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Lizhong Zheng - One of the best experts on this subject based on the ideXlab platform.
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Polynomial spectral decomposition of Conditional Expectation operators
2016 54th Annual Allerton Conference on Communication Control and Computing (Allerton), 2016Co-Authors: Anuran Makur, Lizhong ZhengAbstract:The spectral structure of Conditional Expectation operators underlies several useful notions in statistics and information theory. For instance, the second largest singular value of such an operator is known as maximal correlation, which has received considerable attention in the context of performing non-linear regression and analyzing contraction coefficients. Given source-channel pairs, we study the singular value decompositions of the corresponding Conditional Expectation operators, and derive necessary and sufficient conditions on the Conditional moments that characterize when the Conditional Expectation operators have singular vectors that are orthogonal polynomials. Furthermore, we illustrate that Conditional Expectation operators constructed using well-known natural exponential families with quadratic variance functions and their conjugate priors have orthogonal polynomial singular vectors. In particular, the Gaussian source and Gaussian channel beget the Hermite case, the gamma source and Poisson channel beget the Laguerre case, and the beta source and binomial channel beget the Jacobi case.
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Allerton - Polynomial spectral decomposition of Conditional Expectation operators
2016 54th Annual Allerton Conference on Communication Control and Computing (Allerton), 2016Co-Authors: Anuran Makur, Lizhong ZhengAbstract:The spectral structure of Conditional Expectation operators underlies several useful notions in statistics and information theory. For instance, the second largest singular value of such an operator is known as maximal correlation, which has received considerable attention in the context of performing non-linear regression and analyzing contraction coefficients. Given source-channel pairs, we study the singular value decompositions of the corresponding Conditional Expectation operators, and derive necessary and sufficient conditions on the Conditional moments that characterize when the Conditional Expectation operators have singular vectors that are orthogonal polynomials. Furthermore, we illustrate that Conditional Expectation operators constructed using well-known natural exponential families with quadratic variance functions and their conjugate priors have orthogonal polynomial singular vectors. In particular, the Gaussian source and Gaussian channel beget the Hermite case, the gamma source and Poisson channel beget the Laguerre case, and the beta source and binomial channel beget the Jacobi case.
Yousef Estaremi - One of the best experts on this subject based on the ideXlab platform.
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Decompositions of Weighted Conditional Expectation Type Operators
Mediterranean Journal of Mathematics, 2017Co-Authors: Yousef Estaremi, Massoud AminiAbstract:In this paper, we investigate boundedness, polar decomposition and spectral decomposition of weighted Conditional Expectation type operators on \(L^2(\Sigma )\). Some examples are provided to illustrate concrete application of the main results of the paper.
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Unbounded Weighted Conditional Expectation Operators
Complex Analysis and Operator Theory, 2015Co-Authors: Yousef EstaremiAbstract:In this note some basic properties of unbounded weighted Conditional Expectation operators are investigated. A description of polar decomposition, spectrum, normality, quasi-normality and self-adjointness in this context are provided. Also, we give some necessary and sufficient conditions for $$EM_u$$ to leave invariant its domain. Finally, some examples are provided to illustrate concrete application of the main results of the paper.
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Unbounded weighted Conditional Expectation operators
arXiv: Functional Analysis, 2014Co-Authors: Yousef EstaremiAbstract:In this note basic properties of unbounded weighted Conditional Expectation operators are investigated. A description of polar decomposition and quasinormality in this context are provided. Also, we study hyperexpan- sive weighted Conditional Expectation operators.
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Multiplication Conditional Expectation Type Operators On Orlicz Spaces
arXiv: Functional Analysis, 2013Co-Authors: Yousef EstaremiAbstract:In this paper we consider a generalized Conditional-type Holder- inequality and investigate some classic properties of multiplication Conditional Expectation type operators on Orlicz-spaces.
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Decompositions of Weighted Conditional Expectation Type Operators
arXiv: Functional Analysis, 2013Co-Authors: Yousef EstaremiAbstract:In this paper we investigate boundedness, polar decomposition and spectral decomposition of weighted Conditional Expectation type operators on L^2(\Sigma).
Jeremy Staum - One of the best experts on this subject based on the ideXlab platform.
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Estimating the density of a Conditional Expectation
Electronic Journal of Statistics, 2016Co-Authors: Samuel G. Steckley, Shane G. Henderson, David Ruppert, Ran Yang, Daniel W. Apley, Jeremy StaumAbstract:Given uncertainty in the input model and parameters of a stochastic simulation study, the goal of the study often becomes the estimation of a Conditional Expectation. The Conditional Expectation is expected performance conditioned on the selected model and parameters. The density of this Conditional Expectation describes precisely, and concisely, the impact of input uncertainty on performance prediction. In this paper we estimate the density of a Conditional Expectation using ideas from the field of kernel density estimation. We show that our estimator converges under reasonable conditions and present results on optimal rates of convergence. We present two modifications of this estimator, a local estimator and a bias-corrected estimator. Convergence results are given for these estimators. We study the performance of our estimators on a number of test cases and a call center example in which the arrival process of customers is unknown.
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efficient nested simulation for estimating the variance of a Conditional Expectation
Operations Research, 2011Co-Authors: Yunpeng Sun, Daniel W. Apley, Jeremy StaumAbstract:In a two-level nested simulation, an outer level of simulation samples scenarios, while the inner level uses simulation to estimate a Conditional Expectation given the scenario. Applications include financial risk management, assessing the effects of simulation input uncertainty, and computing the expected value of gathering more information in decision theory. We show that an ANOVA-like estimator of the variance of the Conditional Expectation is unbiased under mild conditions, and we discuss the optimal number of inner-level samples to minimize this estimator's variance given a fixed computational budget. We show that as the computational budget increases, the optimal number of inner-level samples remains bounded. This finding contrasts with previous work on two-level simulation problems in which the inner-and outer-level sample sizes must both grow without bound for the estimation error to approach zero. The finding implies that the variance of a Conditional Expectation can be estimated to arbitrarily high precision by a simulation experiment with a fixed inner-level computational effort per scenario, which we call a one-and-a-half-level simulation. Because the optimal number of inner-level samples is often quite small, a one-and-a-half-level simulation can avoid the heavy computational burden typically associated with two-level simulation.
