Abstract: This paper extends to continuous time the concept of universal portfolio introduced by Cover (1991). Being a performance weighted average of constant rebalanced portfolios, the universal portfolio outperforms constant rebalanced and buy-and-hold portfolios exponentially over the long run. an asymptotic formula summarizing its long-term performance is reported that supplements the one given by Cover. A criterion in terms of long-term averages of instantaneous stock drifts and covariances is found which determines the particular form of the asymptotic growth. A formula for the expected universal wealth is given.

Abstract: Consider scheduling a batch of jobs with stochastic processing times on parallel machines, with minimization of expected weighted flowtime as objective. Smith's Rule, which orders job starts by decreasing ratio of weight to expected processing time, provides a natural heuristic for this problem. In a previous paper we have found an upper bound for the difference between the expected objective under Smith's Rule and under the optimal strategy. In this paper we find an upper bound on the expected number of times that Smith's rule differs from the optimal decision. Under some mild and reasonable assumptions these bounds remain constant when the number of jobs increases. Hence, under these conditions Smith's Rule is asymptotically optimal, and it has a Turnpike Optimality property, that is, Smith's Rule is optimal most of the time.

Abstract: Our aim is to perturb the input so that an algorithm designed under the hypothesis of input non-degeneracy can execute on arbitrary instances. The deterministic scheme of [EmCa] was the first efficient method and was applied to two important predicates. Here it is extended in a consistent manner to another two common predicates, thus making it valid for most algorithms in computational geometry. It is shown that this scheme incurs no extra algebraic complexity over the original algorithm while it increases the bit complexity by a factor roughly proportional to the dimension of the geometric space. The second contribution of this paper is a variant scheme for a restricted class of algorithms that is asymptotically optimal with respect to the algebraic as well as the bit complexity. Both methods are simple to implement and require no symbolic computation. They also conform to certain criteria ensuring that the solution to the original input can be restored from the output on the perturbed input. This is immediate when the input to solution mapping obeys a continuity property and requires some case-specific work otherwise. Finally we discuss extensions and limitations to our approach.

Abstract: We investigate a distributed, state-dependent, dynamic routing strategy for circuit-switched loss networks which we have called Aggregated-Least-Busy-Alternative (ALBA). The networks considered are symmetric and fully connected, the offered calls form Poisson streams and routes have at most two links. In ALBA(K), the states of each link are lumped intoK (K ≥ 2) aggregates and the route of each call is determined by local information on the aggregate-states of the links of the alternate routes. The last aggregate is always the set of states reserved for direct traffic and defined by the trunk reservation parameter. The particular case of ALBA in which there is no aggregation is Least-Busy-Alternative (LBA); ALBA(2) represents the other extreme of aggregation. We consider two separate asymptotic scalings based on Fixed Point Models for ALBA(K) which were obtained and investigated in an earlier paper. In the first, it is assumed that the number of network nodes, the offered traffic and trunk group size are all large; their ratios have been chosen to reflect practical interest. The results show that there exists a threshold which delineates fundamentally different behavior: for offered traffic below the threshold, the network loss probability decreases exponentially with increasing network size, while above the threshold the decrease is only polynomial. In the related second asymptotic scaling, the asymptotically optimum trunk reservation parameter is obtained as the solution of a simple equation. Such asymptotically optimal designs are compared to the outputs of exhaustive numerical searches for some realistically sized networks and found to perform very well.

Abstract: In the stable0–1 sorting problem the task is to sort an array ofn elements with two distinct values such that equal elements retain their relative input order. Recently, Munro, Raman and Salowe gave an algorithm which solves this problem inO(n log*n) time and constant extra space. We show that by a modification of their method the stable0–1 sorting is possible inO(n) time andO(1) extra space. Stable three-way partitioning can be reduced to stable0–1 sorting. This immediately yields a stable minimum space quicksort, which sorts multisets in asymptotically optimal time with high probability.

