Abstract: Distortion-free compressibility of individual pictures, i.e., two-dimensional arrays of data, by finite-state encoders is investigated. For every individual infinite picture I, a quantity ρ(I) is defined, called the compressibility of I, which is shown to be the asymptotically attainable lower bound on the compression-ratio that can be achieved for I by any finite-state, information-lossless encoder. This is demonstrated by means of a constructive coding theorem and its converse that, apart from their asymptotic significance, might also provide useful criteria for finite and practical data-compression tasks. The proposed picture-compressibility is also shown to possess the properties that one would expect and require of a suitably defined concept of two-dimensional entropy for arbitrary probabilistic ensembles of infinite pictures. While the definition of ρ(I) allows the use of different machines for different pictures, the constructive coding theorem leads to a universal compression-scheme that is asymptotically optimal for every picture. The results of this paper are readily extendable to data arrays of any finite dimension. The proofs of the theorems will appear in a forthcoming paper.

Abstract: We give an asymptotic error analysis of the Fourier reconstruction algorithm and show that a modified algorithm is asymptotically optimal. We compare the modified algorithm with standard Fourier and with filtered backprojection using a mathematically generated chest phantom.

Abstract: Estimation of the vector β of the regression coefficients in a multiple linear regressionY=Xβ+e is considered when β has a completely unknown and unspecified distribution and the error-vector e has a multivariate standard normal distribution. The optimal estimator for β, which minimizes the overall mean squared error, cannot be constructed for use in practice. UsingX, Y and the information contained in the observation-vectors obtained fromn independent past experiences of the problem, (empirical Bayes) estimators for β are exhibited. These estimators are compared with the optimal estimator and are shown to be asymptotically optimal. Estimators asymptotically optimal with rates nearO(n−1) are constructed.

Abstract: There are n jobs that have to be processed on one available machine. The ith job arrives at epoch r1 and it takes P1 units of time to carry out its operation. No preemptions are allowed. It is desired to minimize total flow time. A probabilistic analysis is presented for this NP-hard problem known as 1/rj/ΣCj in the scheduling literature. Under very general probability distributions of the problem data, it is shown that the problem can be broken into 2 categories: the undersaturated and the oversaturated case. Asymptotically optimal algorithms are presented for each case.

Abstract: This paper is concerned with the probabilistic behavior of three NP-complete problems: the K-center, the dominating set, and the set covering problem. There are two main contributions: one is the asymptotic values of the optimal solutions and the probability expressions for the interval of values where the optimal solution value lies for every finite problem size. The other contribution is the proof that any random feasible solution is asymptotically optimal with probability that is asymptotically 1.

Abstract: Suppose that for i = 1, …, I, a random variable whose distribution depends on parameter μ i , > 0 is observable from population i. The problem is to estimate θ = IIμ i , using a Bayesian approach with squared error estimation loss in θ and allowing sequential allocation. That is, k observations may be taken one at a time and the decision to take an observation from population i may depend on past observations from all populations. For estimating θ, the best nonrandom allocation scheme, the myopic scheme, and the optimal allocation are discussed. The myopic scheme is typically asymptotically optimal and in several examples allocates in proportion to estimated population coefficients of variation, with the estimates updated at each stage. It is also shown that sequential allocation can improve the Bayes risk (over the best nonrandom allocation) up to 100 (I — 1)/I%.

Abstract: : This paper deals with the problem of selecting good binomial populations compared with a standard or a control through the empirical Bayes approach. Two cases have been studied: one with the pior distribution completely unknown and the other with the prior distribution symmetrical about p = 1/2, but otherwise unknown. In each case, empirical Bayes rules are derived and their rates of convergence are shown to be of order O(exp(-cn)) for some cO, where n is the number of accumulated post experiences at hand. Keywords: Statistical decision theory; Smoothing(Mathematics); Asymptotically optimal. (Author)

Abstract: A test for the equality of two or more two-parameter exponential distributions is suggested. It is developed on an intuitive basis and is obtained by combining two independent tests by the Fisher method (1950, pp. 99-101). The test is simple for application and is optimal asymptotically in the sense of Bahadur efficiency (1960). A numerical example is discussed to illustrate its application in a real-world situation. The Monte Carlo simulation is used for calculating its power which is compared with that of the test suggested by Singh and Narayan (1983). The suggested test is found oftener more powerful.

Abstract: A learning controller is presented for a Markovian decision problem in which the transition probabilities are unknown. This controller, which is designed to be asymptotically optimal with consideration of a conflict between estimation and control, uses a performance criterion incorporating a tradeoff between them explicitly for determination of a control policy. It is shown that this controller achieves asymptotic optimality in the sense that the relative frequency of applying the optimal policy converges to unity.

