Abstract: We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based density estimators. We show that, unlike the analogous situation for kernel density estimators, this MISE formula is relatively unaffected by assumptions of continuity. In particular, it is available for densities which are smooth in only a piecewise sense. Another difference is that in the wavelet case the classical MISE formula is valid only for sufficiently small values of the bandwidth. For larger bandwidths MISE assumes a very different form and hardly varies at all with changing bandwidth. This remarkable property guarantees a high level of robustness against oversmoothing, not encountered in the context of kernel methods. We also use the MISE formula to describe an asymptotically optimal empirical bandwidth selection rule.

Abstract: Consider a class of optimization problems for which the cardinality of the set of feasible solutions is m and the size of every feasible solution is N. We prove in a general probabilistic framework that the value of the optimal solution and the value of the worst solution are asymptotically almost surely (a.s.) the same provided as N and m become large. This result implies that for such a class of combinatorial optimization problems almost ecery algorithm finds asymptotically optimal solution! The quadratic assignment problem, a location problem on graphs, and a pattern matching problem fall into this class

Abstract: For eachd?2, it is possible to placen points ind-space so that, given any two-coloring of the points, a half-space exists within which one color outnumbers the other by as much ascn1/2?1/2d, for some constantc>0 depending ond. This result was proven in a slightly weaker form by Beck and the bound was later tightened by Alexander. It was recently shown to be asymptotically optimal by Matousek. We present a proof of the lower bound, which is based on Alexander's technique but is technically simpler and more accessible. We present three variants of the proof, for three diffrent cases, to provide more intuitive insight into the "large-discrepancy" phenomenon. We also give geometric and probabilistic interpretations of the technique.

Abstract: This paper addresses the problem of tracking random drifting parameters of a linear regression system. The asymptotic properties of several estimation algorithms in the limit of slow drift are studied. The basic tool is the central limit theorem for a class of stochastic difference equations established under weak conditions on disturbances and observations. The estimates of the rate of convergence obtained in the paper allow the asymptotically optimal algorithms to be developed.

Abstract: Let fnλ be the regularised solution of a general, linear operator equation, K f0 = g, from discrete, noisy data yi = g(xi ) + ei, i = 1, …, n, where ei are uncorrelated random errors with variance σ2. In this paper, we consider the two well–known methods – the discrepancy principle and generalised maximum likelihood (GML), for choosing the crucial regularisation parameter λ. We investigate the asymptotic properties as n → ∞ of the “expected” estimates λD and λM corresponding to these two methods respectively. It is shown that if f0 is sufficiently smooth, then λD is weakly asymptotically optimal (ao) with respect to the risk and an L2 norm on the output error. However, λD oversmooths for all sufficiently large n and also for all sufficiently small σ2. If f0 is not too smooth relative to the regularisation space W, then λD can also be weakly ao with respect to a whole class of loss functions involving stronger norms on the input error. For the GML method, we show that if f0 is smooth relative to W (for example f0 xs2208 Wθ, 2, θ > m, if W = Wm, 2), then λM is asymptotically sub-optimal and undersmoothing with respect to all of the loss functions above.

Abstract: Addresses the problem of detecting and isolating abrupt changes in systems with random disturbances. In this paper the author establishes two new criteria of optimality for this problem. The author defines two classes of change detection and isolation algorithms, gives an asymptotically optimal solution to the above problems, investigates the statistical properties of the proposed algorithm and proves optimality theorems for each case.

Abstract: Using a dynamic linear equation that has a conditionally homoskedastic moving average disturbance, we compare two parameterizations of a commonly used instrumental variables estimator (Hansen (1982)) to one that is asymptotically optimal in a class of estimators that includes the conventional one (Hansen (1985)). We find that for some plausible data generating processes, the optimal one is distinctly more efficient asymptotically. Simulations indicate that in samples of size typically available, asymptotic theory describes the distribution of the parameter estimates reasonably well, but that test statistics sometimes are poorly sized.

Abstract: For predicting ∫ G v(x)Z(x) dx, where v is a fixed known function and Z is a stationary random field, a good sampling design should have a greater density of observations where v is relatively large in absolute value. Designs using this idea when G = [0, 1] have been studied for some time. For G a region in two dimensions, very little is known about the statistical properties of cubature rules based on designs with varying density. This work proposes a class of designs that are locally parallelogram lattices but whose densities can vary. The asymptotic variance of the cubature error for these designs is obtained for a class of isotropic random fields and an asymptotically optimal sequence of cubature rules within this class is found. I conjecture that this sequence of cubature rules is asymptotically optimal with respect to all cubature rules.

Abstract: Following Rissanen we consider the statistical model {P θ | as a code-book, θ indexing the codes. To obtain a single code, we first encode some θ and then encode our data x with the code corresponding to this θ. Rissanen's minimum description length principle recommends using the value of θ minimizing the total code length as an estimate of θ given the data x. For some standard statistical models we find easily computable estimators which respect this principle when θ is encoded with the asymptotically optimal coding scheme due to Levin and Chaitin.

Abstract: The optimal universal code for FSMX sources (Rissanen 1981) with respect to Bayes redundancy criterion (Davison 1973) is deduced under the condition that the model, the probabilistic parameters and the initial state are unknown. The algorithm is not only Bayes optimal for FSMX sources but also asymptotically optimal for stationary ergodic sources. Moreover the algorithm is regarded as a generalisation of the Ziv-Lempel algorithm. In the basic CTW algorithm, the algorithm needs the initial context x/sub 1-d/x/sub 2-d/...x/sub 0/ where: a finite constant d is the depth of the context tree, for calculating the coding probability of x/sub 1/. For the problem of the initial situation and the infinite depth tree, the extensions to the CTW algorithm have been proposed in Willems (1994). The optimal algorithm proposed in the present paper gives a solution against these problems from another new point of view.

