56,835 research outputs found
A Note on Multilevel Based Error Estimation
By employing the infinite multilevel representation of the residual, we derive computable bounds to estimate the distance of finite element approximations to the solution of the Poisson equation. If the finite element approximation is a Galerkin solution, the derived error estimator coincides with the standard element and edge based estimator. If Galerkin orthogonality is not satisfied, then the discrete residual additionally appears in terms of the BPX preconditioner. As a by-product of the present analysis, conditions are derived such that the hierarchical error estimation is reliable and efficient
A note on multilevel based error estimation
By employing the infinite multilevel representation of the residual, we derive computable bounds to estimate the distance of finite element approximations to the solution of the Poisson equation. If the finite element approximation is a Galerkin solution, the derived error estimator coincides with the standard element and edge based estimator. If Galerkin orthogonality is not satisfied, then the discrete residual additionally appears in terms of the BPX preconditioner. As a by-product of the present analysis, conditions are derived such that the hierarchical error estimation is the reliable and efficient
Multilevel IRT Modeling in Practice with the Package mlirt
Variance component models are generally accepted for the analysis of hierarchical structured data. A shortcoming is that outcome variables are still treated as measured without an error. Unreliable variables produce biases in the estimates of the other model parameters. The variability of the relationships across groups and the group-effects on individuals' outcomes differ substantially when taking the measurement error in the dependent variable of the model into account. The multilevel model can be extended to handle measurement error using an item response theory (IRT) model, leading to a multilevel IRT model. This extended multilevel model is in particular suitable for the analysis of educational response data where students are nested in schools and schools are nested within cities/countries.\u
A review of R-packages for random-intercept probit regression in small clusters
Generalized Linear Mixed Models (GLMMs) are widely used to model clustered categorical outcomes. To tackle the intractable integration over the random effects distributions, several approximation approaches have been developed for likelihood-based inference. As these seldom yield satisfactory results when analyzing binary outcomes from small clusters, estimation within the Structural Equation Modeling (SEM) framework is proposed as an alternative. We compare the performance of R-packages for random-intercept probit regression relying on: the Laplace approximation, adaptive Gaussian quadrature (AGQ), Penalized Quasi-Likelihood (PQL), an MCMC-implementation, and integrated nested Laplace approximation within the GLMM-framework, and a robust diagonally weighted least squares estimation within the SEM-framework. In terms of bias for the fixed and random effect estimators, SEM usually performs best for cluster size two, while AGQ prevails in terms of precision (mainly because of SEM's robust standard errors). As the cluster size increases, however, AGQ becomes the best choice for both bias and precision
A Continuation Multilevel Monte Carlo algorithm
We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for
weak approximation of stochastic models. The CMLMC algorithm solves the given
approximation problem for a sequence of decreasing tolerances, ending when the
required error tolerance is satisfied. CMLMC assumes discretization hierarchies
that are defined a priori for each level and are geometrically refined across
levels. The actual choice of computational work across levels is based on
parametric models for the average cost per sample and the corresponding weak
and strong errors. These parameters are calibrated using Bayesian estimation,
taking particular notice of the deepest levels of the discretization hierarchy,
where only few realizations are available to produce the estimates. The
resulting CMLMC estimator exhibits a non-trivial splitting between bias and
statistical contributions. We also show the asymptotic normality of the
statistical error in the MLMC estimator and justify in this way our error
estimate that allows prescribing both required accuracy and confidence in the
final result. Numerical results substantiate the above results and illustrate
the corresponding computational savings in examples that are described in terms
of differential equations either driven by random measures or with random
coefficients
Multilevel Models with Stochastic Volatility for Repeated Cross-Sections: an Application to tribal Art Prices
In this paper we introduce a multilevel specification with stochastic
volatility for repeated cross-sectional data. Modelling the time dynamics in
repeated cross sections requires a suitable adaptation of the multilevel
framework where the individuals/items are modelled at the first level whereas
the time component appears at the second level. We perform maximum likelihood
estimation by means of a nonlinear state space approach combined with
Gauss-Legendre quadrature methods to approximate the likelihood function. We
apply the model to the first database of tribal art items sold in the most
important auction houses worldwide. The model allows to account properly for
the heteroscedastic and autocorrelated volatility observed and has superior
forecasting performance. Also, it provides valuable information on market
trends and on predictability of prices that can be used by art markets
stakeholders
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