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13
A nonparametric EM algorithm for multiscale Hawkes processes.
 Journal of Nonparametric Statistics,
, 2011
"... Estimating the conditional intensity of a selfexciting point process is particularly challenging when both exogenous and endogenous effects play a role in clustering. We propose maximum penalized likelihood estimation as a method for simultaneously estimating the background rate and the triggering ..."
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Cited by 21 (1 self)
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Estimating the conditional intensity of a selfexciting point process is particularly challenging when both exogenous and endogenous effects play a role in clustering. We propose maximum penalized likelihood estimation as a method for simultaneously estimating the background rate and the triggering density of Hawkes process intensities that vary over multiple time scales. We compare the accuracy of the algorithm with the recently introduced Model Independent Stochastic Declustering (MISD) algorithm and then use the model to examine selfexcitation in Iraq IED event patterns.
Nonparametric Density Estimation using Wavelets
, 1995
"... Here the problem of density estimation using wavelets is considered. Nonparametric wavelet density estimators have recently been proposed and seem to outperform classical estimators in representing discontinuities and local oscillations. The purpose of this paper is to give a review of di#erent ..."
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Cited by 12 (1 self)
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Here the problem of density estimation using wavelets is considered. Nonparametric wavelet density estimators have recently been proposed and seem to outperform classical estimators in representing discontinuities and local oscillations. The purpose of this paper is to give a review of di#erent types of wavelet density estimators proposed in the literature. Properties, comparisons with classical estimators and applications are stressed. Multivariate extensions are considered. Performances of wavelet estimators are analyzed using a family of normal mixture densities and the Old Faithful Geyser dataset. Key words and phrases: Nonparametric Density Estimation, Wavelets. AMS Subject Classification: 62G07, 42A06. 1 Introduction In nonparametric theory, density estimation is perhaps one of the most investigated topics. Let X 1 , , X n be a sample of size n from an unknown probability density function f . The purpose is to estimate f without any assumption on its form. In this pa...
Estimating Density Functions: A Constrained Maximum Likelihood Approach
, 1998
"... We propose estimating density functions by means of a constrained optimization problem whose criterion function is the maximum likelihood function, and whose constraints model any (prior) information that might be available. The asymptotic justification for such an approach relies on the theory of e ..."
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Cited by 8 (0 self)
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We propose estimating density functions by means of a constrained optimization problem whose criterion function is the maximum likelihood function, and whose constraints model any (prior) information that might be available. The asymptotic justification for such an approach relies on the theory of epiconvergence. A simple numerical example is used to signal the potential of such an approach.
THE MDL PRINCIPLE, PENALIZED LIKELIHOODS, AND STATISTICAL RISK
"... ABSTRACT. We determine, for both countable and uncountable collections of functions, informationtheoretic conditions on a penalty pen(f) such that the optimizer ˆ f of the penalized log likelihood criterion log 1/likelihood(f) + pen(f) has statistical risk not more than the index of resolvability co ..."
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Cited by 4 (3 self)
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ABSTRACT. We determine, for both countable and uncountable collections of functions, informationtheoretic conditions on a penalty pen(f) such that the optimizer ˆ f of the penalized log likelihood criterion log 1/likelihood(f) + pen(f) has statistical risk not more than the index of resolvability corresponding to the accuracy of the optimizer of the expected value of the criterion. If F is the linear span of a dictionary of functions, traditional descriptionlength penalties are based on the number of nonzero terms of candidate fits (the ℓ0 norm of the coefficients) as we review. We specialize our general conclusions to show the ℓ1 norm of the coefficients times a suitable multiplier λ is also an informationtheoretically valid penalty. 1.
A Functional EM Algorithm for Mixing Density Estimation via Nonparameteric Penalized Likelihood Maximization
, 2007
"... When the true mixing distribution is known to be continuous, the nonparametric maximum likelihood estimate of the mixing distribution cannot provide a satisfying answer due to its degeneracy. The estimation of mixing densities is an illposed indirect problem. In this article, we propose to estimate ..."
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When the true mixing distribution is known to be continuous, the nonparametric maximum likelihood estimate of the mixing distribution cannot provide a satisfying answer due to its degeneracy. The estimation of mixing densities is an illposed indirect problem. In this article, we propose to estimate the mixing density by maximizing a penalized likelihood and call the resulting estimate the nonparametric maximum penalized likelihood estimate (NPMPLE). Using theory and methods from the calculus of variations and differential equations, a new functional EM algorithm is derived for computing the NPMPLE of the density. In the algorithm, maximizers in Msteps are found by solving an ordinary differential equation with boundary conditions numerically. Simulation studies show the algorithm outperforms other existing methods such as the popular EMS algorithm and the kernel method. Some theoretical properties of the NPMPLE and the algorithm are also given in the article.
