Bootstrap Prediction Intervals for Small Area Means from Unit-Level Nonlinear Models

2018 ◽  
Vol 7 (3) ◽  
pp. 309-333
Author(s):  
Andreea L Erciulescu ◽  
Wayne A Fuller
Keyword(s):  
Small Area ◽  
Degrees Of Freedom ◽  
Prediction Interval ◽  
Nonlinear Models ◽  
Parametric Bootstrap ◽  
Prediction Intervals ◽  
Water Runoff ◽  
Department Of Agriculture ◽  
Bootstrap Percentile ◽  
T Distribution

Abstract For analyses based on nonlinear models, agencies and policy makers are often interested in prediction intervals for small area means. We give statistics for small area predictions that can be used to construct prediction intervals in the same way that standard errors and degrees of freedom are used to construct prediction intervals based on the Student-t distribution. In a simulation study, the new parametric bootstrap prediction interval has good coverage properties and much better coverage than the bootstrap percentile prediction interval. The methods are applied in a study of soil erosion and water runoff conducted by the US Department of Agriculture.

scholarly journals Parametric bootstrap estimators for hybrid inference in forest inventories

2017 ◽  
Vol 91 (3) ◽  
pp. 354-365 ◽  
Author(s):  
Mathieu Fortin ◽  
Rubén Manso ◽  
Robert Schneider
Keyword(s):  
Multiple Imputation ◽  
Nonlinear Models ◽  
Parametric Bootstrap ◽  
Simulation Studies ◽  
Variance Estimator ◽  
Resampling Methods ◽  
Forest Inventories ◽  
Point Estimator ◽  
Complex Models ◽  
High Level

Abstract In forestry, the variable of interest is not always directly available from forest inventories. Consequently, practitioners have to rely on models to obtain predictions of this variable of interest. This context leads to hybrid inference, which is based on both the probability design and the model. Unfortunately, the current analytical hybrid estimators for the variance of the point estimator are mainly based on linear or nonlinear models and their use is limited when the model reaches a high level of complexity. An alternative consists of using a variance estimator based on resampling methods (Rubin, D. B. (1987). Multiple imputation for nonresponse surveys. John Wiley & Sons, Hoboken, New Jersey, USA). However, it turns out that a parametric bootstrap (BS) estimator of the variance can be biased in contexts of hybrid inference. In this study, we designed and tested a corrected BS estimator for the variance of the point estimator, which can easily be implemented as long as all of the stochastic components of the model can be properly simulated. Like previous estimators, this corrected variance estimator also makes it possible to distinguish the contribution of the sampling and the model to the variance of the point estimator. The results of three simulation studies of increasing complexity showed no evidence of bias for this corrected variance estimator, which clearly outperformed the BS variance estimator used in previous studies. Since the implementation of this corrected variance estimator is not much more complicated, we recommend its use in contexts of hybrid inference based on complex models.

Periodic Steady State Response and Fatigue Analysis of a Nonlinear City Bus Model

2009 ◽  
Author(s):  
George Valsamos ◽  
Christos Theodosiou ◽  
Sotirios Natsiavas
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Nonlinear Models ◽  
Equations Of Motion ◽  
High Stress ◽  
Direct Determination ◽  
Accurate Information ◽  
Steady State Response ◽  
State Response ◽  
Periodic Steady State

Dynamic response related to fatigue prediction of an urban bus is investigated. First, a quite complete model subjected to road excitation is employed in order to extract sufficiently reliable and accurate information in a fast way. The bus model is set up by applying the finite element method, resulting to an excessive number of degrees of freedom. In addition, the bus suspension units involve nonlinear characterstics. A step towards alleviating this difficulty is the application of an appropriate coordinate transformation, causing a drastic reduction in the dimension of the final set of the equations of motion. This allows the application of a systematic numerical methodology leading to direct determination of periodic steady state response of nonlinear models subjected to periodic excitation. Next, typical results were obtained for excitation resulting from selected urban road profiles. These profiles have either a known form or known statistical properties, expressed by an appropriate spatial power spectral density function. In all cases examined, the emphasis was put on investigating ride response. The main attention was focused on identifying areas of the bus suspension and frame subsystems where high stress levels are developed. This information is based on the idea of a nonlinear transfer function and provides the basis for applying suitable criteria in order to perform analyses leading to prediction of fatigue failure.

3 Power and sample size for ABE in the 2× 2 design Here we give formulae for the calculation of the power of the TOST procedure assuming that there are in total n subjects in the 2× 2 trial. Suppose that the null hypotheses given below are to be tested using a 100(1 − α)% two-sided confidence interval and a power of (1 − β) is required to reject these hypotheses when they are false. H :µ ≤− ln 1.25 H :µ ≥ ln 1.25. Let us define x to be a random variable that has a noncentral t-distribution with df degrees of freedom and noncentrality parameter nc, i.e., x ∼ t(df,nc). The cumulative distribution function of x is defined as CDF(t,df,nc) = Pr(x≤ t). Assume that the power is to be calculated using log(AUC). If σ is the common within-subject variance for T and R for log(AUC), and n/2 subjects are allocated to each of the sequences RT and TR, then 1−β = CDF(t , df,nc )−CDF(t , df,nc ) (7.1) where √ n(log(µ )− log(0.8)) nc = 2σ √ n(log(µ )− log(1.25)) nc = 2σ √ (n− 2)(nc −nc = ) df 2t and t is the 100(1 − α)% point of the central t-distribution on n− 2 degrees of freedom. Some SAS code to calculate the power for an ABE 2 × 2 trial is given below, where the required input variables are α, σ value of the ratio µ and n, the total number of subjects in the trial.

