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Showing papers on "Gaussian process published in 1989"


Journal ArticleDOI
TL;DR: Asymptotic normality of the maximum likelihood estimator for the parameters of a long range dependent Gaussian process is proved in this paper, where the limit of the Fisher information matrix is derived for such processes which implies efficiency of the estimator.
Abstract: Asymptotic normality of the maximum likelihood estimator for the parameters of a long range dependent Gaussian process is proved. Furthermore, the limit of the Fisher information matrix is derived for such processes which implies efficiency of the estimator and of an approximate maximum likelihood estimator studied by Fox and Taqqu. The results are derived by using asymptotic properties of Toeplitz matrices and an equicontinuity property of quadratic forms.

891 citations


Journal ArticleDOI
TL;DR: Inverse Gaussian distributions have been used for life testing and reliability as mentioned in this paper, and they have been applied in a variety of applications, such as life testing, reliability, and life assurance.
Abstract: 1. Introduction 2. Properties of the Inverse Gaussian Distribution 3. Genesis 4. Certain Useful Transformations and Characterizations 5. Sampling and Estimation of Parameters 6. Significance Tests 7. Bayesian Inference 8. Regression Analysis 9. Life Testing and Reliability 10. Applications 11. Additional Topics

581 citations


Journal ArticleDOI
TL;DR: In this paper, the authors compared three estimators, namely, the moment method (MM), the maximum likelihood (ML), and the moment/Newton step (MNS), for estimating the parameters of a three-parameter generalized Gaussian distribution.
Abstract: The primary objective of this paper is to compare the large‐sample as well as the small‐sample properties of different methods for estimating the parameters of a three‐parameter generalized Gaussian distribution. Three estimators, namely, the moment method (MM), the maximum‐likelihood (ML), and the moment/Newton‐step (MNS) estimators, are considered. The applicability of general asymptotic optimality results of the efficient ML and MNS estimation techniques is studied in the generalized Gaussian context. The asymptotic normal distributions of the estimators are obtained. The asymptotic relative superiority of the ML estimator or its variant, the MNS estimator, over the moment method is studied in terms of asymptotic relative efficiency. Based on this study, it is concluded that deviations from normality in the underlying distribution of the data necessitate the use of the efficient ML or MNS methods. In the small‐sample case, a detailed comparative study of the estimators is made possible by extensive Monte Carlo simulations. From this study, it is concluded that the maximum‐likelihood method is found to be significantly superior for heavy‐tailed distributions. In a region of the parameter space corresponding to the vicinity of the Gaussian distribution, the moment method compares well with the other methods. Further, the MNS estimator is shown to perform best for light‐tailed distributions. The simulation results are shown to lend support to analytically derived asymptotic results for each of the methods.

324 citations


Journal ArticleDOI
TL;DR: An asymptotic equipartition theorem for nonstationary Gaussian processes is proved and it is proved that the feedback capacity C/sub FB/ in bits per transmission and the nonfeedback capacity C satisfy C > C >.
Abstract: The capacity of time-varying additive Gaussian noise channels with feedback is characterized. Toward this end, an asymptotic equipartition theorem for nonstationary Gaussian processes is proved. Then, with the aid of certain matrix inequalities, it is proved that the feedback capacity C/sub FB/ in bits per transmission and the nonfeedback capacity C satisfy C >

240 citations


Journal ArticleDOI
TL;DR: In this paper, a power covariance with range parameter is proposed for the spatial linear model and a convenient profile likelihood is introduced and studied in view of potential multimodal likelihoods for small samples.
Abstract: SUMMARY A popular covariance scheme used for the spatial linear model in geostatistics has spherical form. However, the likelihood is not twice differentiable with respect to the range parameter, and this raises some questions regarding the unimodality of the likelihood. We compare the likelihoods of the spatial linear model for small samples under this scheme and the doubly geometric scheme. Also, a power covariance with range parameter is proposed. In view of potential multimodal likelihoods for small samples for this model, a convenient profile likelihood is introduced and studied.

