Galois action on the Neron-Severi group of Dwork surfaces

We study the Galois action attached to the Dwrok surfaces $X_{\lambda}:X_0^4+X_1^4+X_2^4+X_3^4-4\lambda X_0X_1X_2X_3=0$ with parameter $\lambda$ in a number field $F$. We show that when $X_{\lambda}$ has geometric Picard number $19$, its N\'eron-Severi group $NS(\overline{X}_{\lambda})\otimes \mathbb{Q}$ is a direct sum of quadratic characters. We provide two proofs to this conclusion in our article. In particular, the geometrically proof determines the conductor of each of quadratic characters. Our result matches the one in \cite{Voight2}. With this decomposition, we give another proof to a result of Wan

Posterior Convergence and Model Estimation in Bayesian Change-point Problems

We study the posterior distribution of the Bayesian multiple change-point regression problem when the number and the locations of the change-points are unknown. While it is relatively easy to apply the general theory to obtain the $O(1/\sqrt{n})$ rate up to some logarithmic factor, showing the exact parametric rate of convergence of the posterior distribution requires additional work and assumptions. Additionally, we demonstrate the asymptotic normality of the segment levels under these assumptions. For inferences on the number of change-points, we show that the Bayesian approach can produce a consistent posterior estimate. Finally, we argue that the point-wise posterior convergence property as demonstrated might have bad finite sample performance in that consistent posterior for model selection necessarily implies the maximal squared risk will be asymptotically larger than the optimal $O(1/\sqrt{n})$ rate. This is the Bayesian version of the same phenomenon that has been noted and studied by other authors

A note on conditional Akaike information for Poisson regression with random effects

A popular model selection approach for generalized linear mixed-effects models is the Akaike information criterion, or AIC. Among others, \cite{vaida05} pointed out the distinction between the marginal and conditional inference depending on the focus of research. The conditional AIC was derived for the linear mixed-effects model which was later generalized by \cite{liang08}. We show that the similar strategy extends to Poisson regression with random effects, where condition AIC can be obtained based on our observations. Simulation studies demonstrate the usage of the criterion.Comment: 7 pages, 1 figur

On posterior distribution of Bayesian wavelet thresholding

We investigate the posterior rate of convergence for wavelet shrinkage using a Bayesian approach in general Besov spaces. Instead of studying the Bayesian estimator related to a particular loss function, we focus on the posterior distribution itself from a nonparametric Bayesian asymptotics point of view and study its rate of convergence. We obtain the same rate as in \citet{abramovich04} where the authors studied the convergence of several Bayesian estimators

Minimax Prediction for Functional Linear Regression with Functional Responses in Reproducing Kernel Hilbert Spaces

In this article, we consider convergence rates in functional linear regression with functional responses, where the linear coefficient lies in a reproducing kernel Hilbert space (RKHS). Without assuming that the reproducing kernel and the covariate covariance kernel are aligned, or assuming polynomial rate of decay of the eigenvalues of the covariance kernel, convergence rates in prediction risk are established. The corresponding lower bound in rates is derived by reducing to the scalar response case. Simulation studies and two benchmark datasets are used to illustrate that the proposed approach can significantly outperform the functional PCA approach in prediction

Shrinkage Tuning Parameter Selection in Precision Matrices Estimation

Recent literature provides many computational and modeling approaches for covariance matrices estimation in a penalized Gaussian graphical models but relatively little study has been carried out on the choice of the tuning parameter. This paper tries to fill this gap by focusing on the problem of shrinkage parameter selection when estimating sparse precision matrices using the penalized likelihood approach. Previous approaches typically used K-fold cross-validation in this regard. In this paper, we first derived the generalized approximate cross-validation for tuning parameter selection which is not only a more computationally efficient alternative, but also achieves smaller error rate for model fitting compared to leave-one-out cross-validation. For consistency in the selection of nonzero entries in the precision matrix, we employ a Bayesian information criterion which provably can identify the nonzero conditional correlations in the Gaussian model. Our simulations demonstrate the general superiority of the two proposed selectors in comparison with leave-one-out cross-validation, ten-fold cross-validation and Akaike information criterion

Empirical Likelihood Confidence Intervals for Nonparametric Functional Data Analysis

We consider the problem of constructing confidence intervals for nonparametric functional data analysis using empirical likelihood. In this doubly infinite-dimensional context, we demonstrate the Wilks's phenomenon and propose a bias-corrected construction that requires neither undersmoothing nor direct bias estimation. We also extend our results to partially linear regression involving functional data. Our numerical results demonstrated the improved performance of empirical likelihood over approximation based on asymptotic normality

Spontaneous Spin Textures in Dipolar Spinor Condensates: A Dirac String Gas Approach

We study the spontaneous spin textures induced by magnetic dipole-dipole interaction in ferromagnetic spinor condensates under various trap geometries. At the mean-field level, we show the system is dual to a Dirac string gas with a negative string tension in which the ground state spin texture can be easily determined. We find that three-dimensional condensates prefer a meron-like vortex texture, quasi one-dimensional condensates prefer the axially polarized flare texture, while condensates in quasi two dimensions exhibit either a meron texture or an in-plane polarized texture.Comment: 7 pages, 5 figure

On rates of convergence for posterior distributions under misspecification

We extend the approach of Walker (2003, 2004) to the case of misspecified models. A sufficient condition for establishing rates of convergence is given based on a key identity involving martingales, which does not require construction of tests. We also show roughly that the result obtained by using tests can also be obtained by our approach, which demonstrates the potential wider applicability of this method.Comment: 8 pages, no figure

Cross Validation for Comparing Multiple Density Estimation Procedures

We demonstrate the consistency of cross validation for comparing multiple density estimators using simple inequalities on the likelihood ratio. In nonparametric problems, the splitting of data does not require the domination of test data over the training/estimation data, contrary to Shao (1993). The result is complementary to that of Yang (2005) and Yang (2006)