Multivariate Bernoulli distribution

In this paper, we consider the multivariate Bernoulli distribution as a model to estimate the structure of graphs with binary nodes. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. Importantly the model can estimate not only the main effects and pairwise interactions among the nodes but also is capable of modeling higher order interactions, allowing for the existence of complex clique effects. We compare the multivariate Bernoulli model with existing graphical inference models - the Ising model and the multivariate Gaussian model, where only the pairwise interactions are considered. On the other hand, the multivariate Bernoulli distribution has an interesting property in that independence and uncorrelatedness of the component random variables are equivalent. Both the marginal and conditional distributions of a subset of variables in the multivariate Bernoulli distribution still follow the multivariate Bernoulli distribution. Furthermore, the multivariate Bernoulli logistic model is developed under generalized linear model theory by utilizing the canonical link function in order to include covariate information on the nodes, edges and cliques. We also consider variable selection techniques such as LASSO in the logistic model to impose sparsity structure on the graph. Finally, we discuss extending the smoothing spline ANOVA approach to the multivariate Bernoulli logistic model to enable estimation of non-linear effects of the predictor variables.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP10 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A sieve M-theorem for bundled parameters in semiparametric models, with application to the efficient estimation in a linear model for censored data

In many semiparametric models that are parameterized by two types of parameters---a Euclidean parameter of interest and an infinite-dimensional nuisance parameter---the two parameters are bundled together, that is, the nuisance parameter is an unknown function that contains the parameter of interest as part of its argument. For example, in a linear regression model for censored survival data, the unspecified error distribution function involves the regression coefficients. Motivated by developing an efficient estimating method for the regression parameters, we propose a general sieve M-theorem for bundled parameters and apply the theorem to deriving the asymptotic theory for the sieve maximum likelihood estimation in the linear regression model for censored survival data. The numerical implementation of the proposed estimating method can be achieved through the conventional gradient-based search algorithms such as the Newton--Raphson algorithm. We show that the proposed estimator is consistent and asymptotically normal and achieves the semiparametric efficiency bound. Simulation studies demonstrate that the proposed method performs well in practical settings and yields more efficient estimates than existing estimating equation based methods. Illustration with a real data example is also provided.Comment: Published in at http://dx.doi.org/10.1214/11-AOS934 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Naturalness and a light Z′Z'

Models with a light, additional gauge boson are attractive extensions of the standard model. Often these models are only considered as effective low energy theory without any assumption about an UV completion. This leaves not only the hierarchy problem of the SM unsolved, but introduces a copy of it because of the new fundamental scalars responsible for breaking the new gauge group. A possible solution is to embed these models into a supersymmetric framework. However, this gives rise to an additional source of fine-tuning compared to the MSSM and poses the question how natural such a setup is. One might expect that the additional fine-tuning is huge, namely, $O(M^2_{\rm SUSY}/m^2_{Z'})$. In this paper we point out that this is not necessarily the case. We show that it is possible to find a focus point behaviour also in the new sector in co-existence to the MSSM focus point. We call this 'Double Focus Point Supersymmetry'. Moreover, we stress the need for a proper inclusion of radiative corrections in the fine-tuning calculation: a tree-level estimate would lead to predictions for the tuning which can be wrong by many orders of magnitude. As showcase, we use the $U(1)_{B-L}$ extended MSSM and discuss possible consequence of the observed $^8\textrm{Be}$ anomaly. However, similar features are expected for other models with an extended gauge group which involve potentially large Yukawa-like interactions of the new scalars.Comment: 11 pages, 4 figures, two column format, reference update

Neutralino Dark Matter in Gauge Mediation After Run I of LHC and LUX

Neutralino can be the dark matter candidate in the gauge-mediated supersymmetry breaking models if the conformal sequestered mechanism is assumed in the hidden sector. In this paper, we study this mechanism by using the current experimental results after the run I of LHC and LUX. By adding new Yukawa couplings between the messenger fields and Higgs fields, we find that this mechanism can predict a neutralino dark matter with correct relic density and a Higgs boson with mass around 125 GeV. All our survived points have some common features. Firstly, the Higgs sector falls into the decoupling limit. So the properties of the light Higgs boson are similar to the predictions of the Standard Model one. Secondly, the correct EWSB hints a relatively small $\mu$-term, which makes the lightest neutralino lighter than the lightest stau. So a bino-higgsino dark matter with correct relic density can be achieved. And the relatively small $\mu$-term results in a small fine-tuning. Finally, this bino-higgsino dark matter can pass all current bounds, including both spin-independent and spin-dependent direct searches. The spin-independent cross section of our points can be examined by further experiments.Comment: Minor changes, version to appear in Phys. Lett.