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16
Valuation of Mortgage Backed Securities Using Brownian Bridges to Reduce Effective Dimension
, 1997
"... The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in hig ..."
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Cited by 100 (15 self)
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The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in high dimensional problems this benefit might not appear at feasible sample sizes. Substantial improvements from quasiMonte Carlo integration have, however, been reported for problems such as the valuation of mortgagebacked securities, in dimensions as high as 360. We believe that this is due to a lower effective dimension of the integrand in those cases. This paper defines the effective dimension and shows in examples how the effective dimension may be reduced by using a Brownian bridge representation. 1 Introduction Simulation is often the only effective numerical method for the accurate valuation of securities whose value depends on the whole trajectory of interest Mathematics Departmen...
Latin Supercube Sampling for Very High Dimensional Simulations
, 1997
"... This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables ..."
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Cited by 84 (8 self)
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This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables are grouped into subsets, and a lower dimensional QMC method is used within each subset. The QMC points are presented in random order within subsets. QMC methods have been observed to lose effectiveness in high dimensional problems. This paper shows that LSS can extend the benefits of QMC to much higher dimensions, when one can make a good grouping of input variables. Some suggestions for grouping variables are given for the motivating examples. Even a poor grouping can still be expected to do as well as LHS. The paper also extends LHS and LSS to infinite dimensional problems. The paper includes a survey of QMC methods, randomized versions of them (RQMC) and previous methods for extending Q...
Projection estimation in multiple regression with application to functional ANOVA models
 Ann. Statist
, 1998
"... A general theory on rates of convergence of the leastsquares projection estimate in multiple regression is developed. The theory is applied to the functional ANOVA model, where the multivariate regression function is modeled as a specified sum of a constant term, main effects Žfunctions of one vari ..."
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Cited by 58 (14 self)
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A general theory on rates of convergence of the leastsquares projection estimate in multiple regression is developed. The theory is applied to the functional ANOVA model, where the multivariate regression function is modeled as a specified sum of a constant term, main effects Žfunctions of one variable. and selected interaction terms Žfunctions of two or more variables.. The leastsquares projection is onto an approximating space constructed from arbitrary linear spaces of functions and their tensor products respecting the assumed ANOVA structure of the regression function. The linear spaces that serve as building blocks can be any of the ones commonly used in practice: polynomials, trigonometric polynomials, splines, wavelets and finite elements. The rate of convergence result that is obtained reinforces the intuition that loworder ANOVA modeling can achieve dimension reduction and thus overcome the curse of dimensionality. Moreover, the components of the projection estimate in an appropriately defined ANOVA decomposition provide consistent estimates of the corresponding components of the regression function. When the regression function does not satisfy the assumed ANOVA form, the projection estimate converges to its best approximation of that form.
Generalized entropy power inequalities and monotonicity properties of information
 IEEE Trans. Inform. Theory
, 2007
"... New families of Fisher information and entropy power inequalities for sums of independent random variables are presented. These inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables, for an arbitrary ..."
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Cited by 53 (8 self)
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New families of Fisher information and entropy power inequalities for sums of independent random variables are presented. These inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables, for an arbitrary collection of subsets. As a consequence, a simple proof of the monotonicity of information in central limit theorems is obtained, both in the setting of i.i.d. summands as well as in the more general setting of independent summands with variancestandardized sums. 1
The Dimension Distribution, and Quadrature Test Functions
"... This paper introduces the dimension distribution for a square integrable function f on [0; 1]^s. The dimension distribution is used to relate several definitions of the effective dimension of a function. Functions of low effective dimension can be easy to integrate numerically. ..."
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Cited by 36 (4 self)
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This paper introduces the dimension distribution for a square integrable function f on [0; 1]^s. The dimension distribution is used to relate several definitions of the effective dimension of a function. Functions of low effective dimension can be easy to integrate numerically.
Estimating Mean Dimensionality
, 2003
"... The mean dimension of a function is defined as a certain weighted combination of its ANOVA mean squares. This paper presents some new identities relating this mean dimension, and some similarly defined higher moments, to the variable importance measures of Sobol' (1993). The methods are applied ..."
