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AbstractHypothesis testing for the Weibull distribution always raises attention in the literature. It is challenging especially in small-sample scenarios. In this paper, we propose a new Kolmogorov-Smirnov type test based on the distance between two empirical functions, one from the data and another from the representative points of the underlying distribution. A bias correction technique is used to estimate unknown parameters in the Weibull distribution. We also discuss how to choose a suitable number of representative points based on the proposed loss function, which is recommended for constructing the empirical function from the Weibull distribution. To compare with tailor-made tests in the literature, a simulation study of the empirical type I error and testing power is conducted. It shows that our proposed test statistic is more powerful in most alternative scenarios, and it significantly improves the power of testing in small-sample scenarios. Finally, two real-world datasets are conducted to further demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Testing multivariate normality in high-dimensional data analysis has been a long-lasting topic in the area of goodness of fit. Numerous methods for this purpose can be found in the literature. Reviews on different methods given by influential researchers show that new methods keep emerging in the literature from different perspectives. The theory of statistical representative points provides a new perspective to construct tests for multivariate normality. To avoid the difficulty and huge computational load in finding the statistical representative points from a high-dimensional probability distribution, we develop an approach to constructing a test for high-dimensional normal distribution based on the representative points of the simple univariate beta distribution. The representative-points-based approach is extended to the the case that the sample size may be smaller than the dimension. A Monte Carlo study shows that the new test is able to control type I error rates fairly well for both large and small sample sizes when faced with a high dimension. The power of the new test against some non-normal distributions is generally or substantially improved for a set of selected alternative distributions. A real-data example is given for a simple application illustration. [ABSTRACT FROM AUTHOR]

MATLAB is a powerful package for numerical computation. MATLAB contains a rich pool of mathematical functions and provides flexible plotting functions for illustrating mathematical solutions. The course of calculus-based business mathematics consists of two major topics: 1) derivative and its applications in business; and 2) integration and its applications in business. Both topics involve various mathematical functions that may scare many business students. Fortunately, MATLAB provides an easy solution to handling the derivative and integration of various mathematical functions. With the help of MATLAB, all elementary formulas for derivative and integration in the textbooks of calculus-based business mathematics become less important to those students who have difficulty in learning and using mathematical formulas to find solutions to quantitative business problems. With the wide availability of various business data bases including the MATLAB package in the Samuel S. Bergami, Jr. Learning Center for Finance and Technology at the University of New Haven, in this paper, we illustrate how to use MATLAB to help students? learning of calculus-based business mathematics and apply the calculus principles to find solutions to real quantitative business problems. Our experience shows that MATLAB greatly saves students? efforts in learning difficult and pure calculus formulas and it is easy for business students to acquire the basic internal functions in using MATLAB.

High-dimensional data often display heteroscedasticity. If the heteroscedasticity is neglected in the regression model, it will produce inefficient inference for the regression coefficients. Quantile regression is not only robust to outliers, but also accommodates heteroscedasticity. This paper aims to simultaneously carry out variable selection and heteroscedasticity identification for the linear location-scale model under a unified framework. We develop a regularized multiple quantile regression approach simultaneously identifying the heteroscedasticity, seeking common features of quantile coefficients and eliminating irrelevant variables. We also establish the theoretical properties of the proposed method under some regularity conditions. Simulation studies are conducted to evaluate the finite sample performance of the proposed method, showing that it is able to identify the covariates that affect the variability of the response. We further apply the proposed method to analyse the Wage data. [ABSTRACT FROM AUTHOR]

Missing observations in r × c contingency tables often occur in medical, bio-pharmaceutical and epidemiological researches. The most common method to analyze incomplete contingency tables simply removes the non-response counts from both variables or depends on an independence assumption, which may be improper and could result in unreliable conclusions because of the under-estimation of the uncertainty. In this article, we first derive the valid sampling distribution of the observed counts by taking three categories missing data into consideration under the assumption of missing at random and the assumption of the total number of observations being fixed. Next, based on the new sampling distribution, the Fisher scoring algorithm for calculating the maximum likelihood estimates of parameters is developed, and the small-sample bootstrap confidence interval method is also provided. In addition, we theoretically compare the proposed sampling distribution with two existing sampling distributions, and conduct some simulations to investigate the performance of the three different sampling distributions in statistical inferences. Finally, two real data sets are analyzed to illustrate the newly proposed sampling distribution and corresponding statistical methods. [ABSTRACT FROM AUTHOR]

Maximum likelihood is commonly used for the estimation of model parameters in the analysis of two-level structural equation models. Constraints on model parameters could be encountered in some situations such as equal factor loadings for different factors. Linear constraints are the most common ones and they are relatively easy to handle in maximum likelihood analysis. Nonlinear constraints could be encountered in complicated applications. The authors develop an EM-type algorithm for estimating model parameters with both linear and nonlinear constraints. The empirical performance of the algorithm is demonstrated by a Monte Carlo study. Application of the algorithm for linear constraints is illustrated by setting up a two-level mean and covariance structure model for a real two-level data set and running an EQS program. (Contains 4 tables and 2 figures.)

