ABSTRACTS

Volume 11 Number 1 December 1998

On The Numerical Computation Of Confluent Hypergeometric Function With Zonal Polynomials Of Order 3
Takakazu Sugiyama (Department of Mathematics, Chuo University, Tokyo 112-8551, Japan)
Yuuichi Takeda (Commun. Engineering Dept., The Mainichi Newspaper, Tokyo 100-8051, Japan)
Masafumi Fukuda (Commun. Engineering Dept., The Mainichi Newspaper, Tokyo 100-8051, Japan)

Recurrence relations of the coefficients are obtained of the generalized hypergeometric function for the argument of order 3. The numerical computation of a distribution function is shown. The Fortran program is available to compute the distribution expressed by the generalized hypergeometric function.

Keywords: generalized hypergeometric function; zonal polynomials; recurrent relations; latent root

Nonparametric Test For Equality Of Intermediate Latent Roots In Non-Normal Distribution
Kenji Ushizawa (Sanno College, 1573 Kamikasuya, Isehara, Kanagawa 259-1197, Japan)
Yoshiharu Sato (Hokkaido University, North 13, West 8, Kita-ku, Sapporo 060-8628, Japan)
Takakazu Sugiyama (Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan)

Two-sample problem is considered to test the equality of the intermediate latent roots of two covariance matrices assuming non-normal distributions. The nonparametric method known as the Moses rank-like test is proposed for principal component scores(PC-scores), and its efficiency is compared with the Ansari-Bradley test and $F$-test by Monte Carlo experiments. This testing procedure turns out to be very useful when the population latent roots are sufficiently distinct and the sample sizes increase.

Keywords: nonparametric test; Moses rank-like test; elliptical distribution; log-normal distribution; latent root; covariance matrix

Two-Stage Testing To Establish Non-Inferiority In The Stratified 2 X 2 Contingency Tables
Takashi Koshimizu (Graduate School of Engineering, Osaka Electro-Communication University, 18-8, Hatsu-cho, Neyagawa, Osaka 572-0833, Japan)
Masaaki Tsujitani (Department of Engineering Informatics, Osaka Electro-Communication University, 18-8, Hatsu-cho, Neyagawa, Osaka 572-0833, Japan)

In this article we propose a two-stage testing procedure for demonstrating non-inferiority of a test drug in stratified $2\times 2$ contingency tables, which includes a preliminary test of homogeneity of odds ratios. In order to prove that a new drug is not inferior to a control by more than a prescribed amount, we use model-based methods such as the weighted least-square (WLS) procedure by Grizzle et al. (1969), and test the null hypothesis of a specified non-unity ratio. Also included are some results of a Monte Carlo study conducted to investigate the accuracy of the proposed test to the null distribution, and power of the test under small and moderate sample size configurations.

Keywords: homogeneity test; stratified 2x2 contingency tables; power; Monte Carlo simulation

Dynamic MDS Methods for three-way asymmetric dissimilarity data
Hiroshi Yadohisa (Kagoshima University, Korimoto, Kagoshima 890, Japan)
Naoto Niki (Science University of Tokyo, Shinjuku-ku, Tokyo 162, JAPAN)
Hiroki Hashiguchi (Science University of Tokyo, Shinjuku-ku, Tokyo 162, Japan)

Two methods for analyzing three-way asymmetric (dis)similarity data are proposed. In both methods, asymmetry found in data is represented as a set of vectors, which is determined from the skew-symmetric parts in the matrices for visualizing the latent structure of the asymmetry. In the first method, an asymmetric matrix called ``super asymmetric (dis)similarity matrix'' is first composed from the given set of data matrices. Making the analysis proposed here of that matrix yields the estimated moving paths of the objects as well as the chronological changes of asymmetric vectors expressing the stress brought by the forced allocation of the points. In the second method, Procrustes transformation is used for analysis and the objects and the asymmetric vectors are represented together in each separate space. Graphical representation of the result in a definite time-space by smooth spline interpolation of trajectories is discussed with some numerical illustrations.

Keywords: multidimensional scaling; INDSCAL; dynamic MDS; asymmetry

Multivariate Bayes Regression Models For Smoothing Of Color Images
Saeko Kusanobu (Graduate School of Engineering, Hiroshima University, Kagamiyama 1-4-1, Higashi-Hiroshima 739-8527, Japan)

We propose a smoothing method for color images based on multivariate Bayes models. This method simultaneously smoothes the images of red, green and blue lights while taking into account their correlations. The performance of the color smoothing technique is applied to test images and the Landsat TM data. We compare the multivariate smoothing with some other techniques which remove noise in monochrome images.