Abstract: It is shown that as rate increases the problem of asymptotically optimal scalar quantization has polynomial-time (or space) encoding complexity if the distribution function corresponding to the one-third power of the source density is polynomial-time (or space) computable in the Turing sense.

Abstract: In this thesis, we study and compare the performance of several distributed channel assignment algorithms (CAAs) in a cellular system. The CAA which is used to assign a channel to a new call greatly influences the amount of traffic the system can support. We are interested in the design and analysis of algorithms which perform well, but at the same time are relatively easy to implement. In this thesis, we have analyzed the performance of a very simple CAA which we call the Timid Algorithm, in the limiting case of a large number of channels. We have been able to show that, under a plausible mathematical hypothesis, the algorithm is asymptotically optimal, where "asymptotically" refers to a system with a large number of channels. This is very surprising as there are algorithms of much higher complexity which provably do not have this property. The Timid Algorithm is asymptotically optimal, but it requires a large number of channels for a satisfactory performance. We looked at some algorithms which retain the simplicity of the Timid algorithm but which can be expected to give a good performance even with a smaller number of channels. We called one such algorithm the Modified DCAA. We present some simulation results which show that this algorithm gives a reasonably good performance even when the number of channels is small. One of the ways to increase the capacity of a cellular system is through the use of micro-cells. The Modified DCAA, because of its distributed nature and low complexity, is particularly suitable for such microcellular systems. We also present a method for computing the upper bound on the performance of any CAA in a cellular system with adjacent channel constraints. The method, although computationally intensive, may be useful for determining how close an algorithm's performance is to the optimal performance. Finally, we discuss ways of obtaining the set of "allowable" states for a system. We also present some "measurement-based" algorithms and compare their performance with "prediction-based" algorithms.

Abstract: The problem of minimizing Egc(Zt) is considered asymptotically, where Z0,Z1,… is a perturbed random walk and gc are convex functions which are minimized at values that approach ∞ as c ↓ 0. It is shown that a first passage time is asymptotically optimal, and the boundary for this time is characterized in terms of gc and the limiting distribution of the excess over the boundary. Applications to change point problems and power one tests are presented.

Abstract: Wojciech Szpankowskit Department of Computer Science Purdue Unlversity W. Lafayette, IN 47907 U.S.A. spa@cs.purdue.edu There has recently been a resurgence of interest in the shortest common superstring problem due to important applications in molecular biology (e.g., recombination of DNA) and data compression. The problem is NP-hard, but it has been known for some time that greedy algorithms work well for this problem. More precisely, it was proved in a recent sequence of papers that in the worst case a greedy algorithm produces a superstring that is at most f3 times (2 ~ f3 ~ 3) worse than optimal. We analyze the problem in a probabilistic framework, and consider the total overlaps O~pl and O~~ prod uced respectively by the optimal algorithm and a greedy one which turn out to be asymptotically equivalent. More precisely, we show that with high probability limn.....oo n~~:'n :::: llmn_oo nfo~11 :::: k where n is the number of original strings, and H is the entropy of the underlying alphabet. Our result holds under a condition that the lengths of all strings are not too short. Finally, we provide several generalizations and extensions of our basic result..

Abstract: A two-parameter generalization of Jaccard's index of similarity is proposed as a class of measures for testing the homogeneity of two independent multinomial samples. The power approach used in modern asymptotic theory of decomposable statistics is applied to the asymptotic analysis of these measures. The asymptotic analysis is amplified by numerical tabulation yielding an asymptotically optimal similarity test in this class of measures.

Abstract: We have synthesized asymptotically optimal detectors of weak and strong signals with random noninformative parameters against a background of non-Gaussian passive clutter. A possibility of using an adaptive approach to synthesis has been considered and an appropriate structure of an optimal detector of signals with an unknown initial phase has been found.

Abstract: | We introduce a new multiaccess coding strategy called stripping CDMA for a LOOK AWGN channel. Stripping CDMA can be viewed as a generalization of the regular CDMA scheme to one that can achieve arbitrarily high bandwidth eeciency. It can also be viewed as a modiication of the pure stripping scheme to one that is suited for multiaccess channel with bursty users. In this scheme, each user is split into M subusers and the decoder consist of M stripping stages using only single-user decoders. All transmitters of all users are identical, making it suitable for bursty user environment where the set of active users is not known in advanced. No synchronization between users at the phase, symbol, or frame level is needed. Although the scheme is only optimal asymptotically in M, it is close to optimal for small M, depending on the bandwidth eeciency. Performance of this scheme using existing non-optimal convolutional code is also investigated.

Abstract: Let ∏1,…,∏k denote k independent populations, where a random observation from population ∏ i has a uniform distribution over the interval (0,θ i ) and θ i is a realization of a random variable having an unknown prior distribution G i . Population ∏ i is said to be a good population if θ i ≥θ0, where θ0 is a given, positive number. This paper provides a sequence of empirical Bayes procedures for selecting the good populationsamong ∏1,…,∏ k . It is shown that these procedures are asymptotically optimal and that the order of associated convergence rates is O(n-r/4) for some r, 0