Optimal design for linear models with correlated observations
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Date
2011-09-29
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Abstract
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. For one parameter
models this condition is also shown to be sufficient in many cases and
for several models optimal designs can be identified explicitly. For the
multi-parameter regression models a simple relation which allows verifying
the necessary optimality condition is established. Moreover, it
is proved that the arcsine distribution is universally optimal for the
polynomial regression model with a correlation structure defined by
the logarithmic potential. It is also shown that for models in which
the regression functions are eigenfunctions of an integral operator induced
by the correlation kernel of the error process, designs satisfying
the necessary conditions of optimality can be found explicitly. 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.
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Keywords
Optimal design, Logarithmic potential, Arcsine distribution, Eigenfunctions, Integral operator, Correlated observations