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Let { X ( t ): t ∈ [ a , b ]} be a Gaussian process with mean μ ∈ L 2 [ a , b ] and continuous covariance K ( s , t ). When estimating μ under the loss ∫ a b (μ(t)−μ(t)) 2 dt the natural estimator X is admissible if K is unknown. If K is known, X is minimax with risk ∫ a b K ( t , t ) dt and admissible if and only if the three by three matrix whose entries are K ( t i , t j ) has a determinant which vanishes identically in t i ∈ [ a , b ], i = 1, 2, 3.

Simultaneous confidence intervals for the p means of a multivariate normal random variable with known variances may be generated by the projection method of Scheffe and by the use of Bonferroni's inequality. It has been conjectured that the Bonferroni intervals are shorter than the Scheffe intervals, at least for the usual confidence levels. This conjecture is proved for all p≥2 and all confidence levels above 50%. It is shown, incidentally, that for all p≥2 Scheffe's intervals are shorter for sufficiently small confidence levels. The results are also applicable to the Bonferroni and Scheffe intervals generated for multinomial proportions.

Let μ be the mean function of an observable stochastic process whose sample paths fall in some Banach space with a basis and assume μ is also in this space. A procedure like Cover's ( Ann. Statist. 1 , 862–871, 1973) is given which has the property that if the last nonzero coordinate of μ is the m th then with probability one this is discovered after at most a finite number of erros. If μ has an infinite number of nonzero coordinates, then with probability one this is discovered after at most a finite number of errors except for a set of μ of prior probability zero.

In extrapolating a function which is close to being a polynimial the least squares estimator combined with the Hoel-Levine optimal design is shown to perform well in terms of mean square error when compared with an optimal spline extrapolator.

Let the process {Y(x,t) : t ϵ T} be observable for each x in some compact set X. Assume that Y(x, t) = θ0f0(x)(t) + … + θkfk(x)(t) + N(t) where fi are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with R the closed convex hull of the set {(φ, f(x))H : ‖φ ‖H ≤ 1, x ϵ X}, where (·, ·)H is the inner product on H. It is shown that if X is convex and fi are linear the design points may be chosen from the extreme points of X. In some problems each linear functional c′θ can be optimally estimated by a design on one point x(c). These problems are completely characterized. An example is worked and some partial results on minimax designs are obtained.

In the experimentation of any of the sciences, physical, biological, or social there arise situations in which the exact conduct of a specified experiment is under the control of an experimenter. The experimenter must therefore make a decision. Where the conduct of an individual experiment may be costly or where there are a large number of possible experiments a premium is placed upon the quality of this decision. This research was concerned with the selection of the proper experiments when the data are 'time' recordings. Of special interest was data which arise as solutions to linear white noise random differential equations. Approximately optimal designs were effectively characterized for this case. Similar characterizations were shown to hold for partial differential operators and random fields. Work was also begun on the identification of experiments which are robust against departures from the assumed model. Results were obtained for both scalar and 'time' recording data. (Author)