Pub Date : 2021-02-22DOI: 10.2200/S01073ED1V01Y202101MAS037
S. Chowdhury
{"title":"Monte Carlo Methods: A Hands-On Computational Introduction Utilizing Excel","authors":"S. Chowdhury","doi":"10.2200/S01073ED1V01Y202101MAS037","DOIUrl":"https://doi.org/10.2200/S01073ED1V01Y202101MAS037","url":null,"abstract":"","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"24 1","pages":"1-133"},"PeriodicalIF":0.9,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73441475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper shows a general weak approximation method for time-inhomogeneous stochastic differential equations (SDEs) using Malliavin weights. A unified approach is introduced to construct a higher order discretization scheme for expectations of non-smooth functionals of solutions of time-inhomogeneous SDEs. Numerical experiments show the validity of the method.
{"title":"High order weak approximation for irregular functionals of time-inhomogeneous SDEs","authors":"T. Yamada","doi":"10.1515/MCMA-2021-2085","DOIUrl":"https://doi.org/10.1515/MCMA-2021-2085","url":null,"abstract":"Abstract This paper shows a general weak approximation method for time-inhomogeneous stochastic differential equations (SDEs) using Malliavin weights. A unified approach is introduced to construct a higher order discretization scheme for expectations of non-smooth functionals of solutions of time-inhomogeneous SDEs. Numerical experiments show the validity of the method.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"27 1","pages":"117 - 136"},"PeriodicalIF":0.9,"publicationDate":"2021-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/MCMA-2021-2085","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42412878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A1,A2,…,Ak{A_{1},A_{2},dots,A_{k}}.
{"title":"On the absorption probabilities and mean time for absorption for discrete Markov chains","authors":"N. Halidias","doi":"10.1515/MCMA-2021-2084","DOIUrl":"https://doi.org/10.1515/MCMA-2021-2084","url":null,"abstract":"Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A1,A2,…,Ak{A_{1},A_{2},dots,A_{k}}.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"27 1","pages":"105 - 115"},"PeriodicalIF":0.9,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/MCMA-2021-2084","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44362253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kekoura Sakouvogui, Saleem Shaik, C. Doetkott, R. Magel
Abstract The efficiency measures of the Stochastic Frontier Analysis (SFA) models are dependent on distributional assumptions of the one-sided error or inefficiency term. Given the intent of earlier researchers in the evaluation of a single inefficiency distribution using Monte Carlo (MC) simulation, much attention has not been paid to the comparative analysis of SFA models. Our paper aims to evaluate the effects of the assumption of the inefficiency distribution and thus compares different SFA model assumptions by conducting a MC simulation. In this paper, we derive the population statistical parameters of truncated normal, half-normal, and exponential inefficiency distributions of SFA models with the objective of having comparable sample mean and sample standard deviation during MC simulation. Thus, MC simulation is conducted to evaluate the statistical properties and robustness of the inefficiency distributions of SFA models and across three different misspecification scenarios, sample sizes, production functions, and input distributions. MC simulation results show that the misspecified truncated normal SFA model provides the smallest mean absolute deviation and mean square error when the true data generating process is a half-normal inefficiency distribution.
