Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Let $(M,\omega)$ be a pseudo-Hermitian space of real dimension $2n+1$, that is \RManBase is a \CR-manifold of dimension $2n+1$ and $\omega$ is a contact form on $M$ giving the Levi distribution $HT(M)\subset TM$. Let $M^\omega\subset T^*M$ be the canonical symplectization of $(M,\omega)$ and $M$ be identified with the zero section of $M^\omega$. Then $M^\omega$ is a manifold of real dimension $2(n+1)$ which admit a canonical foliation by surfaces parametrized by $\mathbb{C}\ni t+i\sigma\mapsto \phi_p(t+i\sigma)=\sigma\omega_{g_t(p)}$, where p\inM is arbitrary and $g_t$ is the flow generated by the Reeb vector field associated to the contact form $\omega$. Let $J$ be an (integrable) complex structure defined in a neighbourhood $U$ of $M$ in $M^\omega$. We say that the pair $(U,J)$ is an {adapted complex tube} on $M^\omega$ if all the parametrizations $\phi_p(t+i\sigma)$ defined above are holomorphic on $\phi_p^{-1}(U)$. In this paper we prove that if $(U,J)$ is an adapted complex tube on $M^\omega$, then the real function $E$ on $M^\omega\subset T^*M$ defined by the condition $\alpha=E(\alpha)\omega_{\pi(\alpha)}$, for each $\alpha\in M^\omega$, is a canonical equation for $M$ which satisfies the homogeneous Monge-Amp\`ere equation $(dd^c E)^{n+1}=0$. We also prove that if $M$ and $\omega$ are real analytic then the symplectization $M^\omega$ admits an unique maximal adapted complex tube.Comment: 6 page

Generalized Quantile Treatment Effect: A Flexible Bayesian Approach Using Quantile Ratio Smoothing

We propose a new general approach for estimating the effect of a binary treatment on a continuous and potentially highly skewed response variable, the generalized quantile treatment effect (GQTE). The GQTE is defined as the difference between a function of the quantiles under the two treatment conditions. As such, it represents a generalization over the standard approaches typically used for estimating a treatment effect (i.e., the average treatment effect and the quantile treatment effect) because it allows the comparison of any arbitrary characteristic of the outcome's distribution under the two treatments. Following Dominici et al. (2005), we assume that a pre-specified transformation of the two quantiles is modeled as a smooth function of the percentiles. This assumption allows us to link the two quantile functions and thus to borrow information from one distribution to the other. The main theoretical contribution we provide is the analytical derivation of a closed form expression for the likelihood of the model. Exploiting this result we propose a novel Bayesian inferential methodology for the GQTE. We show some finite sample properties of our approach through a simulation study which confirms that in some cases it performs better than other nonparametric methods. As an illustration we finally apply our methodology to the 1987 National Medicare Expenditure Survey data to estimate the difference in the single hospitalization medical cost distributions between cases (i.e., subjects affected by smoking attributable diseases) and controls.Comment: Published at http://dx.doi.org/10.1214/14-BA922 in the Bayesian Analysis (http://projecteuclid.org/euclid.ba) by the International Society of Bayesian Analysis (http://bayesian.org/