We provide a new insight of the difficulty of nonparametric estimation of a whole function. A new method is invented for finding a minimax lower bound of globally estimating a function. The idea is to adjust automatically the direction to the nearly hardest I-dimensional subproblem at each location, and to use locally the difficulty of I-dimensional subproblem. In a variety of contexts, our method can give not only attainable global rates, but also constant factors. Comparing with the existing techniques, our method has the advantages of being easily implemented and understood, and can give constant factors as well.

We propose a semiparametric latent Gaussian copula model for modelling mixed multivariate data, which contain a combination of both continuous and binary variables. The model assumes that the observed binary variables are obtained by dichotomizing latent variables that satisfy the Gaussian copula distribution. The goal is to infer the conditional independence relationship between the latent random variables, based on the observed mixed data. Our work has two main contributions: we propose a unified rankbased approach to estimate the correlation matrix of latent variables; we establish the concentration inequality of the proposed rankbased estimator. Consequently, our methods achieve the same rates of convergence for precision matrix estimation and graph recovery, as if the latent variables were observed. The methods proposed are numerically assessed through extensive simulation studies, and real

Kernel density estimates are frequently used, based on a second order kernel. Thus, the bias inherent to the estimates has an order of O(<i>h</i>²_<i>n</i>). In this note, a method of correcting the bias in the kernel density estimates is provided, which reduces the bias to a smaller order. Effectively, this method produces a higher order kernel based on a second order kernel. For a kernel function <i>K</i>, the functions W_k(x)=^k-1₁₌₀(^k_l+1)x^lK^(l)(x)/l! and [1/<i>K</i>^{(<i>k</i> 1)}(<i>x</i>)/<i>x</i> d <i>x</i>]<i>K</i>^{(<i>k</i> 1)}(<i>x</i>)/<i>x</i> are kernels of order <i>k</i>, under some mild conditions.

We develop an expectation-maximization algorithm with local adaptivity for image segmentation and classification. The key idea of our approach is to combine global statistics extracted from the Gaussian mixture model or other proper statistical models with local statistics and geometrical information, such as local probability distribution, orientation, and anisotropy. The combined information is used to design an adaptive local classification strategy that improves the robustness of the algorithm and also keeps fine features in the image. The proposed methodology is flexible and can be easily generalized to deal with other inferred information/quantities and statistical methods/models.

When people in a society want to make inference about some parameter, each person may want to use data collected by other people. Information (data) exchange in social networks is usually costly, so to make reliable statistical decisions, people need to weigh the benefits and costs of information acquisition. Conflicts of interests and coordination problems will arise in the process. Classical statistics does not consider peoples incentives and interactions in the data-collection process. To address this imperfection, this work explores multi-agent Bayesian inference problems with a game theoretic social network model. Motivated by our interest in aggregate inference at the societal level, we propose a new concept, <i>finite population learning</i>, to address whether with high probability, a large fraction of people in a given finite population network can make good inference. Serving as a foundation, this concept enables