Jeff Terpstra

Population structure and genetic differentiation among the USDA common bean (Phaseolus vulgaris L.) core collection

Genetic diversity data were collected from a large population of common bean (Phaseolus vulgaris L.) landraces representing the United States Department of Agriculture core collection. The data were based on microsatellite data from all linkage groups. A procedure was developed to determine if we collected sufficient marker data to adequately estimated pairwise diversity. The diversity data were used to define populations using distance and model-based approaches. Genetic differentiation and genetic isolation by distance data were collected. Diversity was also compared for markers linked and unlinked to domestication loci. Using a model-based approach, the landraces were divided into the traditional Middle American and Andean gene pools. Diversity was greater for the Middle American gene pool. Six Middle American and three Andean subpopulations were defined, and the Middle American subpopulations exhibited strong geographic identity. Unlike other studies, seed size varied considerably with subpopulations, and a number of the subpopulations contained landraces from multiple common bean races. All of the subpopulations were highly differentiated, with the Middle American subpopulations showing the greatest differentiation. Genetic isolation by distance was observed among the Middle American and Andean subpopulations but not among subpopulations within a gene pool. Within each gene pool, diversity was lower for markers linked to domestication loci.

Confidence intervals for a population proportion based on a ranked set sample

Abstract This article examines several approximate methods to formulate confidence intervals for a single population proportion based on a ranked set sample (RSS). All of the intervals correspond to certain test statistics. That is, the confidence intervals are obtained by inverting the Wald, Wilson, score, and likelihood ratio tests. The Wald and Wilson intervals are based on the asymptotic distributions of two point estimators; the method of moments (MM) estimator and the maximum likelihood (ML) estimator. Continuity corrected versions of these intervals are also discussed. The R statistical software program is used to both calculate and evaluate the proposed intervals. For instance, an actual data set is analyzed for the sake of illustration. Furthermore, a simulation study which compares the intervals via expected widths and coverage probabilities is presented. The study indicates that the confidence intervals derived from the ML methodology generally outperform those based on MM procedures. Additionally, the Wilson and score intervals do not yield the same results under RSS as they do under simple random sampling. Lastly, the ML-based Wilson interval (without continuity correction) is recommended for use in practice.

Exact inference for a population proportion based on a ranked set sample

ABSTRACT This article develops and investigates a confidence interval and hypothesis testing procedure for a population proportion based on a ranked set sample (RSS). The inference is exact, in the sense that it is based on the exact distribution of the total number of successes observed in the RSS. Furthermore, this distribution can be readily computed with the well-known and freely available R statistical software package. A data example that illustrates the methodology is presented. In addition, the properties of the inference procedures are compared with their simple random sample (SRS) counterparts. In regards to expected lengths of confidence intervals and the power of tests, the RSS inference procedures are superior to the SRS methods.

Proposed tests for the non-decreasing alternative in a mixed design

Abstract Two versions of a proposed nonparametric test are introduced for the mixed design consisting of a randomized complete block and a completely randomized design. The proposed versions are designed to test for a nondecreasing trend. Both versions are a combination of Page’s test and the Jonckheere-Terpstra test. It is noted that the randomized complete block portion could be replaced by a repeated measures design. A simulation study is conducted comparing the estimated powers of the two proposed test versions along with the estimated powers of Page’s test including only the randomized complete block design portion. Three different underlying distributions are considered. Equal and unequal sample sizes for the completely randomized design are used. When equal sample sizes are used, the sample size, n, is selected so that it is 1/8, 1/4, and 1/2 that of the number of blocks considered. The study considers a variety of increasing location parameter arrangements for three, four, and five populations. At least one of the proposed test versions generally has higher power than the Page’s test which uses only the RCBD portion. This paper shows that both test versions can be written as linear combinations of Page’s test and the Jonckheere-Terpstra test with differing weights.

Nonparametric Tests for Mixed Designs

This article proposes tests for testing the equality of k-medians when the data are a mixture of a randomized complete block, a completely randomized design, and possibly an incomplete block (or blocks) design. The proposed tests are compared to the Friedman test when just the randomized complete block portion of observations is kept. Two tests are also developed for the umbrella alternative with known turning point for a mixed design. The two proposed tests are compared to each other and a test just for the randomized complete block portion (Kim and Kim, 1992). Results are given.

Weighted L 1-estimates for a VAR(p) time series model

The most common method of estimating the parameters of a vector-valued autoregressive time series model is the method of least squares (LS). However, since LS estimates are sensitive to the presence of outliers, more robust techniques are often useful. This paper investigates one such technique, weighted-L 1 estimates. Following traditional methods of proof, asymptotic uniform linearity and asymptotic uniform quadricity results are established. Additionally, the gradient of the objective function is shown to be asymptotically normal. These results imply that the weighted-L 1 parameter estimates for this model are asymptotically normal at rate n −1/2. The results rely heavily on covariance inequalities for geometric absolutely regular processes and a Martingale central limit theorem. Estimates for the asymptotic variance–covariance matrix are also discussed. A finite-sample efficiency study is presented to examine the performance of the weighted-L 1 estimate in the presence of both innovation and additive outliers. Specifically, the classical LS estimate is compared with three versions of the weighted- L 1 estimate. Finally, a quadravariate financial time series is used to demonstrate the estimation procedure. A brief residual analysis is also presented.

A Nonparametric Test for the Ordered Alternative Based on Kendall's Correlation Coefficient

Several methods have been developed for testing the ordered alternative. These include the Jonckheere–Terpstra (JT) test (Jonckheere, 1954; Terpstra, 1952), a modified JT test (MJT) (Tryon and Hettmansperger, 1987), and a test proposed by Terpstra and Magel (TM) (Terpstra and Magel, 2003), among others. This article proposes a new method for testing the ordered alternative. The proposed test is based on Kendalls tau statistic. The asymptotic distribution of the test statistic is given. A Monte Carlo simulation study is conducted comparing the estimated powers of the proposed test with existing tests under a variety of sample sizes and distributions.

Some Illustrative Classroom Examples Regarding Sums of Discrete Random Variables With Finite Support

In many probability and mathematical statistics courses the probability generating function (PGF) is typically overlooked in favor of the more utilized moment generating function. However, for certain types of random variables, the PGF may be more appealing. For example, sums of independent, non-negative, integer-valued random variables with finite support are easily studied via the PGF. In particular, the exact distribution of the sum can easily be calculated. Several illustrative classroom examples, with varying degrees of difficulty, are presented. All of the examples have been implemented using the R statistical software package.