Summary A major challenge in many modern superresolution fluorescence microscopy techniques at the nanoscale lies in the correct alignment of long sequences of sparse but spatially and temporally highly resolved images. This is caused by the temporal drift of the protein structure, e.g. due to temporal thermal inhomogeneity of the object of interest or its supporting area during the observation process. We develop a simple semiparametric model for drift correction in single‐marker switching microscopy. Then we propose an M‐estimator for the drift and show its asymptotic normality. This is used to correct the final image and it is shown that this purely statistical method is competitive with state of the art calibration techniques which require the incorporation of fiducial markers in the specimen. Moreover, a simple bootstrap algorithm allows us to quantify the precision of the drift estimate and its effect on the final image estimation. We argue that purely statistical drift correction is even more robust than fiducial tracking, rendering the latter superfluous in many applications. The practicability of our method is demonstrated by a simulation study and by a single‐marker switching application. This serves as a prototype for many other typical imaging techniques where sparse observations with high temporal resolution are blurred by motion of the object to be reconstructed.

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In practical applications, when testing parametric restrictions for hidden Markov models (HMMs), one frequently encounters non-standard situations such as testing for zero entries in the transition matrix, one-sided tests for the parameters of the transition matrix or for the components of the stationary distribution of the underlying Markov chain, or testing boundary restrictions on the parameters of the state-dependent distributions. In this paper, we briefly discuss how the relevant asymptotic distribution theory for the likelihood ratio test (LRT) when the true parameter is on the boundary extends from the independent and identically distributed situation to HMMs. Then we concentrate on discussing a number of relevant examples. The finite-sample performance of the LRT in such situations is investigated in a simulation study. An application to series of epileptic seizure counts concludes the paper. Copyright (c) Board of the Foundation of the Scandinavian Journal of Statistics 2008.

We review the theory on semiparametric hidden Markov models (HMMs), that is, HMMs for which the state-dependent distributions are not fully parametrically, but rather semi-or nonparametrically specified. We start by reviewing identifiability in such models, where by exploiting the dependence much stronger results can be achieved than for independent finite mixtures. We also discuss estimation, in particular in an algorithmic fashion by using appropriate versions or modifications of the Baum-Welch (or EM) algorithm. We present some simulation results and give an application to modeling bivariate financial time series, where we compare parametric with nonparametric fits for the state-dependent distributions as well as the resulting state-decoding.Conflict of interest: The authors have declared no conflicts of interest for this article. Recently, there has been some interest in a semi-or even fully nonparametric specification of the state-dependent distributions, cf. Refs 12-16 for some applications of such models. We shall call the resulting HMMs as semiparametric HMMs. Semiparametric modeling may be of interest for the following reasons:418

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