5 research outputs found
Consistent order estimation and minimal penalties
Consider an i.i.d. sequence of random variables whose distribution f* lies in
one of a nested family of models M_q, q>=1. The smallest index q* such that
M_{q*} contains f* is called the model order. We establish strong consistency
of the penalized likelihood order estimator in a general setting with penalties
of order \eta(q) log log n, where \eta(q) is a dimensional quantity. Moreover,
such penalties are shown to be minimal. In contrast to previous work, an a
priori upper bound on the model order is not assumed. The results rely on a
sharp characterization of the pathwise fluctuations of the generalized
likelihood ratio statistic under entropy assumptions on the model classes. Our
results are applied to the geometrically complex problem of location mixture
order estimation, which is widely used but poorly understood.Comment: 26 page
Relevant States and Memory in Markov Chain Bootstrapping and Simulation
Markov chain theory is proving to be a powerful approach to bootstrap highly nonlinear time series. In this work we provide a method to estimate the memory of a Markov chain (i.e. its order) and to identify its relevant states. In particular the choice of memory lags and the aggregation of irrelevant states are obtained by looking for regularities in the transition probabilities.
Our approach is based on an optimization model. More specifically we consider two competing objectives that a researcher will in general pursue when dealing with bootstrapping: preserving the “structural” similarity between the original and the simulated series and assuring
a controlled diversification of the latter. A discussion based on information theory is developed to define the desirable properties for such optimal criteria. Two numerical tests are developed to verify the effectiveness of the method proposed here