scholarly journals One-Step Targeted Minimum Loss-based Estimation Based on Universal Least Favorable One-Dimensional Submodels

2016 ◽  
Vol 12 (1) ◽  
pp. 351-378 ◽  
Author(s):  
Mark van der Laan ◽  
Susan Gruber
Keyword(s):  
Probability Distribution ◽  
Minimum Loss ◽  
One Dimensional ◽  
Target Parameter ◽  
Targeted Maximum Likelihood ◽  
One Step ◽  
Influence Curve ◽  
The One ◽  
Asymptotically Efficient ◽  
Curve Equation

AbstractConsider a study in which one observesnindependent and identically distributed random variables whose probability distribution is known to be an element of a particular statistical model, and one is concerned with estimation of a particular real valued pathwise differentiable target parameter of this data probability distribution. The targeted maximum likelihood estimator (TMLE) is an asymptotically efficient substitution estimator obtained by constructing a so called least favorable parametric submodel through an initial estimator with score, at zero fluctuation of the initial estimator, that spans the efficient influence curve, and iteratively maximizing the corresponding parametric likelihood till no more updates occur, at which point the updated initial estimator solves the so called efficient influence curve equation. In this article we construct a one-dimensional universal least favorable submodel for which the TMLE only takes one step, and thereby requires minimal extra data fitting to achieve its goal of solving the efficient influence curve equation. We generalize these to universal least favorable submodels through the relevant part of the data distribution as required for targeted minimum loss-based estimation. Finally, remarkably, given a multidimensional target parameter, we develop a universal canonical one-dimensional submodel such that the one-step TMLE, only maximizing the log-likelihood over a univariate parameter, solves the multivariate efficient influence curve equation. This allows us to construct a one-step TMLE based on a one-dimensional parametric submodel through the initial estimator, that solves any multivariate desired set of estimating equations.

scholarly journals A Generally Efficient Targeted Minimum Loss Based Estimator based on the Highly Adaptive Lasso

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Mark van der Laan
Keyword(s):  
Nuisance Parameter ◽  
Adaptive Lasso ◽  
Nuisance Parameters ◽  
Minimum Loss ◽  
Target Parameter ◽  
Empirical Risk ◽  
Super Learner ◽  
One Step ◽  
Variation Norm ◽  
Asymptotically Efficient

Abstract Suppose we observe $n$ independent and identically distributed observations of a finite dimensional bounded random variable. This article is concerned with the construction of an efficient targeted minimum loss-based estimator (TMLE) of a pathwise differentiable target parameter of the data distribution based on a realistic statistical model. The only smoothness condition we will enforce on the statistical model is that the nuisance parameters of the data distribution that are needed to evaluate the canonical gradient of the pathwise derivative of the target parameter are multivariate real valued cadlag functions (right-continuous and left-hand limits, (G. Neuhaus. On weak convergence of stochastic processes with multidimensional time parameter. Ann Stat 1971;42:1285–1295.) and have a finite supremum and (sectional) variation norm. Each nuisance parameter is defined as a minimizer of the expectation of a loss function over over all functions it its parameter space. For each nuisance parameter, we propose a new minimum loss based estimator that minimizes the loss-specific empirical risk over the functions in its parameter space under the additional constraint that the variation norm of the function is bounded by a set constant. The constant is selected with cross-validation. We show such an MLE can be represented as the minimizer of the empirical risk over linear combinations of indicator basis functions under the constraint that the sum of the absolute value of the coefficients is bounded by the constant: i.e., the variation norm corresponds with this $L_1$-norm of the vector of coefficients. We will refer to this estimator as the highly adaptive Lasso (HAL)-estimator. We prove that for all models the HAL-estimator converges to the true nuisance parameter value at a rate that is faster than $n^{-1/4}$ w.r.t. square-root of the loss-based dissimilarity. We also show that if this HAL-estimator is included in the library of an ensemble super-learner, then the super-learner will at minimal achieve the rate of convergence of the HAL, but, by previous results, it will actually be asymptotically equivalent with the oracle (i.e., in some sense best) estimator in the library. Subsequently, we establish that a one-step TMLE using such a super-learner as initial estimator for each of the nuisance parameters is asymptotically efficient at any data generating distribution in the model, under weak structural conditions on the target parameter mapping and model and a strong positivity assumption (e.g., the canonical gradient is uniformly bounded). We demonstrate our general theorem by constructing such a one-step TMLE of the average causal effect in a nonparametric model, and establishing that it is asymptotically efficient.

