We propose an approach to compute the boundary crossing probabilities for a class of diffusion pr... more We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein-Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear boundaries are obtained, which are useful in evaluation and comparison of various omputational algorithms. Moreover, numerical computation can be easily done by Monte Carlo integration and the approximation errors for general boundaries are automatically calculated. Some numerical examples are presented.
We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or pi... more We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general nonlinear boundaries. The jump process can be any integer-valued process and jump sizes can have general distributions. Moreover, the jump sizes can be even correlated and/or non-identically distributed. The numerical algorithm is straightforward and easy to implement. Some numerical examples are presented.
Generalized method of moments (GMM) is an estimation technique which estimates unknown parameters... more Generalized method of moments (GMM) is an estimation technique which estimates unknown parameters by matching theoretical moments with sample moments. It may provide a poor approximation to the finite sample distribution of the estimator. Moreover, increasing the number of moment conditions requires substantial increase of the sample size. Second-order least squares (SLS) estimation is an extension of the ordinary least squares method by adding to the criterion function the distance of the squared response variable to its second conditional moment. It is shown in this paper that the SLS is asymptotically more efficient than the GMM when both use the same moment conditions. Moreover, Monte Carlo simulation studies show that SLS performs better than the GMM estimators using three or four moment conditions.
Management of financial risks and sound decision making rely on the accurate information and pred... more Management of financial risks and sound decision making rely on the accurate information and predictive models. Drawing useful information efficiently from big data with complex structures and building accurate models are therefore crucial tasks. Most commonly used methods for statistical inference in dynamic panel data models are based on the differencing transformation of data. However, differencing data may cause substantial loss of information, and therefore the subsequent analysis may fail to capture important features in the original level data. This point is demonstrated by a real data example where we use a semiparametrically efficient estimation method on the level data to reach a more favorable model. In particular, we study a second-order least squares approach which is based on the first two conditional moments of the response variable given the explanatory variables. This estimator is root-N consistent and its asymptotic variance reaches a lower bound semiparametric eff...
Abstract A necessary and sufficient condition of local identifiability is derived by using the we... more Abstract A necessary and sufficient condition of local identifiability is derived by using the well-known Rank Theorem. Using this condition a suitable local parametrization is defined. The topological and geometrical properties of this parametrization are investigated. The results obtained generalize the corresponding results of Deistler and Wang (1989), which deal with essentially the systems with linear restrictions. The treatment covers also the systems without a priori restrictions.
This paper is concerned with general nonlinear regression models where the predictor variables ar... more This paper is concerned with general nonlinear regression models where the predictor variables are subject to Berkson-type measurement errors. The measurement errors are assumed to have a general parametric distribution, which is not necessarily normal. In addition, the distribution of the random error in the regression equation is nonparametric. A minimum distance estimator is proposed, which is based on the first two conditional moments of the response variable given the observed predictor variables. To overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals, a simulation-based estimator is constructed. Consistency and asymptotic normality for both estimators are derived under fairly general regularity conditions.
This paper is concerned with general nonlinear regression models where the predictor variables ar... more This paper is concerned with general nonlinear regression models where the predictor variables are subject to Berkson-type measurement errors. The measurement errors are assumed to have a general parametric distribution, which is not necessarily normal. In addition, the distribution of the random error in the regression equation is nonparametric. A minimum distance estimator is proposed, which is based on the first two conditional moments of the response variable given the observed predictor variables. To overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals, a simulation-based estimator is constructed. Consistency and asymptotic normality fo r both estimators are derived under fairly general regularity conditions.
Management of financial risks and sound decision making rely on the accurate information and pred... more Management of financial risks and sound decision making rely on the accurate information and predictive models. Drawing useful information efficiently from big data with complex structures and building accurate models are therefore crucial tasks. Most commonly used methods for statistical inference in dynamic panel data models are based on the differencing transformation of data. However, differencing data may cause substantial loss of information, and therefore the subsequent analysis may fail to capture important features in the original level data. This point is demonstrated by a real data example where we use a semiparametrically efficient estimation method on the level data to reach a more favorable model. In particular, we study a second-order least squares approach which is based on the first two conditional moments of the response variable given the explanatory variables. This estimator is root-N consistent and its asymptotic variance reaches a lower bound semiparametric eff...
