Abstract: SUMMARY We define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression. Our optimal difference sequences do not depend on unknowns, such as the mean function, and provide substantial improvements over the suboptimal sequences commonly used in practice. For example, in the case of normal data the usual variance estimator based on symmetric second-order differences is only 64% efficient relative to the estimator based on optimal second-order differences. The efficiency of an optimal mth-order difference estimator relative to the error sample variance is 2m/(2m+1). Again this is for normal data, and increases as the tails of the error distribution become heavier.

Abstract: In structural equation modeling the statistician needs assumptions inorder (1) to guarantee that the estimates are consistent for the parameters of interest, and (2) to evaluate precision of the estimates and significance level of test statistics. With respect to purpose (1), the typical type of analyses (ML and WLS) are robust against violation of distributional assumptions; i.e., estimates remain consistent or any type of WLS analysis and distribution of z. (It should be noted, however, that (1) is sensitive to structural misspecification.) A typical assumption used for purpose (2), is the assumption that the vector z of observable follows a multivariate normal distribution. In relation to purpose (2), distributional misspecification may have consequences for efficiency, as well as power of test statistics (see Satorra, 1989a); that is, some estimation methods may bemore precise than others for a given specific distribution of z. For instance, ADF-WLS is asymptotically optimal under a variety of distributions of z, while the asymptotic optimality of NT-WLS may be lost when the data is non-normal Violation of a distributional assumption may have consequences for purpose (2). However, recent theory, such as the one described in Sections 7 and 8, showes that asymptotic variances of estimates and asympttic null distributions of test statistics derived under the normality assumption may be correct even when z is non-normal provided certain model conditions hold (the conditions of Theorem 1). That is, in a specific application with z non-normally distributed, the assumption that z is normal play the role of a “working device” that facilitates calculation of the correct distribution of statistics of interest. This corresponds to what in Section 7 and 8 has been called asymptotic robustness. For most of the models considered in practice, replacing the assyumption uncorrelation for the assumption of independence implised reaching the properties of asymptotic robustness; in that case, in order to evaluate the asymptotic behavior of statistics of interest, a NT form for Γ produces correct results even for non-normal data. This robustness result applies regardless of the type of fitting criterion used. Distinction between “uncorrelation’ and ‘independence’ becomes crucial when dealing with the asymptotic robustness issue. Statistical independence among variables of the model guarantee that the distribution of statistics of interest are asymptotically distribution-free of the non-normal variables; thus a NT form for Γ applies. As an example of where such distinction is apparent, consider a simple regression model with a heteroskedastic disturbance term. Here the disturbance term is uncorrelated with the regressor, but the variance varies with the value of the regressor. For a study showing that ADF-WLS protects against heteroskedasticity of erros, while ML wil generally fail, see Mooijaart and Satorra (1987). In regresion analysis the usual method for detecting heteroskedasticity is by looking at residual plots. Presumably, alsi in structural equation modeling, the need to distinguish between uncorrelation and independence will force the researcher to go back to the row data in order to do a similar type of “residuals’ inspection. In concluding, an importance consideration is to compute sampling variability for estimates and test statistics using appropriate formulae, without requiring that the estimation procedure be the ‘best’ in some sense. We have seen that such computations can be carried out correctly using the wrong assumptions with respect to the distribution of the vector of observable variables, provided some additional model conditions hold. Roughly speaking, such additional model conditions amount to strengthen the usual assumption of uncorrelation among some random constituents of the model to the assumption of stochastic independecen.

Abstract: A description is given of an asymptotically-minimum-variance algorithm for estimating the MA (moving-average) and ARMA (autoregressive moving-average) parameters of non-Gaussian processes from sample high-order moments. The algorithm uses the statistical properties (covariances and cross covariances) of the sample moments explicitly. A simpler alternative algorithm that requires only linear operations is also presented. The latter algorithm is asymptotically-minimum-variance in the class of weighted least-squares algorithms. >

Abstract: Recent results on the asymptotically optimal design of sliding windows for virtual circuits in high speed, geographically dispersed data networks in a stationary environment are exploited here in the synthesis of algorithms for adapting windows in realistic, non-stationary environments. The algorithms proposed here require each virtual circuit's source to measure the round trip response times of its packets and to use these measurements to dynamically adjust its window. Our design philosophy is quasi-stationary: we first obtain, for a complete range of parameterized stationary conditions, the relation, called the “design equation”, that exists between the window and the mean response time in asymptotically optimal designs; the adaptation algorithm is simply an iterative algorithm for tracking the root of the design equation as conditions change in a non-stationary environment. A report is given of extensive simulations of networks with data rates of 45 Mbps and propagation delays of up to 47 msecs. The simulations generally confirm that the realizations of the adaptive algorithms give stable, efficient performance and are close to theoretical expectations when these exist.