Calculus-based business mathematics is a required quantitative course for undergraduate business students in most AACSB accredited schools or colleges of business. Many business students, however, have relatively weak mathematical background or even display math-phobia when presented with calculus problems. Because of the popularity of Excel, its ease of learning, and its rich computational functions, we have been teaching our calculus-based business mathematics in computer labs and accumulating feasible experience in employing Excel to assist our students' learning in this course. In this paper we illustrate how to use Excel to enhance students' understanding in difficult and important calculus-based mathematical principles and to find numerical solutions to difficult quantitative business problems by providing them with heuristic examples. Our experience shows that Excel can greatly simplify the interpretation of pure calculus principles and can substantially reduce students' misunderstanding in applying calculus principles in solving quantitative business problems. (Contains 5 tables and 7 figures.)

A Bayesian semiparametric model framework is proposed to analyze multivariate longitudinal data. The new framework leads to simple explicit posterior distributions of model parameters. It results in easy implementation of the MCMC algorithm for estimation of model parameters and demonstrates fast convergence. The proposed model framework associated with the MCMC algorithm is validated by four covariance structures and a real-life dataset. A simple Monte Carlo study of the model under four covariance structures and an analysis of the real dataset show that the new model framework and its associated Bayesian posterior inferential method through the MCMC algorithm perform fairly well in the sense of easy implementation, fast convergence, and smaller root mean square errors compared with the same model without the specified autoregression structure. [ABSTRACT FROM AUTHOR]

Maximum likelihood is an important approach to analysis of two-level structural equation models. Different algorithms for this purpose have been available in the literature. In this paper, we present a new formulation of two-level structural equation models and develop an EM algorithm for fitting this formulation. This new formulation covers a variety of two-level structural equation models. As a result, the proposed EM algorithm is widely applicable in practice. A practical example illustrates the performance of the EM algorithm and the maximum likelihood statistic.

A quantile-quantile (Q–Q) plot is derived from a characterization for the multivariate normal distribution and the Shapiro–Wilk's (1965) S–W statistic. The normal characterization results in some independent spherical distributions. The affine invariance of the S–W statistic and a simple property of spherical distributions are employed to construct the Q–Q plot. Easy simulation of the empirical distribution of the S–W statistic avoids the complicated exact null distribution of S-W statistic. The Q–Q plot can be easily implemented for detecting a possible departure from multivariate normality in high dimensional data analysis. Two examples are illustrated for real application. [ABSTRACT FROM AUTHOR]

By using the idea of principal component analysis, we propose an approach to applying the classical skewness and kurtosis statistics for detecting univariate normality to testing high-dimensional normality. High-dimensional sample data are projected to the principal component directions on which the classical skewness and kurtosis statistics can be constructed. The theory of spherical distributions is employed to derive the null distributions of the combined statistics constructed from the principal component directions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and a simple power comparison with some existing statistics. The effectiveness of the proposed statistics is illustrated by two real-data examples. [ABSTRACT FROM AUTHOR]

The family of lp-norm symmetric distributions was proposed by Yue and Ma and is a natural generalization to the family of l1-norm symmetric distributions studied by Fang et al. In this article, we propose a stochastic representation for the lp-norm symmetric distribution for any constant p > 0. The stochastic representation is expressed through independent and identically distributed uniform U(0, 1) random variables. It is illustrated that the stochastic representation can be applied to statistical simulation and uniform experimental design. [ABSTRACT FROM AUTHOR]

A new dimension-reduction graphical method for testing high-dimensional normality is developed by using the theory of spherical distributions and the idea of principal component analysis. The dimension reduction is realized by projecting high-dimensional data onto some selected eigenvector directions. The asymptotic statistical independence of the plotting functions on the selected eigenvector directions provides the principle for the new plot. A departure from multivariate normality of the raw data could be captured by at least one plot on the selected eigenvector direction. Acceptance regions associated with the plots are provided to enhance interpretability of the plots. Monte Carlo studies and an illustrative example show that the proposed graphical method has competitive power performance and improves the existing graphical method significantly in testing high-dimensional normality. [ABSTRACT FROM AUTHOR]