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Motivated from second-order stochastic dominance, we introduce a risk measure that we call shortfall. We examine shortfall's properties and discuss its relation to such commonly used risk measures as standard deviation, VaR, lower partial moments, and coherent risk measures. We show that the mean-shortfall optimization problem, unlike mean-VaR, can be solved e ciently as a convex optimization problem, while the sample mean-shortfall portfolio optimization problem can be solved very e ciently as a linear optimization problem. We provide empirical evidence (a) in asset allocation, and (b) in a problem of tracking an index using only a limited number of assets that the mean-shortfall approach might have advantages over mean-variance. ?

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There exist several estimators of the memory parameter in long-memory time series models with the spectrum specified only locally near zero frequency. In this paper we give an asymptotic lower bound for the minimax risk of any estimator of the memory parameter as a function of the degree of local smoothness of the spectral density at zero. The lower bound allows one to evaluate and compare different estimators by their asymptotic behaviour, and to claim the rate optimality for any estimator attaining the bound. A log-periodogram regression estimator, analysed by Robinson (Log-periodogram regression of time series with long range dependence. Ann. Stat. 23 (1995), 1048-72), is then shown to attain the lower bound, and is thus rate optimal.