparametric nonlinear regression models based on the concept of least favorable curve, introduced by Severini and Wong (1992). The authors use this framework to drive three kinds of improved approximate confidence regions for the parameter and parameter subset in terms of curvatures. The results obtained by Hamilton et al. (1982), Hamilton (1986) and Wei (1994) are extended to semiparametric nonlinear regression models.展开更多
The purpose of this paper is to study the theory of conservative estimating functions in nonlinear regression model with aggregated data. In this model, a quasi-score function with aggregated data is defined. When thi...The purpose of this paper is to study the theory of conservative estimating functions in nonlinear regression model with aggregated data. In this model, a quasi-score function with aggregated data is defined. When this function happens to be conservative, it is projection of the true score function onto a class of estimation functions. By constructing, the potential function for the projected score with aggregated data is obtained, which have some properties of log-likelihood function.展开更多
Remaining useful life(RUL) prediction is one of the most crucial elements in prognostics and health management(PHM). Aiming at the imperfect prior information, this paper proposes an RUL prediction method based on a n...Remaining useful life(RUL) prediction is one of the most crucial elements in prognostics and health management(PHM). Aiming at the imperfect prior information, this paper proposes an RUL prediction method based on a nonlinear random coefficient regression(RCR) model with fusing failure time data.Firstly, some interesting natures of parameters estimation based on the nonlinear RCR model are given. Based on these natures,the failure time data can be fused as the prior information reasonably. Specifically, the fixed parameters are calculated by the field degradation data of the evaluated equipment and the prior information of random coefficient is estimated with fusing the failure time data of congeneric equipment. Then, the prior information of the random coefficient is updated online under the Bayesian framework, the probability density function(PDF) of the RUL with considering the limitation of the failure threshold is performed. Finally, two case studies are used for experimental verification. Compared with the traditional Bayesian method, the proposed method can effectively reduce the influence of imperfect prior information and improve the accuracy of RUL prediction.展开更多
To better capture the characteristics of asymmetry and structural fluctuations observed in count time series,this study delves into the application of the quantile regression(QR)method for analyzing and forecasting no...To better capture the characteristics of asymmetry and structural fluctuations observed in count time series,this study delves into the application of the quantile regression(QR)method for analyzing and forecasting nonlinear integer-valued time series exhibiting a piecewise phenomenon.Specifically,we focus on the parameter estimation in the first-order Self-Exciting Threshold Integer-valued Autoregressive(SETINAR(2,1))process with symmetry,asymmetry,and contaminated innovations.We establish the asymptotic properties of the estimator under certain regularity conditions.Monte Carlo simulations demonstrate the superior performance of the QR method compared to the conditional least squares(CLS)approach.Furthermore,we validate the robustness of the proposed method through empirical quantile regression estimation and forecasting for larceny incidents and CAD drug call counts in Pittsburgh,showcasing its effectiveness across diverse levels of data heterogeneity.展开更多
In this article, to improve the doubly robust estimator, the nonlinear regression models with missing responses are studied. Based on the covariate balancing propensity score (CBPS), estimators for the regression coef...In this article, to improve the doubly robust estimator, the nonlinear regression models with missing responses are studied. Based on the covariate balancing propensity score (CBPS), estimators for the regression coefficients and the population mean are obtained. It is proved that the proposed estimators are asymptotically normal. In simulation studies, the proposed estimators show improved performance relative to usual augmented inverse probability weighted estimators.展开更多
Triaxial tests,a staple in rock engineering,are labor-intensive,sample-demanding,and costly,making their optimization highly advantageous.These tests are essential for characterizing rock strength,and by adopting a fa...Triaxial tests,a staple in rock engineering,are labor-intensive,sample-demanding,and costly,making their optimization highly advantageous.These tests are essential for characterizing rock strength,and by adopting a failure criterion,they allow for the derivation of criterion parameters through regression,facilitating their integration into modeling programs.In this study,we introduce the application of an underutilized statistical technique—orthogonal regression—well-suited for analyzing triaxial test data.Additionally,we present an innovation in this technique by minimizing the Euclidean distance while incorporating orthogonality between vectors as a constraint,for the case of orthogonal linear regression.Also,we consider the Modified Least Squares method.We exemplify this approach by developing the necessary equations to apply the Mohr-Coulomb,Murrell,Hoek-Brown,andÚcar criteria,and implement these equations in both spreadsheet calculations and R scripts.Finally,we demonstrate the technique's application using five datasets of varied lithologies from specialized literature,showcasing its versatility and effectiveness.展开更多