Abstract: The problem is to select as rapidly as possible a pre-specified number of secretaries (sequentially and without recall) from a proffered queue of applicants. Each candidate makes demands in each of several currencies, and we know the joint probability distribution of these demands. We also have a given total budget in each currency. We consider procedures that minimize the expected number of interviews and derive an asymptotically optimal procedure. Derivation of this procedure depends on the following result. Given a probability measure F on the positive orthant Q and a point ρ0 ∈ Q, consider a region A ⊂ Q whose center of gravity lies below ρ0 coordinatewise. Then F(A) is maximal when A is a simplicial section A = {ρ : ρ · θ ≤ r}, where r ≥ 0 and θ is a unit vector in Q. The original motivation for this problem arose in connection with a scheme for selecting optical fibers.

Abstract: For every individual infinite sequence x, a quantity ρ(x) is defined, called the normalized complexity (or compressibility) of x, which is shown to be the asymptotically attainable lower bound on the compression ratio (i.e., normalized encoded length) that can be achieved for x by any finite-state information lossless encoder. This is demonstrated by a constructive coding theorem and its converse that, apart from their asymptotic significance, also provide a useful performance criteria for finite practical data compression algorithms, and leads to a universal encoding algorithm which is asymptotically optimal for all sequences.

Abstract: It is shown that differentiated SPR tests, which are limiting cases of ordinary SPR tests, are asymptotically optimal in a decision-theoretic formulation for composite hypotheses in one-parameter problems. It is also given some results connected with comparisons of different tests in this case.

Abstract: The problem is to estimate a quantile of a symmetric distribution. The cases of known and unknown center are studied for small and large samples. The estimators for known center are the sample quantile, the symmetrized sample quantile, the sample quantile from flipped over data, the Rao-Blackwellized sample quantile, and a Bayes estimator using a Dirichlet prior. For center unknown, we study the analogues of the first four estimators listed above. For small samples and center known, the Rao-Blackwellized sample quantile performs very well for normal and double exponential distributions while for the Cauchy distribution the flipped over estimator did well. In the center known case, the latter four estimators are asymptotically equivalent, asymptotically optimal in the sense of Hajek's convolution, and asymptotically minimax in the Hajek-LeCam sense. For center unknown, those properties remain true if one uses an adaptive estimator of the center for the symmetrized sample quantile, the flipped over estimator, and the Rao-Blackwell estimator.

Abstract: For the problem of estimating the mean of independent, identically distributed random variables, with loss equal to a linear combination of squared error and sample size, certain sequential procedures have been shown to be asymptotically optimal when compared with the best fixed sample size rule In this paper it is shown that these procedures are asymptotically suboptimal when compared with a closely related optimal stopping rule

Abstract: Former results on BAHADUR efficiency of signed rank tests are carried over to the class of two-sample rank tests. It is shown that the two-sample rank tests are asymptotically optimal at alternatives far away from the hypothesis under fairly general conditions. Surprisingly, the median test appears to be optimal only in case of equal sample sizes.

Abstract: We assume a parallel RAM model which allows both concurrent writes and concurrent reads of global memory. Our algorithms are randomized: each processor is allowed an independent random number generator. However our stated resource bounds hold for worst case input with overwhelming likelihood as the input size grows. We give a new parallel algorithm for integer sorting where the integer keys are restricted to at most polynomial magnitude. Our algorithm costs only logarithmic time and is the first known where the product of the time and processor bounds are bounded by a linear function of the input size. These simultaneous resource bounds are asymptotically optimal. All previous known parallel sorting algorithms required at least a linear number of processors to achieve logarithmic time bounds, and hence were nonoptimal by at least a logarithmic factor.

Abstract: The power of Pearson chi-squared and likelihood ratio goodness-of-fit tests based on different partitions is studied by considering families of densities “between” the null density and fixed alternative densities. For sample sizes n → ∞, local asymptotic theory with respect to the number of classes k is developed for such families. Simple sufficient and almost necessary conditions are derived under which it is asymptotically optimal to let k tend to infinity with n. A numerical study shows that the results of the asymptotic local theory for contamination families agree well with the actual power performance of the tests. For heavy-tailed alternatives, the tests have the best power when k is relatively large. Unbalanced partitions with some small classes in the tails perform surprisingly well, in particular when the alternatives have fairly heavy tails.

Abstract: Let \{X_{n}\} n \geq 1 be a finite Markov chain with transition probability matrix of strictly positive entries. A large deviation theorem is proved for the empirical transition count matrix and is used to get asymptotically optimal critical regions for testing simple hypotheses about the transition matrix. As a corollary, the error exponent in the source coding theorem for \{X_{n}\} is obtained. These results generalize the corresponding results for the independent and identically distributed case.