A monte carlo study of robust and least squares response surface methods

Response surface methodology is useful for exploring a response over a region of factor space and in searching for extrema. Its generality, makes it applicable to a variety of areas. Classical response surface methodology for a continuous response variable is generally based on least squares fitting. The sensitivity of least squares to outlying observations carries over to the surface procedures. To overcome this sensitivity, we propose response surface methodology based on robust procedures for continuous response variables. This robust methodology is analogous to the methodology based on least squares, while being much less sensitive to outlying observations. The results of a Monte Carlo study comparing it and classical surface methodologies for normal and contaminated normal errors are presented. The results show that as the proportion of contamination increases, the robust methodology correctly identifies a higher proportion of extrema than the least squares methods and that the robust estimates of extrema tend to be closer to the true extrema than the least squares methods.     

A monte carlo comparison of regression trimmed means

Two proposals are made for constructing adaptive estimators of the parameters in a linear regression model. These estimators are based on regression trimmed means and use an idea of Jaeckel [(1971) Ann Math Statist 42, 1540-1552] and the bootstrap respectively. These adaptive trimmed means as well as some nonadaptive trimmed means are studied by Monte Carlo. A one-step biweight is also included for comparison purposes.     

The relevance of large sample properties of estimators for the errors-in-variables model:a monte carlo study

By means of a Monte Carlo study it is investigated whether moments of the asymptotic distributions of two estimators for the errors-in-variables model are appropriate for employment in small-sample applications.     

A monte carlo study on the robustness of four MANOVA criterion tests

Four MANOVA tests (Wilk's Lambda, Roy's Largest Root Test, the Hotelling-Lawley Trace and the Pillai-Bartlett Trace) were studied when restricted sample data were drawn from normal populations. Robustness was compared by examining bias at critical points and fluctuations in the standard error of the empirical distributions. The Wilk's Lambda statistic was found to be the least affected by the restricted sampling.     

A monte carlo study of collinearity in linear simultaneous equation models ∗

Although the collinearity issue has been studied in previous simulation studies with a simultaneous system of equations, alternative estimators to circumvent this problem have received little attention. Monte Carlo techniques are used to examine the performance of several estimators under a squared error loss criterion. In particular, this study considers the Vinod–Ullah ridge-type estimators at the first and/or second stage of 2SLS. Ridge regression in the second stage only of 2SLS but not the first stage only, seems to be a practical alternative to 2SLS, especially in situations of strong collinearity. The OLS estimator and the ordinary ridge regression estimator also yield favorable results in situations of moderate to strong collinearity.     

A geometric approach to transdimensional markov chain monte carlo

The authors present theoretical results that show how one can simulate a mixture distribution whose components live in subspaces of different dimension by reformulating the problem in such a way that observations may be drawn from an auxiliary continuous distribution on the largest subspace and then transformed in an appropriate fashion. Motivated by the importance of enlarging the set of available Markov chain Monte Carlo (MCMC) techniques, the authors show how their results can be fruitfully employed in problems such as model selection (or averaging) of nested models, or regeneration of Markov chains for evaluating standard deviations of estimated expectations derived from MCMC simulations.     

A note on bounding monte carlo variances michael lavine

A density bounded class P of probability distributions on a space χ is the set of all probability distributions corresponding to probability densities bounded below by a given subprob-ability density and bounded above by a given superprobability density. Density bounded classes arise in robust Bayesian analysis (Lavine 1991) and also in Monte Carlo integration (Fishman Granovsky and Rubin 1989). Finding upper and lower bounds on the variance over all p? P allows one to bound the Monte Carlo variance. Fishman Granovsky and Rubin (1989) find bounds on the variance over all p ? P and also find the densities in P achieving those bounds in the case where χ is discrete; that is, where P is actually a set of probability mass functions. This article generalizes their result by showing how to bound the variance and find the densities achieving the bounds when χ is continuous.     

Fatal shortcomings of stationary AR (1) exogeneous variables in econometric monte carlo tudies of estimators

The paper first shows that the stationary normal AR(1) process (SNAR1), the most frequently used process for generating exogenous variables in econometric Monte Carlo studies, cannot generate realistic exogenous variables, which are generally trended and similar to those generated by ARIMA (p,d,q) process withd≧1 and positive drift (trend). Then, it illustrates that in the context of AR(1) disturbances,trends in exogenous variables can frequently alter the very ranking of two competing estimators, the ordinary least squares estimator (OLS) and the Cochrane-Orcutt estimators (CO). For three common econometric models—a standard regression model, a dynamic model (i.e., a model with a lagged dependent variable), and a seemingly unrelated regression model, OLS becomes superior in many cases. This is so in spite of the fact that the CO estimator in the study utilizes the true value of the first-order autocorrelation coefficient of the disturbances. The message to be derived from these findings should be ccear. If one accepts the fact that most if not all economic time series are trended, and endorses a proposition that the fundamental if not sole purpose of Monte Carlo studies in econometrics should be to provide useful guidelines to practicing econometricians, then, he must not employ SNARl (nor anyother artificially created nontrended series) as a generator of exogenous variables in a Monte Carlo study, at least in the econometrics of autocorrelated disturbances. Alternative methods of generating stochastic exogenous variables that are trended are suggested in the paper. For almost four decades, the principle of the autoregressive transformation of a regression model with first-order autocorrelated disturbances (the Coestimation priciple) has been taken for granted as a method of correcting for the autocorrelation in the disturbances—be it in the two-stage Cochrane—Orcutt estimator, the iterative Cochrane-Orcutt estimator, or an estimator utilizing nonlinear techniques or search procedures. (Comitting the first observation due to transformation is not considered very crucial in general.) The results of the pertinent Monte Carlo studies appear to justify such a procedure only because most studies have employed SNARl exogenous variables, not trended ones. Thus, Monte Carlo experimenters must be blamed, at least partially, for this prevailining malpractice. It is hoped that they will not commit additional sins by not using realistic data in their future experiments.     

