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# Monte Carlo methods

## Abstract

Monte Carlo methods deal with generating random variates from probability density functions in order to estimate unknown parameters or general functions of unknown parameters and to compute their expected values, variances and covariances. One generally works with the multivariate normal distribution due to the central limit theorem. However, if random variables with the normal distribution and random variables with a different distribution are combined, the normal distribution is not valid anymore. The Monte Carlo method is then needed to get the expected values, variances and covariances for the random variables with distributions different from the normal distribution. The error propagation by Monte Carlo methods is discussed and methods for generating random variates from the multivariate normal distribution and from the multivariate uniform distribution. The Monte Carlo integration is presented leading to the sampling–importance-resampling algorithm. Markov chain Monte Carlo methods provide by the Metropolis algorithm and the Gibbs sampler additional ways of generating random variates. A special topic is the Gibbs sampler for computing and propagating large covariance matrices. This task arises, for instance, when the geopotential is determined from satellite observations. The example of the minimal detectable outlier shows, how the Monte Carlo method is used to determine the power of a hypothesis test.

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## Acknowledgements

The author is indebted to Willi Freeden for his invitation of this paper for GEM and to Jan Martin Brockmann for his valuable comments.

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Correspondence to Karl-Rudolf Koch.

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Koch, KR. Monte Carlo methods. Int J Geomath 9, 117–143 (2018). https://doi.org/10.1007/s13137-017-0101-z

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• DOI: https://doi.org/10.1007/s13137-017-0101-z

### Keywords

• Bayesian statistics
• SIR algorithm
• Metropolis algorithm
• Gibbs sampler
• Markov chain Monte Carlo methods

### Mathematics Subject Classification

• 62 Statistics