{"title":"Sensitivity analysis of stochastic frontier analysis models","authors":"Kekoura Sakouvogui, Saleem Shaik, C. Doetkott, R. Magel","doi":"10.1515/mcma-2021-2083","DOIUrl":"https://doi.org/10.1515/mcma-2021-2083","url":null,"abstract":"Abstract The efficiency measures of the Stochastic Frontier Analysis (SFA) models are dependent on distributional assumptions of the one-sided error or inefficiency term. Given the intent of earlier researchers in the evaluation of a single inefficiency distribution using Monte Carlo (MC) simulation, much attention has not been paid to the comparative analysis of SFA models. Our paper aims to evaluate the effects of the assumption of the inefficiency distribution and thus compares different SFA model assumptions by conducting a MC simulation. In this paper, we derive the population statistical parameters of truncated normal, half-normal, and exponential inefficiency distributions of SFA models with the objective of having comparable sample mean and sample standard deviation during MC simulation. Thus, MC simulation is conducted to evaluate the statistical properties and robustness of the inefficiency distributions of SFA models and across three different misspecification scenarios, sample sizes, production functions, and input distributions. MC simulation results show that the misspecified truncated normal SFA model provides the smallest mean absolute deviation and mean square error when the true data generating process is a half-normal inefficiency distribution.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"27 1","pages":"71 - 90"},"PeriodicalIF":0.9,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mcma-2021-2083","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46281378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we consider the procedure for deriving variable bandwidth in univariate kernel density estimation for nonnegative heavy-tailed (HT) data. These procedures consider the Birnbaum–Saunders power-exponential (BS-PE) kernel estimator and the bayesian approach that treats the adaptive bandwidths. We adapt an algorithm that subdivides the HT data set into two regions, high density region (HDR) and low-density region (LDR), and we assign a bandwidth parameter for each region. They are derived by using a Monte Carlo Markov chain (MCMC) sampling algorithm. A series of simulation studies and real data are realized for evaluating the performance of a procedure proposed.
{"title":"Body tail adaptive kernel density estimation for nonnegative heavy-tailed data","authors":"Y. Ziane, N. Zougab, S. Adjabi","doi":"10.1515/mcma-2021-2082","DOIUrl":"https://doi.org/10.1515/mcma-2021-2082","url":null,"abstract":"Abstract In this paper, we consider the procedure for deriving variable bandwidth in univariate kernel density estimation for nonnegative heavy-tailed (HT) data. These procedures consider the Birnbaum–Saunders power-exponential (BS-PE) kernel estimator and the bayesian approach that treats the adaptive bandwidths. We adapt an algorithm that subdivides the HT data set into two regions, high density region (HDR) and low-density region (LDR), and we assign a bandwidth parameter for each region. They are derived by using a Monte Carlo Markov chain (MCMC) sampling algorithm. A series of simulation studies and real data are realized for evaluating the performance of a procedure proposed.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"27 1","pages":"57 - 69"},"PeriodicalIF":0.9,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mcma-2021-2082","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48276990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-01DOI: 10.1007/978-3-030-98319-2_13
Robert Nasdala, D. Potts
{"title":"A note on transformed Fourier systems for the approximation of non-periodic signals","authors":"Robert Nasdala, D. Potts","doi":"10.1007/978-3-030-98319-2_13","DOIUrl":"https://doi.org/10.1007/978-3-030-98319-2_13","url":null,"abstract":"","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"1 1","pages":"253-271"},"PeriodicalIF":0.9,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78736117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Many pricing problems boil down to the computation of a high-dimensional integral, which is usually estimated using Monte Carlo. In fact, the accuracy of a Monte Carlo estimator with M simulations is given by σM{frac{sigma}{sqrt{M}}}. Meaning that its convergence is immune to the dimension of the problem. However, this convergence can be relatively slow depending on the variance σ of the function to be integrated. To resolve such a problem, one would perform some variance reduction techniques such as importance sampling, stratification, or control variates. In this paper, we will study two approaches for improving the convergence of Monte Carlo using Neural Networks. The first approach relies on the fact that many high-dimensional financial problems are of low effective dimensions. We expose a method to reduce the dimension of such problems in order to keep only the necessary variables. The integration can then be done using fast numerical integration techniques such as Gaussian quadrature. The second approach consists in building an automatic control variate using neural networks. We learn the function to be integrated (which incorporates the diffusion model plus the payoff function) in order to build a network that is highly correlated to it. As the network that we use can be integrated exactly, we can use it as a control variate.
