We consider adaptive sequential prediction of arbitrary binary sequences when the performance is evaluated using a general loss function. The goal is to predict on each individual sequence nearly as well as the best prediction strategy in a given comparison class of (possibly adaptive) prediction strategies, called experts. By using a general loss function, we generalize previous work on universal prediction, forecasting, and data compression. However, here we restrict ourselves to the case when the comparison class is finite. For a given sequence, we define the regret as the total loss on the entire sequence suffered by the adaptive sequential predictor, minus the total loss suffered by the predictor in the comparison class that performs best on that particular sequence. We show that for a large class of loss functions, the minimax regret is either \Theta(log N) or \Omega\Gamma p ` log N ), depending on the loss function, where N is the number of predictors in the comparison class a...

Flajolet and Soria established several central limit theorems for the parameter "number of components" in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This theorem is also applicable to arithmetical functions. Moreover, asymptotic expressions are derived for moments of integral order. Many examples from different applications are discussed.

Motivated by the desire to understand the performance of service-oriented call centers, which often provide low-to-moderate quality of service, this paper investigates the efficiency-driven (ED) limiting regime for many-server queues with abandonments. The starting point is the realization that, in the presence of substantial customer abandonment, call-center service-level agreements (SLA’s) can be met in the ED regime, where the arrival rate exceeds the maximum possible service rate. Mathematically, the ED regime is defined by letting the arrival rate and the number of servers increase together so that the probability of abandonment approaches a positive limit. To obtain the ED regime, it suffices to let the arrival rate and the number of servers increase with the traffic intensity ρ held fixed with ρ> 1 (so that the arrival rate exceeds the maximum possible service rate). Even though the probability of delay necessarily approaches 1 in the ED regime, the ED regime can be realistic because, due to the abandonments, the delays need not be excessively large. This paper establishes ED many-server heavy-traffic limits and develops associated ap-proximations for performance measures in the M/M/s/r + M model, having a Poisson arrival process, exponential service times, s servers, r extra waiting spaces and exponential abandon times (the final +M). In the ED regime, essentially the same limiting behavior occurs when the abandonment rate α approaches 0 as when the number of servers s approaches ∞; in-deed, it suffices to assume that s/α → ∞. The ED approximations are shown to be useful by comparing them to exact numerical results for the M/M/s/r + M model obtained using an algorithm developed in Whitt (2003), which exploits numerical transform inversion.

JEL No. C10 The major contributions of twentieth century econometrics to knowledge were the definition of causal parameters when agents are constrained by resources and markets and causes are interrelated, the analysis of what is required to recover causal parameters from data (the identification problem), and clarification of the role of causal parameters in policy evaluation and in forecasting the effects of policies never previously experienced. This paper summarizes the development of those ideas by the Cowles Commission, the response to their work by structural econometricians and VAR econometricians, and the response to structural and VAR econometrics by calibrators, advocates of natural and social experiments, and by nonparametric econometricians and statisticians.

We study how intermediation and asset prices in over-the-counter markets are affected by illiquidity associated with search and bargaining. We compute explicitly the prices at which investors trade with each other, as well as marketmakers ’ bid and ask prices in a dynamic model with strategic agents. Bid-ask spreads are lower if investors can more easily find other investors, or have easier access to multiple marketmakers. With a monopolistic marketmaker, bid-ask spreads are higher if investors have easier access to the marketmaker. We characterize endogenous search and welfare, and discuss empirical implications. ∗Part of this paper was previously distributed under the title “Valuation in Dynamic Bargaining Markets. ” We are grateful for conversations with Yakov Amihud, Helmut

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We briefly introduce some basic facts about multivariate extreme value theory and present some new results regarding finite aggregates and multivariate extreme value distributions. Based on our results high frequency data can considerably improve quality of estimates of extreme movements in financial markets. Secondly we present an empirical exploration of what the tails really look like for four foreign exchange rates sampled at varying frequencies. Both temporal and spatial dependence is considered. In particular we estimate the spectral measure, which along with the tail index, completely determines the extreme value distribution. Lastly we apply our results to the problem of portfolio optimisation or risk minimization. We analyze how the expected shortfall and VaR scale with time horizon and find that this scaling is not by a factor of square root of time as is frequently used, but by a different power of time. We show that the accuracy of risk estimation can be drast...

We consider a connection-level model of Internet congestion control, introduced by Massoulié and Roberts [36], that represents the randomly varying number of flows present in a network. Here bandwidth is shared fairly amongst elastic document transfers according to a weighted α-fair bandwidth sharing policy introduced by Mo and Walrand [37] (α ∈ (0,∞)). Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [29] by two of the authors. Here we use the long time behavior of the solutions of this fluid model established in [29] to derive a property called multiplicative state space collapse, which loosely speaking shows that in diffusion scale the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process. Under weighted proportional fair sharing of bandwidth (α = 1) and a mild

This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a non-negligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennett's observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost

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