Piezo actuators are widely used in ultra-precision fields because of their high response and nano-scale step length.However,their hysteresis characteristics seriously affect the accuracy and stability of piezo actuato...Piezo actuators are widely used in ultra-precision fields because of their high response and nano-scale step length.However,their hysteresis characteristics seriously affect the accuracy and stability of piezo actuators.Existing methods for fitting hysteresis loops include operator class,differential equation class,and machine learning class.The modeling cost of operator class and differential equation class methods is high,the model complexity is high,and the process of machine learning,such as neural network calculation,is opaque.The physical model framework cannot be directly extracted.Therefore,the sparse identification of nonlinear dynamics(SINDy)algorithm is proposed to fit hysteresis loops.Furthermore,the SINDy algorithm is improved.While the SINDy algorithm builds an orthogonal candidate database for modeling,the sparse regression model is simplified,and the Relay operator is introduced for piecewise fitting to solve the distortion problem of the SINDy algorithm fitting singularities.The Relay-SINDy algorithm proposed in this paper is applied to fitting hysteresis loops.Good performance is obtained with the experimental results of open and closed loops.Compared with the existing methods,the modeling cost and model complexity are reduced,and the modeling accuracy of the hysteresis loop is improved.展开更多
Assume that in the nonlinear regression model, independent variable sequence {xi, i ≥ 1} is a known constant-vector sequence. This article proposes a condition on {xi}, which can be tested and verified easily. The co...Assume that in the nonlinear regression model, independent variable sequence {xi, i ≥ 1} is a known constant-vector sequence. This article proposes a condition on {xi}, which can be tested and verified easily. The condition is essential for proving the consistency and asymptotic normality of the estimator.展开更多
Chaos theory has taught us that a system which has both nonlinearity and random input will most likely produce irregular data. If random errors are irregular data, then random error process will raise nonlinearity （K...Chaos theory has taught us that a system which has both nonlinearity and random input will most likely produce irregular data. If random errors are irregular data, then random error process will raise nonlinearity （Kantz and Schreiber （1997））. Tsai （1986） introduced a composite test for autocorrelation and heteroscedasticity in linear models with AR（1） errors. Liu （2003） introduced a composite test for correlation and heteroscedasticity in nonlinear models with DBL（p, 0, 1） errors. Therefore, the important problems in regression model axe detections of bilinearity, correlation and heteroscedasticity. In this article, the authors discuss more general case of nonlinear models with DBL（p, q, 1） random errors by score test. Several statistics for the test of bilinearity, correlation, and heteroscedasticity are obtained, and expressed in simple matrix formulas. The results of regression models with linear errors are extended to those with bilinear errors. The simulation study is carried out to investigate the powers of the test statistics. All results of this article extend and develop results of Tsai （1986）, Wei, et al （1995）, and Liu, et al （2003）.展开更多
This paper constructs a set of confidence regions of parameters in terms of statistical curvatures for AR(q) nonlinear regression models. The geometric frameworks are proposed for the model. Then several confidence re...This paper constructs a set of confidence regions of parameters in terms of statistical curvatures for AR(q) nonlinear regression models. The geometric frameworks are proposed for the model. Then several confidence regions for parameters and parameter subsets in terms of statistical curvatures are given based on the likelihood ratio statistics and score statistics. Several previous results, such as [1] and [2] are extended to AR(q) nonlinear regression models.展开更多
In this paper,a sem iparam etric regression m odel in w hich errors are i.i.d random variables from an unknow n density f(·) is considered.Based on Hallet al.(1995),a nonlinear w avelet estim ation of f(·)...In this paper,a sem iparam etric regression m odel in w hich errors are i.i.d random variables from an unknow n density f(·) is considered.Based on Hallet al.(1995),a nonlinear w avelet estim ation of f(·) withoutrestrictions ofcontinuity everyw here on f(·) is given,and the convergence rate ofthe estim ators in L2 is obtained.展开更多
