On the Existence and Uniqueness of the Maximum Likelihood Estimators of Normal and Lognormal Population Parameters with Grouped Data

Lognormal distribution has abundant applications in various fields. In literature, most inferences on the two parameters of the lognormal distribution are based on Type-I censored sample data. However, exact measurements are not always attainable especially when the observation is below or above the detection limits, and only the numbers of measurements falling into predetermined intervals can be recorded instead. This is the so-called grouped data. In this paper, we will show the existence and uniqueness of the maximum likelihood estimators of the two parameters of the underlying lognormal distribution with Type-I censored data and grouped data. The proof was first established under the case of normal distribution and extended to the lognormal distribution through invariance property. The results are applied to estimate the median and mean of the lognormal population.

Site specific growth of metal catalyzed silica nanowires for biological and chemical sensing

In this research the integration of nanostructures and micro-scale devices was investigated using silica nanowires to develop a simple yet robust nanomanufacturing technique for improving the detection parameters of chemical and biological sensors. This has been achieved with the use of a dielectric barrier layer, to restrict nanowire growth to site-specific locations which has removed the need for post growth processing, by making it possible to place nanostructures on pre-pattern substrates. Nanowires were synthesized using the Vapor-Liquid-Solid growth method. Process parameters (temperature and time) and manufacturing aspects (structural integrity and biocompatibility) were investigated. Silica nanowires were observed experimentally to determine how their physical and chemical properties could be tuned for integration into existing sensing structures. Growth kinetic experiments performed using gold and palladium catalysts at 1050°C for 60 minutes in an open-tube furnace yielded dense and consistent silica nanowire growth. This consistent growth led to the development of growth model fitting, through use of the Maximum Likelihood Estimation (MLE) and Bayesian hierarchical modeling. Transmission electron microscopy studies revealed the nanowires to be amorphous and X-ray diffraction confirmed the composition to be SiO2 . Silica nanowires were monitored in epithelial breast cancer media using Impedance spectroscopy, to test biocompatibility, due to potential in vivo use as a diagnostic aid. It was found that palladium catalyzed silica nanowires were toxic to breast cancer cells, however, nanowires were inert at 1μg/mL concentrations. Additionally a method for direct nanowire integration was developed that allowed for silica nanowires to be grown directly into interdigitated sensing structures. This technique eliminates the need for physical nanowire transfer thus preserving nanowire structure and performance integrity and further reduces fabrication cost. Successful nanowire integration was physically verified using Scanning electron microscopy and confirmed electrically using Electrochemical Impedance Spectroscopy of immobilized Prostate Specific Antigens (PSA). The experiments performed above serve as a guideline to addressing the metallurgic challenges in nanoscale integration of materials with varying composition and to understanding the effects of nanomaterials on biological structures that come in contact with the human body.

Site Specifc Growth of Metal Catalyzed Silica Nanowires for Biological and Chemical Sensing

In this research the integration of nanostructures and micro-scale devices was investigated using silica nanowires to develop a simple yet robust nanomanufacturing technique for improving the detection parameters of chemical and biological sensors. This has been achieved with the use of a dielectric barrier layer, to restrict nanowire growth to site-specific locations which has removed the need for post growth processing, by making it possible to place nanostructures on pre-pattern substrates. Nanowires were synthesized using the Vapor-Liquid-Solid growth method. Process parameters (temperature and time) and manufacturing aspects (structural integrity and biocompatibility) were investigated. Silica nanowires were observed experimentally to determine how their physical and chemical properties could be tuned for integration into existing sensing structures. Growth kinetic experiments performed using gold and palladium catalysts at 1050 ˚C for 60 minutes in an open-tube furnace yielded dense and consistent silica nanowire growth. This consistent growth led to the development of growth model fitting, through use of the Maximum Likelihood Estimation (MLE) and Bayesian hierarchical modeling. Transmission electron microscopy studies revealed the nanowires to be amorphous and X-ray diffraction confirmed the composition to be SiO2 . Silica nanowires were monitored in epithelial breast cancer media using Impedance spectroscopy, to test biocompatibility, due to potential in vivo use as a diagnostic aid. It was found that palladium catalyzed silica nanowires were toxic to breast cancer cells, however, nanowires were inert at 1µg/mL concentrations. Additionally a method for direct nanowire integration was developed that allowed for silica nanowires to be grown directly into interdigitated sensing structures. This technique eliminates the need for physical nanowire transfer thus preserving nanowire structure and performance integrity and further reduces fabrication cost. Successful nanowire integration was physically verified using Scanning electron microscopy and confirmed electrically using Electrochemical Impedance Spectroscopy of immobilized Prostate Specific Antigens (PSA). The experiments performed above serve as a guideline to addressing the metallurgic challenges in nanoscale integration of materials with varying composition and to understanding the effects of nanomaterials on biological structures that come in contact with the human body.

Bayesian Estimation of Small Proportions Using Binomial Group Test

Group testing has long been considered as a safe and sensible relative to one-at-a-time testing in applications where the prevalence rate p is small. In this thesis, we applied Bayes approach to estimate p using Beta-type prior distribution. First, we showed two Bayes estimators of p from prior on p derived from two different loss functions. Second, we presented two more Bayes estimators of p from prior on π according to two loss functions. We also displayed credible and HPD interval for p. In addition, we did intensive numerical studies. All results showed that the Bayes estimator was preferred over the usual maximum likelihood estimator (MLE) for small p. We also presented the optimal β for different p, m, and k.

Interval and Point Estimators for the Location Parameter of the Three-Parameter Lognormal Distribution

The three-parameter lognormal distribution is the extension of the two-parameter lognormal distribution to meet the need of the biological, sociological, and other fields. Numerous research papers have been published for the parameter estimation problems for the lognormal distributions. The inclusion of the location parameter brings in some technical difficulties for the parameter estimation problems, especially for the interval estimation. This paper proposes a method for constructing exact confidence intervals and exact upper confidence limits for the location parameter of the three-parameter lognormal distribution. The point estimation problem is discussed as well. The performance of the point estimator is compared with the maximum likelihood estimator, which is widely used in practice. Simulation result shows that the proposed method is less biased in estimating the location parameter. The large sample size case is discussed in the paper.