We consider the problem of constructing simultaneous confidence intervals (CIs) for the ranks of n means based on their estimates together with the (known) standard errors of those estimates. We... Show moreWe consider the problem of constructing simultaneous confidence intervals (CIs) for the ranks of n means based on their estimates together with the (known) standard errors of those estimates. We present a generic method based on the partitioning principle in which the parameter space is partitioned into disjoint subsets and then each one of them is tested at level a. The resulting CIs have then a simultaneous coverage of 1 - alpha. We show that any procedure which produces simultaneous CIs for ranks can be written as a partitioning procedure. We present a first example where we test the partitions using the likelihood ratio (LR) test. Then, in a second example we show that a recently proposed method for simultaneous CIs for ranks using Tukey's honest significant difference test has an equivalent procedure based on the partitioning principle. By embedding these two methods inside our generic partitioning procedure, we obtain improved variants. We illustrate the performance of these methods through simulations and real data analysis on hotel ratings. While the novel method that uses the LR test and its variant produce shorter CIs when the number of means is small, the Tukey-based method and its variant produce shorter CIs when the number of means is high. Show less
Pas, S.L. van der; Szabo, B.T.; Vaart, A.W. van der 2017
Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational... Show moreEstimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods. Show less