Taking the nonlinear nature of runoff system into account,and combining auto-regression method and multi-regression method,a Nonlinear Mixed Regression Model (NMR) was established to analyze the impact of temperature ...Taking the nonlinear nature of runoff system into account,and combining auto-regression method and multi-regression method,a Nonlinear Mixed Regression Model (NMR) was established to analyze the impact of temperature and precipitation changes on annual river runoff process. The model was calibrated and verified by using BP neural network with observed meteorological and runoff data from Daiying Hydrological Station in the Chaohe River of Hebei Province in 1956–2000. Compared with auto-regression model,linear multi-regression model and linear mixed regression model,NMR can improve forecasting precision remarkably. Therefore,the simulation of climate change scenarios was carried out by NMR. The results show that the nonlinear mixed regression model can simulate annual river runoff well.展开更多
In order to reduce the influence of outliers on the parameter estimate of the attenuation formula for the blasting vibration velocity,a fuzzy nonlinear regression method of Sadov’s vibration formula was proposed on t...In order to reduce the influence of outliers on the parameter estimate of the attenuation formula for the blasting vibration velocity,a fuzzy nonlinear regression method of Sadov’s vibration formula was proposed on the basis of the fuzziness of blasting engineering,and the algorithm was described in details as well.In accordance with an engineering case,the vibration attenuation formula was regressed by the fuzzy nonlinear regression method and the nonlinear least square method,respectively.The calculation results showed that the fuzzy nonlinear regression method is more suitable to the field test data.It differs from the nonlinear least square method because the weight of residual square in the objective function can be adjusted according to the membership of each data.And the deviation calculation of least square estimate of parameters in the nonlinear regression model verified the rationality of using the membership to assign the weight of residual square.The fuzzy nonlinear regression method provides a calculation basis for estimating Sadov’s vibration formula’s parameters more accurately.展开更多
This paper is devoted to a study of geometric properties of AR(q) nonlinear regression models. We present geometric frameworks for regression parameter space and autoregression parameter space respectively based on th...This paper is devoted to a study of geometric properties of AR(q) nonlinear regression models. We present geometric frameworks for regression parameter space and autoregression parameter space respectively based on the weighted inner product by fisher information matrix. Several geometric properties related to statistical curvatures are given for the models. The results of this paper extended the work of Bates & Watts(1980,1988)[1.2] and Seber & Wild (1989)[3].展开更多
The effects of centering response and explanatory variables as a way of simplifying fitted linear models in the presence of correlation are reviewed and extended to include nonlinear models, common in many biological ...The effects of centering response and explanatory variables as a way of simplifying fitted linear models in the presence of correlation are reviewed and extended to include nonlinear models, common in many biological and economic applications. In a nonlinear model, the use of a local approximation can modify the effect of centering. Even in the presence of uncorrelated explanatory variables, centering may affect linear approximations and related test statistics. An approach to assessing this effect in relation to intrinsic curvature is developed and applied. Mis-specification bias of linear versus nonlinear models also reflects this centering effect.展开更多
In this paper,the authors propose a nonlinear dimension reduction technique based on Fréchet inverse regression to achieve sufficient dimension reduction for responses in metric spaces and predictors in Riemannia...In this paper,the authors propose a nonlinear dimension reduction technique based on Fréchet inverse regression to achieve sufficient dimension reduction for responses in metric spaces and predictors in Riemannian manifolds.The authors rigorously establish statistical properties of the estimators,providing formal proofs of their consistency and asymptotic behaviors.The effectiveness of our method is demonstrated through extensive simulations and applications to real-world datasets which highlight its practical utility for complex data with non-Euclidean structures.展开更多
In this paper, three smoothed empirical log-likelihood ratio functions for the parameters of nonlinear models with missing response are suggested. Under some regular conditions, the corresponding Wilks phenomena are o...In this paper, three smoothed empirical log-likelihood ratio functions for the parameters of nonlinear models with missing response are suggested. Under some regular conditions, the corresponding Wilks phenomena are obtained and the confidence regions for the parameter can be constructed easily.展开更多
文摘As far as the nonlinear regression method is concerned, the condition when both independent and dependent variable take the Fuzzy value, while the parameter, θ∈ΘR m the real value, have been discussed in . But for most of actual conditions, the independent variable generally takes the real value, while both parameter and dependent variable take the Fuzzy value. This paper propounded a method for the latter and its relevant Fuzzy regreession model. In addition the Fuzzy observation, matrix distribution and the rational estimation of modeling parameter have also been discussed. Furthermore, the Max min estimation of modeling parameter and its corresponding calculating sequence have also been offered to and the calculating example shows the method is feasible.
