Equidistant and D-optimal designs for parameters of Ornstein–Uhlenbeck process

Kiseľák, Jozef; Stehlík, Milan

doi:10.1016/j.spl.2007.12.012

Cited by 45 publications

(43 citation statements)

References 9 publications

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“…This was also independently observed by Zagoraiou and Baldi-Antognini (2009). For a proof first notice that an Ornstein-Uhlenbeck process possesses a Markovian property and thus the structure of the inverse covariance matrix is tridiagonal (see for details Kiseľák and Stehlík, 2008). From this fact and the structure of M θ we conjecture Stehlík, 2005).…”

Section: One-dimensional Examplesupporting

confidence: 56%

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Compound optimal spatial designs

Müller

Stehlík

2009

Environmetrics

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A single purpose design may be quite inefficient for handling a real-life problem. Therefore, we often need to incorporate more than one design criterion and a common approach is simply to construct a weighted average, which may depend upon different information matrices. Designs based upon this method have been termed compound designs. The need to satisfy more than one design criterion is particularly relevant in the context of random fields. It is evident that for precise universal kriging it is important not only to efficiently estimate the spatial trend parameters, but also the parameters of the variogram or covariance function. Both tasks could for instance be comprised by applying corresponding design criteria and constructing a compound design from there. Modern techniques for such first and second order characteristics will be suggested and reviewed in the presentation. A new hybrid stochastic exchange type optimization algorithm is proposed and an illustrating example of the design of a water-quality monitoring network is provided. COMPOUND OPTIMAL SPATIAL DESIGNS 355 Case 1. We are interested only in the trend parameters β and consider θ as known or a nuisance. Case 2. We are interested only in the covariance parameters θ. Case 3. We are interested in both sets of parameters. periment. In the following, we will define optimality of a design always strictly in the tradition of Kiefer (see e.g. Kiefer, 1959), where the inputs are selected such, that a prespecified design criterion (e.g. the determinant of the trend parameters variance-covariance matrix, so-called D-optimality) is optimized. The classic Fisher information M(β) = E[(∂ ln f (x, β))/∂β) 2 ], which is the basis for We also use now the notation [C(ξ)] ii = c(x i , x i ; θ) to emphasize the dependence of C on the design, and we assume knowledge of θ.Again all the entries can be condensed to a design criterion . If we now want to determine a Doptimal design for the whole parameter set, we can obviously-due to the orthogonality of the first and second order parameters-simply use the product of the respective determinants as an optimum design

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Section: One-dimensional Examplesupporting

confidence: 56%

“…In Kiseľák and Stehlík (2008), it is shown that the equidistant design is D-optimal in the case of Ornstein-Uhlenbeck process with constant trend. Their Theorem 4 is also providing the form of the FIM for parameter β when the process is OU with constant trend, i.e.…”

Section: One-dimensional Examplementioning

confidence: 99%

Compound optimal spatial designs

Müller

Stehlík

2009

Environmetrics

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“…1 = 0) on X = [0, 2]. It is shown in [42] that for n-point designs with equal interdistances 170 W. MÜLLER AND M. STEHLÍK d = x i+1 − x i <0.5 (the so-called uniform designs), the Fisher information matrix can be expressed as M 0 ( n ) = (2−n +ne d/ 1 )/(1+e d/ 1 ). Then we clearly have M 0 ( 5 ) = M 0 ( 2 )+ M 0 ( 3 ), although such a five-point design unifies such two-and three-point uniform designs by simply 'adding' one edge connecting two neighbouring points.…”

Section: Examplementioning

confidence: 99%

“…Dette et al [46] have proved that if 1 →+∞, then any exact n-point Doptimal design in the linear regression model with exponential semivariogram converges to the equally spaced design. In [42] a thorough study of small sample and asymptotical comparisons of the efficiencies of equidistant designs while taking into account both the parameter of the trend 0 as well as the parameter of the covariance function 1 is provided. If only the trend parameter 0 is of interest, the designs covering more or less uniformly the whole design space are rather efficient.…”

Section: Examplementioning

confidence: 99%

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Issues in the optimal design of computer simulation experiments

Müller

Stehlík

2009

Appl Stoch Models Bus & Ind

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Output from computer simulation experiments is often approximated as realizations of correlated random fields. Consequently, the corresponding optimal design questions must cope with the existence and detection of an error correlation structure, issues largely unaccounted for by traditional optimal design theory. Unfortunately, many of the nice features of well-established design techniques, such as additivity of the information matrix, convexity of design criteria, etc., do not carry over to the setting of interest. This may lead to unexpected, counterintuitive, even paradoxical effects in the design as well as the analysis stage of computer simulation experiments. In this paper we intend to give an overview and some simple but illuminating examples of this behaviour.

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Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process

Zagoraiou

Antognini

2008

Appl Stoch Models Bus & Ind

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This paper deals with optimal designs for Gaussian random fields with constant trend and exponential correlation structure, widely known as the Ornstein-Uhlenbeck process. Assuming the maximum likelihood approach, we study the optimal design problem for the estimation of the trend and the correlation parameter using a criterion based on the Fisher information matrix. For the problem of trend estimation, we give a new proof of the optimality of the equispaced design for any sample size (see Statist. Probab. Lett. 2008; 78:1388-1396. We also show that for the estimation of the correlation parameter, an optimal design does not exist. Furthermore, we show that the optimal strategy for conflicts with the one for , since the equispaced design is the worst solution for estimating the correlation. Hence, when the inferential purpose concerns both the unknown parameters we propose the geometric progression design, namely a flexible class of procedures that allow the experimenter to choose a suitable compromise regarding the estimation's precision of the two unknown parameters guaranteeing, at the same time, high efficiency for both.

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Equidistant and D-optimal designs for parameters of Ornstein–Uhlenbeck process

Cited by 45 publications

References 9 publications

Compound optimal spatial designs

Compound optimal spatial designs

Issues in the optimal design of computer simulation experiments

Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process

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