Abstract: The problem of mapping the tasks of a multitarget tracking algorithm onto parallel computing architectures to maximize speedup is considered An asymptotically optimal mapping algorithm is developed and applied to study the effects of task granularity and processor architectures on the speedup From the simulation results, it is concluded that task granularity and the parallelization of clustering and global hypotheses formation stages of the tracking algorithm are major determinants of speedup >

Abstract: In most vehicle routing problems, a given set of customers is to be partitioned into a collection of regions each of which is assigned to a single vehicle starting at a depot and returning there after visiting all of the region's customers exactly once in a route. In this paper we consider problem settings where the cost of a route may depend on its length ϑ as well as m, the number of points on the route, according to some general function f(ϑ, m), assumed to be nondecreasing and concave in ϑ. We describe a class of O(N logN) or O(N2) heuristics and show under mild probabilistic assumptions that the solutions generated are asymptotically optimal. We also show that lower and upper bounds on the system-wide costs may be computed (with even simpler procedures) and that these bounds are asymptotically tight under the same assumptions.

Abstract: Controlled diffusions depending on an unknown parameter and with small system perturbation are considered in this paper. Two parameter identification methods are proposed and error probabilities are estimated in terms of the small perturbation parameter. These methods are then used to choose among competing filters on successive time intervals. Asymptotically optimal controls based on partial observations are found on successive time intervals by using the best filter identified on the previous time intervals.

Abstract: Distortion-free compressibility of individual pictures by finite-state encoders is investigated. In a recent paper (see IEEE Trans. Inform. Theory, vol.32, no.1, p.1-8, 1986) the compressibility of a given picture I was defined and shown to be the asymptotically attainable lower bound on the compression ratio that can be achieved for I by any finite-state encoder. Here, a different and more direct approach is taken to prove similar results, which are summarized in a converse-to-coding theorem and a constructive-coding-theorem that leads to a universal asymptotically optimal compression algorithm. >

Abstract: For a risk whose annual claim amounts are conditionally i.i.d. with respect to a risk parameter, it is known that the Bayes and credibility premiums are asymptotically optimal in terms of losses. In the present note it is shown that the Bayes and credibility premiums actually converge to the individual premium.

Abstract: The problem of testing linear hypotheses about the parameter vector of an autoregressive process with finite order is considered. Based on the property of local asymptotic normality, we derive asymptotically optimal statistical tests. Additionally, we define and investigate so-called residual rank tests. For these tests we obtain under the null hypothesis an asymptotic distribution which does not depend on the distribution of the innovation.

Abstract: Ak-extremal point set is a point set on the boundary of ak-sided rectilinear convex hull. Given ak-extremal point set of sizen, we present an algorithm that computes a rectilinear Steiner minimal tree in timeO(k 4 n). For constantk, this algorithm runs inO(n) time and is asymptotically optimal and, for arbitraryk, the algorithm is the fastest known for this problem.

Abstract: Quantile and semiparametric M estimation are methods for estimating a censored linear regression model without assuming that the distribution of the random component of the model belongs to a known parametric family. Both methods require estimating derivatives of the unknown cumulative distribution function of the random component. The derivatives can be estimated consistently using kernel estimators in the case of quantile estimation and finite difference quotients in the case of semiparametric M estimation. However, the resulting estimates of derivatives, as well as parameter estimates and inferences that depend on the derivatives, can be highly sensitive to the choice of the kernel and finite difference bandwidths. This paper discusses the theory of asymptotically optimal bandwidths for kernel and difference quotient estimation of the derivatives required for quantile and semiparametric M estimation, respectively. We do not present a fully automatic method for bandwidth selection.

Abstract: This article considers estimation of f(0) of a density function satisfying a Lipshitz condition in a neighborhood of 0. A nonstandard use of the Cramer-Rao inequality yields numerical lower bounds on the minimax squared error risk of any estimator. These bounds are then compared with the minimax risk of the asymptotically optimal kernel-type estimator. The asymptotic bounds obtained (as the sample size n → ∞) are not quite as good as those in Donoho and Liu (in press a, b), but bounds are presented here also for finite values of n, and that paper contains no such bounds for this problem. The numerical results reported in Table 1 show that the asymptotically optimal kernel estimator performs within a factor of 3 of the minimax bound, even for sample size n = 30. As n increases the relative performance improves to its limiting value, although the convergence is fairly slow.

Abstract: A precise charcterization is given of the speed of convergence of the optimally scaled Newton method for the polar decomposition of a nonsingular complex matrix. The results are readily and practically implementable and give a sile way of bounding the number of steps required for a given degree of accuracy. For the matrix sign problem, optimal scaling requires complete knowledge of the eigenvalues of the original matrix. Because this is impractical, we examine spectral scaling, which is asymptotically optimal but slow when the eigenvalues are near the imaginary axis, and the more common determinantal scaling, which does well near the imaginary axis but is not asymptotically optimal. The complementary strengths of these two methods are combined into a new hybrid scaling strategy which is practical and nearly optimal.

