Carl Lee

The beta-Pareto distribution

In this paper, a four-parameter beta-Pareto distribution is defined and studied. Various properties of the distribution are discussed. The distribution is found to be unimodal and has either a unimodal or a decreasing hazard rate. The expressions for the mean, mean deviation, variance, skewness, kurtosis and entropies are obtained. The relationship between these moments and the parameters are provided. The method of maximum likelihood is proposed to estimate the parameters of the distribution. The distribution is applied to two flood data sets.

Constrained optimal designs

Abstract The problem combining optimality criteria using constrained optimization techniques is considered. Constraints may be due to some optimality criteria so that the designs satisfying the constraints will have at least the minimal quality that an investigator wishes to maintain. A necessary and sufficient condition similar to Kiefer (Theorem 1, 1974a) is obtained using Frechet derivatives. Some examples are presented to illustrate some possible applications of the constrained optimality criterion, including Stiglers (1971) C-restricted D-criterion, Lauters (1976) multiresponse modeling problem and Lee (1987), combination of A- and D-criteria.

Weibull-pareto distribution and its applications

In this article, a new distribution, namely, Weibull-Pareto distribution is defined and studied. Various properties of the Weibull-Pareto distribution are obtained. The distribution is found to be unimodal and the shape of the distribution can be skewed to the right or skewed to the left. Results for moments, limiting behavior, and Shannons entropy are provided. The method of modified maximum likelihood estimation is proposed for estimating the model parameters. Several real data sets are used to illustrate the applications of Weibull-Pareto distribution.

Constrained optimal designs for regressiom models

An attempt of combining several optimality criteria simulaneously by using the techniques of nonliear programming is demonstrated. Four constrained D- and G-optimality criteria are introduced, namely, D-restrcted, Ds-restricted, A-restricted and E-restricted D- and G-optimality. The emphasis is particularly on the polynomial regression. Examples for quadratic polynomial regression are investigated to illustrate the applicability of these constrained optimality criteria.

The gamma-normal distribution: Properties and applications

In this paper, some properties of gamma-X family are discussed and a member of the family, the gamma-normal distribution, is studied in detail. The limiting behaviors, moments, mean deviations, dispersion, and Shannon entropy for the gamma-normal distribution are provided. Bounds for the non-central moments are obtained. The method of maximum likelihood estimation is proposed for estimating the parameters of the gamma-normal distribution. Two real data sets are used to illustrate the applications of the gamma-normal distribution.

On generating T-X family of distributions using quantile functions

The cumulative distribution function (CDF) of the T-X family is given by R{W(F(x))}, where R is the CDF of a random variable T, F is the CDF of X and W is an increasing function defined on [0, 1] having the support of T as its range. This family provides a new method of generating univariate distributions. Different choices of the R, F and W functions naturally lead to different families of distributions. This paper proposes the use of quantile functions to define the W function. Some general properties of this T-X system of distributions are studied. It is shown that several existing methods of generating univariate continuous distributions can be derived using this T-X system. Three new distributions of the T-X family are derived, namely, the normal-Weibull based on the quantile of Cauchy distribution, normal-Weibull based on the quantile of logistic distribution, and Weibull-uniform based on the quantile of log-logistic distribution. Two real data sets are applied to illustrate the flexibility of the distributions.

Introductory statistics, college student attitudes and knowledge – a qualitative analysis of the impact of technology-based instruction

This paper presents findings from a qualitative study that compared the learning experiences of a group of students from a technology-based college-level introductory statistics course with the learning experiences of a group of students with non-technology-based instruction. Findings from the study indicate differences with regards to classroom experiences, student enjoyment of statistics, and student understanding of the many roles that technology plays in statistics. However, no significant differences were found between technology-based and non-technology-based instruction on students’ grasp of fundamental statistical concepts. In particular, these findings agree with the findings of several other studies, which indicate that incorporation of statistical software in the introductory statistics classroom might not always be very effective in building student intuitions about important statistical ideas related to statistical inference.

T-normal family of distributions: a new approach to generalize the normal distribution

The idea of generating skewed distributions from normal has been of great interest among researchers for decades. This paper proposes four families of generalized normal distributions using the T-X framework. These four families of distributions are named as T-normal families arising from the quantile functions of (i) standard exponential, (ii) standard log-logistic, (iii) standard logistic and (iv) standard extreme value distributions. Some general properties including moments, mean deviations and Shannon entropy of the T-normal family are studied. Four new generalized normal distributions are developed using the T-normal method. Some properties of these four generalized normal distributions are studied in detail. The shapes of the proposed T-normal distributions can be symmetric, skewed to the right, skewed to the left, or bimodal. Two data sets, one skewed unimodal and the other bimodal, are fitted by using the generalized T-normal distributions.AMS 2010 Subject Classification60E05; 62E15; 62P10

Beginning Statistics A to Z

This volume introduces students to the problem-solving power of statistics in the modern era of the computer. It primarily concentrates on active learning of statistical concepts and methods and presents a data-analytic approach to descriptive statistics that stresses graphic techniques of describing data. Features include: definition of probability in terms of P value; computer exercises throughout (although computer use is not mandatory with the text); statistical control and control charts; and a diskette containing the three real data sets.

Beta-Cauchy Distribution: Some Properties and Applications

Some properties of the four-parameter beta-Cauchy distribution such as the mean deviation and Shannon’s entropy are obtained. The method of maximum likelihood is proposed to estimate the parameters of the distribution. A simulation study is carried out to assess the performance of the maximum likelihood estimates. The usefulness of the new distribution is illustrated by applying it to three empirical data sets and comparing the results to some existing distributions. The beta-Cauchy distribution is found to provide great flexibility in modeling symmetric and skewed heavy-tailed data sets.