Chul Gyu Park

A Simple Method for Generating Correlated Binary Variates

Abstract Correlated binary data are frequently analyzed in studies of repeated measurements, reliability analysis, and others. In such studies correlations among binary variables are usually nonnegative. This article provides a simple algorithm for generating an arbitrary dimensional random vector of non-negatively correlated binary variables. In some frequently encountered situations the algorithm reduces to explicit expressions. The correlated binary variables are generated from correlated Poisson variables. The key idea lies in the property that any Poisson random variable can be expressed as a convolution of other independent Poisson random variables. The binary variables have desired correlations by sharing common independent Poisson variables.

An algorithm for generating correlated random variables in a class of infinitely divisible distributions

A simple algorithm is proposed for generating a set of nonnegatively correlated variables having a specified correlation structure in a certain class of infinitely divisible distributions. These distributions are characterized by a property that they are closed under summation. The correlated variables are expressed as sums of other independent random variables. The algorithm produces simple explicit expressions for typical correlation structures such as AR(1) correlation, banded correlation and symmetric correlation.

A generalized estimating equations approach for testing ordered group effects with repeated measurements.

In repeated measures studies, we are often interested in comparing group effects in which groups are associated with a certain order relation. We propose testing procedures for ordered group effects using the generalized estimating equations (GEE) approach of Liang and Zeger (1986, Biometrika 73, 13-22). The order-constrained GEE estimators of group effects are approximated by the isotonic regression of the unconstrained GEE estimators. Based on these constrained estimators, we construct test statistics for detecting ordered group effects. The limiting distributions of the test statistics are mixtures of chi-square distributions. A Monte Carlo experiment shows improved performances of the proposed tests over the usual chi-square tests in detecting ordered group effects. The proposed test procedures are illustrated by familial polyposis supplementation trial data.

Testing for unimodal dependence in an ordered contingency table with restricted marginal probabilities

Likelihood ratio test is considered for testing unimodal dependence as a null hypothesis in a 2 x k ordered contingency table having order restriction on a marginal distribution. The asymptotic least-favorable distribution of the LRT statistic is derived as a convolution of two chi-bar-square distributions. This testing method is illustrated with blood pressure and heart disease data.

Testing for one-sided group effects in repeated measures study

In repeated measures experiments, one-sided tests for group effects are developed using the likelihood ratio principle. The constrained maximum likelihood estimators under hypotheses of one-sided group effects are approximated by projecting unconstrained maximum likelihood estimators onto the first-quadrant. From these constrained maximum likelihood estimators, likelihood ratio tests are constructed. The limiting null distributions of the proposed test statistics are shown to be chi-bar square distributions. A simulation study shows that the proposed tests have improved power over the usual chi-square statistics for testing group effects. Two real data sets are analyzed to illustrate the proposed statistics.

Least squares estimation of two functions under order restriction in isotonicity

Least squares estimators of two functions are obtained under the restriction that their difference is isotonic. The difference between the restricted estimates of the functions is the isotonic regression of that of the unrestricted estimates. Brief discussion on the practical implication of this ordering is also provided.

Goodness‐of‐fit test for uniform stochastic ordering among several distributions

The likelihood-ratio test (LRT) is considered as a goodness-of-fit test for the null hypothesis that several distribution functions are uniformly stochastically ordered. Under the null hypothesis, H : F1 -< F2 -< ..- -< FN, the asymptotic distribution of the LRT statistic is a convolution of several chi-bar-square distributions each of which depends upon the location parameter. The leastfavourable parameter configuration for the LRT is not unique. It can be two different types and depends on the number of distributions, the number of intervals and the significance level ca. This testing method is illustrated with a data set of survival times of five groups of male fruit flies.

Likelihood ratio test for homogeneity of steady-state availabilities against order restrictions

The likelihood ratio test (LRT) is considered for comparing the steady-state availabilities among several system units within order restrictions. Assuming exponential distributions for failure and repair times, we discuss finding the maximum likelihood estimates of the order restricted parameters, and then derive the asymptotic null distribution of the likelihood ratio test statistic. The test based on this limiting distribution shows fairly good performances even with very small sample sizes. This fact will be investigated through a power analysis at various parameter configurations in the alternative hypothesis.

A weighted geometric regularity from order restricted statistical inference

In Euclideank-space, the cone of vectors x = (x 1,x 2,...,x k ) satisfyingx 1 ≤x 2 ≤ ... ≤x k and\(\sum\nolimits_{j = 1}^k {x_j } = 0\) is generated by the vectorsv j = (j −k, ...,j −k,j, ...,j) havingj −k’s in its firstj coordinates andj’s for the remainingk −j coordinates, for 1 ≤j <k. In this equal weights case, the average angle between v i and v j over all pairs (i, j) with 1 ≤i <j <k is known to be 60°. This paper generalizes the problem by considering arbitrary weights with permutations.

Analysis of ordered covariate effects among groups with repeated measurements

Testing procedures for ordered covariate effects are developed in the repeated measures experiment. The maximum likelihood estimators of covariate effects under the ordered hypothesis are approximated by the isotonic regression of their unconstrained estimators. The asymptotic null distributions of the test statistics are chi-bar-square distributions which are mixtures of chi-square distributions. A Monte-Carlo simulation reveals that the proposed test for ordered covariate effects is seriously more powerful than the usual chi-square test that neglects the information on the order restriction. These testing methods are applied for analyzing the effect of vitamin E diet supplement on growth rate of animals.