基金Project(51275414)supported by the National Natural Science Foundation of ChinaProject(2015JM5204)supported by the Natural Science Foundation of Shaanxi Province,China+1 种基金Project(Z2015064)supported by the Graduate Starting Seed Fund of the Northwestern Polytechnical University,ChinaProject(130-QP-2015)supported by the Research Fund of the State Key Laboratory of Solidification Processing(NWPU),China
文摘In order to study the work-ability and establish the optimum hot formation processing parameters for industrial 1060 pure aluminum, the compressive deformation behavior of pure aluminum was investigated at temperatures of 523?823 K and strain rates of 0.005?10 s?1 on a Gleeble?1500 thermo-simulation machine. The influence rule of processing parameters (strain, strain rate and temperature) on flow stress of pure aluminum was investigated. Nine analysis factors consisting of material parameters and according weights were optimized. Then, the constitutive equations of multilevel series rules, multilevel parallel rules and multilevel series ¶llel rules were established. The correlation coefficients (R) are 0.992, 0.988 and 0.990, respectively, and the average absolute relative errors (AAREs) are 6.77%, 8.70% and 7.63%, respectively, which proves that the constitutive equations of multilevel series rules can predict the flow stress of pure aluminum with good correlation and precision.
文摘A geometric framework is proposed for semiparametric nonlinear regression models based on the concept of least favorable curve, introduced by Severini and Wong (1992). The authors use this framework to drive three kinds of improved approximate confidence regions for the parameter and parameter subset in terms of curvatures. The results obtained by Hamilton et al. (1982), Hamilton (1986) and Wei (1994) are extended to semiparametric nonlinear regression models.
文摘The purpose of this paper is to study the theory of conservative estimating functions in nonlinear regression model with aggregated data. In this model, a quasi-score function with aggregated data is defined. When this function happens to be conservative, it is projection of the true score function onto a class of estimation functions. By constructing, the potential function for the projected score with aggregated data is obtained, which have some properties of log-likelihood function.
基金supported by National Natural Science Foundation of China (61703410,61873175,62073336,61873273,61773386,61922089)。
文摘Remaining useful life(RUL) prediction is one of the most crucial elements in prognostics and health management(PHM). Aiming at the imperfect prior information, this paper proposes an RUL prediction method based on a nonlinear random coefficient regression(RCR) model with fusing failure time data.Firstly, some interesting natures of parameters estimation based on the nonlinear RCR model are given. Based on these natures,the failure time data can be fused as the prior information reasonably. Specifically, the fixed parameters are calculated by the field degradation data of the evaluated equipment and the prior information of random coefficient is estimated with fusing the failure time data of congeneric equipment. Then, the prior information of the random coefficient is updated online under the Bayesian framework, the probability density function(PDF) of the RUL with considering the limitation of the failure threshold is performed. Finally, two case studies are used for experimental verification. Compared with the traditional Bayesian method, the proposed method can effectively reduce the influence of imperfect prior information and improve the accuracy of RUL prediction.
基金supported by Social Science Planning Foundation of Liaoning Province(Grand No.L22ZD065)National Natural Science Foundation of China(Grand Nos.12271231,1247012719,12001229)。
文摘To better capture the characteristics of asymmetry and structural fluctuations observed in count time series,this study delves into the application of the quantile regression(QR)method for analyzing and forecasting nonlinear integer-valued time series exhibiting a piecewise phenomenon.Specifically,we focus on the parameter estimation in the first-order Self-Exciting Threshold Integer-valued Autoregressive(SETINAR(2,1))process with symmetry,asymmetry,and contaminated innovations.We establish the asymptotic properties of the estimator under certain regularity conditions.Monte Carlo simulations demonstrate the superior performance of the QR method compared to the conditional least squares(CLS)approach.Furthermore,we validate the robustness of the proposed method through empirical quantile regression estimation and forecasting for larceny incidents and CAD drug call counts in Pittsburgh,showcasing its effectiveness across diverse levels of data heterogeneity.
