More often than not, we cannot directly measure many phenomena that are crucial to us. However, we usually have access to certain partial observations on the phenomena of interest as well as a mathematical model of them. The filtering problem seeks estimation of the phenomena given all the accumulated partial information. In this thesis, we study several topics concerning the numerical approximation of the filtering problem. First, we study the continuous-time filtering problem. Given high-frequency ob- servations in discrete-time, we perform double discretization of the non-linear filter to allow for filter estimation with particle filter. By using the multilevel strategy, given any ε > 0, our algorithm achieve an MSE level of O(ε2) with a cost of O(ε−3), while the particle filter requires a cost of O(ε−4). Second, we propose a de-bias scheme for the particle filter under the partially observed diffusion model. The novel scheme is free of innate particle filter bias and discretization bias, through a double randomization method of [14]. Our estimator is perfectly parallel and achieves a similar cost reduction to the multilevel particle filter. Third, we look at a high-dimensional linear Gaussian state-space model in con- tinuous time. We propose a novel multilevel estimator which requires a cost of O(ε−2 log(ε)2) compared to ensemble Kalman-Bucy filters (EnKBFs) which requiresO(ε−3) for an MSE target of O(ε2). Simulation results verify our theory for models of di- mension ∼ 106. Lastly, we consider the model estimation through learning an unknown parameter that characterizes the partially observed diffusions. We propose algorithms to provide unbiased estimates of the Hessian and the inverse Hessian, which allows second-order optimization parameter learning for the model.

In the present study, we investigate kernel density estimation (KDE) and its application to the Gumbel probability distribution. We introduce the basic concepts of reliability analysis and estimation in ordinary and Bayesian settings. The robustness of top three kernels used in KDE with respect to three different optimal bandwidths is presented. The parametric, Bayesian, and empirical Bayes estimates of the reliability, failure rate, and cumulative failure rate functions under the Gumbel failure model are derived and compared with the kernel density estimates. We also introduce the concept of target time subject to obtaining a specified reliability. A comparison of the Bayes estimates of the Gumbel reliability function under six different priors, including kernel density prior, is performed. A comparison of the maximum likelihood (ML) and Bayes estimates of the target time under desired reliability using the Jeffrey's non-informative prior and square error loss function is studied. In order to determine which of the two different loss functions provides a better estimate of the location parameter for the Gumbel probability distribution, we study the performance of four criteria, including the non-parametric kernel density criterion. Finally, we apply both KDE and the Gumbel probability distribution in modeling the annual extreme stream flow of the Hillsborough River, FL. We use the jackknife procedure to improve ML parameter estimates. We model quantile and return period functions both parametrically and using KDE, and show that KDE provides a better fit in the tails.

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This dissertation consists of two parts. The first part deals with the performance appraisal of estimation algorithms. The second part focuses on the application of estimation algorithms to target tracking. Performance appraisal is crucial for understanding, developing and comparing various estimation algorithms. In particular, with the evolvement of estimation theory and the increase of problem complexity, performance appraisal is getting more and more challenging for engineers to make comprehensive conclusions. However, the existing theoretical results are inadequate for practical reference. The first part of this dissertation is dedicated to performance measures which include local performance measures, global performance measures and model distortion measure. The second part focuses on application of the recursive best linear unbiased estimation (BLUE) or lineae minimum mean square error (LMMSE) estimation to nonlinear measurement problem in target tracking. Kalman filter has been the dominant basis for dynamic state filtering for several decades. Beyond Kalman filter, a more fundamental basis for the recursive best linear unbiased filtering has been thoroughly investigated in a series of papers by Dr. X. Rong Li. Based on the so-called quasirecursive best linear unbiased filtering technique, the constraints of the Kalman filter Linear-Gaussian assumptions can be relaxed such that a general linear filtering technique for nonlinear systems can be achieved. An approximate optimal BLUE filter is implemented for nonlinear measurements in target tracking which outperforms the existing method significantly in terms of accuracy, credibility and robustness.

