Generalized principal components analysis (GPCA) has been proposed by Gnanadesikan and Wilk in order to analyze data structure by obtaining fitting functions to data points. The present paper treats the invariability of the fitting functions under orthogonal coordinate transformations in GPCA for two variables and points out that some methods of GPCA do not give the same results under such transformations. Numerical examples are also shown to compare invariant methods with others.

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A poker hand consists of five cards which are drawn from a deck of 52 cards. There are 2, 598, 960 (=_??_) different poker hands. The total number of the hands which (i) contain two or more cards of the same rank, or (ii) consist of the same suit, or (iii) consist of circularly consecutive ranks ((ii) and (iii) are not necessarily exclusive) is 1, 299, 480=_??_/2. In this paper we consider a problem arising from this fact. Let us suppose that there is a deck of nk cards of n ranks and k suits, and a hand consists of r cards. Let S(n, k, r) be the total number of the hands which satisfy at least one of (i), (ii) and (iii). Is there any triplet (n, k, r) for r_??_3 other than (13, 4, 5) for which S(n, k, r)=_??_/2 holds. One partial answer is “No” if n_??_10, 000.

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In this paper, we consider rank statistics to measure the degree of association of ranked data. The association we want to consider is not only the monotone association but the departure from the uniformity of the distribution of ranks. Hence, the ordinary linear rank statistics are not necessarily adequate. Two classes of statistics are to be considered. The statistics which belong to one class are based on the rank regression coefficients. The statistics in the other class are extended versions of the rank serial correlation coefficient. The asymptotic as well as exact properties of our statistics are developed under the hypothesis of no association and numerical examples are given.

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This paper investigates the improper solutions in latent class analysis and suggests a new procedure. Using this procedure, it is shown by a Monte Carlo study that the solutions are more stable than the usual procedure.

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The purpose of this paper is to derive three kinds of decompositions of McCullagh's [4] conditional symmetry model and new decompositions for the symmetry model in a square contingency table. Moreover, a test theory for the decomposed models is given.

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In a bivariate linear structural relationship, we obtain the exact density of the maximum likelihood estimator of the slope coefficient. The asymptotic expansion of the distribution of the estimator is also derived when the sample size goes to infinity, and the accuracy is discussed.

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A class of unbiased estimators of the correlation coefficient of a bivariate normal distribution with known means and variances is constructed in terms of the incomplete sufficient statistic, and the variances of these estimators are obtained analytically for any sample size. Moreover, some properties of the estimators in the case that the sample size is equal to unity are investigated both analytically and numerically, and then an unbiased estimator of the correlogram of a stationary Gaussian process with known mean and variance is proposed.

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Uniformly minimum variance unbiased (UMVU) estimators of parameters of a semi-lognormal distribution and their variances are obtained. The estimators and the variances are expressed by using generalized hypergeometric functions.

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In this paper, we study the following two cases for partially balanced fractional 2m1+m2 factorial (2m1+m2-PBFF) designs of resolution IV derived from simple partially balanced arrays (SPB-arrays): One is that the general mean, all m1+m2 main effects, all (m12) two-factor interactions and some linear combinations of the (m22) two-factor interactions and of the m1m2 ones are estimable for m1≥2 and m2≥4, and the other is that the general mean, all main effects and some linear combinations of the (mk2) two-factor interactions (k=1, 2) and of the m1m2 ones are estimable for m1, m2≥4. In addition, generalized trace optimal (GT-optimal) designs as defined by Shirakura [10] are presented.

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This gives new insights concerning conditions for the validity of Fisher's ineauality. The theorem derived here includes some useful and combinatorially interesting results.

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In this article, we improve and extend the Padgett and Wei's paper [2]. That is, we improve their Lemma 1 and Theorem 1, and derive the average sampling time.

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同時方程式モデルの構造形パラメータに関する非報知事前分布として,いくつかのタイプが提案されている.ベイズ流の全システム法である完全情報法,そしてベイズ流の単一方程式法である制限情報法の両者に関して,それら事前分布に対応する事後分布を導出し,事後分布のモーメント存在条件を検討し,さらに数値例によって事前分布が健全であるか否かを示している.その結果,非報知事前分布がそこから選択されるのが適当と考えられる事前分布のクラスを明らかにしている.

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