State space models are well-known for their versatility in modeling dynamic systems that arise in various scientific disciplines. Although parametric state space models are wellstudied, nonparametric approaches are much less explored in comparison. In this article we propose a novel Bayesian nonparametric approach to state space modeling assuming that both the observational and evolutionary functions are unknown and are varying with time;crucially, we assume that the unknown evolutionary equation describes dynamic evolution of some latent circular random variable.Based on appropriate kernel convolution of the standard Weiner process we model the time-varying observational and evolutionary functions as suitable Gaussian processes that take both linear and circular variables as arguments. Additionally, for the time-varying evolutionary function, we wrap the Gaussian process thus constructed around the unit circle to form an appropriate circular Gaussian process. We show that our process thus created satisfies desirable properties.For the purpose of inference we develop an MCMC based methodology combining Gibbs sampling and Metropolis-Hastings algorithms. Applications to a simulated dataset, a real wind speed dataset and a real ozone dataset demonstrated quite encouraging performances of our model and methodologies.

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The median absolute deviation from the median (MAD) is an important robust univariate spread measure. It also plays important roles with multivariate data through statistics based on the univariate projections of the data, in which case a modified sample MAD introduced by Tyler (1994) and Gather and Hilker (1997) is used to gain increased robustness. Here we establish for the modified sample MAD the same almost sure convergence to the population MAD shown by Hall and Welsh (1985) and Welsh (1986) for the usual sample MAD, and at the same time we eliminate the regularity assumptions imposed in the previous results. Our method is to establish for the sample MAD and modified versions an exponential probability inequality which yields the desired almost sure convergence and also carries independent interest. Further, the asymptotic joint normality of the sample median and the sample MAD established by Falk (1997) is extended to the modified sample MAD. Besides eliminating some regularity conditions, these results provide theoretical validation for use of the more general form of sample MAD.

The median absolute deviation about the median (MAD) is an important univariate spread measure having wide appeal due to its highly robust sample version. A powerful tool in treating the asymptotics of a statistic is a linearization, i.e., a Bahadur representation. Here we establish both strong and weak Bahadur representations for the sample MAD. The strong version is the first in the literature, while the weak version improves upon previous treatments by reducing regularity conditions. Our results also apply to a modified version of sample MAD (Tyler, 1994, andGather andHilker, 1997) introduced to obtain improved robustness for statistical procedures using sample median and MAD combinations over the univariate projections of multivariate data. The strong version yields the law of iterated logarithm for the sample MAD and supports study of almost sure properties of randomly trimmed means based on the MAD and development of robust sequential nonparametric confidence intervals for the MAD. The weak version is needed to simplify derivations of the asymptotic joint distributions of vectors of dependent sample median and sample MAD combinations, which arise in constructing nonparametric multivariate outlyingness functions via projection pursuit.

Actualistic studies are important for evaluating the fidelity of fossil assemblages in representing the living community. Poor live-dead (LD) fidelity in molluscan assemblages may result from transport-induced mixing. Large-scale mixing is more common in siliciclastic settings with a narrow shelf, high sedimentation rate, and those that are frequented by episodically high-energy events. Chandipur-on-sea, on the east coast of India has an optimal setting to promote such conditions. By studying the LD fidelity and modeling size-frequency distribution (SFD) of the fauna, we attempted to evaluate the contribution of “out-of-habitat” versus “within-habitat” mixing in developing the molluscan death assemblage. The correlation between the composition of live (LA) and death assemblages (DA) was insufficient; unlike LAs, the DAs do not show environmental partitioning in ordination space. A numerical simulation of the shell size frequency distribution (SFD) for DAs from LAs was compared with the observed SFD of the DAs. The results of this simulation indicate that DAs are not likely to be a product of within-habitat mixing. DAs probably received considerable input via regional transport, facilitated by frequent tropical cyclones affecting the coast of Odisha. Chandipur receives a large proportion of cyclones originating above 15°N, which causes a high degree of lateral transport and shell mixing between 15° to 21°N, explained by the high compositional similarity of species within this latitudinal extent. Our study highlights the significance of out-of-habitat transport in shaping the regional distribution of marine fossil assemblages, especially in storm dominated siliciclastic shallow-marine settings.