Breast cancer pathologic complete response (pCR) to neoadjuvant chemotherapy (NAC) varies with tu... more Breast cancer pathologic complete response (pCR) to neoadjuvant chemotherapy (NAC) varies with tumor subtype. The purpose of this study was to identify an early treatment window for predicting pCR based on tumor subtype, pretreatment total hemoglobin (tHb) level, and early changes in tHb following NAC. Twenty-two patients (mean age 56Â years, range 34-74Â years) were assessed using a near-infrared imager coupled with an Ultrasound system prior to treatment, 7Â days after the first treatment, at the end of each of the first three cycles, and before their definitive surgery. Pathologic responses were dichotomized by the Miller-Payne system. Tumor vascularity was assessed from tHb; vascularity changes during NAC were assessed from a percentage tHb normalized to the pretreatment level (%tHb). After training the logistic prediction models using the previous study data, we assessed the early treatment window for predicting pathological response according to their tumor subtype (human epiderm...
The Ensemble Transform Kalman Filter (ETKF) assimilation scheme has recently seen rapid developme... more The Ensemble Transform Kalman Filter (ETKF) assimilation scheme has recently seen rapid development and wide application. As a specific implementation of the Ensemble Kalman Filter (EnKF), the ETKF is computationally more efficient than the conventional EnKF. However, the current implementation of the ETKF still has some limitations when the observation operator is strongly nonlinear. One problem is that the nonlinear operator and its tangent-linear operator are iteratively calculated in the minimization of a nonlinear objective function similar to 4DVAR, which may be computationally expensive. Another problem is that it uses the tangent-linear approximation of the observation operator to estimate the multiplicative inflation factor of the forecast errors, which may not be sufficiently accurate. <br><br> This study seeks a way to avoid these problems. First, we apply the second-order Taylor approximation of the nonlinear observation operator to avoid iteratively calculat...
ABSTRACT We study a linear mixed effects model for longitudinal data, where the response variable... more ABSTRACT We study a linear mixed effects model for longitudinal data, where the response variable and covariates with fixed effects are subject to measurement error. We propose a method of moment estimation that does not require any assumption on the functional forms of the distributions of random effects and other random errors in the model. For a classical measurement error model we apply the instrumental variable approach to ensure identifiability of the parameters. Our methodology, without instrumental variables, can be applied to Berkson measurement errors. Using simulation studies, we investigate the finite sample performances of the estimators and show the impact of measurement error on the covariates and the response on the estimation procedure. The results show that our method performs quite satisfactory, especially for the fixed effects with measurement error (even under misspecification of measurement error model). This method is applied to a real data example of a large birth and child cohort study.
Volume 2: 29th Design Automation Conference, Parts A and B, 2003
ABSTRACT The presence of black-box functions in engineering design, which are usually computation... more ABSTRACT The presence of black-box functions in engineering design, which are usually computation-intensive, demands efficient global optimization methods. This work proposes a new global optimization method for black-box functions. The global optimization method is based on a novel mode-pursuing sampling (MPS) method which systematically generates more sample points in the neighborhood of the function mode while statistically covers the entire search space. Quadratic regression is performed to detect the region containing the global optimum. The sampling and detection process iterates until the global optimum is obtained. Through intensive testing, this method is found to be effective, efficient, robust, and applicable to both continuous and discontinuous functions. It supports simultaneous computation and applies to both unconstrained and constrained optimization problems. Because it does not call any existing global optimization tool, it can be used as a standalone global optimization method for inexpensive problems as well. Limitation of the method is also identified and discussed.
Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this... more Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.
This paper studies a minimum distance moment estimator for general nonlinear regression models wi... more This paper studies a minimum distance moment estimator for general nonlinear regression models with Berkson-type measurement errors in predictor variables. The estimator is based on the first two conditional moments of the re-sponse variable given the observed predictor variable. It is shown that under gen-eral regularity conditions the proposed estimator is consistent and asymptotically normally distributed.
We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or pi... more We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general nonlinear boundaries. The jump process can be any integer-valued process and jump sizes can have general distributions. Moreover, the jump sizes can be even correlated and/or non-identically distributed. The numerical algorithm is straightforward and easy to implement. Some numerical examples are presented.