Relation Between Surface Crystallography and Surface Electron Structure of the Superlattice

2003 ◽  
Vol 10 (02n03) ◽  
pp. 195-199 ◽  
Author(s):  
I. Bartoš ◽  
T. Strasser ◽  
W. Schattke
Keyword(s):  
Energy Distribution ◽  
Surface State ◽  
Electron Structure ◽  
Surface Electron ◽  
One Dimensional ◽  
One Step ◽  
Surface Brillouin Zone ◽  
Topmost Layer ◽  
The One ◽  
Periodic Crystal

Profound gradual changes of surface state energies were predicted for varying surface terminations of the periodic crystal potential in one-dimensional models.1 This situation can be realized in superlattices with different thicknesses of topmost layers. For the ideally terminated (100) surface of a very thin superlattice (GaAs)2(AlAs)2, the shift of the energy of the surface state over the whole minigap in the lower part of the valence band has been found for different terminations of the topmost layer. In the center of the surface Brillouin zone the surface state shift follows model trends. The changes of the energy distribution of photoemitted electrons as determined from the one-step photoemission calculation2 indicate that experimental observation by the surface-sensitive technique of angle-resolved photoemission should be feasible, and preliminary data indicate this. The results show a straigthforward tuning of surface electron structure by geometrical means.

scholarly journals One-Step Leapfrog LOD-BOR-FDTD Algorithm with CPML Implementation

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Yi-Gang Wang ◽  
Yun Yi ◽  
Bin Chen ◽  
Hai-Lin Chen ◽  
Kang Luo ◽  
...  
Keyword(s):  
Time Domain ◽  
Finite Difference Time Domain ◽  
Algebraic Manipulation ◽  
Body Of Revolution ◽  
One Dimensional ◽  
Electric And Magnetic Fields ◽  
One Step ◽  
Cartesian Coordinate ◽  
The One ◽  
Difference Time

An unconditionally stable one-step leapfrog locally one-dimensional finite-difference time-domain (LOD-FDTD) algorithm towards body of revolution (BOR) is presented. The equations of the proposed algorithm are obtained by the algebraic manipulation of those used in the conventional LOD-BOR-FDTD algorithm. The equations forz-direction electric and magnetic fields in the proposed algorithm should be treated specially. The new algorithm obtains a higher computational efficiency while preserving the properties of the conventional LOD-BOR-FDTD algorithm. Moreover, the convolutional perfectly matched layer (CPML) is introduced into the one-step leapfrog LOD-BOR-FDTD algorithm. The equation of the one-step leapfrog CPML is concise. Numerical results show that its reflection error is small. It can be concluded that the similar CPML scheme can also be easily applied to the one-step leapfrog LOD-FDTD algorithm in the Cartesian coordinate system.

The One-Dimensional Optimum Hydrodynamic Gas Slider Bearing

1968 ◽  
Vol 90 (1) ◽  
pp. 281-284 ◽  
Author(s):  
C. J. Maday
Keyword(s):  
Film Thickness ◽  
Load Capacity ◽  
Maximum Load ◽  
Slider Bearing ◽  
Compression Process ◽  
One Dimensional ◽  
One Step ◽  
The Difference ◽  
The One ◽  
Bearing Number

Bounded variable methods of the calculus of variations are used to determine the optimum or maximum load capacity hydrodynamic one-dimensional gas slider bearing. A lower bound is placed on the minimum film thickness in order to keep the load finite, and also to satisfy the boundary conditions. Using the Weierstrass-Erdmann corner conditions and the Weierstrass E-function it is found that the optimum gas slider bearing is stepped with a convergent leading section and a uniform thickness trailing section. The step location and the leading section film thickness depend upon the bearing number and compression process considered. It is also shown that the bearing contains one and only one step. The difference in the load capacity and maximum film pressure between the isothermal and adiabatic cases increases with increasing bearing number.

Exploring the Multidimensional Facets of Authoritarianism: Authoritarian Aggression and Social Dominance Orientation

2008 ◽  
Vol 67 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Stefano Passini
Keyword(s):  
Social Dominance ◽  
Factor Model ◽  
Three Dimensional ◽  
Social Dominance Orientation ◽  
Model Fit ◽  
Regression Analyses ◽  
One Dimensional ◽  
Confirmatory Factor ◽  
The One ◽  
Better Than

The relation between authoritarianism and social dominance orientation was analyzed, with authoritarianism measured using a three-dimensional scale. The implicit multidimensional structure (authoritarian submission, conventionalism, authoritarian aggression) of Altemeyer’s (1981, 1988) conceptualization of authoritarianism is inconsistent with its one-dimensional methodological operationalization. The dimensionality of authoritarianism was investigated using confirmatory factor analysis in a sample of 713 university students. As hypothesized, the three-factor model fit the data significantly better than the one-factor model. Regression analyses revealed that only authoritarian aggression was related to social dominance orientation. That is, only intolerance of deviance was related to high social dominance, whereas submissiveness was not.

Adiabatic large polarons in anisotropic molecular crystals

2011 ◽  
Vol 35 (1) ◽  
pp. 15-27
Author(s):  
Zoran Ivić ◽  
Željko Pržulj
Keyword(s):  
Energy Transfer ◽  
Effective Mass ◽  
Molecular Crystals ◽  
Single Chain ◽  
Proper Understanding ◽  
One Dimensional ◽  
Interchain Coupling ◽  
Large Polaron ◽  
The One

Adiabatic large polarons in anisotropic molecular crystals We study the large polaron whose motion is confined to a single chain in a system composed of the collection of parallel molecular chains embedded in threedimensional lattice. It is found that the interchain coupling has a significant impact on the large polaron characteristics. In particular, its radius is quite larger while its effective mass is considerably lighter than that estimated within the one-dimensional models. We believe that our findings should be taken into account for the proper understanding of the possible role of large polarons in the charge and energy transfer in quasi-one-dimensional substances.

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