Performance of Empirical Bayes Estimators of Level-2 Random Parameters in Multilevel Analysis: A Monte Carlo Study for Longitudinal Designs

2003 ◽  
Vol 28 (2) ◽  
pp. 169-194 ◽  
Author(s):  
Math J. J. M. Candel ◽  
Bjorn Winkens
Keyword(s):  
Monte Carlo ◽  
Normal Distribution ◽  
Multilevel Analysis ◽  
Empirical Bayes ◽  
Ordinary Least Squares ◽  
Random Coefficients ◽  
Bayes Estimator ◽  
Performance Criteria ◽  
Random Parameters ◽  
T Distribution

Multilevel analysis is a useful technique for analyzing longitudinal data. To describe a person’s development across time, the quality of the estimates of the random coefficients, which relate time to individual changes in a relevant dependent variable, is of importance. The present study compares three estimators of the random coefficients: the Bayes estimator (BE), the empirical Bayes estimator (EBE), and the ordinary least squares estimator (OLSE). Using MLwiN, Monte Carlo simulations are carried out to study the performance of the estimators, while systematically varying the size of the sample as well as the number of measurement occasions. First, we examine for normally distributed random coefficients to what extent the EBE performs better than the OLSE and to what extent the EBE preserves the good properties of the BE. Second, we examine the robustness of the EBE which is based on a normal distribution of the random parameters, by comparing its performance to the OLSE for data being generated from two distributions other than the normal distribution: a modified t-distribution and a modified exponential distribution. As performance criteria we examine the Bayes risk as well as a criterion based on the frequentist notion of mean squared error.

scholarly journals No Need to Turn Bayesian in Multilevel Analysis with Few Clusters: How Frequentist Methods Provide Unbiased Estimates and Accurate Inference

2016 ◽  
Author(s):  
Martin Elff ◽  
Jan Paul Heisig ◽  
Merlin Schaeffer ◽  
Susumu Shikano
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood ◽  
Multilevel Analysis ◽  
Degrees Of Freedom ◽  
Multilevel Models ◽  
Maximum Likelihood Estimators ◽  
Monte Carlo Study ◽  
Bayesian Techniques ◽  
Upper Level ◽  
T Distribution

Comparative political science has long worried about the performance of multilevel models when the number of upper-level units is small. Exacerbating these concerns, an influential Monte Carlo study by Stegmueller (2013) suggests that frequentist methods yield biased estimates and severely anti-conservative inference with small upper-level samples. Stegmueller recommends Bayesian techniques, which he claims to be superior in terms of both bias and inferential accuracy. In this paper, we reassess and refute these results. First, we formally prove that frequentist maximum likelihood estimators of coefficients are unbiased. The apparent bias found by Stegmueller is simply a manifestation of Monte Carlo Error. Second, we show how inferential problems can be overcome by using restricted maximum likelihood estimators for variance parameters and a t-distribution with appropriate degrees of freedom for statistical inference. Thus, accurate multilevel analysis is possible without turning to Bayesian methods, even if the number of upper-level units is small.

5. Exploiting Spatial Dependence to Improve Measurement of Neighborhood Social Processes

2009 ◽  
Vol 39 (1) ◽  
pp. 151-183 ◽  
Author(s):  
Natalya Verbitsky Savitz ◽  
Stephen W. Raudenbush
Keyword(s):  
Spatial Dependence ◽  
Empirical Bayes ◽  
Mean Squared Error ◽  
Ordinary Least Squares ◽  
Social Processes ◽  
Bayes Estimator ◽  
Random Errors ◽  
Improve Measurement ◽  
Empirical Bayes Estimator ◽  
Bayes Approach

A number of recent studies have used surveys of neighborhood informants and direct observation of city streets to assess aspects of community life such as collective efficacy, the density of kin networks, and social disorder. Raudenbush and Sampson (1999a) have coined the term “ecometrics” to denote the study of the reliability and validity of such assessments. Random errors of measurement will attenuate the associations between these assessments and key outcomes. To address this problem, some studies have used empirical Bayes methods to reduce such biases, while assuming that neighborhood random effects are statistically independent. In this paper we show that the precision and validity of ecometric measures can be considerably improved by exploiting the spatial dependence of neighborhood social processes within the framework of empirical Bayes shrinkage. We compare three estimators of a neighborhood social process: the ordinary least squares estimator (OLS), an empirical Bayes estimator based on the independence assumption (EBE), and an empirical Bayes estimator that exploits spatial dependence (EBS). Under our model assumptions, EBS performs better than EBE and OLS in terms of expected mean squared error loss. The benefits of EBS relative to EBE and OLS depend on the magnitude of spatial dependence, the degree of neighborhood heterogeneity, as well as neighborhood's sample size. A cross-validation study using the original 1995 data from the Project on Human Development in Chicago Neighborhoods and a replication of that survey in 2002 show that the empirical benefits of EBS approximate those expected under our model assumptions; EBS is more internally consistent and temporally stable and demonstrates higher concurrent and predictive validity. A fully Bayes approach has the same properties as does the empirical Bayes approach, but it is preferable when the number of neighborhoods is small.

scholarly journals Value at Risk in the South African equity market: a view from the tails

2010 ◽  
Vol 13 (3) ◽  
pp. 345-361 ◽  
Author(s):  
C Milwidsky ◽  
Eben Mare
Keyword(s):  
At Risk ◽  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Normal Distribution ◽  
South African ◽  
Value At Risk ◽  
Equity Market ◽  
Financial Returns ◽  
The South ◽  
T Distribution

Traditional parametric Value at Risk (VaR) estimates assume normality in financial returns data.  However, it is well known that this distribution, while convenient and simple to implement, underestimates the kurtosis demonstrated in most financial returns.  Huisman, Koedijk and Pownall (1998) replace the normal distribution with the Student’s t distribution in modelling financial returns for calculation of VaR.  In this paper we extend their approach to the Monte Carlo simulation of VaR on both linear and non-linear instruments with application to the South African equity market.  We show, via backtesting, that the t-distribution produces superior results to the normal one.

scholarly journals Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

2013 ◽  
Vol 125 (3) ◽  
pp. 569-600 ◽  
Author(s):  
A. L. Teckentrup ◽  
R. Scheichl ◽  
M. B. Giles ◽  
E. Ullmann
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Random Coefficients ◽  
Elliptic Pdes ◽  
Multilevel Monte Carlo

Monte Carlo simulation and analysis of free-form surface registration

1997 ◽  
Vol 211 (8) ◽  
pp. 605-617 ◽  
Author(s):  
D Brujic ◽  
M Ristic
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Confidence Intervals ◽  
Computer Aided Design ◽  
Maximum Error ◽  
Surface Registration ◽  
Performance Criteria ◽  
Free Form ◽  
Registration Method ◽  
Free Form Surfaces

Accurate dimensional inspection and error analysis of free-form surfaces requires accurate registration of the component in hand. Registration of surfaces defined as non-uniform rational B-splines (NURBS) has been realized through an implementation of the iterative closest point method (ICP). The paper presents performance analysis of the ICP registration method using Monte Carlo simulation. A large number of simulations were performed on an example of a precision engineering component, an aero-engine turbine blade, which was judged to possess a useful combination of geometric characteristics such that the results of the analysis had generic significance. Data sets were obtained through CAD (computer aided design)-based inspection. Confidence intervals for estimated transformation parameters, maximum error between a measured point and the nominal surface (which is extremely important for inspection) mean error and several other performance criteria are presented. The influence of shape, number of measured points, measurement noise and some less obvious, but not less important, factors affecting confidence intervals are identified through statistical analysis.

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