L-moments are defined as linear combinations of expected values of order statistics of a variable.(Hosking 1990) L-moments are estimated from samples using functions of weighted means of order statistics. The advantages of L-moments over classical moments are: able to characterize a wider range of distributions; L-moments are more robust to the presence of outliers in the data when estimated from a sample; and L-moments are less subject to bias in estimation and approximate their asymptotic normal distribution more closely. Hosking (1990) obtained an asymptotic result specifying the sample L-moments have a multivariate normal distribution as n approaches infinity. The standard deviations of the estimators depend however on the distribution of the variable. So in order to be able to build confidence intervals we would need to know the distribution of the variable. Bootstrapping is a resampling method that takes samples of size n with replacement from a sample of size n. The idea is to use the empirical distribution obtained with the subsamples as a substitute of the true distribution of the statistic, which we ignore. The most common application of bootstrapping is building confidence intervals without knowing the distribution of the statistic. The research question dealt with in this work was: How well do bootstrapping confidence intervals behave in terms of coverage and average width for estimating L-moments and ratios of L-moments? Since Hosking's results about the normality of the estimators of L-moments are asymptotic, we are particularly interested in knowing how well bootstrap confidence intervals behave for small samples. There are several ways of building confidence intervals using bootstrapping. The most simple are the standard and percentile confidence intervals. The standard confidence interval assumes normality for the statistic and only uses bootstrapping to estimate the standard error of the statistic. The percentile methods work with the (α/2)th and (1-α/2)th percentiles of the empirical sampling distribution. Comparing the performance of the three methods was of interest in this work. The research question was answered by doing simulations in Gauss. The true coverage of the nominal 95% confidence interval for the L-moments and ratios of L-moments were found by simulations.

The beta distribution is useful in modeling continuous random variables that lie between 0 and 1, such as proportions and percentages. The beta distribution takes on many different shapes and may be described by two shape parameters, alpha and beta, that can be difficult to estimate. Maximum likelihood and method of moments estimation are possible, though method of moments is much more straightforward. We examine both of these methods here, and compare them to three more proposed methods of parameter estimation: 1) a method used in the Program Evaluation and Review Technique (PERT), 2) a modification of the two-sided power distribution (TSP), and 3) a quantile estimator based on the first and third quartiles of the beta distribution. We find the quantile estimator performs as well as maximum likelihood and method of moments estimators for most beta distributions. The PERT and TSP estimators do well for a smaller subset of beta distributions, though they never outperform the maximum likelihood, method of moments, or quantile estimators. We apply these estimation techniques to two data sets to see how well they approximate real data from Major League Baseball (batting averages) and the U.S. Department of Energy (radiation exposure). We find the maximum likelihood, method of moments, and quantile estimators perform well with batting averages (sample size 160), and the method of moments and quantile estimators perform well with radiation exposure proportions (sample size 20). Maximum likelihood estimators would likely do fine with such a small sample size were it not for the iterative method needed to solve for alpha and beta, which is quite sensitive to starting values. The PERT and TSP estimators do more poorly in both situations. We conclude that in addition to maximum likelihood and method of moments estimation, our method of quantile estimation is efficient and accurate in estimating parameters of the beta distribution.

The lognormal distribution is useful in modeling continuous random variables which are greater than or equal to zero. Example scenarios in which the lognormal distribution is used include, among many others: in medicine, latent periods of infectious diseases; in environmental science, the distribution of particles, chemicals, and organisms in the environment; in linguistics, the number of letters per word and the number of words per sentence; and in economics, age of marriage, farm size, and income. The lognormal distribution is also useful in modeling data which would be considered normally distributed except for the fact that it may be more or less skewed (Limpert, Stahel, and Abbt 2001). Appropriately estimating the parameters of the lognormal distribution is vital for the study of these and other subjects. Depending on the values of its parameters, the lognormal distribution takes on various shapes, including a bell-curve similar to the normal distribution. This paper contains a simulation study concerning the effectiveness of various estimators for the parameters of the lognormal distribution. A comparison is made between such parameter estimators as Maximum Likelihood estimators, Method of Moments estimators, estimators by Serfling (2002), as well as estimators by Finney (1941). A simulation is conducted to determine which parameter estimators work better in various parameter combinations and sample sizes of the lognormal distribution. We find that the Maximum Likelihood and Finney estimators perform the best overall, with a preference given to Maximum Likelihood over the Finney estimators because of its vast simplicity. The Method of Moments estimators seem to perform best when σ is less than or equal to one, and the Serfling estimators are quite accurate in estimating μ but not σ in all regions studied. Finally, these parameter estimators are applied to a data set counting the number of words in each sentence for various documents, following which a review of each estimator's performance is conducted. Again, we find that the Maximum Likelihood estimators perform best for the given application, but that Serfling's estimators are preferred when outliers are present.

