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733
Stochastic Perturbation Theory
, 1988
"... . In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variatio ..."
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Cited by 678 (33 self)
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. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variation in the perturbed quantity. Up to the higherorder terms that are ignored in the expansion, these statistics tend to be more realistic than perturbation bounds obtained in terms of norms. The technique is applied to a number of problems in matrix perturbation theory, including least squares and the eigenvalue problem. Key words. perturbation theory, random matrix, linear system, least squares, eigenvalue, eigenvector, invariant subspace, singular value AMS(MOS) subject classifications. 15A06, 15A12, 15A18, 15A52, 15A60 1. Introduction. Let A be a matrix and let F be a matrix valued function of A. Two principal problems of matrix perturbation theory are the following. Given a matrix E, pr...
The Jackknife and the Bootstrap for General Stationary Observations
, 1989
"... this paper we will always consider statistics TN of the form TN (X 1 ; :::; XN ) = T (ae ..."
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Cited by 286 (2 self)
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this paper we will always consider statistics TN of the form TN (X 1 ; :::; XN ) = T (ae
Testing ContinuousTime Models of the Spot Interest Rate
 Review of Financial Studies
, 1996
"... Different continuoustime models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuoustime model by discrete approximations, even though the data are rec ..."
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Cited by 232 (9 self)
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Different continuoustime models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuoustime model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then meanreverts strongly when far away from the mean. The volatility is higher when away from the mean. The continuoustime financial theory has developed extensive tools to price derivative securities when the underlying traded asset(s) or nontraded factor(s) follow stochastic differential equations [see Merton (1990) for examples]. However, as a practical matter, how to specify an appropriate stochastic differential equation is for the most part an unanswered question. For example, many different continuoustime The comments and suggestions of Kerry Back (the editor) and an anonymous referee were very helpful. I am also grateful to George Constantinides,
An efficient Semiparametric Estimator for Binary Response Models
 Econometrica
, 1993
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 118 (2 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Continuous Record Asymptotics for Rolling Sample Variance Estimators
 Econometrica
, 1996
"... It is widely known that conditional covariances of asset returns change over time. ..."
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Cited by 104 (0 self)
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It is widely known that conditional covariances of asset returns change over time.
Mean and Variance of Implicitly Defined Biased Estimators (such as Penalized Maximum Likelihood): Applications to Tomography
 IEEE Tr. Im. Proc
, 1996
"... Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear leastsquares estimation. For such estimators, exact analyti ..."
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Cited by 88 (30 self)
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Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear leastsquares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore investigators usually resort to numerical simulations to examine properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson sta...
Unsupervised Learning of Distributions on Binary Vectors Using Two Layer Networks
, 1994
"... this paper is related to both of these lines of work and has some advantages over each of them. If we find a good model of the distribution, we can tackle other interesting learning problems, such as the problem of estimating the conditional distribution on certain components of the vector ~x when p ..."
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Cited by 65 (1 self)
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this paper is related to both of these lines of work and has some advantages over each of them. If we find a good model of the distribution, we can tackle other interesting learning problems, such as the problem of estimating the conditional distribution on certain components of the vector ~x when provided with the values for the other components (a kind of regression problem), or predicting the actual values for certain components of ~x based on the values of the other components (a kind of pattern completion task). In the example of the binary images presented above, this would amount to the task of recovering the value of a pixel whose value has been corrupted. We can often also use the distribution model to help us in a supervised learning task. This is because it is often easier to express the mapping of an instance to the correct label by using "features" that are correlation patterns among the bits of the instance. For example, it is easier to describe each of the ten digits in terms of patterns such as lines and circles, rather than in terms of the values of individual pixels, that are more likely to change between different instances of the same digit. The process of learning an unknown distribution from examples is usually called density estimation or
Prediction in dynamic models with time dependent conditional heteroskedasticity, Working paper no
, 1990
"... This paper considers forecasting the conditional mean and variance from a singleequation dynamic model with autocorrelated disturbances following an ARMA process, and innovations with timedependent conditional heteroskedasticity as represented by a linear GARCH process. Expressions for the minimum ..."
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Cited by 65 (9 self)
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This paper considers forecasting the conditional mean and variance from a singleequation dynamic model with autocorrelated disturbances following an ARMA process, and innovations with timedependent conditional heteroskedasticity as represented by a linear GARCH process. Expressions for the minimum MSE predictor and the conditional MSE are presented. We also derive the formula for all the theoretical moments of the prediction error distribution from a general dynamic model with GARCHtl, 1) innovations. These results are then used in the construction of ex ante prediction confidence intervals by means of the CornishFisher asymptotic expansion. An empirical example relating to the uncertainty of the expected depreciation of foreign exchange rates illustrates the usefulness of the results. 1.
General Notions of Statistical Depth Function
, 2000
"... Statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data. Here we introduce several general structures for depth functions, classify many existing examples as special cases, and establish results on the possession, or lack th ..."
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Cited by 62 (27 self)
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Statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data. Here we introduce several general structures for depth functions, classify many existing examples as special cases, and establish results on the possession, or lack thereof, of four key properties desirable for depth functions in general. Roughly speaking, these properties may be described as: affine invariance, maximality at center, monotonicity relative to deepest point, and vanishing at infinity. This provides a more systematic basis for selection of a depth function. In particular, from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors.