In this paper we introduce a smooth version of local linear regression estimators and address their advantages. The MSE and MISE of the estimators are computed explicitly. It turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency-they are not only efficient in rates but also nearly efficient in constant factors. In the nonparametric regression context, the asymptotic minimax lower bound is developed via the heuristic of the" hardest onedimensional subproblem" of Donoho and Liu. Connections of the minimax risk with the modulus of continuity are made. The lower bound is also applicable for estimating conditional mean (regression) and conditional quantiles for both fixed and random design regression problems.

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This paper studies the eect of risk-aversion in the competitive newsvendor game. Multiple newsvendors with risk-averse preferences face a random demand and the demand is allocated proportionally to their inventory levels. Each newsvendor aims to maximize his expected utility instead of his expected prot. Assuming a general form of risk-averse utility function, we prove that there exists a pure Nash equilibrium in this game, and it is also unique under certain conditions. We nd that the order quantity of each newsvendor is decreasing in the degree of risk-aversion and increasing in the initial wealth. Newsvendors with moderate preferences of risk-aversion make more prots compared with the risk-neutral situation. We also discuss the joint eect of risk-aversion and competition. If the eect of risk-aversion is strong enough to dominate the eect of competition, the total inventory level under competition will be lower than that under centralized decision-making.

The classical Fej\'er's theorem is a criterion for pointwise convergence of Fourier series on the unit circle. We generalize it to locally compact groups.