We propose an approach to compute the boundary crossing probabilities for a class of diffusion pr... more We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein-Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear boundaries are obtained, which are useful in evaluation and comparison of various omputational algorithms. Moreover, numerical computation can be easily done by Monte Carlo integration and the approximation errors for general boundaries are automatically calculated. Some numerical examples are presented.
We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or pi... more We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general nonlinear boundaries. The jump process can be any integer-valued process and jump sizes can have general distributions. Moreover, the jump sizes can be even correlated and/or non-identically distributed. The numerical algorithm is straightforward and easy to implement. Some numerical examples are presented.
Generalized method of moments (GMM) is an estimation technique which estimates unknown parameters... more Generalized method of moments (GMM) is an estimation technique which estimates unknown parameters by matching theoretical moments with sample moments. It may provide a poor approximation to the finite sample distribution of the estimator. Moreover, increasing the number of moment conditions requires substantial increase of the sample size. Second-order least squares (SLS) estimation is an extension of the ordinary least squares method by adding to the criterion function the distance of the squared response variable to its second conditional moment. It is shown in this paper that the SLS is asymptotically more efficient than the GMM when both use the same moment conditions. Moreover, Monte Carlo simulation studies show that SLS performs better than the GMM estimators using three or four moment conditions.
Management of financial risks and sound decision making rely on the accurate information and pred... more Management of financial risks and sound decision making rely on the accurate information and predictive models. Drawing useful information efficiently from big data with complex structures and building accurate models are therefore crucial tasks. Most commonly used methods for statistical inference in dynamic panel data models are based on the differencing transformation of data. However, differencing data may cause substantial loss of information, and therefore the subsequent analysis may fail to capture important features in the original level data. This point is demonstrated by a real data example where we use a semiparametrically efficient estimation method on the level data to reach a more favorable model. In particular, we study a second-order least squares approach which is based on the first two conditional moments of the response variable given the explanatory variables. This estimator is root-N consistent and its asymptotic variance reaches a lower bound semiparametric eff...
Abstract A necessary and sufficient condition of local identifiability is derived by using the we... more Abstract A necessary and sufficient condition of local identifiability is derived by using the well-known Rank Theorem. Using this condition a suitable local parametrization is defined. The topological and geometrical properties of this parametrization are investigated. The results obtained generalize the corresponding results of Deistler and Wang (1989), which deal with essentially the systems with linear restrictions. The treatment covers also the systems without a priori restrictions.
This paper is concerned with general nonlinear regression models where the predictor variables ar... more This paper is concerned with general nonlinear regression models where the predictor variables are subject to Berkson-type measurement errors. The measurement errors are assumed to have a general parametric distribution, which is not necessarily normal. In addition, the distribution of the random error in the regression equation is nonparametric. A minimum distance estimator is proposed, which is based on the first two conditional moments of the response variable given the observed predictor variables. To overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals, a simulation-based estimator is constructed. Consistency and asymptotic normality for both estimators are derived under fairly general regularity conditions.
This paper is concerned with general nonlinear regression models where the predictor variables ar... more This paper is concerned with general nonlinear regression models where the predictor variables are subject to Berkson-type measurement errors. The measurement errors are assumed to have a general parametric distribution, which is not necessarily normal. In addition, the distribution of the random error in the regression equation is nonparametric. A minimum distance estimator is proposed, which is based on the first two conditional moments of the response variable given the observed predictor variables. To overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals, a simulation-based estimator is constructed. Consistency and asymptotic normality fo r both estimators are derived under fairly general regularity conditions.
Management of financial risks and sound decision making rely on the accurate information and pred... more Management of financial risks and sound decision making rely on the accurate information and predictive models. Drawing useful information efficiently from big data with complex structures and building accurate models are therefore crucial tasks. Most commonly used methods for statistical inference in dynamic panel data models are based on the differencing transformation of data. However, differencing data may cause substantial loss of information, and therefore the subsequent analysis may fail to capture important features in the original level data. This point is demonstrated by a real data example where we use a semiparametrically efficient estimation method on the level data to reach a more favorable model. In particular, we study a second-order least squares approach which is based on the first two conditional moments of the response variable given the explanatory variables. This estimator is root-N consistent and its asymptotic variance reaches a lower bound semiparametric eff...