2003 ◽  
pp. 361-361
Keyword(s):  
Distribution Function ◽  
Confidence Interval ◽  
Degrees Of Freedom ◽  
Random Variable ◽  
Noncentrality Parameter ◽  
Cumulative Distribution ◽  
Total N ◽  
The Common ◽  
Input Variables ◽  
T Distribution

scholarly journals Integration of Nonlinear Models of Flexible Body Deformation in Multibody System Dynamics

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Martin Schulze ◽  
Stefan Dietz ◽  
Bernhard Burgermeister ◽  
Andrey Tuganov ◽  
Holger Lang ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Multibody System ◽  
Nonlinear Models ◽  
Equations Of Motion ◽  
Flexible Structures ◽  
General Purpose ◽  
Cosserat Rod ◽  
Rod Model ◽  
Flexible Deformation ◽  
Run Up

Current challenges in industrial multibody system simulation are often beyond the classical range of application of existing industrial simulation tools. The present paper describes an extension of a recursive order-n multibody system (MBS) formulation to nonlinear models of flexible deformation that are of particular interest in the dynamical simulation of wind turbines. The floating frame of reference representation of flexible bodies is generalized to nonlinear structural models by a straightforward transformation of the equations of motion (EoM). The approach is discussed in detail for the integration of a recently developed discrete Cosserat rod model representing beamlike flexible structures into a general purpose MBS software package. For an efficient static and dynamic simulation, the solvers of the MBS software are adapted to the resulting class of MBS models that are characterized by a large number of degrees of freedom, stiffness, and high frequency components. As a practical example, the run-up of a simplified three-bladed wind turbine is studied where the dynamic deformations of the three blades are calculated by the Cosserat rod model.

AN EXPERIMENT IN CONTROL OF BRACKEN FERN AT TUTAKI, MURCHISON

1962 ◽  
pp. 84-89
Author(s):  
R. Inch
Keyword(s):  
Experimental Work ◽  
Small Area ◽  
Department Of Agriculture ◽  
Bracken Fern ◽  
Degree Of Control ◽  
River Valleys

The Murchison district has only a small area of flat land along its river valleys and at their confluence. Steep hills cover the remainder of the country. Rainfall is approximately 60 in. per year. When the land was originally cleared bracken fern (Pteridium esccrlentum) became a menace. The struggle over the years to control this invasion has been long and strenuous and, except on land that can be cultivated, largely unrewarded. Some degree of control has been achieved on the closer hills, but generally results have not been good. Murchison Federated Farmers in 1958 asked the Department of Agriculture to conduct some experimental work to find out how this weed could be handled economically.

scholarly journals Linear Models and Empirical Bayes Methods for Assessing Differential Expression in Microarray Experiments

2004 ◽  
Vol 3 (1) ◽  
pp. 1-25 ◽  
Author(s):  
Gordon K Smyth
Keyword(s):  
Differential Expression ◽  
Degrees Of Freedom ◽  
Empirical Bayes ◽  
Linear Models ◽  
Single Channel ◽  
Posterior Odds ◽  
Microarray Experiments ◽  
Study Results ◽  
Standard Deviations ◽  
T Distribution

The problem of identifying differentially expressed genes in designed microarray experiments is considered. Lonnstedt and Speed (2002) derived an expression for the posterior odds of differential expression in a replicated two-color experiment using a simple hierarchical parametric model. The purpose of this paper is to develop the hierarchical model of Lonnstedt and Speed (2002) into a practical approach for general microarray experiments with arbitrary numbers of treatments and RNA samples. The model is reset in the context of general linear models with arbitrary coefficients and contrasts of interest. The approach applies equally well to both single channel and two color microarray experiments. Consistent, closed form estimators are derived for the hyperparameters in the model. The estimators proposed have robust behavior even for small numbers of arrays and allow for incomplete data arising from spot filtering or spot quality weights. The posterior odds statistic is reformulated in terms of a moderated t-statistic in which posterior residual standard deviations are used in place of ordinary standard deviations. The empirical Bayes approach is equivalent to shrinkage of the estimated sample variances towards a pooled estimate, resulting in far more stable inference when the number of arrays is small. The use of moderated t-statistics has the advantage over the posterior odds that the number of hyperparameters which need to estimated is reduced; in particular, knowledge of the non-null prior for the fold changes are not required. The moderated t-statistic is shown to follow a t-distribution with augmented degrees of freedom. The moderated t inferential approach extends to accommodate tests of composite null hypotheses through the use of moderated F-statistics. The performance of the methods is demonstrated in a simulation study. Results are presented for two publicly available data sets.

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