140 citations


Journal ArticleDOI
TL;DR: In this paper, the identifiability conditions under which the MA coefficient matrices, the input statistics, and the order q can be uniquely determined are studied. But the selection of a unique representative from the equivalence class corresponding to a given cumulant structure involves fewer restrictions than that corresponding to the given covariance structure.
Abstract: Given cumulants of a stationary, perhaps noisy, non-Gaussian r-variate moving average, MA(q) process, identifiability conditions are studied, under which the MA coefficient matrices, the input statistics, and the order q can be uniquely determined. The selection of a unique representative from the equivalence class corresponding to a given cumulant structure involves fewer restrictions than that corresponding to a given covariance structure. Two algorithms are derived for estimating the (possibly) nonminimum-phase MA coefficient matrices. >

104 citations


Journal ArticleDOI
TL;DR: It is shown that, if the process is Gaussian and B/sub k/( tau ) is a Fourier integral with respect to a density function g/ sub k/( lambda ), a two-dimensional periodogram can be smoothed along a line of constant difference frequency to provide a consistent estimator for g/sub g/( lambda ).
Abstract: Correlation functions of continuous-time periodically correlated processes can be represented by a Fourier series with coefficient functions. It is shown that the usual estimator for stationary covariances, formed from a single sample path of the process, can be simply modified to provide a consistent (in quadratic mean) estimator for any of the coefficient functions resulting from the aforementioned representation. It is shown that, if the process is Gaussian and B/sub k/( tau ) is a Fourier integral with respect to a density function g/sub k/( lambda ), a two-dimensional periodogram, formed from a single sample function, can be smoothed along a line of constant difference frequency to provide a consistent estimator for g/sub k/( lambda ). This natural extension of the well-known procedure for stationary processes provides a method for nonparametric spectral analysis of periodically correlated processes. >

96 citations


Journal ArticleDOI
TL;DR: It is shown that a given finite-duration sequence rho can be extended to be the covariance of a periodic stationary processes whenever the Toeplitz matrix R generated by this sequence is strictly positive definite.
Abstract: The class of nonnegative definite Toeplitz matrices that can be embedded in nonnegative definite circulant matrices of a larger size is characterized. An equivalent characterization in terms of the spectrum of the underlying process is also presented, together with the corresponding extremal processes. It is shown that a given finite-duration sequence rho can be extended to be the covariance of a periodic stationary processes whenever the Toeplitz matrix R generated by this sequence is strictly positive definite. The sequence rho =1, cos alpha , cos 2 alpha with ( alpha / pi ) irrational, which has a unique nonperiodic extension as a covariance sequence, demonstrates that the strictness is needed. A simple constructive proof supplies a bound on the abovementioned period in terms of the minimal eigenvalue of R. It also yields, under the same conditions, an extension of rho to covariances that eventually decay to zero. For the maximum-likelihood estimate of the covariance of a stationary Gaussian process, the extension length required for using the estimate-maximize iterative algorithm is determined. >

87 citations


Journal ArticleDOI
TL;DR: In this article, the authors derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum.
Abstract: Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t 1(1 + t 2), .

76 citations


Book ChapterDOI
TL;DR: In this article, the authors adapted the bootstrap method, proposed by Efron (1979, 1982), to the present situation, and established weak convergence of mean residual life, total time of test, Lorenz and Goldie processes to appropriate Gaussian processes.
Abstract: In our previous sections we have established weak convergence of mean residual life, total time of test, Lorenz and Goldie processes to appropriate Gaussian processes. Apart from a few special cases (cf. Section 8), these limiting processes are functions of the underlying distributions. Consequently when testing for statistical hypotheses for example, one would have to compute the resulting limiting distribution for each F of interest. The same is true when trying to use our results for constructing confidence bands for the theoretical functionals of the said empirical processes. This type of problems can be solved by adapting the bootstrap method, proposed by Efron (1979, 1982), to the present situation. As we will see, the bootstrap method is simply a Monte Carlo simulation determined by the given observations.

67 citations


Journal ArticleDOI
Peter Hänggi1
TL;DR: In this article, a path integral solution for a nonlinear stochastic flow driven by Markovian or non-Markovian colored noise ζ(t) is presented.
Abstract: For a nonlinear stochastic flow driven by Markovian or non-Markovian colored noise ζ(t) we present the path integral solution for the single-event probabilityp(x,t). The solution has the structure of a complex-valued double path integral. Explicit formulas for the action functional, i.e., the non-Markovian Onsager-Machlup functional, are derived for the case that ζ(t) is characterized by a stationary Gaussian process. Moreover, we derive explicit results for (generalized) Poissonian colored shot noise ζ(t). The use of the path integral solution is elucidated by a weak noise analysis of the WKB-type. As a simple application, we consider stochastic bistability driven by colored noise with an extremely long correlation time.