{"title":"Automatic control variates for option pricing using neural networks","authors":"Zineb El Filali Ech-Chafiq, J. Lelong, A. Reghai","doi":"10.1515/MCMA-2020-2081","DOIUrl":"https://doi.org/10.1515/MCMA-2020-2081","url":null,"abstract":"Abstract Many pricing problems boil down to the computation of a high-dimensional integral, which is usually estimated using Monte Carlo. In fact, the accuracy of a Monte Carlo estimator with M simulations is given by σM{frac{sigma}{sqrt{M}}}. Meaning that its convergence is immune to the dimension of the problem. However, this convergence can be relatively slow depending on the variance σ of the function to be integrated. To resolve such a problem, one would perform some variance reduction techniques such as importance sampling, stratification, or control variates. In this paper, we will study two approaches for improving the convergence of Monte Carlo using Neural Networks. The first approach relies on the fact that many high-dimensional financial problems are of low effective dimensions. We expose a method to reduce the dimension of such problems in order to keep only the necessary variables. The integration can then be done using fast numerical integration techniques such as Gaussian quadrature. The second approach consists in building an automatic control variate using neural networks. We learn the function to be integrated (which incorporates the diffusion model plus the payoff function) in order to build a network that is highly correlated to it. As the network that we use can be integrated exactly, we can use it as a control variate.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"27 1","pages":"91 - 104"},"PeriodicalIF":0.9,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/MCMA-2020-2081","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43144945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we investigate the estimation of the unknown parameters of a competing risk model based on a Weibull distributed decreasing failure rate and an exponentially distributed constant failure rate, under right censored data. The Bayes estimators and the corresponding risks are derived using various loss functions. Since the posterior analysis involves analytically intractable integrals, we propose a Monte Carlo method to compute these estimators. Given initial values of the model parameters, the maximum likelihood estimators are computed using the expectation-maximization algorithm. Finally, we use Pitman’s closeness criterion and integrated mean-square error to compare the performance of the Bayesian and the maximum likelihood estimators.
{"title":"Bayesian estimation of a competing risk model based on Weibull and exponential distributions under right censored data","authors":"H. Talhi, H. Aiachi, N. Rahmania","doi":"10.1515/mcma-2022-2112","DOIUrl":"https://doi.org/10.1515/mcma-2022-2112","url":null,"abstract":"Abstract In this paper, we investigate the estimation of the unknown parameters of a competing risk model based on a Weibull distributed decreasing failure rate and an exponentially distributed constant failure rate, under right censored data. The Bayes estimators and the corresponding risks are derived using various loss functions. Since the posterior analysis involves analytically intractable integrals, we propose a Monte Carlo method to compute these estimators. Given initial values of the model parameters, the maximum likelihood estimators are computed using the expectation-maximization algorithm. Finally, we use Pitman’s closeness criterion and integrated mean-square error to compare the performance of the Bayesian and the maximum likelihood estimators.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"28 1","pages":"163 - 174"},"PeriodicalIF":0.9,"publicationDate":"2021-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42702069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Order statistics arising from 𝑚 independent but not identically distributed random variables are typically constructed by arranging some X 1 , X 2 , … , X m X_{1},X_{2},ldots,X_{m} , with X i X_{i} having distribution function F i ( x ) F_{i}(x) , in increasing order denoted as X ( 1 ) ≤ X ( 2 ) ≤ ⋯ ≤ X ( m ) X_{(1)}leq X_{(2)}leqcdotsleq X_{(m)} . In this case, X ( i ) X_{(i)} is not necessarily associated with F i ( x ) F_{i}(x) . Assuming one can simulate values from each distribution, one can generate such “non-iid” order statistics by simulating X i X_{i} from F i F_{i} , for i = 1 , 2 , … , m i=1,2,ldots,m , and arranging them in order. In this paper, we consider the problem of simulating ordered values X ( 1 ) , X ( 2 ) , … , X ( m ) X_{(1)},X_{(2)},ldots,X_{(m)} such that the marginal distribution of X ( i ) X_{(i)} is F i ( x ) F_{i}(x) . This problem arises in Bayesian principal components analysis (BPCA) where the X i X_{i} are ordered eigenvalues that are a posteriori independent but not identically distributed. We propose a novel coupling-from-the-past algorithm to “perfectly” (up to computable order of accuracy) simulate such order-constrained non-iid order statistics. We demonstrate the effectiveness of our approach for several examples, including the BPCA problem.
