We provide a detailed, introductory exposition of the Metropolis-Hastings algorithm, a powerful Markov chain method to simulate multivariate distributions. A simple, intuitive derivation of this method is given along with guidance on implementation. Also discussed are two applications of the algorithm, one for implementing acceptance-rejection sampling when a blanketing function is not available and the other for implementing the algorithm with block-at-a-time scans. In the latter situation, many different algorithms, including the Gibbs sampler, are shown to be special cases of the Metropolis-Hastings algorithm. The methods are illustrated with examples.

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Understanding the Metropolis-Hastings Algorithm

A survey of maintenance policies of deteriorating systems

An introduction to survival and event history analysis.- Stochastic processes in event history analysis.- Nonparametric analysis of survival and event history data.- Regression models.- Parametric counting process models.- Unobserved heterogeneity: The odd effects of frailty.- Multivariate frailty models.- Marginal and dynamic models for recurrent events and clustered survival data.- Causality.- First passage time models: Understanding the shape of the hazard rate.- Diffusion and L#x00E9 vy process models for dynamic frailty.

https://www.springer.com/productFlyer_978-0-387-20287-7.pdf?SGWID=0-0-1297-52088484-0

Survival and Event History Analysis: A Process Point of View

Invited reviewImperfect maintenance

New product warranty: A literature review

Author's preface. Algorithmic conventions. Glossary. Simulation and random variate generation random number sequences general methods for generating random variates from unvariate distributions methods of generation from specific continuous distributions discrete distributions multivariate distributions miscellaneous topics, including the generation of order statistics, simulation of stochastic processes, sampling from distribution tails. Appendices: 1 - FORTRAN 77 routines for generating variates from selected distributions 2 - graphical methods for sampling from standardized gama distributions and the standardized normal distributions. References. Index.

Principles of Random Variate Generation

Using the ‘ratio’ method an easily implemented algorithm is derived for the generalised inverse Gaussian distribution. Computer timings and efficiency calculations show that the procedure is fast over a wide range of distribution parameter values.

An Easily Implemented Generalised Inverse Gaussian Generator

Simulation and Monte Carlo

An optimal imperfect maintenance policy over a warranty period

Preface. Glossary. 1 Introduction to simulation and Monte Carlo. 1.1 Evaluating a definite integral. 1.2 Monte Carlo is integral estimation. 1.3 An example. 1.4 A simulation using Maple. 1.5 Problems. 2 Uniform random numbers. 2.1 Linear congruential generators. 2.2 Theoretical tests for random numbers. 2.3 Shuffled generator. 2.4 Empirical tests. 2.5 Combinations of generators. 2.6 The seed(s) in a random number generator. 2.7 Problems. 3 General methods for generating random variates. 3.1 Inversion of the cumulative distribution function. 3.2 Envelope rejection. 3.3 Ratio of uniforms method. 3.4 Adaptive rejection sampling. 3.5 Problems. 4 Generation of variates from standard distributions. 4.1 Standard normal distribution. 4.2 Lognormal distribution. 4.3 Bivariate normal density. 4.4 Gamma distribution. 4.5 Beta distribution. 4.6 Chi-squared distribution. 4.7 Student's t distribution. 4.8 Generalized inverse Gaussian distribution. 4.9 Poisson distribution. 4.10 Binomial distribution. 4.11 Negative binomial distribution. 4.12 Problems. 5 Variance reduction. 5.1 Antithetic variates. 5.2 Importance sampling. 5.3 Stratified sampling. 5.4 Control variates. 5.5 Conditional Monte Carlo. 5.6 Problems. 6 Simulation and finance. 6.1 Brownian motion. 6.2 Asset price movements. 6.3 Pricing simple derivatives and options. 6.4 Asian options. 6.5 Basket options. 6.6 Stochastic volatility. 6.7 Problems. 7 Discrete event simulation. 7.1 Poisson process. 7.2 Time-dependent Poisson process. 7.3 Poisson processes in the plane. 7.4 Markov chains. 7.5 Regenerative analysis. 7.6 Simulating a G/G/1 queueing system using the three-phase method. 7.7 Simulating a hospital ward. 7.8 Problems. 8 Markov chain Monte Carlo. 8.1 Bayesian statistics. 8.2 Markov chains and the Metropolis-Hastings (MH) algorithm. 8.3 Reliability inference using an independence sampler. 8.4 Single component Metropolis-Hastings and Gibbs sampling. 8.5 Other aspects of Gibbs sampling. 8.6 Problems. 9 Solutions. 9.1 Solutions 1. 9.2 Solutions 2. 9.3 Solutions 3. 9.4 Solutions 4. 9.5 Solutions 5. 9.6 Solutions 6. 9.7 Solutions 7. 9.8 Solutions 8. Appendix 1: Solutions to problems in Chapter 1. Appendix 2: Random Number Generators. Appendix 3: Computations of acceptance probabilities. Appendix 4: Random variate generators (standard distributions). Appendix 5: Variance Reduction. Appendix 6: Simulation and Finance. Appendix 7: Discrete event simulation. Appendix 8: Markov chain Monte Carlo. References. Index.

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John Dagpunar

Papers

Principles of Random Variate Generation

An Easily Implemented Generalised Inverse Gaussian Generator

Simulation and Monte Carlo

An optimal imperfect maintenance policy over a warranty period

Simulation and Monte Carlo: With Applications in Finance and MCMC