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Cited by 11 (1 self)
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The mean dimension of a function is defined as a certain weighted combination of its ANOVA mean squares. This paper presents some new identities relating this mean dimension, and some similarly defined higher moments, to the variable importance measures of Sobol' (1993). The methods are applied to study some functions arising in the bootstrap, in computational finance, and in extreme value theory.
Tail behaviour of multiple random integrals and Ustatistics ∗
"... Abstract: This paper contains sharp estimates about the distribution of multiple random integrals of functions of several variables with respect to a normalized empirical measure, about the distribution of Ustatistics and multiple Wiener–Itô integrals with respect to a white noise. It also contains ..."
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Cited by 10 (0 self)
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Abstract: This paper contains sharp estimates about the distribution of multiple random integrals of functions of several variables with respect to a normalized empirical measure, about the distribution of Ustatistics and multiple Wiener–Itô integrals with respect to a white noise. It also contains good estimates about the supremum of appropriate classes of such integrals or Ustatistics. The proof of most results is omitted, I have concentrated on the explanation of their content and the picture behind them. I also tried to explain the reason for the investigation of such questions. My goal was to yield such a presentation of the results which a nonexpert also can understand, and not only on a formal level.
The L2 Rate Of Convergence For Event History Regression With TimeDependent Covariates
 SCAND. J. STATIST
, 1996
"... Consider repeated events of multiple kinds that occur according to a rightcontinuous semiMarkov process whose transition rates are influenced by one or more timedependent covariates. The logarithms of the intensities of the transitions from one state to another are modeled as members of an arbit ..."
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Cited by 6 (3 self)
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Consider repeated events of multiple kinds that occur according to a rightcontinuous semiMarkov process whose transition rates are influenced by one or more timedependent covariates. The logarithms of the intensities of the transitions from one state to another are modeled as members of an arbitrary linear function space, which may be finite or infinitedimensional. Maximum likelihood estimates are used, where the maximizations are taken over suitably chosen finitedimensional approximating spaces. It is shown that the L 2 rates of convergence of the maximum likelihood estimates are determined by the approximationpower and dimension of the approximating spaces. The theory is applied to a functional ANOVA model, where the logarithms of the intensities are approximated by functions having the form of a specified sum of a constant term, main effects (functions of one variable), and interaction terms (functions of two or more variables). It is shown that the curse of dimensionality c...
Modelling and calibration of logarithmic CMOS image sensors
 in 1982 and the Ph.D. degree from the University of
, 2002
"... Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be ..."
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Cited by 5 (2 self)
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Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the author. Logarithmic CMOS image sensors capture high dynamic range scenes without saturation or loss of perceptible detail but problems exist with image quality. This thesis develops and applies methods of modelling and calibration to understand and improve the fixed pattern noise (FPN) and colour rendition of logarithmic imagers. Chapter 1 compares CCD and CMOS image sensors and, within the latter category, compares linear and logarithmic pixel designs. Chapter 2 reviews the literature on multilinear algebra, unifying and extending approaches for analytic and numeric manipulation of multiindex arrays, which are the generalisation of scalars, vectors and matrices. Chapter 3 defines and solves the problem of multilinear regression with linear constraints for the calibration of a sensor array, permitting models with linear relationships of parameters
LOCAL ANTITHETIC SAMPLING WITH SCRAMBLED NETS
, 811
"... We consider the problem of computing an approximation to the integral I = ∫ [0,1] d f(x)dx. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of O(n −1/2) from n independent random function evaluations. By contrast, quasiMonte Carlo (QMC) sampling using carefully equispac ..."
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Cited by 4 (1 self)
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We consider the problem of computing an approximation to the integral I = ∫ [0,1] d f(x)dx. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of O(n −1/2) from n independent random function evaluations. By contrast, quasiMonte Carlo (QMC) sampling using carefully equispaced evaluation points can attain the rate O(n −1+ε) for any ε> 0 and randomized QMC (RQMC) can attain the RMSE O(n −3/2+ε), both under mild conditions on f. Classical variance reduction methods for MC can be adapted to QMC. Published results combining QMC with importance sampling and with control variates have found worthwhile improvements, but no change in the error rate. This paper extends the classical variance reduction method of antithetic sampling and combines it with RQMC. One such method is shown to bring a modest improvement in the RMSE rate, attaining O(n −3/2−1/d+ε) for any ε> 0, for smooth enough f.