Exact monte carlo tests using several statistics

Concepts of ranking and boundary of multivariate statistics are discussed and applied to the simultaneous use of several test statistics calculated for data and simulated replicates. An example of residual analysis in regression is given using layer ranks and supplementary simulation with a stopping rule.     

Volume estimation by monte carlo methods *

In estimating a multiple integral, it is known that Monte Carlo methods are more efficient than analytical techniques when the number of dimensions is beyond seven. In general, the sample-mean method is better than the hit-or-miss Monte Carlo method. However, when the volume of a domain in a high-dimensional space is of interest, the hit-or-miss method is usually preferred. It is because of the difficulty in generalizing the sample-mean method for the computation of the volume of a domain. This paper develops a technique to make such a generalization possible. The technique can be interpreted as a volume-preserving transformation procedure. A volume-preserving transformation is first performed to transform the concerned domain into a hypersphere. The volume of the domain is then evaluated by computing the volume of the hypersphere.     

A monte carlo analysis of two spectral tests of the martingale hypothesis

Summary The size, power, and robustness properties of the Kolmogorov-Smirnov and Cramér-von Mises spectral tests of the martingale (difference) hypothesis are investigated by Monte Carlo methods. The results highlight a marked superiority of the Cramér-von Mises with respect to the Kolmogorov-Smirnov test. The paper also shows that the Cramér-von Mises test is simple to compute, more general and more powerful than other converntionally used tests.     

The relative performances of improved ridge estimators and an empirical bayes estimator: some monte carlo results

The relative 'performances of improved ridge estimators and an empirical Bayes estimator are studied by means of Monte Carlo simulations. The empirical Bayes method is seen to perform consistently better in terms of smaller MSE and more accurate empirical coverage than any of the estimators considered here. A bootstrap method is proposed to obtain more reliable estimates of the MSE of ridge esimators. Some theorems on the bootstrap for the ridge estimators are also given and they are used to provide an analytical understanding of the proposed bootstrap procedure. Empirical coverages of the ridge estimators based on the proposed procedure are generally closer to the nominal coverage when compared to their earlier counterparts. In general, except for a few cases, these coverages are still less accurate than the empirical coverages of the empirical Bayes estimator.     

Robust regression estimators compared via monte carlo

This paper presents results of a Monte Carlo simulation of eight families of robust regression estimators in various situations. The effects studied include long-tailed error terms, measurement error in the independent variables, various spacings of the independent variables, different sample sizes and correlation between the independent variables. An estimator that combines the best features of several of the estimators is recommended for further study.     

Markov-chain monte carlo: Some practical implications of theoretical results

We review and discuss some recent progress in the theory of Markov-chain Monte Carlo applications, particularly oriented to applications in statistics. We attempt to assess the relevance of this theory for practical applications.     

Monte carlo sampling approach to testing nonnested hypothesis: monte carlo results

Alternative ways of using Monte Carlo methods to implement a Cox-type test for separate families of hypotheses are considered. Monte Carlo experiments are designed to compare the finite sample performances of Pesaran and Pesaran's test, a RESET test, and two Monte Carlo hypothesis test procedures. One of the Monte Carlo tests is based on the distribution of the log-likelihood ratio and the other is based on an asymptotically pivotal statistic. The Monte Carlo results provide strong evidence that the size of the Pesaran and Pesaran test is generally incorrect, except for very large sample sizes. The RESET test has lower power than the other tests. The two Monte Carlo tests perform equally well for all sample sizes and are both clearly preferred to the Pesaran and Pesaran test, even in large samples. Since the Monte Carlo test based on the log-likelihood ratio is the simplest to calculate, we recommend using it.     

A monte carlo evaluation of response surface analysis based on paired-comparison data

The use of the logit transformation on paired-comparison data in the weighted least squares analysis of response surfaces for aesthetic qualities of products is discussed. Monte Carlo simulations are employed to investigate the small sample properties of the estimators and test statistics. A secondary objective of the Monte Carlo simulations is the comparison of two transformation procedures. The simulations are of standard-item paired-compar-ison experiments in which ties are not allowed.     

Evaluating the efficiency of fractional integration parameter estimators

This article deals with the efficiency of fractional integration parameter estimators. This study was based on Monte Carlo experiments involving simulated stochastic processes with integration orders in the range ]-1,1[. The evaluated estimation methods were classified into two groups: heuristics and semiparametric/maximum likelihood (ML). The study revealed that the comparative efficiency of the estimators, measured by the lesser mean squared error, depends on the stationary/non-stationary and persistency/anti-persistency conditions of the series. The ML estimator was shown to be superior for stationary persistent processes; the wavelet spectrum-based estimators were better for non-stationary mean reversible and invertible anti-persistent processes; the weighted periodogram-based estimator was shown to be superior for non-invertible anti-persistent processes.