Abstract: Results obtained by the authors (1991) worst-case/deterministic H/sub infinity / identification of discrete-time plants are extended to continuous-time plants. The problem involves identification of the transfer function of a stable strictly proper continuous-time plant from a finite number of noisy point samples of the plant frequency response. The assumed information consists of a lower bound on the relative stability of the plant, an upper bound on a certain gain associated with the plant, an upper bound on the roll-off rate of the plant, and an upper bound on the noise level. Concrete plans of identification algorithms are provided for this problem. Explicit worst-case/deterministic error bounds for each algorithm establish that they are robustly convergent and (essentially) asymptotically optimal. Additionally, these bounds provide an a priori computable H/sub infinity / uncertainty specification, corresponding to the resulting identified plant transfer function, as an explicit function of the plant and noise prior information and the data cardinality. >

Abstract: We present the following results about IDA * and related algorithms: • We show that IDA * is not asymptotically optimal in all of the cases where it was thought to be so. In particular, there are trees satisfying all of the conditions previously thought to guarantee asymptotic optimality for IDA *, such that IDA * will expand more than O(N) nodes, where N is the number of nodes eligible for expansion by A*. • We present a new set of necessary and sufficient conditions to guarantee that IDA * expands O(N) nodes on trees. • On trees not satisfying the above conditions, there is no best-first admissible tree search algorithm that runs in S = N/Ψ(N) (where Ψ(N) ≠ O(1)) memory and always expands O(N) nodes. • There are acyclic graphs on which IDA * expands Ω(22N) nodes.

Abstract: Explicit formulas are derived for the coding gain of hierarchical transforms for a given number of stages, and the asymptotic gain as this number goes to infinity. The intuitive result that hierarchical transforms are not asymptotically optimal, that is, their coding gains do not approach the inverse of the spectral flatness measure as the number of stages goes to infinity, is confirmed. Examples comparing the limits for hierarchical transforms and M-band parallel systems (filter banks) for AR(1) signals are presented. >

Abstract: Suppose $X_1,X_2,\ldots,X_n$ are i.i.d. with unknown density $f$. There is a well-known expression for the asymptotic mean integrated squared error (MISE) in estimating $f$ by a kernel estimate $\hat{f}_n$, under certain conditions on $f$, the kernel and the bandwidth. Suppose that one would like to choose a sample size so that the MISE is smaller than some preassigned positive number $w$. Based on the asymptotic expression for the MISE, one can identify an appropriate sample size to solve this problem. However, the appropriate sample size depends on a functional of the density that typically is unknown. In this paper, a stopping rule is proposed for the purpose of bounding the MISE, and this rule is shown to be asymptotically efficient in a certain sense as $w$ approaches zero. These results are obtained for data-driven bandwidths that are asymptotically optimal as $n$ goes to infinity.

Abstract: A processor-efficient systolic algorithm for the dynamic programming approach to the knapsack problem is presented in this paper. The algorithm is implemented on a linear systolic array where the number of the cells q, the cell memory storage α and the input/output requirements are design parameters. These are independent of the problem size given by the number of the objects m and the knapsack capacity c. The time complexity of the algorithm is Θ(mc/q + m) and both the time speedup and the processor efficiency are asymptotically optimal.

Abstract: A vigorous branch of automatic learning is directed at the task of locating a global minimum of an unknown multimodal function f( theta ) on the basis of noisy observations L( theta (i))=f( theta (i))+W( theta (i)) taken at sequentially-chosen control points ( theta (i)). In all preceding convergence deviations known to the authors, the noise is postulated to depend on the past only through control selection. Here they allow the observation noise sequence to be stochastically dependent, in particular, a function of an unknown underlying Markov decision process, the observations being the stagewise losses. In a sense, in order to be made precise, the algorithm offered is shown to attain asymptotically optimal performance, and rates are assured. A motivating example from queueing theory is offered, and connections with classical problems of Markov control theory and other disciplines are mentioned. >

Abstract: We present a simple, efficient, robust plane-sweep algorithm that solves 2-dimensional nearest-neighbor problems in asymptotically optimal timeO(n logn). A “foolproof” implementation guarantees an exact result at the cost of using triple-precision integer arithmetic at some key steps.

Abstract: This article is concerned with the problem of allocating a fixed number of trials between two independent normal populations with unknown means θ and ω, in order to estimate the product θω with squared error loss. Introducing independent normal priors on θ and ω a two stage procedure is introduced and shown to be asymptotically optimal. This problem arises, most obviously, in situations of deter-mining area based on measurements of length and width.

