Testing the equality of matrix distributions

Statistical Methods & Applications - While matrices are usually used as the basic data structure for experiments with repeated measurements or longitudinal data, testing methods for the...     

Compressed Natural Gas Bus Safety: A Quantitative Risk Assessment

This study assesses the fire safety risks associated with compressed natural gas (CNG) vehicle systems, comprising primarily a typical school bus and supporting fuel infrastructure. The study determines the sensitivity of the results to variations in component failure rates and consequences of fire events. The components and subsystems that contribute most to fire safety risk are determined. Finally, the results are compared to fire risks of the present generation of diesel-fueled school buses. Direct computation of the safety risks associated with diesel-powered vehicles is possible because these are mature technologies for which historical performance data are available. Because of limited experience, fatal accident data for CNG bus fleets are minimal. Therefore, this study uses the probabilistic risk assessment (PRA) approach to model and predict fire safety risk of CNG buses. Generic failure data, engineering judgments, and assumptions are used in this study. This study predicts the mean fire fatality risk for typical CNG buses as approximately 0.23 fatalities per 100-million miles for all people involved, including bus passengers. The study estimates mean values of 0.16 fatalities per 100-million miles for bus passengers only. Based on historical data, diesel school bus mean fire fatality risk is 0.091 and 0.0007 per 100-million miles for all people and bus passengers, respectively. One can therefore conclude that CNG buses are more prone to fire fatality risk by 2.5 times that of diesel buses, with the bus passengers being more at risk by over two orders of magnitude. The study estimates a mean fire risk frequency of 2.2 x 10(-5) fatalities/bus per year. The 5% and 95% uncertainty bounds are 9.1 x 10(-6) and 4.0 x 10(-5), respectively. The risk result was found to be affected most by failure rates of pressure relief valves, CNG cylinders, and fuel piping.     

Hybrid test for the hypothesis of symmetry

In recent years, McWilliams and Tajuddin have proposed new and more powerful non-parametric tests of symmetry for continuous distributions about a known center. In this paper, we propose a simple non-parametric two-stage procedure based on the sign test and a percentile-modified two-sample Wilcoxon test. The small-sample properties of this test, Tajuddin's test, McWilliams' test and a modified runs test of Modarres and Gastwirth are investigated in a Monte Carlo simulation study. The simulations indicate that, for a wide variety of asymmetric alternatives in the lambda family, the hybrid test is more powerful than are existing tests in the literature.     

Testing the Equality and Exchangeability of Bivariate Distributions

We consider three methods (oments, cut-points, and ranks) for testing the hypotheses of equality of two bivariate distribution functions (H _0a ) and exchangeability (H _0b ). To test H _0a , the asymptotic normality of the vector of mixed moments provides a statistic with an asymptotic chi-square distribution. With every observation, method of cut-points associates three 2 × 2 tables to record the proportions of the X, Y, and the combined samples that fall in the four regions around the observation. We measure the total squared deviations of the proportions in the combined sample from X and Y samples. The two methods are compared with the method of ranks based on the Puri and Sen (1971 Puri , M. L. , Sen , P. K. ( 1971 ). Nonparametric Methods in Multivariate Analysis . New York : John Wiley and Sons . [Google Scholar]) multivariate two-sample rank test for location.

To test H _0b we identify two bivariate distributions, one above and the other below the line of symmetry X = Y, to which a test of H _0a is applied. Under H _0b , matrix of mixed moments is symmetric and a quadratic form in differences of (r,s)-th and (s, r)-th mixed moments provides an asymptotic chi-square distribution. A permutation test is devised to apply the method of cut-points to the observations above and below the line of symmetry after they are folded. We also describe an adaption of the Puri-Sen rank test to assess H _0b . To estimate the power of the above methods under different types of alternatives and compare them to existing tests, we report on a Monte Carlo experiment that evaluates the finite-sample performance of these methods under the Plackett's family of bivariate distributions.     

Cryptosporidiosis Susceptibility and Risk: A Case Study

Regional estimates of cryptosporidiosis risks from drinking water exposure were developed and validated, accounting for AIDS status and age. We constructed a model with probability distributions and point estimates representing Cryptosporidium in tap water, tap water consumed per day (exposure characterization); dose response, illness given infection, prolonged illness given illness; and three conditional probabilities describing the likelihood of case detection by active surveillance (health effects characterization). The model predictions were combined with population data to derive expected case numbers and incidence rates per 100,000 population, by age and AIDS status, borough specific and for New York City overall in 2000 (risk characterization). They were compared with same-year surveillance data to evaluate predictive ability, assumed to represent true incidence of waterborne cryptosporidiosis. The predicted mean risks, similar to previously published estimates for this region, overpredicted observed incidence-most extensively when accounting for AIDS status. The results suggest that overprediction may be due to conservative parameters applied to both non-AIDS and AIDS populations, and that biological differences for children need to be incorporated. Interpretations are limited by the unknown accuracy of available surveillance data, in addition to variability and uncertainty of model predictions. The model appears sensitive to geographical differences in AIDS prevalence. The use of surveillance data for validation and model parameters pertinent to susceptibility are discussed.     

A technical note on “A fuzzy set approach for R&D portfolio selection using a real options valuation model” by Wang and Hwang (2007)

In this paper, three critical issues with the paper “A fuzzy set approach for R&D portfolio selection using a real options valuation model”, coauthored by Wang and Hwang and published in Omega 2007 are addressed. Shortcomings of the original work are highlighted and corrective measures to improve the approach are proposed.     

Efficiency of t-Test and Hotelling's T 2-Test After Box-Cox Transformation

Early investigations of the effects of non-normality indicated that skewness has a greater effect on the distribution of t-statistic than does kurtosis. When the distribution is skewed, the actual p-values can be larger than the values calculated from the t-tables. Transformation of data to normality has shown good results in the case of univariate t-test. In order to reduce the effect of skewness of the distribution on normal-based t-test, one can transform the data and perform the t-test on the transformed scale. This method is not only a remedy for satisfying the distributional assumption, but it also turns out that one can achieve greater efficiency of the test. We investigate the efficiency of tests after a Box-Cox transformation. In particular, we consider the one sample test of location and study the gains in efficiency for one-sample t-test following a Box-Cox transformation. Under some conditions, we prove that the asymptotic relative efficiency of transformed t-test and Hotelling's T ²-test of multivariate location with respect to the same statistic based on untransformed data is at least one.     

Testing the equality of correlation matrices

We present results that extend an existing test of equality of correlation matrices. A new test statistic is proposed and is shown to be asymptotically distributed as a linear combination of independent x ² random variables. This new formulation allows us to find the power of the existing test and our extensions by deriving the distribution under the alternative using a linear combination of independent non-central x ² random variables. We also investigate the null and the alternative distribution of two related statistics. The first one is a quadratic form in deviations from a control group with which the remaining k-1 groups are to be compared. The second test is designed for comparing adjacent groups. Several approximations for the null and the alternative distribution are considered and two illustrative examples are provided.     

Local Box–Cox transformation on time-varying parametric models for smoothing estimation of conditional CDF with longitudinal data

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies, such as epidemiological studies and longitudinal clinical trials. Estimation approaches without any structural assumptions may lead to inadequate and numerically unstable estimators in practice. We propose in this paper a nonparametric approach based on time-varying parametric models for estimating the conditional distribution functions with a longitudinal sample. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model after local Box–Cox transformation. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Applications of our two-step estimation method have been demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through a simulation study. Application and simulation results show that smoothing estimation from time-variant parametric models outperforms the existing kernel smoothing estimator by producing narrower pointwise bootstrap confidence band and smaller root mean squared error.