文摘In this article, to improve the doubly robust estimator, the nonlinear regression models with missing responses are studied. Based on the covariate balancing propensity score (CBPS), estimators for the regression coefficients and the population mean are obtained. It is proved that the proposed estimators are asymptotically normal. In simulation studies, the proposed estimators show improved performance relative to usual augmented inverse probability weighted estimators.
基金Supported by National High Technology Research and Development Program of China(863 Program)(2006AA040308)National Natural Science Foundation of China(60736021)the National Creative Research Groups Science Foundation of China(60721062)
文摘Triaxial tests,a staple in rock engineering,are labor-intensive,sample-demanding,and costly,making their optimization highly advantageous.These tests are essential for characterizing rock strength,and by adopting a failure criterion,they allow for the derivation of criterion parameters through regression,facilitating their integration into modeling programs.In this study,we introduce the application of an underutilized statistical technique—orthogonal regression—well-suited for analyzing triaxial test data.Additionally,we present an innovation in this technique by minimizing the Euclidean distance while incorporating orthogonality between vectors as a constraint,for the case of orthogonal linear regression.Also,we consider the Modified Least Squares method.We exemplify this approach by developing the necessary equations to apply the Mohr-Coulomb,Murrell,Hoek-Brown,andÚcar criteria,and implement these equations in both spreadsheet calculations and R scripts.Finally,we demonstrate the technique's application using five datasets of varied lithologies from specialized literature,showcasing its versatility and effectiveness.
基金National Natural Science Foundation of China(62203118)。
文摘Piezo actuators are widely used in ultra-precision fields because of their high response and nano-scale step length.However,their hysteresis characteristics seriously affect the accuracy and stability of piezo actuators.Existing methods for fitting hysteresis loops include operator class,differential equation class,and machine learning class.The modeling cost of operator class and differential equation class methods is high,the model complexity is high,and the process of machine learning,such as neural network calculation,is opaque.The physical model framework cannot be directly extracted.Therefore,the sparse identification of nonlinear dynamics(SINDy)algorithm is proposed to fit hysteresis loops.Furthermore,the SINDy algorithm is improved.While the SINDy algorithm builds an orthogonal candidate database for modeling,the sparse regression model is simplified,and the Relay operator is introduced for piecewise fitting to solve the distortion problem of the SINDy algorithm fitting singularities.The Relay-SINDy algorithm proposed in this paper is applied to fitting hysteresis loops.Good performance is obtained with the experimental results of open and closed loops.Compared with the existing methods,the modeling cost and model complexity are reduced,and the modeling accuracy of the hysteresis loop is improved.
文摘Assume that in the nonlinear regression model, independent variable sequence {xi, i ≥ 1} is a known constant-vector sequence. This article proposes a condition on {xi}, which can be tested and verified easily. The condition is essential for proving the consistency and asymptotic normality of the estimator.
文摘Chaos theory has taught us that a system which has both nonlinearity and random input will most likely produce irregular data. If random errors are irregular data, then random error process will raise nonlinearity （Kantz and Schreiber （1997））. Tsai （1986） introduced a composite test for autocorrelation and heteroscedasticity in linear models with AR（1） errors. Liu （2003） introduced a composite test for correlation and heteroscedasticity in nonlinear models with DBL（p, 0, 1） errors. Therefore, the important problems in regression model axe detections of bilinearity, correlation and heteroscedasticity. In this article, the authors discuss more general case of nonlinear models with DBL（p, q, 1） random errors by score test. Several statistics for the test of bilinearity, correlation, and heteroscedasticity are obtained, and expressed in simple matrix formulas. The results of regression models with linear errors are extended to those with bilinear errors. The simulation study is carried out to investigate the powers of the test statistics. All results of this article extend and develop results of Tsai （1986）, Wei, et al （1995）, and Liu, et al （2003）.
