Optimal adaptive sampling for a symmetric two-state continuous time Markov chain

Abstract

We consider the optimal sampling times for a symmetric two-state continuous time Markov chain. We first consider sampling times of the form $t_i=i\tau$ and find the optimal τ to minimize the asymptotic variance of our estimated parameter. This optimal τ depends upon the true unknown parameters and so it is infeasible in practice. To address this, we consider propose an adaptive scheme which we requires no knowledge of the true underlying parameter, we show that this method is asymptotically equivalent to the optimal fixed time design.

Keywords:

• Continuous time Markov chain
• martingales
• optimal design

JEL Classification codes:

• C22
• C12
• C9

Acknowledgements

The author would like to thank Jason Blevins for suggesting this topic along with numerous helpful discussions. The author would also like to thank Robert de Jong for helpful comments.

Notes

1 This can also be easily seen from the objective function by first maximizing over $e^{-2\tau \alpha }$ and using that this is a monotone function $\alpha .$