Consider a process that jumps back and forth between two states, with random times spent in between. Suppose the durations of subsequent on and off states are i.i.d. and that the process has started far in the past, so it has achieved stationary. We estimate the sojourn distributions through maximum likelihood when data consist of several realizations observed over windows of fixed length. For discrete and continuous time Markov chains, we also examine if there is any loss of efficiency incurred when ignoring the stationarity structure in the estimation.
In this paper we propose a family of robust estimates for isotonic regression: isotonic M-estimators. We show that their asymptotic distribution is, up to an scalar factor, the same as that of Brunk 's classical isotonic estimator. We also derive the influence function and the breakdown point of these estimates. Finally we perform a Monte Carlo study that shows that the proposed family includes estimators that are simultaneously highly efficient under gaussian errors and highly robust when the error distribution has heavy tails.
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