文摘This paper constructs a set of confidence regions of parameters in terms of statistical curvatures for AR(q) nonlinear regression models. The geometric frameworks are proposed for the model. Then several confidence regions for parameters and parameter subsets in terms of statistical curvatures are given based on the likelihood ratio statistics and score statistics. Several previous results, such as [1] and [2] are extended to AR(q) nonlinear regression models.
文摘In this paper,a sem iparam etric regression m odel in w hich errors are i.i.d random variables from an unknow n density f(·) is considered.Based on Hallet al.(1995),a nonlinear w avelet estim ation of f(·) withoutrestrictions ofcontinuity everyw here on f(·) is given,and the convergence rate ofthe estim ators in L2 is obtained.
基金Under the auspices of National Natural Science Foundation of China (No. 50809004)
文摘Taking the nonlinear nature of runoff system into account,and combining auto-regression method and multi-regression method,a Nonlinear Mixed Regression Model (NMR) was established to analyze the impact of temperature and precipitation changes on annual river runoff process. The model was calibrated and verified by using BP neural network with observed meteorological and runoff data from Daiying Hydrological Station in the Chaohe River of Hebei Province in 1956–2000. Compared with auto-regression model,linear multi-regression model and linear mixed regression model,NMR can improve forecasting precision remarkably. Therefore,the simulation of climate change scenarios was carried out by NMR. The results show that the nonlinear mixed regression model can simulate annual river runoff well.
基金Supported by the National Natural Science Foundation of China(10272109)。
文摘In order to reduce the influence of outliers on the parameter estimate of the attenuation formula for the blasting vibration velocity,a fuzzy nonlinear regression method of Sadov’s vibration formula was proposed on the basis of the fuzziness of blasting engineering,and the algorithm was described in details as well.In accordance with an engineering case,the vibration attenuation formula was regressed by the fuzzy nonlinear regression method and the nonlinear least square method,respectively.The calculation results showed that the fuzzy nonlinear regression method is more suitable to the field test data.It differs from the nonlinear least square method because the weight of residual square in the objective function can be adjusted according to the membership of each data.And the deviation calculation of least square estimate of parameters in the nonlinear regression model verified the rationality of using the membership to assign the weight of residual square.The fuzzy nonlinear regression method provides a calculation basis for estimating Sadov’s vibration formula’s parameters more accurately.
基金Supported by the NSSFC(02BTJ001) Supported by the NSSFC(04BTJ002) Supported by the Grant for Post-Doctorial Fellows in Southeast University
文摘This paper is devoted to a study of geometric properties of AR(q) nonlinear regression models. We present geometric frameworks for regression parameter space and autoregression parameter space respectively based on the weighted inner product by fisher information matrix. Several geometric properties related to statistical curvatures are given for the models. The results of this paper extended the work of Bates & Watts(1980,1988)[1.2] and Seber & Wild (1989)[3].
文摘The effects of centering response and explanatory variables as a way of simplifying fitted linear models in the presence of correlation are reviewed and extended to include nonlinear models, common in many biological and economic applications. In a nonlinear model, the use of a local approximation can modify the effect of centering. Even in the presence of uncorrelated explanatory variables, centering may affect linear approximations and related test statistics. An approach to assessing this effect in relation to intrinsic curvature is developed and applied. Mis-specification bias of linear versus nonlinear models also reflects this centering effect.
文摘In this paper,the authors propose a nonlinear dimension reduction technique based on Fréchet inverse regression to achieve sufficient dimension reduction for responses in metric spaces and predictors in Riemannian manifolds.The authors rigorously establish statistical properties of the estimators,providing formal proofs of their consistency and asymptotic behaviors.The effectiveness of our method is demonstrated through extensive simulations and applications to real-world datasets which highlight its practical utility for complex data with non-Euclidean structures.
文摘In this paper, three smoothed empirical log-likelihood ratio functions for the parameters of nonlinear models with missing response are suggested. Under some regular conditions, the corresponding Wilks phenomena are obtained and the confidence regions for the parameter can be constructed easily.