Journal ArticleDOI
TL;DR: In this article, a higher order technique is presented to compute the probability distribution function of the forced response of a mistuned bladed disk assembly, where the modal stiffness of each blade is assumed to be a random variable with a Gaussian distribution.

Journal ArticleDOI
TL;DR: In this paper, it is shown that consistent estimates of the optimal bandwidths for kernel estimators of the location and size of a peak of a regression function are available and that these estimates yield the same joint asymptotic distribution of locations and sizes of the peak as the bandwidths themselves.
Abstract: It is shown that consistent estimates of the optimal bandwidths for kernel estimators of location and size of a peak of a regression function are available. Such estimates yield the same joint asymptotic distribution of location and size of a peak as the optimal bandwidths themselves. Therefore data-adaptive efficient estimation of peaks is possible. In order to prove this result, the weak convergence of a two-dimensional stochastic process with appropriately scaled bandwidths as arguments to a Gaussian limiting process is shown. A practical method which leads to consistent estimates of the optimal bandwidths and is therefore asymptotically efficient is proposed and its finite sample properties are investigated by simulation.

Journal ArticleDOI
TL;DR: In this paper, a Slepian model for the local behaviour near the level upcrossings of a chi-µ-process with dependent Gaussian components is presented, and this model is shown to take on a rather simple form, thereby simplifying earlier results by Aronowich and Adler.
Abstract: A Slepian model for the local behaviour near the level upcrossings of a chi²-process with dependent Gaussian components is presented. In case of independent components, this model is shown to take on a rather simple form, thereby simplifying earlier results by Aronowich and Adler. The Slepian model is applied to the envelope of a stationary Gaussian process and used to approximate the probability of "empty" envelope upcrossings, i.e. the probability that an envelope upcrossing is not followed by a level crossing in the original process.

Journal ArticleDOI
TL;DR: Any linear statistic defined on a random-matrix ensemble is shown to be Gaussian distributed, which supports the prediction of weak-disorder perturbation theory in the diffusive, metallic limit for the distribution of conductance.
Abstract: Any linear statistic defined on a random-matrix ensemble is shown to be Gaussian distributed. This supports the prediction of weak-disorder perturbation theory in the diffusive, metallic limit for the distribution of conductance, since conductance is a linear statistic on the ensemble of transfer matrices.

Journal ArticleDOI
TL;DR: In this article, the authors derived limit processes for sequences of stochastic processes defined by partial sums of linear functions of regression residuals, which are Gaussian and are functions of standard Brownian motion.


Journal ArticleDOI
TL;DR: The bootstrap statistical method is applied to the discrepancy in the one-charged-particle decay modes of the tau lepton, eliminating questions about the correctness of the errors ascribed to the branching-fraction measurements and the use of Gaussian error distributions for systematic errors.
Abstract: The bootstrap statistical method is applied to the discrepancy in the one-charged-particle decay modes of the tau lepton. This eliminates questions about the correctness of the errors ascribed to the branching-fraction measurements and the use of Gaussian error distributions for systematic errors. The discrepancy is still seen when the results of the bootstrap analysis are combined with other measurements and with deductions from theory. But the bootstrap method assigns less statistical significance to the discrepancy compared to a method using Gaussian error distributions.

Journal ArticleDOI
TL;DR: In this paper, the seafloor is modeled as a stationary, zero-mean, Gaussian random field completely specified by its autocovariance function, which is used for quantifying ensemble properties of small scale bathymetric features such as abyssal hills.
Abstract: Stochastic methods of analysis are useful for quantifying ensemble properties of small-scale bathymetric features such as abyssal hills. In this paper we model the seafloor as a stationary, zero-mean, Gaussian random field completely specified by its autocovariance function. We formulate an anisotropic autocovariance function that has five free parameters describing the amplitude, anisotropic orientation and aspect ratio, characteristic length, and Hausdorff (fractal) dimension of seafloor topography. Parameters estimated from various seafloor regions by an inversion of Sea Beam data indicate that the seafloor exhibits a wide range of stochastic characteristics within the constraints of the model. Synthetic topography can be generated at arbitrary scale and resolution from the Gaussian model using a Fourier method. Color images of these synthetics are useful for illustrating the stochastic behavior of the model.