The purpose of this study was to measure the change in joint energy of the hip,knee and ankle of the right and left leg, in the sagittal plane during the delivery phase of the rotational shot put. We hypothesized that (1) throwers who produced a greater total hip energy change would have greater horizontal displacement and (2) throwers who produced a higher ratio of hip energy, in each leg independently, would produce greater horizontal displacement. Subjects (n = 8) must have been right-handed, collegiate or post collegiate level throwers trained in the rotational technique. Vicon Nexus System (Denver, CO, USA) used six MX13+, two F20, two T20 cameras recorded at 240 Hz, and the body Plug-in Gait model to track the body position during each trial. Two AMTI force plates (OR-6, Watertown, MA, USA) were used for collecting ground reaction force data at 960 Hz. A linear regression analysis was performed to determine a relationship between total hip energy change and horizontal displacement. A mixed model regression was used to determine any correlation between horizontal distance and left and right energy change ratios. Athletes who produced a greater total hip energy change had the greatest horizontal displacement (p = .022). Also throwers who produced a higher ratio of left hip energy change to total left leg energy produced the greatest horizontal displacement (p = .02). The ratio of right hip energy change to right leg energy change was found to not be significant to horizontal displacement (p = .955). We feel the findings on the left leg energy change are an attempt by the athlete to both accelerate the shot put as well as stop the rotational progression to allow the athlete to complete a fair throw. The athlete extending both the right and the left hip rapidly during the delivery phase can help explain the combined right and left hip energy change. This action accelerates the ball in a proximal-distal sequence, which allows athletes to reach high final shot put velocities. The higher the final velocity on the shot put positively correlates with the horizontal displacement.

A portable special purpose computer (s.p.c.) is described which provides decimal readouts of the first four moments of a fluctuating voltage v on four separate registers. A fifth register provides a readout of the measuring time which can be within the range 10 ms. to 30 Hrs. The s.p.c. can be switched to another mode which provides a measure of the cumulative amplitude distribution of v within sixteen positive and negative levels. Salient characteristics of the s.p.c. are as follows: (a) There are no low frequency limitations. The upper frequency limit, established by error considerations, is about 5 kHz with 99.73% confidence that the error is within 1%. (b) At the end of the measuring time T, all the four moments are immediately available in magnitude and sign. (c) The outputs can be available in any code, the only change necessary being in the code of the counting readout registers. (d) All computations for a sample are completed before the next sample arrives so that programming and unnecessary storage facilities are eliminated. The voltage input v is rectified and sampled systematically by an equi-interval a.d. converter. The samples, together with the sign bit, are fed into special purpose digital multipliers based on a "weighted feed" principle. The outputs from these multipliers, with the sign bit, arc fed to accumulators via parallel adders for each of the moments. The overflows of these accumulators are shown to be contributions to the various moments and are fed to the decimal display registers. Direct computation of the standard deviation (σ) of the input, from measured first and second moments has also been investigated. A theoretical analysis of the various errors which occur in such an s.p.c. has been made. Results indicate that for most signals the overall error is within 1% for all four moments. Finally, the development of a universal arithmetic cell, for use in iterative and near-iterative arrays, is reported in this thesis. It is shown that use of such arrays in the arithmetic units of the s.p.c. can lead to a considerably simplified design. / Thesis / Doctor of Philosophy (PhD)

The hands are essential for our ability to complete tasks. Quantifying the many forces acting on the entire hand is important to improve our understanding of hand function and hand-related musculoskeletal disorders. Biomechanical models of the hand used to compute internal tissue loads typically simplify the applied forces into a single point of force applied at the centre of mass of the distal phalanx. Accounting for the distributed loads across the hands and fingers is a needed step in understanding the loads acting on and inside the body. Therefore, the purpose of this thesis was to use a pressure mapping system to examine the effects of distributed loads on net joint moments and muscle activations in the hands during common tasks. Twenty-three right-handed participants completed a series of finger presses, power grips, and pinch tasks. A pressure mapping system measured pressure on 17 regions of the hand. Three- dimensional hand kinematics was collected using a 72-marker setup. Forces were also measured with a six degrees of freedom force transducer to ensure participants matched specified exertion levels. Pressure distribution, kinematics, and kinetics were used to calculate internal net joint moments at the fingers (distal phalangeal flexion, proximal phalangeal flexion, metacarpal flexion, metacarpal abduction) and muscle activations for 22 forearm and hand muscles using an OpenSim model. External loads were represented in three manners: (1) Centre of Mass Model (COM) distributed the forces over segments that contributed to the force production and placed loads at the centre of mass; (2) Centre of Pressure Model (COP) distributed the forces over segments that contributed to the force production and placed loads at the centre of pressure; (3) Single Point Model (SP) placed a single load at the distal phalanx or the centre of mass of the hand. Results of equivalence tests indicate differences in all net joint moments between COM-SP and COP-SP comparisons. There were no differences between COM and COP. COM and COP moments during all tasks were larger in digits with a larger percentage of total force compared to SP. Due to the larger moments in those conditions, COM and COP calculated larger muscle activities compared to SP. Both internal net joint moments and muscle activations were most affected by the pressure distribution and hand posture. Overall, these findings indicate that representing external forces using distributed loads provide increased fidelity of forces at the hand and fingers. Distributed loads provide more information on internal loads of the hand and digits, and in turn, quantify individual differences that can lead to injury in occupational settings. / Thesis / Master of Science in Kinesiology

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The primitive hut, which is regarded as the initial model of architecture, came from humans' desire to survive and be sheltered from the severe weather outside. Architecture lays its root in practical use but when mankind thrives and begins to flourish, the practical use is no longer the only objective architecture needs to fulfill. My thesis is an exploration of processing architectural design when the practical program becomes the minor factor. The architect's inner pursuit for aesthetic satisfaction becomes the major motive in my thesis. Instead of a project that has an end, I consider my thesis as a continuing research of recording daily inspirations. Inspirations come from memories and could stimulate creations. Different architectural moments that come from my inspirations are pictured in this project in the aim of inspiring other people. I am exploring my own identity as a person who studies architecture. / Master of Architecture