Fisher Information Test of Normality
, 1998
"... An extremal property of normal distributions is that they have the smallest Fisher Information for location among all distributions with the same variance. A new test of normality proposed by Terrell (1995) utilizes the above property by finding that density of maximum likelihood constrained on havi ..."
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An extremal property of normal distributions is that they have the smallest Fisher Information for location among all distributions with the same variance. A new test of normality proposed by Terrell (1995) utilizes the above property by finding that density of maximum likelihood constrained on having the expected Fisher Information under normality based on the sample variance. The test statistic is then constructed as a ratio of the resulting likelihood against that of normality. Since the asymptotic distribution of this test statistic is not available, the critical values for n = 3 to 200 have been obtained by simulation and smoothed using polynomials. An extensive power study shows that the test has superior power against distributions that are symmetric and leptokurtic (longtailed). Another advantage of the test over existing ones is the direct depiction of any deviation from normality in the form of a density estimate. This is evident when the test is applied to several real data sets. Testing of normality in residuals is also investigated. Various approaches in dealing with residuals being possibly heteroscedastic and correlated suffer from a loss of power. The approach with the fewest undesirable features is to use the Ordinary Least
Penalized maximum likelihood estimation with l 1 penalty
"... Abstract We focus on density estimation using penalized loglikelihood method. We aim at building an adaptive estimator in the sense that it converges at the optimal rate of convergence without prior knowledge of its regularity. For this, we penalize the loglikelihood by a function, which depends on ..."
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Abstract We focus on density estimation using penalized loglikelihood method. We aim at building an adaptive estimator in the sense that it converges at the optimal rate of convergence without prior knowledge of its regularity. For this, we penalize the loglikelihood by a function, which depends on the roughness of the density: the l 1 norm of the wavelet coefficients of the logdensity. In this setting, we prove adaptivity for l 2 norm over a certain class of sets, Besov spaces.
MDL, Penalized Likelihood, and Statistical Risk
"... AbstractWe determine, for both countable and uncountable collections of functions, informationtheoretic conditions on a penalty pen(f ) such that the optimizerf of the penalized log likelihood criterion log 1/likelihood(f )+pen(f ) has risk not more than the index of resolvability corresponding t ..."
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AbstractWe determine, for both countable and uncountable collections of functions, informationtheoretic conditions on a penalty pen(f ) such that the optimizerf of the penalized log likelihood criterion log 1/likelihood(f )+pen(f ) has risk not more than the index of resolvability corresponding to the accuracy of the optimizer of the expected value of the criterion. If F is the linear span of a dictionary of functions, traditional descriptionlength penalties are based on the number of nonzero terms (the 0 norm of the coefficients). We specialize our general conclusions to show the 1 norm of the coefficients times a suitable multiplier λ is also an informationtheoretically valid penalty.
Uncertainty Quantification using Exponential EpiSplines
"... ABSTRACT: We quantify uncertainty in complex systems by a flexible, nonparametric framework for estimating probability density functions of output quantities of interest. The framework systematically incorporates soft information about the system from engineering judgement and experience to improve ..."
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ABSTRACT: We quantify uncertainty in complex systems by a flexible, nonparametric framework for estimating probability density functions of output quantities of interest. The framework systematically incorporates soft information about the system from engineering judgement and experience to improve the estimates and ensure that they are consistent with prior knowledge. The framework is based on a maximum likelihood criterion, with episplines facilitating rapid solution of the resulting optimization problems. In four numerical examples with few realizations of the system output, we identify the main features of output densities even for nonsmooth and discontinuous system function and highdimensional inputs. 1
WHAT DO KERNEL DENSITY ESTIMATORS OPTIMIZE?
"... Abstract. Some linkages between kernel and penalty methods of density estimation are explored. It is recalled that classical Gaussian kernel density estimation can be viewed as the solution of the heat equation with initial condition given by data. We then observe that there is a direct relationship ..."
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Abstract. Some linkages between kernel and penalty methods of density estimation are explored. It is recalled that classical Gaussian kernel density estimation can be viewed as the solution of the heat equation with initial condition given by data. We then observe that there is a direct relationship between the kernel method and a particular penalty method of density estimation. For this penalty method, solutions can be characterized as a weighted average of Gaussian kernel density estimates, the average taken with respect to the bandwidth parameter. A Laplace transform argument shows that this weighted average of Gaussian kernel estimates is equivalent to a fixed bandwidth kernel estimate using a Laplace kernel. Extensions to higher order kernels are considered and some connections to penalized likelihood density estimators are made in the concluding sections. 1.