Shan Sun

Optimal control, statistics and path planning

In this paper, some of the relationships between optimal control and statistics are examined. In a series of earlier papers, we examined the relationship between optimal control and conventional splines and between optimal control and the statistical theory of smoothing splines. In this paper, we present a unified treatment of these two problems and extend the same framework to include the concept of dynamic time warping, which is being seen as an important statistical tool as well as being of importance in physics. We show that these three major problems unite to give a satisfactory solution to the problem of trajectory or path planning.

Limiting behavior of the perturbed empirical distribution functions evaluated at U-statistics for strongly mixing sequences of random variables

We prove the almost sure representation, a law of the iterated logarithm and an invariance principle for the statistic Fˆn(Un) for a class of strongly mixing sequences of random variables {Xi,i≥1}. Stationarity is not assumed. Here Fˆn is the perturbed empirical distribution function and Un is a U-statistic based on X1,…,Xn.

Necessary and sufficient conditions for the asymptotic normality of perturbed sample quantiles

Abstract We deal with perturbed sample quantiles based on a kernel k and a sequence of window-width an > 0. Under minimal assumptions on the underlying cumulative distribution and the kernel k, necessary and sufficient conditions for the central limit theorem to hold for these quantiles are found for the sequence {an}. Our results (i) generalize the central limit theorem of Nadaraya (1964), and (ii) extend results of Chanda (1975) and Falk (1985). Several applications are included.

A class of adaptive distribution-free procedures☆

Abstract The adaptive nonparametric procedures developed in Hill et al. (J. Roy. Statist. Soc. Ser. C 37 (1988) 205–218) for the problems of testing for ordered alternatives and multiple comparisons, in one-way analysis of variance, are further expanded to include the problem of ties and the related estimation problems. Some applications are provided. The supremacy of these procedures over the usual parametric procedure based on the sample means, and the usual nonparametric procedure (based on ranks) is established.

Bandwidth Selection for Kernel Quantile Estimation

In this article, we summarize some quantile estimators and related bandwidth selection methods and give two new bandwidth selection methods. By four distributions: standard normal, exponential, double exponential and log normal we simulated the methods and compared their efficiencies to that of the empirical quantile. It turns out that kernel smoothed quantile estimators, with no matter which bandwidth selection method used, are more efficient than the empirical quantile estimator in most situations. And when sample size is relatively small, kernel smoothed estimators are especially more efficient than the empirical quantile estimator. However, no one method can beat any other methods for all distributions.

Function approximation and neural-fuzzy approach to machining process selection

The integration of neural networks and fuzzy logic provides an unique tool to improve the performance of solving ill-defined, nonlinear problems. In this paper, we first show a theoretical result that a class of fuzzy systems is a function approximator. This result extends Wang-Mendels work which is based on the Stone-Weierstrass theorem to a broader class of functions. Then we propose a neural-fuzzy technique for machining process selection (MPS), which usually is a crucial step in a semiconductor manufacturing environment and it constitutes a critical link between computer-aided design (CAD) and computer-aided manufacturing (CAM). Given the complexity of MPS process, a direct mathematical formulation and optimization to meet design specifications and cost constraints can be difficult or even formidable. By incorporating artificial neural networks learning and adaptation capability with fuzzy logics structured knowledge manipulation and reasoning, we are able to reduce the neural network training time and improve its prediction accuracy. Primary experimentation confirms the theoretical analysis and shows that the proposed technique is promising and has potential to be adopted in a real manufacturing environment.

Smooth quantile processes from right censored data and construction of simultaneous confidence bands

The smooth nonparametric estimator of a quantile function Q(p) is defined as the solution of , where is the distribution function corresponding to a kernel estimator of a density function. The asymptotic properties of the smooth quantile process, , based on randomly right censored lifetime data are studied. The bootstrap approaches to approximate the distributions of the smooth quantile processes are investigated and are used to construct simultaneous confidence bands for quantile functions. Data-based selection of the bandwidth required for computing is also investigated using bootstrap methods. A Monte Carlo simulation is carried out to assess small sample performance of the proposed confidence bands. An application to construct confidence bands for the quantile function of the time between a manuscripts submission and its first review is provided using a JASA data set. The developed results can be applied to construct simultaneous confidence bands for the difference of two quantile functions and to check whether there is a location shift or scale change for two distributions under study.

Cumulative Distribution Estimation via Control Theoretic Smoothing Splines

In this paper, we explore the relationship between control theory and statistics. Specifically, we consider the use of cubic monotone control theoretic smoothing splines in estimating the cumulative distribution function (CDF) defined on a finite interval [0,T]. The spline construction is obtained by imposing an infinite dimensional, non-negativity constraint on the derivative of the optimal curve. The main theorem of this paper states that the optimal curve y(t) is a piecewise polynomial of known degree with y(0) = 0 and y(T) = 1. The solution is determined through dynamic programming which takes advantage of a finite reparametrization of the problem.

Perturbed empirical distribution functions and quantiles under dependence

AbstractIn this note we consider the perturbed empirical distribution functions of the formn