Journal ArticleDOI
TL;DR: In this article, an approach to quantum dynamics based entirely on Cartesian coordinates, which covers vibrational as well as rotational motion, is presented, where the initial state is represented in terms of multidimensional Gaussian wave packets and subsequent dynamics can be determined from the dynamics of Gaussians corresponding to just one of these orientations.
Abstract: We present an approach to quantum dynamics, based entirely on Cartesian coordinates, which covers vibrational as well as rotational motion. The initial state is represented in terms of multidimensional Gaussian wave packets. Rotational adaptation to angular momentum eigenstates is done by using angular momentum projection operators. This gives an initial state represented as a weighted superposition of Gaussians with different average orientation in space. It is shown that the subsequent dynamics can be determined from the dynamics of Gaussians corresponding to just one of these orientations. An application to the 3D photodissociation dynamics of ICN is presented. All six degrees of freedom which describe the internal motion of the triatomic are included, the only approximation introduced in the present calculation being the thawed Gaussian wave packet approximation for the dynamics. The total absorption spectrum out of vibrational–rotational eigenstates of ICN as well as fully resolved final product dist...

Journal ArticleDOI
TL;DR: In this article, a theoretically consistent definition of the wind response spectrum based upon the equivalent wind spectrum technique is proposed, a calculation procedure by means of which wind is schematized as a stochastic stationary Gaussian process characterized by a mean velocity profile on which an equivalent turbulent fluctuation, perfectly coherent in space, is superimposed.
Abstract: This paper formulates a theoretically consistent definition of the wind response spectrum based upon the equivalent wind spectrum technique, a calculation procedure by means of which wind is schematized as a stochastic stationary Gaussian process characterized by a mean velocity profile on which an equivalent turbulent fluctuation, perfectly coherent in space, is superimposed. The method presented herein allows the evaluation of the dynamic along‐wind respohse of structures, as well as of the structural behavior to the seismic ground motion, by the well‐known response spectrum technique. This procedure, parallelly applied to wind and earthquake actions, reveals significant conceptual and formal analogies, leading to results characterized by the same order of approximation.

Journal ArticleDOI
TL;DR: In this article, the fluctuations of the finite-size corrections to the free energy per site of the random energy model (REM) and the generalized random energy models (GREM) are investigated.
Abstract: The fluctuations of the finite-size corrections to the free energy per site of the random energy model (REM) and the generalized random energy model (GREM) are investigated. Almost sure behavior for the corrections of order (logN)/N is given. We also prove convergence in distribution for the corrections of order 1/N.

Journal ArticleDOI
TL;DR: This covariant Gaussian approximation is represented, in full analogy with the classical approximation, as an initial truncation of the Dyson-Schwinger equations followed by functional differentiation of the effective action.
Abstract: The variational Gaussian approximation is generalized to the time-dependent approach capable of giving time-dependent Green's functions. This covariant Gaussian approximation is represented, in full analogy with the classical approximation, as an initial truncation of the Dyson-Schwinger equations followed by functional differentiation of the effective action. Intuitively simple Schr\"odinger and Heisenberg pictures of the approximation are also discussed.

Journal ArticleDOI
TL;DR: In this paper, the Curie-Weiss mean field model of ferromagnetism was shown to be Gaussian at the critical temperature, i.e., the fluctuations within the fluctuating field are Gaussian.
Abstract: It has been known for some time that the fluctuations of the Curie-Weiss mean field model of ferromagnetism are non-Gaussian at the critical temperature. Here we establish the presence of a substantial Gaussian element in the critical model by showing that the internal fluctuations, i.e. the fluctuations “within” the fluctuating field, are Gaussian. We show for instance that, as the number n of sites increases to infinity, the system, with the appropriate space and spin scalings, behaves asymptotically as a Brownian motion with randomised drift, and that the distribution of the drift has the non-Gaussian form familiar in this theory. With the more conventional space scaling 1/n it is known that the system degenerates asymptotically to a straight line with random slope. We prove here that the error from the straight line, when appropriately amplified, converges in distribution to a Brownian bridge.

Journal ArticleDOI
TL;DR: In this paper, the special case of a one-parameter first-order conditional process on a rectangular lattice is considered in detail, and formulae are compared with numerical results.
Abstract: Formulae are given for the Fisher information loss on parameters for the mean and the variance when some values of a Gaussian process are not observed. The special case of a one-parameter first-order conditional process on a rectangular lattice is considered in detail, and formulae are compared with numerical results.

