scholarly journals Second-Order Inference for the Mean of a Variable Missing at Random

2016 ◽  
Vol 12 (1) ◽  
pp. 333-349 ◽  
Author(s):  
Iván Díaz ◽  
Marco Carone ◽  
Mark J. van der Laan
Keyword(s):  
Coverage Probability ◽  
Convergence Rates ◽  
Kernel Smoothing ◽  
Missing At Random ◽  
Second Order ◽  
Finite Sample ◽  
Case Scenario ◽  
First Order ◽  
Order Expansion ◽  
The Mean

Abstract We present a second-order estimator of the mean of a variable subject to missingness, under the missing at random assumption. The estimator improves upon existing methods by using an approximate second-order expansion of the parameter functional, in addition to the first-order expansion employed by standard doubly robust methods. This results in weaker assumptions about the convergence rates necessary to establish consistency, local efficiency, and asymptotic linearity. The general estimation strategy is developed under the targeted minimum loss-based estimation (TMLE) framework. We present a simulation comparing the sensitivity of the first and second-order estimators to the convergence rate of the initial estimators of the outcome regression and missingness score. In our simulation, the second-order TMLE always had a coverage probability equal or closer to the nominal value 0.95, compared to its first-order counterpart. In the best-case scenario, the proposed second-order TMLE had a coverage probability of 0.86 when the first-order TMLE had a coverage probability of zero. We also present a novel first-order estimator inspired by a second-order expansion of the parameter functional. This estimator only requires one-dimensional smoothing, whereas implementation of the second-order TMLE generally requires kernel smoothing on the covariate space. The first-order estimator proposed is expected to have improved finite sample performance compared to existing first-order estimators. In the best-case scenario of our simulation study, the novel first-order TMLE improved the coverage probability from 0 to 0.90. We provide an illustration of our methods using a publicly available dataset to determine the effect of an anticoagulant on health outcomes of patients undergoing percutaneous coronary intervention. We provide R code implementing the proposed estimator.

scholarly journals Mean Empirical Likelihood Inference for Response Mean with Data Missing at Random

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Hanji He ◽  
Guangming Deng
Keyword(s):  
Empirical Likelihood ◽  
Missing At Random ◽  
Confidence Regions ◽  
Likelihood Inference ◽  
High Dimensional ◽  
Finite Sample ◽  
The Mean ◽  
Data Missing ◽  
Consistency And Asymptotic Normality ◽  
The Impact

We extend the mean empirical likelihood inference for response mean with data missing at random. The empirical likelihood ratio confidence regions are poor when the response is missing at random, especially when the covariate is high-dimensional and the sample size is small. Hence, we develop three bias-corrected mean empirical likelihood approaches to obtain efficient inference for response mean. As to three bias-corrected estimating equations, we get a new set by producing a pairwise-mean dataset. The method can increase the size of the sample for estimation and reduce the impact of the dimensional curse. Consistency and asymptotic normality of the maximum mean empirical likelihood estimators are established. The finite sample performance of the proposed estimators is presented through simulation, and an application to the Boston Housing dataset is shown.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

The Second-Order Master Equation

2019 ◽  
pp. 128-158
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Mean Field ◽  
Second Order ◽  
Well Posedness ◽  
Mean Field Game ◽  
First Order ◽  
Common Noise ◽  
System P ◽  
The Mean

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

Second Order Perturbation Analysis of a Forced Nonlinear Mathieu Equation

2012 ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mathieu Equation ◽  
Second Order ◽  
Order Perturbation ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Order Expansion ◽  
Direct Forcing

The present study deals with the response of a forced nonlinear Mathieu equation. The equation considered has parametric excitation at the same frequency as direct forcing and also has cubic nonlinearity and damping. A second-order perturbation analysis using the method of multiple scales unfolds numerous resonance cases and system behavior that were not uncovered using first-order expansions. All resonance cases are analyzed. We numerically plot the frequency response of the system. The existence of a superharmonic resonance at one third the natural frequency was uncovered analytically for linear system. (This had been seen previously in numerical simulations but was not captured in the first-order expansion.) The effect of different parameters on the response of the system previously investigated are revisited.

Applicability of second-order expansion in s13 to explore δCP in small and medium baseline ν experiments

2016 ◽  
Vol 31 (09) ◽  
pp. 1650037
Author(s):  
Mandip Singh
Keyword(s):  
Series Expansion ◽  
Second Order ◽  
Mixing Angle ◽  
First Order ◽  
Long Baseline ◽  
Order Expansion ◽  
Matrix Formula ◽  
Small Baseline ◽  
Reactor Mixing ◽  
Baseline Experiment

The series expansion of neutrino evolution matrix “[Formula: see text]”, up to first-order in small reactor mixing angle [Formula: see text] is very useful formalism to study experiments quantitatively. The formalism has been used especially to investigate CP-violating phase [Formula: see text]. In order to perform a broad investigation for the possible measurement of [Formula: see text] phase, we will study small baseline experiments: Chooz [Formula: see text], T2K [Formula: see text] and ESS [Formula: see text], medium baseline experiment: NO[Formula: see text]A [Formula: see text] and long baseline experiment: LBNE [Formula: see text].

Perturbation Analysis of Texture Effects on Lubricated Devices

2005 ◽  
Author(s):  
Gustavo C. Buscaglia ◽  
Mohammed Jai ◽  
Sorin Ciuperca
Keyword(s):  
Friction Coefficient ◽  
Friction Force ◽  
Perturbation Analysis ◽  
Mathematical Analysis ◽  
Load Capacity ◽  
Second Order ◽  
Numerical Investigations ◽  
First Order ◽  
The Mean

Given a bearing of some specified shape, what is the effect of texturing its surfaces uniformly? Experimental and numerical investigations on this question have recently been pursued, which we complement here with a mathematical analysis. Assuming the texture length to be much smaller than the bearing’s length, we combine homogenization techniques with perturbation analysis. This allows us to consider arbitrary, 2D texture shapes. The results show that both the load capacity and the friction force depend, to first order in the amplitude, just on the mean depth/height of the texture. The dependence of the friction coefficient is thus of second order.

