• The Korean Journal of Applied Statistics
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• v.10 no.1
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• pp.69-84
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• 1997

We consider a linear calibration problem, $y_i = $$\alpha + \beta (x_i - x_0) + \epsilon_i$, $i=1, 2, {\cdot}{\cdot},n$ $y_f = \alpha + \beta (x_f - x_0) + \epsilon, $ where we observe $(x_i, y_i)$'s for the controlled calibration experiments and later we make inference about $x_f$ from a new observation $y_f$. The objective of the calibration design problem is to find the optimal design $x = (x_i, \cdots, x_n$ that gives the best estimates for $x_f$. We compare Kim(1989)'s Bayesian design which minimizes the expected value of the posterior variance of $x_f$ and some optimal designs from literature. Kim suggested the Bayesian optimal design based on the analysis of the characteristics of the expected loss function and numerical must be equal to the prior mean and that the sum of squares be as large as possible. The designs to be compared are (1) Buonaccorsi(1986)'s AV optimal design that minimizes the average asymptotic variance of the classical estimators, (2) D-optimal and A-optimal design for the linear regression model that optimize some functions of $M(x) = \sum x_i x_i'$, and (3) Hunter & Lamboy (1981)'s reference design from their paper. In order to compare the designs which are optimal in some sense, we consider two criteria. First, we compare them by the expected posterior variance criterion and secondly, we perform the Monte Carlo simulation to obtain the HPD intervals and compare the lengths of them. If the prior mean of $x_f$ is at the center of the finite design interval, then the Bayesian, AV optimal, D-optimal and A-optimal designs are indentical and they are equally weighted end-point design. However if the prior mean is not at the center, then they are not expected to be identical.In this case, we demonstrate that the almost Bayesian-optimal design was slightly better than the approximate AV optimal design. We also investigate the effects of the prior variance of the parameters and solution for the case when the number of experiments is odd.

• The Korean Journal of Applied Statistics
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• v.20 no.2
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• pp.281-290
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• 2007

Linear estimator, a weighted sum of the sample observation, is commonly adopted to estimate the finite population parameters such as population totals in survey sampling. The weight for a sampled unit is often constructed by multiplying the base weight, which is the inverse of the first-order inclusion probability, by an adjustment term that takes into account of the auxiliary information obtained throughout the population. The linear estimator using the weight adjustment is often more efficient than the one using only the bare weight, but its valiance estimation is more complicated. We discuss variance estimation for a general class of weight-adjusted estimator. By identifying that the weight-adjusted estimator can be viewed as a function of estimated nuisance parameters, where the nuisance parameters were used to incorporate the auxiliary information, we derive a linearization of the weight-adjusted estimator using a Taylor expansion. The method proposed here is quite general and can be applied to wide class of the weight-adjusted estimators. Some examples and results from a simulation study are presented.