Breast cancer pathologic complete response (pCR) to neoadjuvant chemotherapy (NAC) varies with tu... more Breast cancer pathologic complete response (pCR) to neoadjuvant chemotherapy (NAC) varies with tumor subtype. The purpose of this study was to identify an early treatment window for predicting pCR based on tumor subtype, pretreatment total hemoglobin (tHb) level, and early changes in tHb following NAC. Twenty-two patients (mean age 56Â years, range 34-74Â years) were assessed using a near-infrared imager coupled with an Ultrasound system prior to treatment, 7Â days after the first treatment, at the end of each of the first three cycles, and before their definitive surgery. Pathologic responses were dichotomized by the Miller-Payne system. Tumor vascularity was assessed from tHb; vascularity changes during NAC were assessed from a percentage tHb normalized to the pretreatment level (%tHb). After training the logistic prediction models using the previous study data, we assessed the early treatment window for predicting pathological response according to their tumor subtype (human epiderm...
The Ensemble Transform Kalman Filter (ETKF) assimilation scheme has recently seen rapid developme... more The Ensemble Transform Kalman Filter (ETKF) assimilation scheme has recently seen rapid development and wide application. As a specific implementation of the Ensemble Kalman Filter (EnKF), the ETKF is computationally more efficient than the conventional EnKF. However, the current implementation of the ETKF still has some limitations when the observation operator is strongly nonlinear. One problem is that the nonlinear operator and its tangent-linear operator are iteratively calculated in the minimization of a nonlinear objective function similar to 4DVAR, which may be computationally expensive. Another problem is that it uses the tangent-linear approximation of the observation operator to estimate the multiplicative inflation factor of the forecast errors, which may not be sufficiently accurate. <br><br> This study seeks a way to avoid these problems. First, we apply the second-order Taylor approximation of the nonlinear observation operator to avoid iteratively calculat...
ABSTRACT We study a linear mixed effects model for longitudinal data, where the response variable... more ABSTRACT We study a linear mixed effects model for longitudinal data, where the response variable and covariates with fixed effects are subject to measurement error. We propose a method of moment estimation that does not require any assumption on the functional forms of the distributions of random effects and other random errors in the model. For a classical measurement error model we apply the instrumental variable approach to ensure identifiability of the parameters. Our methodology, without instrumental variables, can be applied to Berkson measurement errors. Using simulation studies, we investigate the finite sample performances of the estimators and show the impact of measurement error on the covariates and the response on the estimation procedure. The results show that our method performs quite satisfactory, especially for the fixed effects with measurement error (even under misspecification of measurement error model). This method is applied to a real data example of a large birth and child cohort study.
Volume 2: 29th Design Automation Conference, Parts A and B, 2003
ABSTRACT The presence of black-box functions in engineering design, which are usually computation... more ABSTRACT The presence of black-box functions in engineering design, which are usually computation-intensive, demands efficient global optimization methods. This work proposes a new global optimization method for black-box functions. The global optimization method is based on a novel mode-pursuing sampling (MPS) method which systematically generates more sample points in the neighborhood of the function mode while statistically covers the entire search space. Quadratic regression is performed to detect the region containing the global optimum. The sampling and detection process iterates until the global optimum is obtained. Through intensive testing, this method is found to be effective, efficient, robust, and applicable to both continuous and discontinuous functions. It supports simultaneous computation and applies to both unconstrained and constrained optimization problems. Because it does not call any existing global optimization tool, it can be used as a standalone global optimization method for inexpensive problems as well. Limitation of the method is also identified and discussed.
Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this... more Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.
This paper studies a minimum distance moment estimator for general nonlinear regression models wi... more This paper studies a minimum distance moment estimator for general nonlinear regression models with Berkson-type measurement errors in predictor variables. The estimator is based on the first two conditional moments of the re-sponse variable given the observed predictor variable. It is shown that under gen-eral regularity conditions the proposed estimator is consistent and asymptotically normally distributed.
We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or pi... more We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general nonlinear boundaries. The jump process can be any integer-valued process and jump sizes can have general distributions. Moreover, the jump sizes can be even correlated and/or non-identically distributed. The numerical algorithm is straightforward and easy to implement. Some numerical examples are presented.
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Papers by Liqun Wang