Keywords: difference matrix; Landsat images; smoothness prior

A Power Approximation of the Test of Independence in s x r Contingency Tables Based on a Normalizing Transformation
Yuri Sekiya (Kushiro Campus, Hokkaido University of Education, Shiroyama, Kushiro, 085-8580, Japan)
Nobuhiro Taneichi (Obihiro University of Agriculture and Veterinary Medicine, Inada-cho, Obihiro, 080-8555, Japan)
Hideyuki Imai (Division of Systems and Information Engineering, Hokkaido University, Sapporo, 060-8628, Japan)

Read and Cressie(1988) introduced a class of the power-divergence statistics $\vect{R}^a$ for the test of independence in $s \times r$ contingency tables. This class includes Pearson's $\chi^2$ statistic (when $a=1$) and the loglikelihood ratio statistic (when $a=0$). All $\vect{R}^a$ have the same chi-squared limiting null distribution. All $\vect{R}^a$ have the same noncentral chi-squared limiting distribution under local alternatives, whence the power of the class is the same for all $a$ asymptotically. Applying the power approximation methods for the multinomial goodness-of-fit test developed by Broffitt and Randles(1977) and Drost et al.(1989), Taneichi and Sekiya(1995) proposed three approximations to the power of $\vect{R^a}$ that vary with the statistic chosen. In this paper we propose a new approximation to the power of $\vect{R}^a$. The new approximation is a normal approximation based on normalizing transformations of the statistics. The proposed approximation and the other approximations are compared numerically. As a result of comparison, we find that the proposed approximation is very effective for $\vect{R}^{-1}$ and $\vect{R}^{-2}$ when all marginal probabilities are equal. We also find that the approximation is effective for the statistics $\vect{R}^0,\vect{R}^{2/3}$, and $\vect{R}^1$.

Keywords: contingency table; normalizing transformation; test of independence

A shrinkage estimator of the bivariate normal mean with interval restrictions
Hea-Jung Kim (Department of Statistics, Dongguk University, Seoul 100-715, Korea)
Koichi Inada (Department of Mathematics and Computer Science, Kagoshima University, Kagoshima 890-0065, Japan)
Hiroshi Yadohisa(Department of Mathematics and Computer Science, Kagoshima University, Kagoshima 890-0065, Japan)

This study is concerned with estimating the bivariate normal mean vector ($\bmu = (\mu_{1}\; \mu_{2})'$) for the case where one has a prior information about the mean vector in the form of preliminary conjectured intervals, $\mu_{i} \in [\lambda_{i} - \delta_{i}, \lambda_{i} + \delta_{i}]$, for $\delta_{i} > 0,\; i= 1, 2$. It is based on the minimum discrimination information(MDI) approach, intended to propose and develop an estimator that has lower risk than a usual estimator (m.l.e.) in or beyond the conjectured intervals. The MDI estimator is obtained for the constrained estimation. This yields a shrinkage type estimator that shrinks towards the preliminary conjectured intervals. Its risk is evaluated and compared with the usual estimator under a quadratic loss function. Favorable properties of the proposed estimator are noted and recommendations for its use are also made.

Keywords: prior interval information; minimum discrimination information approach; shrinkage type estimator

Inferences Based On The Grouped Observations From The Bivariate Power-Normal Distribution
Toshimitsu Hamasaki (Clinical Statistics Group, Biometrics, Central Research Clinical Development Japan, Pfizer Pharmaceuticals Inc., P. O. Box 226, Mitsui Bldg., 2-1-1 Nishi-Shinjuku, Shinjuku-ku, Tokyo, 163-0461, Japan)
Masashi Goto (Department of Informatics and Mathematical Science, Graduate School of Engineering Sciences, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan)

In this paper, we have investigated the relationship between the two variables in which observations of the two variables were grouped into several intervals, when the bivariate power-normal distribution proposed by Goto {\it et al}. (1980) were fitted to these observations. The results of some numerical examples showed that the bivariate power-normal distribution would be helpful to "regularize" the observations even when the strict bivariate normality was not achieved. Also, a bivariate normal problem was considered in the framework developed in fitting the bivariate power-normal distribution to the grouped observations.

Keywords bivariate normal regression; correlation table; power-transformation; simple regression

Validity And Applicability Of Tests For Ordered Alternatives With Binary Response For A Clinical Dose-Response Study
Hiroyuki Uesaka (Medical Statistics, Japan Clinical Research, Lilly Research Laboratories, Eli Lilly Japan K.K. Sannomiya Plaza BLDG. 1-5, Isogamidori 7-chome, Chuo-ku, Kobe 651-0086, Japan)

This paper examines type I error rates and powers of several tests for ordered alternatives of a clinical dose-response study@of@a drug with dichotomous responses. The tests are analogues of tests for ordered alternatives for means of normal populations with known common variances. The tests are derived assuming asymptotic normality of estimates of response rates and are called large sample approximate tests. Type I error rates and powers of these tests are investigated through exact enumeration of the product binomial probabilities. The results show that Bartholomew's chi-bar test and Williams type test can be used in relatively small samples and also give high power for broad ordered alternatives.

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