A Stochastic Model for Primary Settlers

1970 ◽  
Vol 5 (1) ◽  
pp. 129-170
Author(s):  
D.P. Singh ◽  
A.W. Bryson ◽  
P.L. Silveston
Keyword(s):  
Stochastic Model ◽  
Stochastic Models ◽  
Suspended Solids ◽  
Sewage Treatment ◽  
Sewage Treatment Plant ◽  
Treatment Plant ◽  
Second Order ◽  
Order Model ◽  
First Order ◽  
The Mean

Abstract Stochastic models of process units are useful where the flow and concentrations in a feed stream vary appreciably over long time periods in a random way. Models yield not only the mean, but provide a measure of the variation around the mean. Assuming sedimentation can be described by a rate equation, stochastic models are developed for zero, first and second order rate processes. The zero order model can be rejected because it cannot be made to fit plant data, while the second order model was not developed further because of its complexity. The rate parameter for the first order model was evaluated from 1968 suspended solids data for the Kitchener Sewage Treatment Plant and found to have zero variance. Testing the model against 1966 and 1967 data and shorter period for 1968 showed that the model predicted suspended solids and BOD removals differing on the average from plant results by 10%. The first order stochastic model gives, thus, a satisfactory representation of primary settler performance.

A Pedestal Blocks the Perception of Non-Fourier Motion

Perception ◽  
1997 ◽  
Vol 26 (1_suppl) ◽  
pp. 17-17 ◽  
Author(s):  
O I Ukkonen ◽  
A M Derrington
Keyword(s):  
Modulation Depth ◽  
Discrimination Performance ◽  
Spatial Variations ◽  
Second Order ◽  
First Order ◽  
Direction Discrimination ◽  
Forced Choice Task ◽  
The Mean ◽  
Texture Contrast ◽  
Moving Pattern

We wanted to know whether the mechanisms that discriminate the motion of first-order patterns (defined by spatial variations of luminance) differ from those that detect the motion of non-Fourier or second-order patterns (defined by spatial variations of contrast). To address this question we tested whether motion discrimination performance of first-order and second-order patterns was affected by a pedestal (Lu and Sperling, 1995 Vision Research35 2697 – 2722). A pedestal is a static replica of a moving pattern. We used pedestals with contrast or modulation depth twice the value at which it becomes possible to discriminate the direction of a moving pattern. A two-interval forced-choice task was used to determine how direction discrimination varies with contrast of sine gratings (1 cycle deg−1) and modulation depth of amplitude-modulated gratings presented either alone or with a pedestal. The amplitude-modulated gratings had a 5 cycles deg−1 carrier modulated at 1 cycle deg−1. Three different temporal frequencies (1, 3, and 12 Hz) were studied. Performance with sine gratings was unaffected by the pedestal at all temporal frequencies tested. For amplitude-modulated gratings the pedestal raised the modulation depth at which it became possible to discriminate the direction of motion. This elevation in threshold decreased when the mean contrast of the pattern was high. This result shows that immunity to pedestals of texture-contrast patterns (Lu and Sperling, 1996 Journal of the Optical Society of America13 2305 – 2318) does not generalise to other non-Fourier motion stimuli.

Asymmetric Masking: Luminance Gratings Mask Second-Order Gratings, but Not Vice Versa

Perception ◽  
1997 ◽  
Vol 26 (1_suppl) ◽  
pp. 345-345
Author(s):  
A J Schofield ◽  
M A Georgeson
Keyword(s):  
Modulation Depth ◽  
Broad Band ◽  
Gain Control ◽  
High Amplitude ◽  
Second Order ◽  
Human Vision ◽  
First Order ◽  
Contrast Gain ◽  
Spatiotemporal Information ◽  
The Mean

Human vision can detect spatiotemporal information conveyed by first-order modulations of luminance and by second-order, non-Fourier modulations of image contrast. Models for second-order motion have suggested two filtering stages separated by a rectifying nonlinearity. We explore here the encoding of stationary first-order and second-order gratings, and their interaction. Stimuli consisted of 2-D broad-band static visual noise sinusoidally modulated in luminance (first-order, LM) or contrast (second-order, CM). Modulation thresholds were measured in a two-interval forced-choice staircase procedure. With increasing noise contrast, first-order sensitivity decreased (owing to masking) but sensitivity to contrast modulation increased. Weak background gratings present in both intervals produced order-specific facilitation: LM background facilitated LM detection (the ‘dipper function’) and CM facilitated CM detection. LM did not facilitate CM, nor vice versa, and this is strong evidence that LM and CM are detected via different mechanisms. Nevertheless, suprathreshold LM gratings masked CM detection, but not vice versa. High-amplitude CM masks had little or no effect on CM or LM detection. A broadly tuned divisive gain-control mechanism applied to the first-order filtering stage has been proposed by Foley (1994 Journal of the Optical Society of America A11 1710 – 1719) to account for masking of luminance gratings, and this might also explain the masking of second-order by first-order stimuli. First-order maskers would drive down the effective contrast of the carrier, thus reducing second-order sensitivity. But for second-order maskers the mean contrast, and hence contrast gain, remained constant, independent of modulation depth. Thus second-order gratings would produce no masking effects, as observed.

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