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Statistical challenges with high dimensionality: Feature selection in knowledge discovery
 Proceedings of the International Congress of Mathematicians
, 2006
"... Abstract. Technological innovations have revolutionized the process of scientific research and knowledge discovery. The availability of massive data and challenges from frontiers of research and development have reshaped statistical thinking, data analysis and theoretical studies. The challenges of ..."
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Cited by 35 (9 self)
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Abstract. Technological innovations have revolutionized the process of scientific research and knowledge discovery. The availability of massive data and challenges from frontiers of research and development have reshaped statistical thinking, data analysis and theoretical studies. The challenges of highdimensionality arise in diverse fields of sciences and the humanities, ranging from computational biology and health studies to financial engineering and risk management. In all of these fields, variable selection and feature extraction are crucial for knowledge discovery. We first give a comprehensive overview of statistical challenges with high dimensionality in these diverse disciplines. We then approach the problem of variable selection and feature extraction using a unified framework: penalized likelihood methods. Issues relevant to the choice of penalty functions are addressed. We demonstrate that for a host of statistical problems, as long as the dimensionality is not excessively large, we can estimate the model parameters as well as if the best model is known in advance. The persistence property in risk minimization is also addressed. The applicability of such a theory and method to diverse statistical problems is demonstrated. Other related problems with highdimensionality are also discussed.
ANOVA FOR DIFFUSIONS AND ITO PROCESSES
 SUBMITTED TO THE ANNALS OF STATISTICS
"... Ito processes are the most common form of continuous semimartingales, and include diffusion processes. The paper is concerned with the nonparametric regression relationship between two such Ito processes. We are interested in the quadratic variation (integrated volatility) of the residual in this re ..."
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Cited by 25 (11 self)
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Ito processes are the most common form of continuous semimartingales, and include diffusion processes. The paper is concerned with the nonparametric regression relationship between two such Ito processes. We are interested in the quadratic variation (integrated volatility) of the residual in this regression, over a unit of time (such as a day). A main conceptual finding is that this quadratic variation can be estimated almost as if the residual process were observed, the difference being that there is also a bias which is of the same asymptotic order as the mixed normal error term. The proposed methodology, “ANOVA for diffusions and Ito processes”, can be used to measure the statistical quality of a parametric model, and, nonparametrically, the appropriateness of a oneregressor model in general. On the other hand, it also helps quantify and characterize the trading (hedging) error in the case of financial applications.
Confidence bands in nonparametric time series regression
, 2008
"... We consider nonparametric estimation of mean regression and conditional variance (or volatility) functions in nonlinear stochastic regression models. Simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct. The imposed dependence structure ..."
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Cited by 10 (3 self)
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We consider nonparametric estimation of mean regression and conditional variance (or volatility) functions in nonlinear stochastic regression models. Simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct. The imposed dependence structure allows applications in many linear and nonlinear autoregressive processes. The results are applied to the S&P 500 Index data. 1. Introduction. There
A datadriven method for estimating conditional densities
, 2003
"... this article, we extend the idea of crossvalidation (CV) for choosing the smoothing parameter of the "doublekernel" local linear regression for estimating a conditional density. Our selection rule optimizes the estimated conditional density function by minimizing the integrated square error (ISE). ..."
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Cited by 7 (3 self)
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this article, we extend the idea of crossvalidation (CV) for choosing the smoothing parameter of the "doublekernel" local linear regression for estimating a conditional density. Our selection rule optimizes the estimated conditional density function by minimizing the integrated square error (ISE). We also discuss three other bandwidth selection rules. The first is an adhoc method used by Fan, Yao and Tong (FYT, 1996). The second rule, as suggested by Hall, Wol# and Yao (HWY, 1999), employs the idea of bootstrap for the bandwidth selection in the estimation of conditional distribution functions. We modify the HWY approach to suit the bandwidth selection for the conditional density function. The last is the rule of thumb approach proposed by Hyndman and Yao (2002). The performance of the newly proposed CV approach is compared with these three methods by simulation studies, and our method performs outstandingly. The method is also illustrated by application to two sets of time series
M.: Asymptotic statistical equivalence for scalar ergodic diffusions
 Probab. Theory Rel. Fields
, 2006
"... Abstract. For scalar diffusion models with unknown drift function asymptotic equivalence in the sense of Le Cam’s deficiency between statistical experiments is considered under longtime asymptotics. A local asymptotic equivalence result is established with an accompanying sequence of simple Gaussia ..."
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Cited by 7 (2 self)
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Abstract. For scalar diffusion models with unknown drift function asymptotic equivalence in the sense of Le Cam’s deficiency between statistical experiments is considered under longtime asymptotics. A local asymptotic equivalence result is established with an accompanying sequence of simple Gaussian shift experiments. Corresponding globally asymptotically equivalent experiments are obtained as compound experiments. The results are extended in several directions including time discretisation. An explicit transformation of decision functions from the Gaussian to the diffusion experiment is constructed. 1.
SHARP ADAPTIVE ESTIMATION OF THE DRIFT FUNCTION FOR ERGODIC DIFFUSIONS
, 2006
"... The global estimation problem of the drift function is considered for a large class of ergodic diffusion processes. The unknown drift S(·) is supposed to belong to a nonparametric class of smooth functions of order k ≥ 1, but the value of k is not known to the statistician. A fully datadriven proce ..."
