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The Least Squares Mixture Decomposition Estimator (LSMDE) is a new nonparametric density estimation technique developed by modifying the ordinary kernel density estimators. While the ordinary kernel density estimator assumes equal weight (l/<i>n</i>) for each data point, LSMDE assigns the optimized weight to each data point via the quadratic programming under the Mean Integrated Squared Error (MISE) criterion. As results, we find out that the optimized weights for a given data set are far different from l/<i>n</i> for a reasonable smoothing parameter and, furthermore, many data points are assigned to zero weights after the optimization. This implies that LSMDE decomposes the underlying density function to a finite mixture distribution of <i>p</i> (< n) kernel functions. LSMDE turns out to be more informative, especially in multi-dimensional cases when the visualization of the density function is difficult, than the ordinary kernel density estimator by suggesting the underlying structure of a given data set. / Ph. D.

ABSTRACT ROBUST CONTROL CHARTS &Ccedil / etiny&uuml / rek, Aysun M. Sc., Department of Statistics Supervisor: Dr. BariS S&uuml / r&uuml / c&uuml / Co-Supervisor: Assoc. Prof. Dr. Birdal Senoglu December 2006, 82 pages Control charts are one of the most commonly used tools in statistical process control. A prominent feature of the statistical process control is the Shewhart control chart that depends on the assumption of normality. However, violations of underlying normality assumption are common in practice. For this reason, control charts for symmetric distributions for both long- and short-tailed distributions are constructed by using least squares estimators and the robust estimators -modified maximum likelihood, trim, MAD and wave. In order to evaluate the performance of the charts under the assumed distribution and investigate robustness properties, the probability of plotting outside the control limits is calculated via Monte Carlo simulation technique.

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The speed control of a Brushless DC Motor Drive in the presence of load disturbance is investigated. Firstly some practical results are presented where a simple proportional-integral speed controller is used in the presence of a large step input speed demand as well as load disturbance. The wind-up problem caused by the saturation of the controller is discussed. In order to improve the performance of the proportional-integral speed controller in the presence of load variation, a load estimator is used with torque feedforward control. The results presented show the speed holding capability in the presence of load variation is significantly improved. A genetic algorithm is used on line to optimise the controller for different conditions such as large and small step input speed demand and load disturbance. The results presented show that a genetic algorithm is capable of finding the tuning of the controller for optimal performance. Single-input single-output and two-input two-output fuzzy speed controllers are also used and the results compared to a proportional-integral controller. Results are presented showing that a single-input single-output fuzzy controller works as a proportional controller with variable gain whereas the two-input two-output fuzzy controller is capable of driving the motor at variable speed and load torque with excellent performance. The robustness of the fuzzy controllers is compared to the proportional-integral controller and the results presented show that the fuzzy one is more robust then the proportional-integral. A genetic algorithm is also used on line for the optimisation of the two-input twooutput fuzzy speed controller and the results show that despite the large number of parameters to be optimised, the tuning for optimal performance is also possible.

<p> A family of symmetric distributions is introduced. The means, variances and covariances of ordered observations from the family are calculated. The best linear unbiased estimators of the mean and standard deviation are constructed for complete and censored samples. A computer technique is developed to evaluate range-dependent double integrals. </p> / Thesis / Master of Science (MSc)

We present a theoretical approach to the statistical analysis of the dependence of the gap time length between consecutive recurrent events, on a set of explanatory random variables and in the presence of right censoring. The dependence is expressed through regression-like and overdispersion parameters, estimated via estimating functions and equations. The mean and variance of the length of each gap time, conditioned on the observed history of prior events and other covariates, are known functions of parameters and covariates, and are part of the estimating functions. Under certain conditions on censoring, we construct normalized estimating functions that are asymptotically unbiased and contain only observed data. We then use modern mathematical techniques to prove the existence, consistency and asymptotic normality of a sequence of estimators of the parameters. Simulations support our theoretical results.

The normal distribution is a commonly seen distribution in nature, education, and business. Data that are mounded or bell shaped are easily found across various fields of study. Although there is high utility with the normal distribution; often the full range can not be observed. The truncated normal distribution accounts for the inability to observe the full range and allows for inferring back to the original population. Depending on the amount of truncation, the truncated normal has several distinct shapes. A simulation study evaluating the performance of the maximum likelihood estimators and method of moment estimators is conducted and a comparison of performance is made. The α Likelihood Ratio Test (LRT) is derived for testing the null hypothesis of equal population means for truncated normal data. A simulation study evaluating the power of the LRT to detect absolute standardized differences between the two population means with small sample size was conducted and the power curves were approximated. Another simulation study evaluating the power of the LRT to detect absolute differences for testing the hypothesis with large unequal sample sizes was conducted. The α LRT was extended to a k population hypothesis test for equal population means. A simulation study examining the power of the k population LRT for detecting absolute standardized differences when one of the population means is different than the others was conducted and the power curve approximated. Stat~221 is the largest introductory statistics course at BYU serving about 4,500 students a year. Every section of Stat 221 shares common homework assignments and tests. This controls for confounding when making comparisons between sections. Historically grades have been thought to be bell shaped, but with grade inflation and other factors, the upper tail is lost because of the truncation at 100. It is reasonable to assume that grades follow a truncated normal distribution. Inference using the final grades should be done recognizing the truncation. Performance of the different Stat 221 sections was evaluated using the LRTs derived.

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