Roger W. Johnson

An Introduction to the Bootstrap

This article presents bootstrap methods for estimation, using simple arguments. Minitab macros for implementing these methods are given.

Fitting Percentage of Body Fat to Simple Body Measurements

Percentage of body fat, age, weight, height, and ten body circumference measurements (e.g., abdomen) are recorded for 252 men. Body fat, one measure of health, has been accurately estimated by an u...

Exploring Relationships in Body Dimensions

Body girth measurements and skeletal diameter measurements, as well as age, weight, height and gender, are given for 507 physically active individuals – 247 men and 260 women. These data can be used to provide statistics students practice in the art of data analysis. Such analyses range from simple descriptive displays to more complicated multivariate analyses such as multiple regression and discriminant analysis.

The Bias and Error in Moment Estimators for Parameters of Drop Size Distribution Functions: Sampling from Gamma Distributions

Abstract This paper complements an earlier one that demonstrated the bias in the method-of-moments (MM) estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions. Here the authors consider both the bias and the errors in MM estimators applied to samples from known gamma DSDs (of which the exponential DSD treated in the earlier paper is a special case). The samples were generated using a similar Monte Carlo simulation procedure. The skewness in the sampling distributions of the DSD moments that causes this bias is less pronounced for narrower population DSDs, and therefore the bias problems (and also the errors) diminish as the gamma shape parameter increases. However, the bias still increases with the order of the moments used in the MM procedures; thus it is stronger when higher-order moments (such as the radar reflectivity) are used. The simulation results also show that the errors of the estimates of the DSD param...

L-Moment Estimators as Applied to Gamma Drop Size Distributions

Abstract The traditional approach with experimental raindrop size data has been to use the method of moments in the fitting procedure to estimate the parameters for the raindrop size distribution function. However, the moment method is known to be biased and can have substantial errors. Therefore, the L-moment method, which is widely used by hydrologists, was investigated as an alternative. The L-moment method was applied, along with the moment and maximum likelihood methods, to samples taken from simulated gamma raindrop populations. A comparison of the bias and the errors involved in the L-moments, moments, and maximum likelihood procedures shows that, with samples covering the full range of drop sizes, L-moments and maximum likelihood outperform the method of moments. For small sample sizes the moment method gives a large bias and large error while the L-moment method gives results close to the true population values, outperforming even maximum likelihood results. Because the goal of this work is to un...

Comparison of Estimators for Parameters of Gamma Distributions with Left-Truncated Samples

Abstract When fitting a raindrop size distribution using a gamma model from data collected by a disdrometer, some consideration needs to be given to the small drops that fail to be recorded (typical disdrometer minimum size thresholds being in the 0.3–0.5-mm range). To this end, a gamma estimation procedure using maximum likelihood estimation has recently been published. The current work adds another procedure that accounts for the left-truncation problem in the data; in particular, an L-moments procedure is developed. These two estimation procedures, along with a traditional method-of-moments procedure that also accounts for data truncation, are then compared via simulation of volume samples from known gamma drop size distributions. For the range of gamma distributions considered, the maximum likelihood and L-moments procedures—which perform comparably—are found to outperform the procedure of method-of-moments. As these three procedures do not yield simple estimates in closed form, salient details of the...

Quantifying second generation ethanol inhibition: Design of Experiments approach and kinetic model development

While softwoods represent a potential feedstock for second generation ethanol production, compounds present in their hydrolysates can inhibit fermentation. In this study, a novel Design of Experiments (DoE) approach was used to identify significant inhibitory effects on Saccharomyces cerevisiae D5A for the purpose of guiding kinetic model development. Although acetic acid, furfural and 5-hydroxymethyl furfural (HMF) were present at potentially inhibitory levels, initial factorial experiments only identified ethanol as a significant rate inhibitor. It was hypothesized that high ethanol levels masked the effects of other inhibitors, and a subsequent factorial design without ethanol found significant effects for all other compounds. When these non-ethanol effects were accounted for in the kinetic model, R¯(2) was significantly improved over an ethanol-inhibition only model (R¯(2)=0.80 vs. 0.76). In conclusion, when ethanol masking effects are removed, DoE is a valuable tool to identify significant non-ethanol inhibitors and guide kinetic model development.

Using Games to Teach Markov Chains

ABSTRACT Games are promoted as examples for classroom discussion of stationary Markov chains. In a game context Markov chain terminology and results are made concrete, interesting, and entertaining. Game length for several-player games such as Hi Ho! Cherry-O and Chutes and Ladders is investigated and new, simple formulas are given. Slight modifications to Monopoly rules are discussed to fit it into a Markov chain model with the ultimate goal of determining the long-run frequencies for visiting the various game properties.

Modeling Raindrop Size.

Being able to characterize the size of raindrops is useful in a number of fields including meteorology, hydrology, agriculture and telecommunications. Associated with this article are data sets containing surface (i.e. ground-level) measurements of raindrop size from two different instruments and two different geographical locations. Students may begin to develop some sense of the character of raindrop size distributions through some basic exploratory data analysis of these data sets. Teachers of mathematical statistics students will find an example useful for discussing the beta, gamma, lognormal and Weibull probability density models, as well as fitting these by maximum likelihood and assessing the quality of fit. R software is provided by the authors to assist students in these investigations.

Record Values and Surviving Glacial Moraines

Summary Estimating the number of glacial advances from surviving glacial debris can, amazingly enough, be cast as a records problem for which there is a substantial statistical literature. As a result, instructors may illustrate method of moments and maximum-likelihood estimation techniques with an interesting problem in the physical sciences.