Asymptotically optimal designs for approximating the path of a stochastic process with respect to the $L^∞$-norm

Rok, strany: 1996, 87 - 95

We study the problem of approximating the path of a Gaussian process $Y=\{Y(t):0≤ t≤ 1\}$ with known covariance function $R$ on the basis of finitely many observations with respect to the $p$th moment $(p≤ 1)$ of the maximum distance of the path of $Y$ and its estimate. Dependent on the amount of information about the mean function $μ$ of $Y$, we (i) use the optimal approximation if $μ$ is known, (ii) ignore $μ$ if $μ(t)=\int₀¹R(s,t)φ(s)ds$, (iii) use the best linear unbiased predictors for $Y(t)$, $0≤ t≤ 1$, if $μ=βψ$ with $ψ(t)=\int₀¹R(s, t)φ(s)ds,ψ$ known. For covariance functions which are of product type or satisfy certain linearity conditions, we construct sequences of designs which perform asymptotically optimal in all three cases.

ISO 690:

Müller-Gronbach, T. 1996. Asymptotically optimal designs for approximating the path of a stochastic process with respect to the $L^∞$-norm. In Tatra Mountains Mathematical Publications, vol. 7, no.1, pp. 87-95. 1210-3195.

APA:

Müller-Gronbach, T. (1996). Asymptotically optimal designs for approximating the path of a stochastic process with respect to the $L^∞$-norm. Tatra Mountains Mathematical Publications, 7(1), 87-95. 1210-3195.