Monte Carlo Standard Error Research Articles

Markov Chain Monte Carlo: Can We Trust the Third Significant Figure?

Current reporting of results based on Markov chain Monte Carlo computations could be improved. In particular, a measure of the accuracy of the resulting estimates is rarely reported. Thus we have little ability to objectively assess the quality of the reported estimates. We address this issue in that we discuss why Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they can be used to decide when to stop the simulation. We compare their use to a popular alternative in the context of two examples.

Modelling heterogeneity for bivariate survival data by the log-normal distribution

We propose a bivariate Weibull regression model with heterogeneity (frailty or random effect) which is generated by log-normal distribution. We assume that the bivariate survival data follow bivariate Weibull of [Hanagal, D.D., 2004. Parametric bivariate regression analysis based on censored samples: A Weibull model. Economic Quality Control 19, 83—90]. There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behaviour (which is unknown and random) of patients follows known frailty distribution. These are the situations which motivate to study this particular model. We propose two-stage maximum likelihood estimation for hierarchical likelihood in the proposed model. We present a small simulation study to compare these estimates with the true value of the parameters and it is observed that these estimates are very close to the true values of the parameters. We also compare theoretical standard errors with Monte Carlo standard errors and theoretical coverage probabilities with Monte Carlo coverage probabilities.

8. Permutation Models for Relational Data

A common problem in sociology, psychology, biology, geography, and management science is the comparison of dyadic relational structures (i.e., graphs). Where these structures are formed on a common set of elements, a natural question that arises is whether there is a tendency for elements that are strongly connected in one set of structures to be more—or less—strongly connected within another set. We may ask, for instance, whether there is a correspondence between golf games and business deals, trade and warfare, or spatial proximity and genetic similarity. In each case, the data for such comparisons may be continuous or discrete, and multiple relations may be involved simultaneously (e.g., when comparing multiple measures of international trade volume with multiple types of political interactions). We propose here an exponential family of permutation models that is suitable for inferring the direction and strength of association among dyadic relational structures. A linear-time algorithm is shown for MCMC simulation of model draws, as is the use of simulated draws for maximum likelihood estimation (MCMC-MLE) and/or estimation of Monte Carlo standard errors. We also provide an easily performed maximum pseudo-likelihood estimation procedure for the permutation model family, which provides a reasonable means of generating seed models for the MCMC-MLE procedure. Use of the modeling framework is demonstrated via an application involving relationships among managers in a high-tech firm.

Fixed-Width Output Analysis for Markov Chain Monte Carlo

Markov chain Monte Carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages. A fundamental question is when sampling should stop; that is, at what point the ergodic averages are good estimates of the desired quantities. We consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user-specified value. Hence calculating a Monte Carlo standard error is a critical step in assessing the simulation output. We consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution. We give sufficient conditions for the strong consistency of both methods and investigate their finite-sample properties in various examples.

Spectroscopic bilinear least-squares methods exploiting the second-order advantage. Theoretical and experimental study concerning accuracy, sensitivity and prediction error

The singular value decomposition (SVD) calibration method followed by least-squares (LS) prediction is discussed. Four different algorithms were applied to second-order instrumental data, all capable of achieving the second-order advantage. Both numerical simulations and experimental data were employed to assess the accuracy and precision of these algorithms, concerning the determination of two therapeutic drugs in spiked human urine and serum samples by fluorescence excitation–emission matrices. Prediction uncertainties were experimentally estimated by sample replication, and also numerically by Monte Carlo noise-addition. Both experimental and Monte Carlo standard errors were found to be in agreement with values provided by error propagation theory, involving the so-called HCD net analyte signal approach, after Ho, Christian and Davidson.

Exact tests for the rasch model via sequential importance sampling

Rasch proposed an exact conditional inference approach to testing his model but never implemented it because it involves the calculation of a complicated probability. This paper furthers Rasch’s approach by (1) providing an efficient Monte Carlo methodology for accurately approximating the required probability and (2) illustrating the usefulness of Rasch’s approach for several important testing problems through simulation studies. Our Monte Carlo methodology is shown to compare favorably to other Monte Carlo methods proposed for this problem in two respects: it is considerably faster and it provides more reliable estimates of the Monte Carlo standard error.

A Comparison between the Stochastic Intensity SSRD Model and the Market Model for CDS Options Pricing

In this paper we investigate implied volatility patterns in the Shifted Square Root Diffusion (SSRD) model as functions of the model parameters. We begin by recalling the Credit Default Swap (CDS) options market model that is consistent with a market Black-like formula, thus introducing a notion of implied volatility for CDS options. We examine implied volatilies coming from SSRD prices and characterize the qualitative behavior of implied volatilities as functions of the SSRD model parameters. We introduce an analytical approximation for the SSRD implied volatility that follows the same patterns in the model parameters and that can be used to have a first rough estimate of the implied volatility following a calibration. We compute numerically the CDS-rate volatility smile for the adopted SSRD model. We find a decreasing pattern of SSRD implied volatilities in the interest-rate/intensity correlation. We check whether it is possible to assume zero correlation after the option maturity in computing the option price and provide an upper bound for the Monte Carlo standard error in cases where this is not possible.

Geometric Ergodicity of van Dyk and Meng's Algorithm for the Multivariate Student's t Model

Let π denote the posterior distribution that results when a random sample of size n from a d-dimensional location-scale Student's t distribution (with ν degrees of freedom) is combined with the standard noninformative prior. van Dyk and Meng developed an efficient Markov chain Monte Carlo (MCMC) algorithm for sampling from π and provided considerable empirical evidence to suggest that their algorithm converges to stationarity much faster than the standard data augmentation algorithm. In addition to its practical importance, this algorithm is interesting from a theoretical standpoint because it is based upon a Markov chain that is not positive recurrent. In this article, we formally analyze the relevant sub-Markov chain underlying van Dyk and Meng's algorithm. In particular, we establish drift and minorization conditions that show that, for many (d, ν, n) triples, the sub-Markov chain is geometrically ergodic. This is the first general, rigorous analysis of an MCMC algorithm based upon a nonpositive recurrent Markov chain. Moreover, our results are important from a practical standpoint because (1) geometric ergodicity guarantees the existence of central limit theorems that can be used to construct Monte Carlo standard errors and (2) the drift and minorization conditions themselves allow for the calculation of exact upper bounds on the total variation distance to stationarity. The results are illustrated using a simple numerical example.

Computation of High Breakdown Nonlinear Regression Parameters

Most algorithms for estimating high breakdown regression estimators in linear regression rely on finding the least squares fit to many p-point elemental sets, where p is the dimension of the X matrix. Such an approach is computationally infeasible in nonlinear regression. This article presents a new algorithm for computing high breakdown estimates in nonlinear regression that requires only a small number of least squares fits to p points. The algorithm is used to compute Rousseeuw's least median of squares (LMS) estimate and Yohai's MM estimate in both simulations and examples. It is also used to compute bootstrapped and Monte Carlo Standard error estimates for MM estimates, which are compared with asymptotic Standard errors (ASE's). Using the PROGRESS algorithm for a two-parameter nonlinear model with sample size 30 would require finding the least squares fit to 435 two-point subsets of the data. In the settings considered in this article, the proposed algorithm performs just as well with 25 as with 435 ...