Abstract: The problem of choosing a local value of the bandwidth h for a kernel density estimate is considered. Estimates of the density f at a given point are needed in the estimation of the asymptotic standard error or sample quantiles and in some kernel regression estimators based on random design points. The value of the bandwidth that minimizes the asymptotic MSE of the kernel estimate at the point x involves both f(x) and f″(x). In this paper we show that the “solve-the-equation” bandwidth selection method of Sheather (1986) produces an estimate of the asymptotically optimal h which has relative rate of convergence of n-2/9. We also show how higher order kernel estimates of f″ can be used to improve this rate to n-2/5. How much reliance can be placed on these theoretical results is investigated through a simulation study which compares the performance of a number of different selection methods

Abstract: A discrete-time stochastic system depending on an unknown parameter and with small observation noise is considered in this paper A quadratic variation test is proposed to detect the unknown system parameter Then, based on the test result, asymptotic filters and computable adaptive asymptotically optimal controls are constructed Finally, numerical experiments are undertaken regarding the above aspects

Abstract: A scheduling problem of an exponential single server with a finite queueing capacity that serves customers from n heterogeneous classes is considered. Arrivals are Poissonian and every class has its own rate and its own finite waiting room. The waiting rooms can be of arbitrary size. Arriving customers that find a full queue are lost. Of particular interest is finding a scheduling policy that allows service preemption and has a weighted throughput which is close enough to the optimal one. As an optimal scheduling policy is extremely hard to find, a different methodology is used to tackle the problem. First, the optimal weighted throughput is bound from above, and the asymptotically optimal policy is found. Then, based on the bounding technique and the asymptotically optimal policy, a new policy, the overflow scheduling policy, that provides a weighted throughput which is very close to the upper bound is proposed. The quality of the policy is demonstrated by various examples. >

Abstract: Lower bound for the integral risk of density function estimates by A. Samarov On nonparametric estimation of functions satisfying differential inequalities by A. S. Nemirovskii Asymptotically minimax image reconstruction problems by A. P. Korostelev and A. B. Tsybakov On problems of adaptive estimation in white Gaussian noise by O. V. Lepskii On stochastic approximation with arbitrary noise (the KW-case) by B. T. Polyak and A. B. Tsybakov Pseudovalues and minimax filtering algorithms for the nonparametric median by E. N. Belitser and A. P. Korostelev On large deviations for ergodic process empirical measures by A. Yu. Veretennikov On asymptotically optimal sequential experimental design by V. G. Spokoinyi.

Abstract: Steel [14] proposed a distribution-free many-one method for comparing k≧2 test treatments with a control or standard in the one-way layout based on Wilcoxon [16] rank sum statistics. An efficient method is presented for computing the necessary probability points for this technique and an expanded table of points is given. The method also works for generalizations of Steel's procedure where the rank sum statistics are replaced by general linear rank statistics. Asymptotic probability points are found for many-one methods based on all linear rank statistics satisfying the conditions of the Chernoff-Savage [2] Theorem. It is shown that the asymptotically optimal design for these many-one methods takes approximately k 1/2 times as many observations from the control as from each test treatment. The asymptotic relative efficiency of two many-one procedures is identical to that of their two-sample counterparts.

Abstract: An empirical linear Bayes estimator is asymptotically optimal in the usual sense if its average risk converges to the risk of the corresponding linear Bayes estimator. The present paper demonstrates that the following result holds for the most commonly used models: If the unknown (structural) parameters are estimated in such a way that their mean square error converges at a certain rate, then the corresponding empirical linear Bayes estimator is asymptotically optimal with the same rate of risk convergence. In particular, this is the case for the random coefficient regression model, and for hierarchical models in the univariate case.

Abstract: The authors present an asymptotic analysis of hierarchical production planning in a manufacturing system with two tandem machines that are subject to breakdown and repair. The buffer between the two machines is assumed finite. Therefore, the number of parts in that buffer needs to be nonnegative and bounded above by the buffer size. As the rate of change in machine state approaches infinity, the analysis results in a limiting problem in which the stochastic machine capacity is replaced by the equilibrium mean capacity. The value function for the original problem is proved to converge to the value function of the limiting problem. Controls for the original problem are constructed from near optimal controls of the limiting problem in a way which guarantees their asymptotic optimality. The convergence rate of the value function of the original problem to that of the limiting problem together with the error estimate for the constructed asymptotically optimal controls are obtained. >

Abstract: This article discusses the design and bootstrap construction of asymptotically optimal prediction regions. The emphasis is on devising simultaneous one-sided prediction intervals. A good solution to this problem implies constructions for simultaneous two-sided prediction intervals and for multivariate prediction regions.

Abstract: The authors introduce a bin-packing heuristic that is well-suited for implementation on massively parallel SIMD (single-instruction multiple-data) or MIMD (multiple-instruction multiple-data) computing systems. The average-case behavior (and the variance) of the packing technique can be predicted when the input data have a symmetric distribution. The method is asymptotically optimal, yields perfect packings, and achieves the best possible average case behavior with high probability. The analytical result improves upon any online algorithms previously reported in the literature and is identical to the best results reported so far for offline algorithms. >