<p>In recent years, two estimators have been proposed to correct the bias exhibited by the leastsquares (LS) estimator of the lagged dependent variable (LDV) coefficient in dynamic regression models when the sample is finite. They have been termed as &lsquo / mean-unbiased&rsquo / and &lsquo / medianunbiased&rsquo / estimators. Relative to other similar procedures in the literature, the two locationbased estimators have the advantage that they offer an exact and uniform methodology for LS estimation of the LDV coefficient in a first order autoregressive model with or without exogenous regressors i.e. ARX(1).</p> <p><br /> However, no attempt has been made to accurately establish and/or compare the statistical properties among these estimators, or relative to those of the LS estimator when the LDV coefficient is restricted to realistic values. Neither has there been an attempt to&nbsp / compare their performance in terms of their mean squared error (MSE) when various forms of the exogenous regressors are considered. Furthermore, only implicit confidence intervals have been given for the &lsquo / medianunbiased&rsquo / estimator. Explicit confidence bounds that are directly usable for inference are not available for either estimator. In this study a new estimator of the LDV coefficient is proposed / the &lsquo / most-probably-unbiased&rsquo / estimator. Its performance properties vis-a-vis the existing estimators are determined and compared when the parameter space of the LDV coefficient is restricted. In addition, the following new results are established: (1) an explicit computable form for the density of the LS estimator is derived for the first time and an efficient method for its numerical evaluation is proposed / (2) the exact bias, mean, median and mode of the distribution of the LS estimator are determined in three specifications of the ARX(1) model / (3) the exact variance and MSE of LS estimator is determined / (4) the standard error associated with the determination of same quantities when simulation rather than numerical integration method is used are established and the methods are compared in terms of computational time and effort / (5) an exact method of evaluating the density of the three estimators is described / (6) their exact bias, mean, variance and MSE are determined and analysed / and finally, (7) a method of obtaining the explicit exact confidence intervals from the distribution functions of the estimators is proposed.</p> <p><br /> The discussion and results show that the estimators are still biased in the usual sense: &lsquo / in expectation&rsquo / . However the bias is substantially reduced compared to that of the LS estimator. The findings are important in the specification of time-series regression models, point and interval estimation, decision theory, and simulation.</p>

<p>In recent years, two estimators have been proposed to correct the bias exhibited by the leastsquares (LS) estimator of the lagged dependent variable (LDV) coefficient in dynamic regression models when the sample is finite. They have been termed as &lsquo / mean-unbiased&rsquo / and &lsquo / medianunbiased&rsquo / estimators. Relative to other similar procedures in the literature, the two locationbased estimators have the advantage that they offer an exact and uniform methodology for LS estimation of the LDV coefficient in a first order autoregressive model with or without exogenous regressors i.e. ARX(1).</p> <p><br /> However, no attempt has been made to accurately establish and/or compare the statistical properties among these estimators, or relative to those of the LS estimator when the LDV coefficient is restricted to realistic values. Neither has there been an attempt to&nbsp / compare their performance in terms of their mean squared error (MSE) when various forms of the exogenous regressors are considered. Furthermore, only implicit confidence intervals have been given for the &lsquo / medianunbiased&rsquo / estimator. Explicit confidence bounds that are directly usable for inference are not available for either estimator. In this study a new estimator of the LDV coefficient is proposed / the &lsquo / most-probably-unbiased&rsquo / estimator. Its performance properties vis-a-vis the existing estimators are determined and compared when the parameter space of the LDV coefficient is restricted. In addition, the following new results are established: (1) an explicit computable form for the density of the LS estimator is derived for the first time and an efficient method for its numerical evaluation is proposed / (2) the exact bias, mean, median and mode of the distribution of the LS estimator are determined in three specifications of the ARX(1) model / (3) the exact variance and MSE of LS estimator is determined / (4) the standard error associated with the determination of same quantities when simulation rather than numerical integration method is used are established and the methods are compared in terms of computational time and effort / (5) an exact method of evaluating the density of the three estimators is described / (6) their exact bias, mean, variance and MSE are determined and analysed / and finally, (7) a method of obtaining the explicit exact confidence intervals from the distribution functions of the estimators is proposed.</p> <p><br /> The discussion and results show that the estimators are still biased in the usual sense: &lsquo / in expectation&rsquo / . However the bias is substantially reduced compared to that of the LS estimator. The findings are important in the specification of time-series regression models, point and interval estimation, decision theory, and simulation.</p>

