Kolmogorov–Smirnov,Fluctuation, and Z g Tests for Convergence of Markov Chain Monte Carlo Draws

We examine the sizes and powers of three tests of convergence of Markov Chain Monte Carlo draws: the Kolmogorov–Smirnov test, fluctuation test, and Geweke's test. We show that the sizes and powers are sensitive to the existence of autocorrelation in the draws. We propose a filtered test that is corrected for autocorrelation. We present a numerical illustration using the Federal funds rate.     

Volatility and GMM — Monte Carlo studies and empirical estimations

In this article we examine small sample properties of a generalized method of moments (GMM) estimation using Monte Carlo simulations. We assume that the generated time series describe the stochastic variance rate of a stock index. we use a mean reverting square-root process to simulate the dynamics of this instantaneous variance rate. The time series obtained are used to estimate the parameters of the assumed variance rate process by applying GMM. Our results are described and compared to estimates from empirical data which consist of volatility as well as daily volume data of the German stock market. One of our main findings is that estimates of the mean reverting parameter that are not significantly different from zero do not necessarily imply a rejection of the hypothesis of a mean reverting behavior of the underlying stochastic process.     

A Monte Carlo comparison of some variance estimators of the Horvitz–Thompson estimator

At the design and estimation stage of a survey, large survey organization often uses auxiliary information. This article discusses various procedures for improving variance estimation of the Horvitz–Thompson estimator of a finite population total with the aid of auxiliary information. To study the design-based properties of the proposed variance estimators relative to the standard one, a small scale Monte Carlo study is performed.     

Annealing evolutionary stochastic approximation Monte Carlo for global optimization

In this paper, we propose a new algorithm, the so-called annealing evolutionary stochastic approximation Monte Carlo (AESAMC) algorithm as a general optimization technique, and study its convergence. AESAMC possesses a self-adjusting mechanism, whose target distribution can be adapted at each iteration according to the current samples. Thus, AESAMC falls into the class of adaptive Monte Carlo methods. This mechanism also makes AESAMC less trapped by local energy minima than nonadaptive MCMC algorithms. Under mild conditions, we show that AESAMC can converge weakly toward a neighboring set of global minima in the space of energy. AESAMC is tested on multiple optimization problems. The numerical results indicate that AESAMC can potentially outperform simulated annealing, the genetic algorithm, annealing stochastic approximation Monte Carlo, and some other metaheuristics in function optimization.     

The homotopy perturbation method for the Black–Scholes equation

The homotopy perturbation method is designed to obtain a quick and accurate solution to the Black–Scholes equation and boundary conditions for a European option pricing problem. The problem of pricing a European option can be cast a partial differential equation. The analytical solution of the equation is calculated in the form of a convergent power series with easily computable components.     

Discussion on “A comparison of Monte Carlo methods for computing marginal likelihoods of item response theory models”

In this discussion, the sensitivity of the result by the choice of parameters a and b in one of approaches reviewed by the authors to calculate the marginal likelihood for 2 parameter logistic item response theory model is investigated using a small simulation study.     

Testing for co-directional interactions using union–intersection and intersection–union methods

When interaction terms exist in a two-factor, factorial experiment, the consideration and analysis of main effects are often restricted to those situations where the interaction between factors is not significant. Hinkelman and Kempthorne [4 Hinkelmann, K. and Kempthorne, O. 1994. Design and Analysis of Experiments. Volume 1: Introduction to Experimental Design, New York: Wiley.  [Google Scholar]] softened that stance somewhat and advocate testing main effects when the interaction is deemed co-directional but not anti-directional. A test for the main effects in that situation may be pragmatic to the practitioner and appealing to researchers in other disciplines. Intersection–union and union–intersection methods are examined for assessing the directional nature of significant interactions so that the main effects in a two-factor factorial may be evaluated. The tests suggested are conceptually straightforward and practical and maintain the nominal Type-I error rate. Examples are provided to illustrate the methods.     

A Monte Carlo comparison of Jarque–Bera type tests and Henze–Zirkler test of multivariate normality

In the paper, tests for multivariate normality (MVN) of Jarque-Bera type, based on skewness and kurtosis, have been considered. Tests proposed by Mardia and Srivastava, and the combined tests based on skewness and kurtosis defined by Jarque and Bera have been taken into account. In the Monte Carlo simulations, for each combination of p = 2, 3, 4, 5 number of traits and n = 10(5)50(10)100 sample sizes 10,000 runs have been done to calculate empirical Type I errors of tests under consideration, and empirical power against different alternative distributions. Simulation results have been compared to the Henze–Zirkler’s test. It should be stressed that no test yet proposed is uniformly better than all the others in every combination of conditions examined.     

