Donald Ylvisaker

Minimax and maximin distance designs

Abstract Beginning with an arbitrary set and a distance defined on it, we develop the notions of minimax and maximin distance sets (designs). These are intended for use in the selection-of-sites problem when the underlying surface is modeled by a prior distribution and observations are made without error. It is shown that such designs have quite general asymptotically optimum (and dual) characteristics under what are termed the G- and D-criteria. There are many examples given, dealing espeacially with the unit square and with k factors at two levels.

Bayesian Design and Analysis of Computer Experiments: Use of Derivatives in Surface Prediction

This article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain T, given values of yo and all of its first derivatives at a set of design sites (points) in T. The intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in T represents a particular configuration of the input parameters. It is assumed that the first derivatives are already available (e.g., from a sensitivity analysis) or can be produced by the code that implements the model. A Bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary Gaussian stochastic process is used. The calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example. The issue of experimental design is also discussed, in particular the criterion of maximizing the reduction in entropy...

Confidence Bands for Regression Functions

Abstract A method for obtaining (conservative) confidence bands for regression functions in low dimensions is given. Of particular importance is the applicability of the method to nonparametric regression. Computations, examples, and comparisons indicate the effectiveness of the method.

Limit distributions for the maxima of stationary Gaussian processes

Let {Xn} be a stationary Gaussian sequence with E{X0} = 0, {X20} = 1 and E{X0Xn} = rnn Let cn = (2ln n), bn = cn- c-1n ln(4[pi] ln n), and set Mn = max0 [less-than-or-equals, slant]k[less-than-or-equals, slant]nXk. A classical result for independent normal random variables is that P[cn(Mn-bn)[less-than-or-equals, slant]x]-->exp[-e-x] as n --> [infinity] for all x. Berman has shown that (1) applies as well to dependent sequences provided rnlnn = o(1). Suppose now that {rn} is a convex correlation sequence satisfying rn = o(1), (rnlnn)-1 is monotone for large n and o(1). Then P[rn-1/2(Mn - (1-rn)1/2bn)[less-than-or-equals, slant]x] --> F(x) for all x, where F is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {Mn}, there are others. In particular, the limit distribution is given below when rn is (sufficiently close to) [gamma]/ln n. We further exhibit a collection of limit distributions which can arise when rn decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).

Minimax distance designs in two-level factorial experiments

Abstract A minimax distance criterion was set forth in Johnson et al. (1990) for the purpose of selection among experimental designs. Unlike the usual design criteria such as D-, E- or G-optimality, minimax distance presumes no underlying model and, in turn, is not concerned with the rank of an associated design matrix. In situations where either the model is unknown or it is not possible to run enough experiments to estimate all parameters of an assumed model, this criterion is considered as a viable tool in the task of design selection. This paper deals with the design space associated with n factors, each of which can take two levels. We exhibit minimax distance designs that compare favorably with designs chosen to do well on classical grounds.

Existence of smoothed stationary processes on an interval

The paper identifies the class of stationary processes on an interval which share a given stationary Gaussian process as kth derivative. The membership requirement involves the norm in the reproducing kernel space associated with the process sought as derivative. Some explicit results are obtained when working with the Ornstein-Uhlenbeck and the linear kernels. These latter facts are useful in an adaptive Bayesian modelling of computer experiments; some remarks are given about this type of analysis.

Model robust confidence intervals

Abstract Confidence intervals are constructed for real-valued parameter estimation in a general regression model with normal errors. When the error variance is known these intervals are optimal (in the sense of minimizing length subject to guaranteed probability of coverage) among all intervals estimates which are centered at a linear estimate of the parameter. When the error variance is unknown and the regression model is an approximately linear model (a class of models which permits bounded systematic departures from an underlying ideal model) then an independent estimate of variance is found and the intervals can then be appropriately scaled.

Asymptotically optimum experimental designs for prediction of deterministic functions given derivative information

Abstract The paper considers problems of optimal design for deterministic response surface experiments, specifically computer experiments, when derivative information as well as the response itself are available at each design site. We adopt a Bayesian perspective. The considerable difficulty in finding G -, D - or A -optimum designs when a fixed prior is placed on the response space then leads us to investigate these problems asymptotically for a sequence of priors in which intersite correlations become progressively weaker. In two settings carrying notions of distance, we show that for prediction of the response and its derivatives, asymptotically D -optimum designs are necessarily miximin distance designs. When predicting only the response, asymptotically G -optimum designs are necessarily minimax distance designs. For prediction of the response, our asymptotic D - and A -optimality conditions agree but, as the condition is not really useful, we do no more than state what it is.

After 50+ Years in Statistics, An Exchange

This is an exchange between Jerome Sacks and Donald Ylvisaker covering their career paths along with some related history and philosophy of Statistics.

Another look at adaptation on the average

We give a short proof of the following fact: sequential choice of linear functional observations in a Hilbert space context yields no benefit for mean square error prediction of a linear operator if the underlying distribution is elliptically contoured.