## Abstract

In this paper we present and discuss two methods for efficient sampling from the Watson distribution.

The first approach adapts the rejection sampling algorithm from Kent et al. (2018), which originally

samples Bingham distribution using angular central Gaussian envelopes. We show that for the

case of the Watson distribution, this allows for a closed form expression for the parameters that

maximize the efficiency of the sampling procedure, which is then further investigated and bounded

by derived asymptotic results. What is more, we present a sampling algorithm that removes the

curse of dimensionality by a smart matrix inversion, which allows for fast runtimes even for complex

problems with high dimension. The second method relates to Saw (1978), and simulates from a

projected distribution using adaptive rejection sampling. Also for this sampling procedure, the derived

algorithm offers fast sampling for large dimension. This is not the case for similar algorithms in the

field, which usually require an expensive rotation of the sampled results using a QR-decomposition.

Finally, we propose some simple generators for the trivial cases and compare the two main methods

in a simulation study.

The first approach adapts the rejection sampling algorithm from Kent et al. (2018), which originally

samples Bingham distribution using angular central Gaussian envelopes. We show that for the

case of the Watson distribution, this allows for a closed form expression for the parameters that

maximize the efficiency of the sampling procedure, which is then further investigated and bounded

by derived asymptotic results. What is more, we present a sampling algorithm that removes the

curse of dimensionality by a smart matrix inversion, which allows for fast runtimes even for complex

problems with high dimension. The second method relates to Saw (1978), and simulates from a

projected distribution using adaptive rejection sampling. Also for this sampling procedure, the derived

algorithm offers fast sampling for large dimension. This is not the case for similar algorithms in the

field, which usually require an expensive rotation of the sampled results using a QR-decomposition.

Finally, we propose some simple generators for the trivial cases and compare the two main methods

in a simulation study.

Originalsprache | Englisch |
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Erscheinungsort | Vienna |

Herausgeber | WU Vienna University of Economics and Business |

Publikationsstatus | Veröffentlicht - 2022 |

### Publikationsreihe

Name | Research Report Series / Department of Statistics and Mathematics |
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Nr. | 134 |

## WU Working Paper Reihe

- Research Report Series / Department of Statistics and Mathematics