Proceedings ArticleDOI
23 May 1989
TL;DR: The authors describe the development of a deterministic algorithm for obtaining the global maximum a posteriori probability (MAP) estimate from an image corrupted by additive Gaussian noise that finds the global MAP estimate in a small number of iterations.
Abstract: The authors describe the development of a deterministic algorithm for obtaining the global maximum a posteriori probability (MAP) estimate from an image corrupted by additive Gaussian noise. The MAP algorithm requires the probability density function of the original undegraded image and the corrupting noise. It is assumed that the original image is represented by a compound model consisting of a 2-D noncausal Gaussian-Markov random field (GMRF) to represent the homogeneous regions and a line process model to represent the discontinuities. The MAP algorithm is written in terms of the compound GMRF model parameters. The solution to the MAP equations is realized by a deterministic relaxation algorithm that is an extension of the graduated nonconvexity (GNC) algorithm and finds the global MAP estimate in a small number of iterations. As a byproduct, the line process configuration determined by the MAP estimate produces an accurate edge map without any additional cost. Experimental results are given to illustrate the usefulness of the method. >

Book ChapterDOI
01 Jan 1989
TL;DR: In this paper, it was shown that the abstract definition of Gaussian measures is equivalent to the usual concrete definitions for ℝn-valued Gaussian random variables and Gaussian stochastic processes.
Abstract: A pervasive undercurrent in the study of Gaussian measures is that they are the class of probability measures which it is natural to study if one requires that we see probabilistic properties which are consonant with the linearity and orthogonality properties of the spaces on which the measures are defined. For instance, one entry point into the theory of Gaussian random variables on an arbitrary real vector space with suitable measurable structure is to define a random variable X as being Gaussian if whenever X1, X2 are two independent copies of X, then the pair (α11X1 + α12X2, α21X1+ α22 X2) has the same law as (X1, X2) for each pair of orthonormal vectors (α11, α12), (α21,α22) ∈ ℝ2. It can be shown that, in the appropriate special cases, this abstract definition is equivalent to the usual concrete definitions for ℝn-valued Gaussian random variables and Gaussian stochastic processes.

Journal ArticleDOI
TL;DR: The authors deal with the problem of deconvolution of Bernoulli-Gaussian random processes observed through linear systems, which corresponds to situations frequently encountered in areas such as geophysics, ultrasonic imaging, and nondestructive evaluation.
Abstract: The authors deal with the problem of deconvolution of Bernoulli-Gaussian random processes observed through linear systems. This corresponds to situations frequently encountered in areas such as geophysics, ultrasonic imaging, and nondestructive evaluation. Deconvolution of such signals is a detection-estimation problem that does not allow purely linear data processing, and the nature of the difficulties greatly depends on the type of representation chosen for the linear system. A MA degenerate state-space representation is used. It presents interesting algorithmic properties and simplifies implementation problems. To obtain a globally recursive procedure, a detection step is inserted in an estimation loop by Kalman filtering. Two recursive detectors based on maximum a posteriori and maximum-likelihood criteria, respectively, are derived and compared. >

Journal ArticleDOI
TL;DR: In this article, the distribution of the large values of the supremum of a sample bounded Gaussian process having a constant variance is estimated using the entropy function of the parameter space endowed with the pseudo-metric induced by the L2-norm of the increments of the process.
Abstract: We obtain an estimate of the distribution of the large values of the supremum of a sample bounded Gaussian process having a constant variance. This estimate uses the entropy function of the parameter space endowed, as usual, with the pseudo-metric induced by the L2-norm of the increments of the process.

Journal ArticleDOI
TL;DR: In this article, it is shown that the complete set of equations of motion as derived by the minimum error method (MEM) cannot be used in practical calculations because of numerical problems, and the conditions for an accurate propagation within that assumption are developed, and a simple method is devised to identify those states, which are propagated accurately.
Abstract: The Gaussian wave packet method has been developed for the simulation of processes like molecular collisions, photodissociation of molecules, and laser excitations of molecules. So far a necessary condition for an accurate result is that the fragment states are propagated accurately. We have considered one‐dimensional bound states described by a Morse potential, and carried out a systematic study of the ability of the Gaussian wave packet method to propagate the stationary states. It is found that the complete set of equations of motion as derived by the minimum error method (MEM) cannot be used in practical calculations because of numerical problems. These are eliminated by the introduction of simplifications such as the independent Gaussian approximation (IGA), where each wave packet is propagated independently. The conditions for an accurate propagation within that assumption are developed, and a simple method is devised to identify those states, which are propagated accurately. This procedure may be u...