Abstract: We consider the finite sample performance of a new nonparametric method for bioassay and benchmark analysis in risk assessment, which averages isotonic MLEs based on disjoint subgroups of dosages, and whose asymptotic behavior is essentially optimal (Bhattacharya and Lin, Stat Probab Lett 80:1947–1953, 2010). It is compared with three other methods, including the leading kernel-based method, called DNP, due to Dette et al. (J Am Stat Assoc 100:503–510, 2005) and Dette and Scheder (J Stat Comput Simul 80(5):527–544, 2010). In simulation studies, the present method, termed NAM, outperforms the DNP in the majority of cases considered, although both methods generally do well. In small samples, NAM and DNP both outperform the MLE.

Abstract: By choosing a species sampling random probability measure for the distribution of the basis coefficients, a general class of nonparametric Bayesian methods for clustering of functional data is developed. Allowing the basis functions to be unknown, one faces the problem of posterior simulation over a high-dimensional space of semiparametric models. To address this problem, we propose a novel Metropolis-Hastings algorithm for moving between models, with a nested generalized collapsed Gibbs sampler for updating the model parameters. Focusing on Dirichlet process priors for the distribution of the basis coefficients in multivariate linear spline models, we apply the approach to the problem of clustering of hormone trajectories. This approach allows the number of clusters and the shape of the trajectories within each cluster to be unknown. The methodology can be applied broadly to allow uncertainty in variable selection in semiparametric Bayes hierarchical models.

Abstract: We define three types of nondegenerate Wiener-Poisson functionals. Then, for each type we show that the (weighted) characteristic function of nondegenerate functional is of polynomial decay. Our discussion is based on the analysis of Wiener-Poisson functional (Malliavin calculus) developed by Ishikawa and Kunita (2006). Then we apply the decay property to solutions of Ito’s SDE with jumps. We show that if the SDE is nondegenerate, the law of the solution has a rapidly decreasing C∞-density function. Further, we show that transition function Ps,tϕ(x) of the jump-diffusion determined by the SDE is extended to a tempered distribution Φ such that Ps,tΦ(x) is a C∞-function of x, through which we show that the transition density function ps,t(x, y) is a C∞-function of x and y, and is rapidly decreasing with respect to y.

Abstract: Defining a location parameter as a generalization of the median, a robust test is proposed for (a) the nonparametric Behrens-Fisher problem, where the underlying distributions may have different scales and could be skewed, and (b) the generalized Behrens-Fisher problem, where the distributions may even have different shapes. We propose to bootstrap a signed rank statistic based on differences of sample values and derive rigorous bootstrap central limit theorems for its probabilistic justification, allowing for the so-called m-out-of-n bootstrap. The location parameter of interest is the pseudo-median of the distribution of the difference between a control measurement and an observation from the treatment group. It reduces to (a) the shift in the two sample location model and (b) the difference between the centers of symmetry in the nonparametric Behrens-Fisher model, under the additional assumption that the distributions are symmetric. Due to its importance for applications, we also extend our results to an ANOVA design where each treatment is compared with the control group. Finally, we compare our test with competitors on the basis of theory as well as simulation studies. It turns out that our approach yields a substantial improvement for distributions close to the generalized extreme value type, which makes it attractive for applications in engineering as well as finance. Several heteroscedastic data sets from electrical engineering, astro physics, energy research, analytical chemistry and psychology are used to illustrate our solution.

Abstract: Malliavin (1978) introduced new methods for the study of functionals defined on Wiener space. Concepts, such as smoothness and non-degeneracy of maps of Wiener space into Euclidean spaces were studied and regularity of the induced distributions of smooth non-degenerate maps was established through integration by parts formulae. In the beginning, this was limited to Wiener space and could handle solutions of stochastic differential equations driven by Brownian motion and could show that their distributions have smooth densities under appropriate ellipticity or hypo-ellipticity conditions.