On the semantics of the motif “three hares chasing each other in a circle” on Jewish monuments in Eastern Europe

“Three hares chasing each other in a circle” is one of the most common and peculiar motifs in the murals of Eastern European wooden synagogues. This motif was not an invention of Jewish craftsmen, but rather was borrowed and adapted by them from European art. In the late eighteenth to early nineteenth century this motif appears in various Ashkenazi ritual objects and on the tombstones of Eastern European Jews in the same region, and even in those places where there were painted synagogues. What is the reason for the appearance of this strange motif on Jewish monuments? On some monuments the motif of the “three hares” paradoxically replaces the “three overlapping fish” which are depicted as the zodiacal sign of the month Nisan. On the tombstones the image hints at the name of the deceased. But its central place in the composition of murals is evidence that this motif has an important universal meaning. This analysis of the motif shows that it became an integral part of an artistic tradition and that its semantics were determined within a well-defined geographical and chronological framework.     

Authors’ response

In this rejoinder, we address all or almost all comments that have been provided by the discussants. In particular, we include additional references to literature related to change-point detection as well as to intrusion detection; analyze distributions of stopping times of the CUSUM and Shiryaev–Roberts detection procedures under the no change hypothesis in more detail; and provide an overview of the detection-isolation problem.     

Convergence faible sur des espaces topologiques uniformes et suites aléatoires d'éléments aléatoires

Following some results on weak convergence, theorems are given on the selection of sample elements on uniform spaces. Our results are tied in with a theorem of Fernandez (1971, p. 1740).     

Universal codes as a basis for time series testing

We suggest a new approach to hypothesis testing for ergodic and stationary processes. In contrast to standard methods, the suggested approach gives a possibility to make tests, based on any lossless data compression method even if the distribution law of the codeword lengths is not known. We apply this approach to the following four problems: goodness-of-fit testing (or identity testing), testing for independence, testing of serial independence and homogeneity testing and suggest nonparametric statistical tests for these problems. It is important to note that practically used so-called archivers can be used for suggested testing.     

Modelling stochastic volatility using generalized t distribution

In modelling financial return time series and time-varying volatility, the Gaussian and the Student-t distributions are widely used in stochastic volatility (SV) models. However, other distributions such as the Laplace distribution and generalized error distribution (GED) are also common in SV modelling. Therefore, this paper proposes the use of the generalized t (GT) distribution whose special cases are the Gaussian distribution, Student-t distribution, Laplace distribution and GED. Since the GT distribution is a member of the scale mixture of uniform (SMU) family of distribution, we handle the GT distribution via its SMU representation. We show this SMU form can substantially simplify the Gibbs sampler for Bayesian simulation-based computation and can provide a mean of identifying outliers. In an empirical study, we adopt a GT–SV model to fit the daily return of the exchange rate of Australian dollar to three other currencies and use the exchange rate to US dollar as a covariate. Model implementation relies on Bayesian Markov chain Monte Carlo algorithms using the WinBUGS package.     

Estimation and inference for exponential smooth transition nonlinear volatility models

A family of threshold nonlinear generalised autoregressive conditionally heteroscedastic models is considered, that allows smooth transitions between regimes, capturing size asymmetry via an exponential smooth transition function. A Bayesian approach is taken and an efficient adaptive sampling scheme is employed for inference, including a novel extension to a recently proposed prior for the smoothing parameter that solves a likelihood identification problem. A simulation study illustrates that the sampling scheme performs well, with the chosen prior kept close to uninformative, while successfully ensuring identification of model parameters and accurate inference for the smoothing parameter. An empirical study confirms the potential suitability of the model, highlighting the presence of both mean and volatility (size) asymmetry; while the model is favoured over modern, popular model competitors, including those with sign asymmetry, via the deviance information criterion.