Abstract: SUMMARY A new class of noniterative estimators of a common odds ratio is presented. It is shown that the principle of construction leads to asymptotically optimal estimators, that is estimators which are consistent, asymptotically normally distributed and asymptotically efficient.

Abstract: We are concerned with discrete-time stochastic control models for which the random disturbances are independent with a common unknown distribution. When the state space is compact, we prove that mild continuity conditions are sufficient to obtain adaptive policies which are asymptotically optimal with respect to the discounted reward criterion.

Abstract: The problem of designing an experiment to estimate the difference between the means of two populations from the exponential family is considered. The main results determine a lower bound for the Bayes risk, and procedures which are asymptotically optimal. Simulation studies indicate that the fully sequential procedure performs much better than the best fixed allocation procedure.

Abstract: The authors investigate detectors for multisensor distributed detection, considering memoryless detector structures which generalize nonparametric tests and exhibit nonparametric properties in special cases. In general, such structures are convenient to implement. The optimal design of nonlinear transformations and linear weighting functions is discussed, and some optimal solutions are obtained. Asymptotically optimal implementation of the maximum likelihood detector is derived, and the resulting error probabilities are derived or bounded. Asymptotic relative efficiencies are calculated in some cases. The robust design of multisensor detection systems for a large number of samples is also discussed and solved in one case. >

Abstract: Distortion-free compressibility of individual pictures by finite-state encoders is investigated. In a recent paper [2], the compressibility of a given picture I was defined and shown to be the asymptotically attainable lower bound on the compression ratio that can be achieved for I by any finite-state encoder. In this paper, a different and more direct approach is taken to prove similar results, which are summarized in a converse-to-coding-theorem and a constructive-coding-theorem that leads to a universal and asymptotically optimal compression algorithm.

Abstract: This paper is concerned with a design method for optimizing dynamic compensators of Pearson’s type. Optimal parameter matrices are obtained by use of a parameter matching technique and an arbitrary pole placement technique. The controlled system has the optimal LQ modes and the modes with arbitrarily quick damping. The presented compensator works as the optimal regulator with observer and performs about the same control as the optimal regulator. And it is designed not in two steps; observer, regulator, but in one step; optimization of output feedback gain without considering any state estimation.

Abstract: In this dissertation, we study the approximation of stochastic operators. We are interested in bounds for the error of the algorithms and the complexity of the problem. We characterize optimal information/sampling for continuous linear operators, assuming that arbitrary linear functionals may be used in the sampling. The approximation and integration problem, of functions of several variables, is considered on a Wiener space. Tight error and complexity bounds are obtained for the approximation problem. These bounds are compared to those of the worst case. For the integration problem we show an improved deterministic sampling scheme. Turning our attention to nonlinear operators, we study the average solution of nonlinear equations. In contrast to the situation in the worst case, we show that the bisection method is not optimal on the average. Under general conditions on the measure, we derive average error bounds for bisection. However, if the sampling is restricted to the signs (positive/negative) of the functions, or if the measure on the function space is not known, we show that the bisection method is asymptotically optimal.

Abstract: In this paper a derivation of the Akaike's Information Criterion (AIC) is presented to select the number of bins of a histogram given only the data, showing that AIC strikes a balance between the “bias” and “variance” of the histogram estimate. Consistency of the criterion is discussed, an asymptotically optimal histogram bin width for the criterion is derived and its relationship to penalized likelihood methods is shown. A formula relating the optimal number of bins for a sample and a sub-sample obtained from it is derived. A number of numerical examples are presented.

Abstract: For regression functions on [0, 1] with bounded fourth derivatives, a complete cubic spline estimate is proposed and shown to have an asymptotically optimal error rate among all estimates. The error is measured by the supremum norm.

Abstract: A k-extremal point set is a point set on the boundary of a k-sided rectilinear convex hull Given a k-extremal point set of size n, we present an algorithm that computes a rectilinear Steiner minimal tree in time O (k5+k4n) For constant k, this algorithm runs in O(n) time and is asymptotically optimal, and for arbitrary k, the algorithm is the fastest known for this problem

Abstract: : The notion of approximately integrable linear statistical models is introduced to analyze the higher order optimality properties of some common nonparametric estimators. The approximately integrable models suggest a useful approach to a unified treatment of both regular and irregular non-parameter problems. It is shown that with such models any rate of improvement ranging from (log n) to the alpha power/squared to 1/(log...log n) to the alpha power), alpha > 0, of the classical non-parametric procedures can be anticipated. Both an example of a first order asymptotically optimal estimator with the unusual rate 1/n log n and an estimator with an extremely slow unimprovable rate of convergence 1(log...log n) the alpha power are presented.