Magister Scientiae - MSc / In recent years, two estimators have been proposed to correct the bias exhibited by the leastsquares (LS) estimator of the lagged dependent variable (LDV) coefficient in dynamic regression models when the sample is finite. They have been termed as &lsquo;mean-unbiased&rsquo; and &lsquo;medianunbiased&rsquo; estimators. Relative to other similar procedures in the literature, the two locationbased estimators have the advantage that they offer an exact and uniform methodology for LS estimation of the LDV coefficient in a first order autoregressive model with or without exogenous regressors i.e. ARX(1). However, no attempt has been made to accurately establish and/or compare the statistical properties among these estimators, or relative to those of the LS estimator when the LDV coefficient is restricted to realistic values. Neither has there been an attempt to&nbsp; compare their performance in terms of their mean squared error (MSE) when various forms of the exogenous regressors are considered. Furthermore, only implicit confidence intervals have been given for the &lsquo;medianunbiased&rsquo; estimator. Explicit confidence bounds that are directly usable for inference are not available for either estimator. In this study a new estimator of the LDV coefficient is proposed; the &lsquo;most-probably-unbiased&rsquo; estimator. Its performance properties vis-a-vis the existing estimators are determined and compared when the parameter space of the LDV coefficient is restricted. In addition, the following new results are established: (1) an explicit computable form for the density of the LS estimator is derived for the first time and an efficient method for its numerical evaluation is proposed; (2) the exact bias, mean, median and mode of the distribution of the LS estimator are determined in three specifications of the ARX(1) model; (3) the exact variance and MSE of LS estimator is determined; (4) the standard error associated with the determination of same quantities when simulation rather than numerical integration method is used are established and the methods are compared in terms of computational time and effort; (5) an exact method of evaluating the density of the three estimators is described; (6) their exact bias, mean, variance and MSE are determined and analysed; and finally, (7) a method of obtaining the explicit exact confidence intervals from the distribution functions of the estimators is proposed. The discussion and results show that the estimators are still biased in the usual sense: &lsquo;in expectation&rsquo;. However the bias is substantially reduced compared to that of the LS estimator. The findings are important in the specification of time-series regression models, point and interval estimation, decision theory, and simulation. / South Africa

KUBÍNOVÁ, Zuzana. The effect of elevated CO2 concentration and irradiation on quantitative parameters of mesophyll cells of Norway spruce. Prague, 2010. 74 p. Master's degree thesis. Faculty of Science, Charles University in Prague. Abstract The aim of the present thesis was to choose and adjust a suitable methodology for counting particles in 3D space, which would be suitable for unbiased estimation of chloroplast number in needle mesophyll cells. The disector method was used for the first time to determine the number of chloroplasts. This method enables unbiased estimation of chloroplast number in needle volume from optical sections captured from fresh free-hand sections by confocal microscope. The sections did not need any pre-processing. Another aim was to compare selected photosynthetic and anatomical characteristics of sun and shade Norway spruce needles, which were grown under different CO2 concentration. The trees were grown for eight years in ambient (during the experiment increasing from 357 up to 370 µmol CO2 ∙ mol-1 ) CO2 concentration or elevated (700 µmol ∙ mol-1 ) CO2 concentration in special glass domes on an experimental research site of the Institute of Systems Biology and Ecology, Academy of Sciences of the Czech Republic at Bílý Kříž in Moravskoslezské Beskydy mountains. The sun needles...