Statistical similarities among the latest long expansion in the U.S.and some other past expansions, in particular that of the 1960s, are examined. Corresponding to the definition of statistical similarity, a test based on the covariance matrices of business cycle component variables for the different expansions is proposed. Among available tests, the test based on partial common principal component analysis is argued to be most appropriate. The test is applied to the components of both GDP and the coincident composite index. As a result, the 1990s expansion is concluded to be statistically similar to that of the 1960s. Using the same method we also examine the statistical similarities of whole cycles(defined on a peak-to-peak basis).

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We consider the unbalanced one-way layout for comparing 3 treatment effects which can be assumed to satisfy the simple ordering μ₁ &le; μ₂ &le; μ₃. This problem arises often in the comparison of three dose levels in medical trials, for example. The one-sided studentised range test provides a set of simultaneous confidence intervals for the ordered pairwise differences μ_j - μ_i, i < j which are one-sided and have infinite upper bounds. The new confidence interval procedure proposed in this article maintains the sensitivity of the one-sided studentised range test in detecting each of the ordered differences μ_j - μ_i > 0 while at the same time providing two-sided finite confidence intervals. Consequently, this new procedure has the advantage of allowing the same directional inferences as the one-sided studentised range test while additionally providing upper bounds on the differences among the treatment effects.

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This paper deals with problems of recovering a causal structure by using not only conditional independence relationships but also prior knowledge when data are generated according to the causal structure among variables. Although some algorithms for recovering a causal structure based on independencies have been developed, the influence of prior knowledge on the recovery algorithms has not been clarified. In this paper, a necessary and sufficient condition for the existence on unidentified arrows in a recovered diagram is given in terms of graph structure. Also, it is shown that a causal structure such that a recovered diagram is a forest can be recovered by recognizing exogenous variables in a causal diagram completely. The result enables us to elucidate enough prior information to determine a causal diagram uniquely.

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We give a transformation such that, for a k-dimensional finite normal mixture, the number of components and the mixing ratios are preserved on each marginal density through the transformation. Furthermore, under the only assumption of a k-dimensional finite normal mixture, we construct a one-dimensional random variable with a finite normal mixture of the true number of components and the true mixing ratios.

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In this paper, we consider a Polya urn model containing balls of m different labels under a general replacement scheme, which is characterized by an m × m addition matrix of integers without constraints on the values of these m² integers other than non-negativity. This urn model includes some important urn models treated before. By a method based on the probability generating functions, we consider the exact joint distribution of the numbers of balls with particular labels which are drawn within n draws. As a special case, for m = 2, the univariate distribution, the probability generating function and the expected value are derived exactly. We present methods for obtaining the probability generating functions and the expected values for all n exactly, which are very simple and suitable for computation by computer algebra systems. The results presented here develop a general workable framework for the study of Polya urn models and attract our attention to the importance of the exact analysis. Our attempts are very useful for understanding non-classical urn models. Finally, numerical examples are also given in order to illustrate the feasibility of our results.

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On the goodness-of-fit test for multinomial distribution, Zografos et al.(1990)proposed the φ-divergence family of statistics, which includes the power divergence family of statistics as a special case. They showed that under null hypothesis, the members of the φ-divergence family of statistics all have an asymptotically equivalent chi-square distribution. Furthermore, Menendez et al.(1997)derived an asymptotic expansion for the null distribution of the φ-divergence statistic. In this paper, we derive an approximation for the distribution of the φ-divergence statistic under local alternatives. The approximation is based on the continuous term of the asymptotic expansion for the distribution of the φ-divergence statistic. By using the approximation, we propose a new approximation for the power of the statistic. The results are generalizations of those derived by Taneichi et al.which discussed the power divergence statistic. We numerically investigate the accuracy of the approximation when two types of concrete φ-divergence statistics are applied. By the numerical investigation, we find that the present approximation performs better than the other approximations.

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Associated with an estimable parameter, we consider a linear combination of U-statistics. V-statistic and LB-statistic are its special cases. In this paper the kernel of the estimable parameter is assumed to be non-degenerate. For the difference between this linear combination and the well-known U-statistic, we derive upper bounds of its absolute moments about the origin and the mean. Using these upper bounds we show Berry-Esseen bound for the linear combination on U-statistics with exact expressions.

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A unified treatment, of the estimation of a mean vector in the normal and the inverse Gaussian distributions is discussed. A mean vector in the exponential dispersion model is reparametrized into two orthogonal components; the norm component and the direction. We point out first that the optimum(shrinkage)factor is obtained in an explicit form, when the norm component is known. Then several candidate estimators of a mean vector are discussed in relation with this optimum factor, when the norm component is unknown. The results in the case of the normal distribution provide us with a novel view of the James-Stein estimator and the positive-part Stein estimator. Parallel treatments are possible in estimating a mean vector in the inverse Gaussian case. Extensions to the gamma case are discussed to some extent.

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For a sum of independent discrete random variables, its higher order large-deviation approximation is discussed. An approximation to the tail probability of the distribution of the sum is provided, and its numerical comparison with other approximations is done in the binomial case. Consequently, the approximation formula is seen to be more accurate.

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