ABSTRACTS

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Generating Artificial Data with Preassigned Degree of Multicollinearity by Using Singular Value Decomposition

H.B.Kim (Graduate School of Natural Science and Technology, Okayama University, Tsushima, Okayama 700, Japan)
Y.Tanaka (Department of Environmental and Mathematical Sciences. Okayama University, Tsushima, Okayama 700, Japan)

A method is proposed for generating artificial data with preassigned degree of multicollinearity. The method is based on the singular value decomposition and the degree of multicollinearity is assigned with a set of singular values. Numerical examples are given to show the performance of the proposed method.
Keywords: singular value decomposition, multicollinearity, condition number

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A Diagnostic Method of Detecting Outliers in Testing the Equality of Two Covariance Matrices

Y.H.Park (Department of Mathematics. KAIST, Taejon, Korea.)
M.G.Kim (Department of Applied Statistics, Seowon University, Chongju, Chung-Buk, Korea.)
B.C.Kim (Department of Industrial Management, KAIST, Taejon, Korea.)

The local influence method introduced by Cook (1986) is adapted to the detection of influential observations in testing the equality of two covariance matrices. The method is very informative in that it clearly identifies individually and jointly influential observations. An illustrative example is given to show the effectiveness of the method.
Keywords: Local influence; Covariance; Influential observations; Outliers; Curvature; Perturbation; Case-deletion.

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Higher Order Accurate Confidence Intervals for Smooth Functions of Means with Application to the Correlation Coefficient

J.F.Wang (The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan)
S.Ohuchi (Kisarazu National College of Technology, 2-11-1, Kiyomidai-Higashi, Kisarazu-shi 292, Japan)
M.Taguri (Department of Mathematics and Informatics, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba-shi 263, Japan)

Higher order asymptotic expansions for studentized statistics under smooth function model are developed. The emphasis is on constructing higher order accurate confidence intervals, for parameters being smooth functions of means, based on the third order inverse Edgeworth expansions. The asymptotic results developed here are compared with similar results previously derived for the delete-one jackknife-$t$ statistics. Extensive Monte Carlo studies are carried out in the case of estimating the correlation coefficient from bivariate normal populations. Comparisons are also made with the standard confidence intervals based on Fisher's normal approximation, which serves as a good bench mark in estimating the correlation coefficient.
Keywords: Cornish-Fisher expansion, Coverage error, Edgeworth expansion, Inverse Edgeworth expansion, Jackknife, Smooth function model

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Sojourn Time Test of $m$-sequences with Characteristic Pentanomials

K.Takashima (Division of Mathematical Sciences, Osaka Kyoiku University, Kashiwara, Osaka, 582)

Sojourn time tests are applied to $m$-sequences with characteristic primitive pentanomials. Some of the tested $m$-sequences with characteristic pentanomials are evidently rejected by the sojourn time tests, but their graphs of simulation do not show explicit gaps. This makes clear contrast with the case of $m$-sequences with characteristic trinomials, which showed explicit graphical gaps. We further propose a combination of chi-square test and Kolmogorov-Smirnov test.
Keywords: m-sequence: sojourn time; random walk; Kolmogorov-Smirnov test.

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On the Statistical Properties of Random Digits Generated by the Physical Device

S. Kishimoto (Tsuyama National College of Technology, Tsuyama, Okayama, 708, Japan.)

About $1 \times 10^{8}$ random digits were generated by a new physical means. Statistical tests such as frequency test, gap test, run(sign) test and poker test were carried out to check the randomness of the random digits. Frequency test for the parts of the random digits was also carried out to examine the local fluctuation of uniformity. The results of the tests proved that the random digits have good statistical quality.
Keywords: Random number; Physical random number; Physical random digit.

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Biased Cross-Validation in a Kernel Regression Estimation

J. C .Oh (Department of Computer Science and Statistics, College of Natural Sciences, Kunsan National University, Kunsan. Korea)
B. C. Kim (Department of Industrial Management, KAIST, Taejon, Korea)
J. S. Lee (Samsung Data System, Seoul, Korea)
B. U. Park (Department of Computer Science and Statistics, College of Natural Sciences, Seoul National University, Seoul, Korea)

This article is concerned with the problem of choosing a bandwidth for nonparametric regression. We consider a method based on an biased estimate of mean average squared error. It is seen that the bandwidth chosen by biased cross-validation method, is asymptotically optimal and has small sample variability. In a simulation study, we show that this bandwidth is closer to optimum bandwidth than other bandwidths when the underlying regression function is sufficiently smooth.
Keywords: Nonparametric regression; Kernel estimation; Optimal bandwidth estimator; Biased cross-validation method.