Browsing by Subject "measurement error model"
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Item Deconvolution in Random Effects Models via Normal Mixtures(2010-10-12) Litton, Nathaniel A.This dissertation describes a minimum distance method for density estimation when the variable of interest is not directly observed. It is assumed that the underlying target density can be well approximated by a mixture of normals. The method compares a density estimate of observable data with a density of the observable data induced from assuming the target density can be written as a mixture of normals. The goal is to choose the parameters in the normal mixture that minimize the distance between the density estimate of the observable data and the induced density from the model. The method is applied to the deconvolution problem to estimate the density of $X_{i}$ when the variable $% Y_{i}=X_{i}+Z_{i}$, $i=1,\ldots ,n$, is observed, and the density of $Z_{i}$ is known. Additionally, it is applied to a location random effects model to estimate the density of $Z_{ij}$ when the observable quantities are $p$ data sets of size $n$ given by $X_{ij}=\alpha _{i}+\gamma Z_{ij},~i=1,\ldots ,p,~j=1,\ldots ,n$, where the densities of $\alpha_{i} $ and $Z_{ij}$ are both unknown. The performance of the minimum distance approach in the measurement error model is compared with the deconvoluting kernel density estimator of Stefanski and Carroll (1990). In the location random effects model, the minimum distance estimator is compared with the explicit characteristic function inversion method from Hall and Yao (2003). In both models, the methods are compared using simulated and real data sets. In the simulations, performance is evaluated using an integrated squared error criterion. Results indicate that the minimum distance methodology is comparable to the deconvoluting kernel density estimator and outperforms the explicit characteristic function inversion method.Item Investigation of Simple Linear Measurement Error Models (SLMEMS) with Correlated Data(2014-12-06) Lu, MingThe primary goal of this research is to develop statistical methods to determine if observed real responses are adequately modeled by (possibly stochastic) simulation models that incorporate first-order autoregressive measurement errors. We assume the measurement errors are normally distributed to allow development of likelihood-based methods of inference. Simulated true responses are modeled as a simple linear regression on the true response values. That is, we wish to detect if either additive or multiplicative biases exist in the simulation model. Efficient score and likelihood ratio tests using observed real process data are developed to test the joint null hypothesis that no significant additive or multiplicative biases exist in the stochastic simulation model. Tests for adequacy of both stochastic and deterministic simulation models are developed using, respectively, structural and functional simple linear measurement error models that allow the measurement errors to satisfy normal first-order autoregressive processes. A byproduct of this research is developments of analogous tests of the null hypothesis that errors of measurement are independent. Such tests would be of use if the real process is not a times series and there was uncertainty whether the simulation model should allow for correlated measurement errors. Analytic and simulation results show that all maximum likelihood estimators (MLEs) of model parameters MLEs are consistent under the structural model, but some MLEs of parameters are inconsistent under functional model. Test statistics developed under the structural model are shown to be asymptotically distributed as chi-squared random variables with two degrees of freedom when testing for additive and multiplicative biases in the simulation model having correlated measurement errors. Test statistics developed under the structural model are shown to be asymptotically distributed as chi-squared random variables with one degree of freedom when testing for independence of the measurement errors. However, for functional models, the corresponding test statistics are asymptotically distributed as random variables that are two times the chi-squared distributions. Empirical power curves are plotted under different parameter configurations. Behaviors of test statistics and power curves are found to be affected by the sample size, signal to noise ratio and strength of correlations among measurement errors.