由𝑚独立但非同分布的随机变量产生的序统计量通常是通过排列一些x1, x2，…，X m X＿来构造的{1}， x＿{2}，ldots， x＿{m} ， X i X＿{I} 它的分布函数是F i∑(x) F＿{I}(x)，按递增顺序表示为x(1)≤x(2)≤⋯≤x (m) X＿{（1）}leq x＿{（2）}leqcdotsleq x＿{(m)} 。在这种情况下，X (i) X＿{(i)} 不一定与F i¹(x) F＿相关{I}(x)。假设可以模拟每个分布的值，可以通过模拟X i X＿来生成这种“非id”顺序统计量{I} 从F到F{I} ，对于I =1,2，…，m I =1,2，ldots，m，并按顺序排列它们。本文考虑了模拟有序值X (1)， X(2)，…，X (m) X＿的问题{（1）}， x＿{（2）}，ldots， x＿{(m)} 使得X (i)的边际分布为{(i)} F i乘以(x)是F＿{I}(x)。这个问题出现在贝叶斯主成分分析(BPCA)中，其中X i X＿{I} 是后验独立但不同分布的有序特征值。我们提出了一种新的从过去的耦合算法来“完美地”(达到可计算的精度顺序)模拟这种顺序约束的非id顺序统计量。我们通过几个例子展示了我们的方法的有效性，包括BPCA问题。
{"title":"Controlled accuracy Gibbs sampling of order-constrained non-iid ordered random variates","authors":"J. Corcoran, Caleb Miller","doi":"10.1515/mcma-2022-2121","DOIUrl":"https://doi.org/10.1515/mcma-2022-2121","url":null,"abstract":"Abstract Order statistics arising from 𝑚 independent but not identically distributed random variables are typically constructed by arranging some X 1 , X 2 , … , X m X_{1},X_{2},ldots,X_{m} , with X i X_{i} having distribution function F i ( x ) F_{i}(x) , in increasing order denoted as X ( 1 ) ≤ X ( 2 ) ≤ ⋯ ≤ X ( m ) X_{(1)}leq X_{(2)}leqcdotsleq X_{(m)} . In this case, X ( i ) X_{(i)} is not necessarily associated with F i ( x ) F_{i}(x) . Assuming one can simulate values from each distribution, one can generate such “non-iid” order statistics by simulating X i X_{i} from F i F_{i} , for i = 1 , 2 , … , m i=1,2,ldots,m , and arranging them in order. In this paper, we consider the problem of simulating ordered values X ( 1 ) , X ( 2 ) , … , X ( m ) X_{(1)},X_{(2)},ldots,X_{(m)} such that the marginal distribution of X ( i ) X_{(i)} is F i ( x ) F_{i}(x) . This problem arises in Bayesian principal components analysis (BPCA) where the X i X_{i} are ordered eigenvalues that are a posteriori independent but not identically distributed. We propose a novel coupling-from-the-past algorithm to “perfectly” (up to computable order of accuracy) simulate such order-constrained non-iid order statistics. We demonstrate the effectiveness of our approach for several examples, including the BPCA problem.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"28 1","pages":"279 - 292"},"PeriodicalIF":0.9,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45372736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-18DOI: 10.1007/978-3-030-98319-2_5
C. Lemieux, J. Wiart
{"title":"On the Distribution of Scrambled (0, m, s)-Nets Over Unanchored Boxes","authors":"C. Lemieux, J. Wiart","doi":"10.1007/978-3-030-98319-2_5","DOIUrl":"https://doi.org/10.1007/978-3-030-98319-2_5","url":null,"abstract":"","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"11 1","pages":"87-130"},"PeriodicalIF":0.9,"publicationDate":"2020-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90780487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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