Modelling heterogeneous space–time occurrences of earthquakes and its residual analysis

Summary. Earthquake intensities are modelled as a function of previous activity whose specific form is based on established empirical laws in seismology, but whose parameter values can vary from place to place. This model is used for characterizing regional features of seismic activities in and around Japan, and also for exploring regions where the actual seismicity rate systematically deviates from that of the modelled rate.     

Empirical Bayes Age–Period–Cohort Analysis of Retrospective Incidence Data

We analyse the (age, time)-specific incidence of diabetes based on retrospective data obtained from a prevalent cohort only including survivors to a particular date. From underlying point processes with intensities corresponding to the (age, time)-specific incidence rates the observed point pattern is assumed to be generated by an independent thinning process with parameters (assumed known) depending on population density and survival probability to the sampling date. A Bayesian procedure is carried out for the optimal adjustment and comparison of isotropic and anisotropic smoothing priors for the intensity functions, as well as for the decomposition of the intensity on the (time, age) Lexis diagram into the three factors of age, period and cohort.     

Comparison of Two Methods for Calculating the Partition Functions of Various Spatial Statistical Models

Likelihood computation in spatial statistics requires accurate and efficient calculation of the normalizing constant (i.e. partition function) of the Gibbs distribution of the model. Two available methods to calculate the normalizing constant by Markov chain Monte Carlo methods are compared by simulation experiments for an Ising model, a Gaussian Markov field model and a pairwise interaction point field model.     

Improved estimation for the autocovariances of a Gaussian stationary process

For a Gaussian stationary process with mean μ and autocovariance function γ(·), we consider to improve the usual sample autocovariances with respect to the mean squares error (MSE) loss. For the cases μ=0 and μ≠0, we propose sort of empirical Bayes type estimators Γ? and Γ?, respectively. Then their MSE improvements upon the usual sample autocovariances are evaluated in terms of the spectral density of the process. Concrete examples for them are provided. We observe that if the process is near to a unit root process the improvement becomes quite large. Thus, consideration for estimators of this type seems important in many fields, e.g., econometrics.     

AN EMPIRICAL LIKELIHOOD APPROACH FOR NON‐GAUSSIAN VECTOR STATIONARY PROCESSES AND ITS APPLICATION TO MINIMUM CONTRAST ESTIMATION

We develop the empirical likelihood approach for a class of vector‐valued, not necessarily Gaussian, stationary processes with unknown parameters. In time series analysis, it is known that the Whittle likelihood is one of the most fundamental tools with which to obtain a good estimator of unknown parameters, and that the score functions are asymptotically normal. Motivated by the Whittle likelihood, we apply the empirical likelihood approach to its derivative with respect to unknown parameters. We also consider the empirical likelihood approach to minimum contrast estimation based on a spectral disparity measure, and apply the approach to the derivative of the spectral disparity. This paper provides rigorous proofs on the convergence of our two empirical likelihood ratio statistics to sums of gamma distributions. Because the fitted spectral model may be different from the true spectral structure, the results enable us to construct confidence regions for various important time series parameters without assuming specified spectral structures and the Gaussianity of the process.