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97
Statistical Behavior and Consistency of Classification Methods based on Convex Risk Minimization
, 2001
"... We study how close the optimal Bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function. The measurement of closeness is characterized by the loss function used in the estimation. ..."
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Cited by 163 (6 self)
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We study how close the optimal Bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function. The measurement of closeness is characterized by the loss function used in the estimation. We show that such a classification scheme can be generally regarded as a (non maximumlikelihood) conditional inclass probability estimate, and we use this analysis to compare various convex loss functions that have appeared in the literature. Furthermore, the theoretical insight allows us to design good loss functions with desirable properties. Another aspect of our analysis is to demonstrate the consistency of certain classification methods using convex risk minimization.
Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities
 Ann. Statist
, 2001
"... We study the rates of convergence of the maximum likelihood estimator (MLE) and posterior distribution in density estimation problems, where the densities are location or locationscale mixtures of normal distributions with the scale parameter lying between two positive numbers. The true density is ..."
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Cited by 62 (10 self)
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We study the rates of convergence of the maximum likelihood estimator (MLE) and posterior distribution in density estimation problems, where the densities are location or locationscale mixtures of normal distributions with the scale parameter lying between two positive numbers. The true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having subGaussian tails. We obtain bounds for Hellinger bracketing entropies for this class, and from these bounds, we deduce the convergence rates of (sieve) MLEs in Hellinger distance. The rate turns out to be �log n � κ / √ n, where κ ≥ 1 is a constant that depends on the type of mixtures and the choice of the sieve. Next, we consider a Dirichlet mixture of normals as a prior on the unknown density. We estimate the prior probability of a certain KullbackLeibler type neighborhood and then invoke a general theorem that computes the posterior convergence rate in terms the growth rate of the Hellinger entropy and the concentration rate of the prior. The posterior distribution is also seen to converge at the rate �log n � κ / √ n in, where κ now depends on the tail behavior of the base measure of the Dirichlet process. 1. Introduction. A
Varianceoptimal hedging for processes with stationary and independent increments. The Annals of Applied Probability
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Local limit of labelled trees and expected volume growth in a random quadrangulation
, 2005
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Analysis of Fourier transform valuation formulas and applications
 Applied Mathematical Finance
"... Abstract. The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the ..."
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Cited by 35 (9 self)
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Abstract. The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the process arises naturally. We also extend these results to the multidimensional case, and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in Lévy and stochastic volatility models.
AFFINE PROCESSES ON POSITIVE SEMIDEFINITE MATRICES
, 910
"... Abstract. This paper provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. These matrixvalued affine processes have arisen from a large and growing range of useful applications in finance, including multiasset o ..."
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Cited by 29 (11 self)
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Abstract. This paper provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. These matrixvalued affine processes have arisen from a large and growing range of useful applications in finance, including multiasset option pricing with stochastic volatility and correlation structures, and fixedincome models with stochastically correlated risk factors and default intensities.
The feedback capacity of the firstorder moving average Gaussian channel. Accepted by
 IEEE Trans. Inform. Theory
, 2006
"... Abstract—Despite numerous bounds and partial results, the feedback capacity of the stationary nonwhite Gaussian additive noise channel has been open, even for the simplest cases such as the firstorder autoregressive Gaussian channel studied by Butman, Tiernan and Schalkwijk, Wolfowitz, Ozarow, and ..."
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Cited by 25 (2 self)
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Abstract—Despite numerous bounds and partial results, the feedback capacity of the stationary nonwhite Gaussian additive noise channel has been open, even for the simplest cases such as the firstorder autoregressive Gaussian channel studied by Butman, Tiernan and Schalkwijk, Wolfowitz, Ozarow, and more recently, Yang, Kavčić, and Tatikonda. Here we consider another simple special case of the stationary firstorder moving average additive Gaussian noise channel and find the feedback capacity in closed form. Specifically, the channel is given by = + =12... where the input satisfies a power constraint and the noise is a firstorder moving average Gaussian process defined by = 1 + 1 with white Gaussian innovations =0 1... We show that the feedback capacity of this channel is. We wish to communicate a message index reliably over the channel. The channel output is causally fed back to the transmitter. We specify a code with the codewords1 satisfying the expected power constraint The proband decoding function ability of error is defined by FB = log 0 where 0 is the unique positive root of the equation
WienerHopf factorization for Lévy processes having positive jumps with rational transforms
 Journal of Applied Probability Vol.45
"... We give the closed form of the ruin probability for a Lévy processes, possibly killed at a constant rate, with completely arbitrary positive distributed jumps, and finite intensity negative jumps with distribution characterized by having a rational Laplace or Fourier transform. Abbreviated Title: ..."
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Cited by 24 (0 self)
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We give the closed form of the ruin probability for a Lévy processes, possibly killed at a constant rate, with completely arbitrary positive distributed jumps, and finite intensity negative jumps with distribution characterized by having a rational Laplace or Fourier transform. Abbreviated Title: WHfactors of Lévy processes with rational jumps. 1
Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectrum, Analysis and PDE
"... Abstract. By combining some ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for OPRL in the a.c. spectral region is implied by convergence of 1 n Kn(x, x) for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. ..."
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Cited by 23 (7 self)
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Abstract. By combining some ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for OPRL in the a.c. spectral region is implied by convergence of 1 n Kn(x, x) for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. We then show that these hypotheses are always valid for ergodic Jacobi matrices with a.c. spectrum and prove that the limit of 1 n Kn(x, x) is ρ∞(x)/w(x) where ρ ∞ is the density of zeros and w is the a.c. weight of the spectral measure. 1.