Economic and economic–statistical designs of VSI Bayesian control chart using Monte Carlo method and ABC algorithm

The usual practice in using a Bayesian control chart to monitor a process is done by taking samples from the process with fixed sampling intervals. Recent studies on traditional control charts have shown that variable sampling interval (VSI) scheme compared to classical scheme (fixed ratio sampling, FRS) helps practitioners to detect process shifts more quickly. In this paper, the effectiveness of VSI scheme on performance of Bayesian control chart has been studied, based on economic (ED) and economic–statistical designs (ESD). Monte Carlo method and artificial bee colony algorithm have been utilized to obtain optimal design parameters of Bayesian control chart (sample size, sampling intervals, warning limit and control limit) since the statistic of this approach does not have any specified distribution. Finally, VSI Bayesian control chart has been compared to FRS Bayesian and VSI X-bar approaches based on ED and ESD, separately. According to the results, it has been found that the performance of VSI Bayesian scheme is better than FRS Bayesian and VSI X-bar approaches.     

Comparison of estimation methods for the Marshall–Olkin extended Lindley distribution

The aim of this paper is to compare the parameters' estimations of the Marshall–Olkin extended Lindley distribution obtained by six estimation methods: maximum likelihood, ordinary least-squares, weighted least-squares, maximum product of spacings, Cramér–von Mises and Anderson–Darling. The bias, root mean-squared error, average absolute difference between the true and estimate distributions' functions and the maximum absolute difference between the true and estimate distributions' functions are used as comparison criteria. Although the maximum product of spacings method is not widely used, the simulation study concludes that it is highly competitive with the maximum likelihood method.     

Testing structural breaks versus long memory with the Box–Pierce statistics: a Monte Carlo study

Several studies have found that occasional-break processes may produce realizations with slowly decaying autocorrelations, which is hardly distinguished from the long memory phenomenon. In this paper we suggest the use of the Box–Pierce statistics to discriminate long memory and occasional-break processes. We conduct an extensive Monte Carlo experiment to examine the finite sample properties of the Box–Pierce and other simple tests statistics in this framework. The results allow us to infer important guidelines for applied statistics in practice.     

Adaptive evolutionary Monte Carlo algorithm for optimization with applications to sensor placement problems

In this paper, we present an adaptive evolutionary Monte Carlo algorithm (AEMC), which combines a tree-based predictive model with an evolutionary Monte Carlo sampling procedure for the purpose of global optimization. Our development is motivated by sensor placement applications in engineering, which requires optimizing certain complicated “black-box” objective function. The proposed method is able to enhance the optimization efficiency and effectiveness as compared to a few alternative strategies. AEMC falls into the category of adaptive Markov chain Monte Carlo (MCMC) algorithms and is the first adaptive MCMC algorithm that simulates multiple Markov chains in parallel. A theorem about the ergodicity property of the AEMC algorithm is stated and proven. We demonstrate the advantages of the proposed method by applying it to a sensor placement problem in a manufacturing process, as well as to a standard Griewank test function.     

Estimating Heston's and Bates’ models parameters using Markov chain Monte Carlo simulation

Heston's model and Bates’ model are very important in option pricing. It is mentioned in Mendoza's paper [Bayesian estimation and option mispricing (job market paper). Cambridge, MA: Massachusetts Institute of Technology; 2011] that Mexican Stock Exchange introduced options over its main index (the Índice de Precios y Cotizaciones) in 2004 which used Heston's model to price options on days when there was no trading. The estimation of the parameters in both models is not easy. One of the methods is Markov chain Monte Carlo algorithm (MCMC for short). In this paper, we adopt Li, Wells and Yu's MCMC algorithm [A Bayesian analysis of return dynamics with levy jumps. Rev Financ Stud. 2008;21(5):2345–2377]. We provide the necessary derivation utilizing prior distributions since they are otherwise unavailable in the literature. As Li et al. used their model to analyse S&P 500 data from 2 January 1980 to 29 December 2000, we likewise recreate their analysis, this time using data from 1987 to 2012. We would like to involve the financial crisis and analyse how stable the method is while applying to the financial crisis. Unlike Li et al., we find that the estimation is very sensitive to the prior distribution assumption. In addition, we have R-code available by request. We hope to offer tools for people doing empirical research in financial mathematics or quantitative finance.     

