Topic
Gaussian process
About: Gaussian process is a research topic. Over the lifetime, 18944 publications have been published within this topic receiving 486645 citations. The topic is also known as: Gaussian stochastic process.
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TL;DR: In this article, the authors develop an asymptotic theory of inference for an unrestricted two-regime TAR model with an autoregressive unit root, which is based on a new set of tools that combine unit root and empirical process methods.
Abstract: This paper develops an asymptotic theory of inference for an unrestricted two-regime Ž. threshold autoregressive TAR model with an autoregressive unit root. We find that the asymptotic null distribution of Wald tests for a threshold are nonstandard and different from the stationary case, and suggest basing inference on a bootstrap approximation. We also study the asymptotic null distributions of tests for an autoregressive unit root, and find that they are nonstandard and dependent on the presence of a threshold effect. We propose both asymptotic and bootstrap-based tests. These tests and distribution theory Ž. Ž allow for the joint consideration of nonlinearity thresholds and nonstationary unit . roots . Our limit theory is based on a new set of tools that combine unit root asymptotics with empirical process methods. We work with a particular two-parameter empirical process that converges weakly to a two-parameter Brownian motion. Our limit distributions involve stochastic integrals with respect to this two-parameter process. This theory is entirely new and may find applications in other contexts. We illustrate the methods with an application to the U.S. monthly unemployment rate. We find strong evidence of a threshold effect. The point estimates suggest that the threshold effect is in the short-run dynamics, rather than in the dominate root. While the conventional ADF test for a unit root is insignificant, our TAR unit root tests are arguably significant. The evidence is quite strong that the unemployment rate is not a unit root process, and there is considerable evidence that the series is a stationary TAR process.
719 citations
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TL;DR: In this paper, a functional central limit theorem for empirical processes indexed by classes of functions was established, which depends on a simple new sufficient condition for a Gaussian process to achieve its maximum almost surely at a unique point.
Abstract: We establish a new functional central limit theorem for empirical processes indexed by classes of functions. In a neighborhood of a fixed parameter point, an $n^{-1/3}$ rescaling of the parameter is compensated for by an $n^{2/3}$ rescaling of the empirical measure, resulting in a limiting Gaussian process. By means of a modified continuous mapping theorem for the location of the maximizing value, we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure. These statistics include the short, Rousseeuw's least median of squares estimator, Manski's maximum score estimator, and the maximum likelihood estimator for a monotone density. The limit theory depends on a simple new sufficient condition for a Gaussian process to achieve its maximum almost surely at a unique point.
718 citations
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01 Feb 1999TL;DR: The main aim of this paper is to provide a tutorial on regression with Gaussian processes, starting from Bayesian linear regression, and showing how by a change of viewpoint one can see this method as a Gaussian process predictor based on priors over functions, rather than on prior over parameters.
Abstract: The main aim of this paper is to provide a tutorial on regression with Gaussian processes. We start from Bayesian linear regression, and show how by a change of viewpoint one can see this method as a Gaussian process predictor based on priors over functions, rather than on priors over parameters. This leads in to a more general discussion of Gaussian processes in section 4. Section 5 deals with further issues, including hierarchical modelling and the setting of the parameters that control the Gaussian process, the covariance functions for neural network models and the use of Gaussian processes in classification problems.
712 citations
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TL;DR: In this paper, the authors consider density estimates of the usual type generated by a weight function and obtain limit theorems for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity.
Abstract: We consider density estimates of the usual type generated by a weight function Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved
703 citations
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TL;DR: An information-theoretic perspective on optimum transmitter strategies, and the gains obtained by employing them, for systems with transmit antenna arrays and imperfect channel feedback is provided.
Abstract: The use of channel feedback from receiver to transmitter is standard in wireline communications. While knowledge of the channel at the transmitter would produce similar benefits for wireless communications as well, the generation of reliable channel feedback is complicated by the rapid time variations of the channel for mobile applications. The purpose of this paper is to provide an information-theoretic perspective on optimum transmitter strategies, and the gains obtained by employing them, for systems with transmit antenna arrays and imperfect channel feedback. The spatial channel, given the feedback, is modeled as a complex Gaussian random vector. Two extreme cases are considered: mean feedback, in which the channel side information resides in the mean of the distribution, with the covariance modeled as white, and covariance feedback, in which the channel is assumed to be varying too rapidly to track its mean, so that the mean is set to zero, and the information regarding the relative geometry of the propagation paths is captured by a nonwhite covariance matrix. In both cases, the optimum transmission strategies, maximizing the information transfer rate, are determined as a solution to simple numerical optimization problems. For both feedback models, our numerical results indicate that, when there is a moderate disparity between the strengths of different paths from the transmitter to the receiver, it is nearly optimal to employ the simple beamforming strategy of transmitting all available power in the direction which the feedback indicates is the strongest.
703 citations