We generalize to a broader class of decoupled measures a result of Ziv and Merhav on universal estimation of the specific cross (or relative) entropy, originally for a pair of multilevel Markov measures. Our generalization focuses on abstract decoupling conditions and covers pairs of suitably regular g-measures and pairs of equilibrium measures arising from the “small space of interactions” in mathematical statistical mechanics.

We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. Finally we use simulated fields to test the proposed estimator in different anisotropic and isotropic cases. Results show that the estimator behaves similarly in all cases and is able to detect anisotropy quiteaccurately.

Wind direction is a circular variable. This makes the algorithms used to find its standard deviation different from that of the linear variables. In particular, the requirement for storing all the data points before the standard deviation can be computed limits the storage capacity and puts great strain on remote data acquisition systems. Various algorithms have therefore been developed to estimate the standard deviation in order to reduce the number of terms stored. The following work consists of a comparative analysis of such estimators together with the parameters used. It emerges that some of the assumptions adopted to produce the equations being analysed do not hold in practice, even though this does not affect significantly the performance of the estimators that depend on them. On the other hand, the parameter that has the best trend with the algorithm adopted is the magnitude of the vector to the centre of gravity of the system. However, such a result gives rise to some concerns since it does not account for the ‘vectorial’ nature of the angle being treated.

Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V∗(t1, t2) of the variance function V(t1, t2) of a two-dimensional process is defined, and its first- and second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form where and for 0 < α i < 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t′i are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two.

The second-moment structure of an estimator V*(t) of the variance-time curve V(t) of a weakly stationary point process is obtained in the case where the process is Poisson. This result is used to establish the weak convergence of a class of estimators of the form Tβ(V*(tTα) – V(tTα)), 0 < α < 1, to a non-stationary Gaussian process. Similar results are shown to hold when α = 0 and in the case where V(tTα) is replaced by a suitable estimator.