The Frisch–Waugh–Lovell theorem for the lasso and the ridge regression

The Frisch–Waugh–Lovell (FWL) (partitioned regression) theorem is essential in regression analysis. This is partly because it is quite useful to derive theoretical results. The lasso regression and the ridge regression, both of which are penalized least-squares regressions, have become popular statistical techniques. This article describes that the FWL theorem remains valid for these penalized least-squares regressions. More precisely, we demonstrate that the covariates corresponding to unpenalized regression parameters in these penalized least-squares regression can be projected out. Some other results related to the FWL theorem in such penalized least-squares regressions are also presented.     

Comparisons of ten estimation methods for the parameters of Marshall–Olkin extended exponential distribution

The aim of this article is to compare via Monte Carlo simulations the finite sample properties of the parameter estimates of the Marshall–Olkin extended exponential distribution obtained by ten estimation methods: maximum likelihood, modified moments, L-moments, maximum product of spacings, ordinary least-squares, weighted least-squares, percentile, Crámer–von-Mises, Anderson–Darling, and Right-tail Anderson–Darling. The bias, root mean-squared error, absolute and maximum absolute difference between the true and estimated distribution functions are used as criterion of comparison. The simulation study reveals that the L-moments and maximum products of spacings methods are highly competitive with the maximum likelihood method in small as well as in large-sized samples.     

A comparison of the Hosmer–Lemeshow,Pigeon–Heyse,and Tsiatis goodness-of-fit tests for binary logistic regression under two grouping methods

Algebraic relationships between Hosmer–Lemeshow (HL), Pigeon–Heyse (J²), and Tsiatis (T) goodness-of-fit statistics for binary logistic regression models with continuous covariates were investigated, and their distributional properties and performances studied using simulations. Groups were formed under deciles-of-risk (DOR) and partition-covariate-space (PCS) methods. Under DOR, HL and T followed reported null distributions, while J² did not. Under PCS, only T followed its reported null distribution, with HL and J² dependent on model covariate number and partitioning. Generally, all had similar power. Of the three, T performed best, maintaining Type-I error rates and having a distribution invariant to covariate characteristics, number, and partitioning.     

Mann–Whitney test with empirical likelihood methods for pretest–posttest studies

Pretest–posttest studies are an important and popular method for assessing the effectiveness of a treatment or an intervention in many scientific fields. While the treatment effect, measured as the difference between the two mean responses, is of primary interest, testing the difference of the two distribution functions for the treatment and the control groups is also an important problem. The Mann–Whitney test has been a standard tool for testing the difference of distribution functions with two independent samples. We develop empirical likelihood-based (EL) methods for the Mann–Whitney test to incorporate the two unique features of pretest–posttest studies: (i) the availability of baseline information for both groups; and (ii) the structure of the data with missing by design. Our proposed methods combine the standard Mann–Whitney test with the EL method of Huang, Qin and Follmann [(2008), ‘Empirical Likelihood-Based Estimation of the Treatment Effect in a Pretest–Posttest Study’, Journal of the American Statistical Association, 103(483), 1270–1280], the imputation-based empirical likelihood method of Chen, Wu and Thompson [(2015), ‘An Imputation-Based Empirical Likelihood Approach to Pretest–Posttest Studies’, The Canadian Journal of Statistics accepted for publication], and the jackknife empirical likelihood method of Jing, Yuan and Zhou [(2009), ‘Jackknife Empirical Likelihood’, Journal of the American Statistical Association, 104, 1224–1232]. Theoretical results are presented and finite sample performances of proposed methods are evaluated through simulation studies.     

Second-order stochastic differential equation model as an alternative for the ALT and CALT models

The paper first discusses the autoregressive latent trajectory (ALT) model and presents in detail its improved version, the continuous-time autoregressive latent trajectory (CALT) model. Next, serious problems related to the linear components in the ALT and CALT models are dealt with. As an alternative for the linear component, the first-order derivative in a second-order stochastic differential equation model is proposed. This is applied to Marital Satisfaction data, collected in four consecutive years (2002–2005). It is pointed out that the first-order derivative as explanatory variable has none of the problems associated with the linear component.