154 research outputs found
Optimal designs in regression with correlated errors
This paper discusses the problem of determining optimal designs for
regression models, when the observations are dependent and taken on an
interval. A complete solution of this challenging optimal design problem is
given for a broad class of regression models and covariance kernels.
We propose a class of estimators which are only slightly more complicated
than the ordinary least-squares estimators. We then demonstrate that we can
design the experiments, such that asymptotically the new estimators achieve the
same precision as the best linear unbiased estimator computed for the whole
trajectory of the process. As a by-product we derive explicit expressions for
the BLUE in the continuous time model and analytic expressions for the optimal
designs in a wide class of regression models. We also demonstrate that for a
finite number of observations the precision of the proposed procedure, which
includes the estimator and design, is very close to the best achievable. The
results are illustrated on a few numerical examples.Comment: 38 pages, 5 figure
Efficient experimental designs for sigmoidal growth models
For the Weibull- and Richards-regression model robust designs are determined by maximizing a minimum of D- or D1-efficiencies, taken over a certain range of the non-linear parameters. It is demonstrated that the derived designs yield a satisfactory solution of the optimal design problem for this type of model in the sense that these designs are efficient and robust with respect to misspecification of the unknown parameters. Moreover, the designs can also be used for testing the postulated form of the regression model against a simplified sub-model. --Sigmoidal growth,Weibull regression model,exponential regression model,Richards-regression model,logistic regression model
Analytic moment and Laplace transform formulae for the quasi-stationary distribution of the Shiryaev diffusion on an interval
We derive analytic closed-form moment and Laplace transform formulae for the
quasi-stationary distribution of the classical Shiryaev diffusion restricted to
the interval with absorption at a given .Comment: 24 pages, 2 figure
Optimal design for linear models with correlated observations
In the common linear regression model the problem of determining optimal
designs for least squares estimation is considered in the case where the
observations are correlated. A necessary condition for the optimality of a
given design is provided, which extends the classical equivalence theory for
optimal designs in models with uncorrelated errors to the case of dependent
data. If the regression functions are eigenfunctions of an integral operator
defined by the covariance kernel, it is shown that the corresponding measure
defines a universally optimal design. For several models universally optimal
designs can be identified explicitly. In particular, it is proved that the
uniform distribution is universally optimal for a class of trigonometric
regression models with a broad class of covariance kernels and that the arcsine
distribution is universally optimal for the polynomial regression model with
correlation structure defined by the logarithmic potential. To the best
knowledge of the authors these findings provide the first explicit results on
optimal designs for regression models with correlated observations, which are
not restricted to the location scale model.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1079 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Optimal Discrimination Designs for Exponential Regression Models
We investigate optimal designs for discriminating between exponential regression models of different complexity, which are widely used in the biological sciences; see, e.g., Landaw (1995) or Gibaldi and Perrier (1982). We discuss different approaches for the construction of appropriate optimality criteria, and find sharper upper bounds on the number of support points of locally optimal discrimination designs than those given by Caratheodory?s Theorem. These results greatly facilitate the numerical construction of optimal designs. Various examples of optimal designs are then presented and compared to different other designs. Moreover, to protect the experiment against misspecifications of the nonlinear model parameters, we adapt the design criteria such that the resulting designs are robust with respect to such misspecifications and, again, provide several examples, which demonstrate the advantages of our approach. --Compartmental Model,Model Discrimination,Discrimination Design,Locally Optimal Design,Robust Optimal Design,Maximin Optimal Design
SSA analysis and forecasting of records for Earth temperature and ice extents
In this paper, we continued the research started in [6, 7]. We
applied the so-called Singular Spectrum Analysis (SSA) to forecast
the Earth temperature records, to examine cross-correlations between
these records, the Arctic and Antarctic sea ice extents and the Oceanic
Nino Index (ONI). We have concluded that that the pattern observed
in the last 15 years for the Earth temperatures is not going to change
much, found very high cross-correlations between a lagged ONI index
and some Earth temperature series and noticed several signifi-
cant cross-correlations between the ONI index and the sea ice extent
anomalies; these cross-correlations do not seem to be well-known to
the specialists on Earth climate
Efficient experimental designs for sigmoidal growth models
For the Weibull- and Richards-regression model robust designs are determined by maximizing a minimum of D- or D1-efficiencies, taken over a certain range of the non-linear parameters. It is demonstrated that the derived designs yield a satisfactory solution of the optimal design problem for this type of model in the sense that these designs are efficient and robust with respect to misspecification of the unknown parameters. Moreover, the designs can also be used for testing the postulated form of the regression model against a simplified sub-model
Optimal designs for 3D shape analysis with spherical harmonic descriptors
We determine optimal designs for some regression models which are frequently used for describing 3D shapes. These models are based on a Fourier expansion of a function defined on the unit sphere in terms of spherical harmonic basis functions. In particular it is demonstrated that the uniform distribution on the sphere is optimal with respect to all p-criteria proposed by Kiefer (1974) and also optimal with respect to a criterion which maximizes a p-mean of the r smallest eigenvalues of the variance-covariance matrix. This criterion is related to principal component analysis, which is the common tool for analyzing this type of image data. Moreover, discrete designs on the sphere are derived, which yield the same information matrix in the spherical harmonic regression model as the uniform distribution and are therefore directly implementable in practice. It is demonstrated that the new designs are substantially more efficient than the commonly used designs in 3D-shape analysis. --Shape analysis,spherical harmonic descriptors,optimal designs,quadrature formulas,principal component analysis,3D-image data
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