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a confidence interval for tail Conditional Expectation via two level simulation
Winter Simulation Conference, 2007Co-Authors: Hai Lan, Barry L Nelson, Jeremy StaumAbstract:We develop and evaluate a two-level simulation procedure that produces a confidence interval for tail Conditional Expectation, otherwise known as Conditional tail Expectation. This risk measure is closely related to Conditional value-at-risk, expected shortfall, and worst Conditional Expectation. The outer level of simulation generates risk factors and the inner level estimates each expected loss Conditional on the risk factor. Our procedure uses the statistical theory of empirical likelihood to construct a confidence interval, and it uses tools from the ranking-and-selection literature to make the simulation efficient.
Anuran Makur - One of the best experts on this subject based on the ideXlab platform.
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Polynomial spectral decomposition of Conditional Expectation operators
2016 54th Annual Allerton Conference on Communication Control and Computing (Allerton), 2016Co-Authors: Anuran Makur, Lizhong ZhengAbstract:The spectral structure of Conditional Expectation operators underlies several useful notions in statistics and information theory. For instance, the second largest singular value of such an operator is known as maximal correlation, which has received considerable attention in the context of performing non-linear regression and analyzing contraction coefficients. Given source-channel pairs, we study the singular value decompositions of the corresponding Conditional Expectation operators, and derive necessary and sufficient conditions on the Conditional moments that characterize when the Conditional Expectation operators have singular vectors that are orthogonal polynomials. Furthermore, we illustrate that Conditional Expectation operators constructed using well-known natural exponential families with quadratic variance functions and their conjugate priors have orthogonal polynomial singular vectors. In particular, the Gaussian source and Gaussian channel beget the Hermite case, the gamma source and Poisson channel beget the Laguerre case, and the beta source and binomial channel beget the Jacobi case.
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Allerton - Polynomial spectral decomposition of Conditional Expectation operators
2016 54th Annual Allerton Conference on Communication Control and Computing (Allerton), 2016Co-Authors: Anuran Makur, Lizhong ZhengAbstract:The spectral structure of Conditional Expectation operators underlies several useful notions in statistics and information theory. For instance, the second largest singular value of such an operator is known as maximal correlation, which has received considerable attention in the context of performing non-linear regression and analyzing contraction coefficients. Given source-channel pairs, we study the singular value decompositions of the corresponding Conditional Expectation operators, and derive necessary and sufficient conditions on the Conditional moments that characterize when the Conditional Expectation operators have singular vectors that are orthogonal polynomials. Furthermore, we illustrate that Conditional Expectation operators constructed using well-known natural exponential families with quadratic variance functions and their conjugate priors have orthogonal polynomial singular vectors. In particular, the Gaussian source and Gaussian channel beget the Hermite case, the gamma source and Poisson channel beget the Laguerre case, and the beta source and binomial channel beget the Jacobi case.
Shane G. Henderson - One of the best experts on this subject based on the ideXlab platform.
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Estimating the density of a Conditional Expectation
Electronic Journal of Statistics, 2016Co-Authors: Samuel G. Steckley, Shane G. Henderson, David Ruppert, Ran Yang, Daniel W. Apley, Jeremy StaumAbstract:Given uncertainty in the input model and parameters of a stochastic simulation study, the goal of the study often becomes the estimation of a Conditional Expectation. The Conditional Expectation is expected performance conditioned on the selected model and parameters. The density of this Conditional Expectation describes precisely, and concisely, the impact of input uncertainty on performance prediction. In this paper we estimate the density of a Conditional Expectation using ideas from the field of kernel density estimation. We show that our estimator converges under reasonable conditions and present results on optimal rates of convergence. We present two modifications of this estimator, a local estimator and a bias-corrected estimator. Convergence results are given for these estimators. We study the performance of our estimators on a number of test cases and a call center example in which the arrival process of customers is unknown.
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simulation input modeling a kernel approach to estimating the density of a Conditional Expectation
Winter Simulation Conference, 2003Co-Authors: Samuel G. Steckley, Shane G. HendersonAbstract:Given uncertainty in the input model and parameters of a simulation study, the goal of the simulation study often becomes the estimation of a Conditional Expectation. The Conditional Expectation is expected performance Conditional on the selected model and parameters. The distribution of this Conditional Expectation describes precisely, and concisely, the impact of input uncertainty on performance prediction. In this paper we estimate the density of a Conditional Expectation using ideas from the field of kernel density estimation. We present a result on asymptotically optimal rates of convergence and examine a number of numerical examples.
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Winter Simulation Conference - Simulation input modeling: a kernel approach to estimating the density of a Conditional Expectation
2003Co-Authors: Samuel G. Steckley, Shane G. HendersonAbstract:Given uncertainty in the input model and parameters of a simulation study, the goal of the simulation study often becomes the estimation of a Conditional Expectation. The Conditional Expectation is expected performance Conditional on the selected model and parameters. The distribution of this Conditional Expectation describes precisely, and concisely, the impact of input uncertainty on performance prediction. In this paper we estimate the density of a Conditional Expectation using ideas from the field of kernel density estimation. We present a result on asymptotically optimal rates of convergence and examine a number of numerical examples.