This work is concerned with marginal sure independence feature screening for ultrahigh dimensional discriminant analysis. The response variable is categorical in discriminant analysis. This enables us to use the conditional distribution function to construct a new index for feature screening. In this article, we propose a marginal feature screening procedure based on empirical conditional distribution function. We establish the sure screening and ranking consistency properties for the proposed procedure without assuming any moment condition on the predictors. The proposed procedure enjoys several appealing merits. First, it is model-free in that its implementation does not require specification of a regression model. Second, it is robust to heavy-tailed distributions of predictors and the presence of potential outliers. Third, it allows the categorical response having a diverging number of classes in the order of O(n^κ ) with some κ ≥ 0. We assess the finite sample property of the proposed procedure byMonte Carlo simulation studies and numerical comparison.We further illustrate the proposed methodology by empirical analyses of two real-life datasets. Supplementary materials for this article are available online.

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Consider a normal model with unknown mean bounded by a known constant. This paper deals with minimax estimation of the squared mean. We establish an expression for the asymptotic minimax risk. This result is applied in nonparametric estimation of quadratic functionals.

The availability of high-frequency intraday data allows us to accurately estimate stock volatility. This paper employs a bivariate diffusion to model the price and volatility of an asset and investigates kernel type estimators of spot volatility based on high-frequency return data. We establish both pointwise and global asymptotic distributions for the estimators.

Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve 1, 2. Understanding evolutionary game dynamics in structured populations remains difficult. Mathematical results are known for special structures in which all individuals have the same number of neighbours 3, 4, 5, 6, 7, 8. The general case, in which the number of neighbours can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class that suggests there is no efficient algorithm 9. Whether a simple solution for weak selection exists has remained unanswered. Here we provide a solution for weak selection that applies to any graph or network. Our method relies on calculating the coalescence times 10, 11 of random walks 12. We evaluate large numbers of diverse population structures for their propensity to favour cooperation. We study how small