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Cited by 7 (0 self)
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The global estimation problem of the drift function is considered for a large class of ergodic diffusion processes. The unknown drift S(·) is supposed to belong to a nonparametric class of smooth functions of order k ≥ 1, but the value of k is not known to the statistician. A fully datadriven procedure of estimating the drift function is proposed, using the estimated risk minimization method. The sharp adaptivity of this procedure is proven up to an optimal constant, when the quality of the estimation is measured by the integrated squared error weighted by the square of the invariant density. 1. Introduction. 1.1. The problem. In this paper we consider the statistical problem of estimating the drift function of a diffusion process X, given as the solution of the stochastic differential equation
Dynamic Integration of Time and StateDomain Methods for Volatility Estimation.” Manuscript
, 2005
"... Time and statedomain methods are two common approaches to nonparametric prediction. Whereas the former uses data predominantly from recent history, the latter relies mainly on historical information. Combining these two pieces of valuable information is an interesting challenge in statistics. We s ..."
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Cited by 5 (2 self)
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Time and statedomain methods are two common approaches to nonparametric prediction. Whereas the former uses data predominantly from recent history, the latter relies mainly on historical information. Combining these two pieces of valuable information is an interesting challenge in statistics. We surmount this problem by dynamically integrating information from both the time and state domains. The estimators from these two domains are optimally combined based on a datadriven weighting strategy, which provides a more efficient estimator of volatility. Asymptotic normality is separately established for the time domain, the state domain, and the integrated estimators. By comparing the efficiency of the estimators, we demonstrate that the proposed integrated estimator uniformly dominates the other two estimators. The proposed dynamic integration approach is also applicable to other estimation problems in time series. Extensive simulations are conducted to demonstrate that the newly proposed procedure outperforms some popular ones, such as the RiskMetrics and historical simulation approaches, among others. In addition, empirical studies convincingly endorse our integration method.
A TEST FOR MODEL SPECIFICATION OF DIFFUSION PROCESSES
, 2008
"... We propose a test for model specification of a parametric diffusion process based on a kernel estimation of the transitional density of the process. The empirical likelihood is used to formulate a statistic, for each kernel smoothing bandwidth, which is effectively a Studentized L2distance between ..."
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Cited by 4 (0 self)
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We propose a test for model specification of a parametric diffusion process based on a kernel estimation of the transitional density of the process. The empirical likelihood is used to formulate a statistic, for each kernel smoothing bandwidth, which is effectively a Studentized L2distance between the kernel transitional density estimator and the parametric transitional density implied by the parametric process. To reduce the sensitivity of the test on smoothing bandwidth choice, the final test statistic is constructed by combining the empirical likelihood statistics over a set of smoothing bandwidths. To better capture the finite sample distribution of the test statistic and data dependence, the critical value of the test is obtained by a parametric bootstrap procedure. Properties of the test are evaluated asymptotically and numerically by simulation and by a real data example. 1. Introduction. Let X1,...,Xn+1 be n+1 equally spaced (with spacing
Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case
, 2008
"... ergodic diffusion, Gaussian shift, heteroskedastic regression Abstract. Asymptotic local equivalence in the sense of Le Cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of Gaussian shift experiments. The nonparametric local neighbourho ..."
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Cited by 4 (1 self)
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ergodic diffusion, Gaussian shift, heteroskedastic regression Abstract. Asymptotic local equivalence in the sense of Le Cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of Gaussian shift experiments. The nonparametric local neighbourhoods can be attained for any dimension, provided the regularity of the drift is sufficiently large. In addition, a heteroskedastic Gaussian regression experiment is given, which is also locally asymptotically equivalent and which does not depend on the centre of localisation. For one direction of the equivalence an explicit Markov kernel is constructed. 1.
Penalized Sieve Estimation and Inference of Seminonparametric Dynamic Models: A Selective Review
, 2011
"... In this selective review, we …rst provide some empirical examples that motivate the usefulness of seminonparametric techniques in modelling economic and …nancial time series. We describe popular classes of seminonparametric dynamic models and some temporal dependence properties. We then present pe ..."
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Cited by 3 (1 self)
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In this selective review, we …rst provide some empirical examples that motivate the usefulness of seminonparametric techniques in modelling economic and …nancial time series. We describe popular classes of seminonparametric dynamic models and some temporal dependence properties. We then present penalized sieve extremum (PSE) estimation as a general method for seminonparametric models with crosssectional, panel, time series, or spatial data. The method is especially powerful in estimating di ¢ cult illposed inverse problems such as seminonparametric mixtures or conditional moment restrictions. We review recent advances on inference and large sample properties of the PSE estimators, which include (1) consistency and convergence rates of the PSE estimator of the nonparametric part; (2) limiting distributions of plugin PSE estimators of functionals that are either smooth (i.e., rootn estimable) or nonsmooth (i.e., slower than rootn estimable); (3) simple criterionbased inference for plugin PSE estimation of smooth or nonsmooth functionals; and (4) rootn asymptotic normality of semiparametric twostep estimators and their consistent variance estimators. Examples from dynamic asset pricing, nonlinear spatial VAR, semiparametric GARCH,