The extended Nelson-Plosser data on the historical US macro-economic time series is re-analyzed from the view of the Dickey-Fuller type auto-regressive (AR) unit root test as well as from the view of the recently developed moving average (MA) unit root test (Hatanaka and Koto, 1995). The analysis included all MA errors but not breaks in the deterministic trend. An example of the test proposed by Hatanaka and Koto is illustrated when a time series is doubted as to whether it is a stationary ARMA about trend or a non-stationary ARIMA about drift. Search for the best ARMA regressions always includes an MA error term since long lags in the AR process often cause over-differencing phenomena. The adjusted likelihood ratio test is used for arriving at the best ARMA regression. Search for the best ARIMA regressions is carried out in a similar way as that applied for selecting the best ARMA regression. The MA unit root test is calculated for the best ARIMA regression. The AR and MA unit root tests associated with the best ARMA and the best ARIMA regressions, respectively, are compared with each other in order to classify the series into ARMA or ARIMA process as done previously by Hatanaka and Koto. Various lag lengths are examined and overall judgments are drawn.

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This paper discusses the problem of classifying an observed stretch X=(x₁, …, x_r) into Π₁ or Π₂, where Π_i is a Gaussian stationary process with zero mean and spectral density f_θi(λ). We propose a new discriminant statistic based on some estimator θ=θ(X) of a spectral parameter. The statistic D[θ, W] is motivated by a spectral measure with divergence function W. Most of the work presented is devoted to higher order asymptotic theory when θ₂ is contiguous to θ₁, in order to study the asymptotic difference between different D[θ, W]. In particular, it is shown that for any choice of W, D[θ, W] has the same second order averaged risk as the optimal likelihood ratio (LR) if θ belongs to an appropriate class of asymptotically efficient estimators, and the third order term of the averaged risk is minimized by the (bias-adjusted) maximum likelihood estimator (MLE). We also examine the case of the rule based on the MLE without bias adjustment.

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The problem of constructing fixed-size confidence regions for estimating the multinormal mean is considered when auxiliary information about the covariance matrix exists. Incorporating such information, we propose a two-stage procedure which meets the requirement and is asymptotically efficient. Furthermore, it turns out that the procedure requires less observations than the usual one.

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This paper develops an asymptotic theory for CUSUM tests for structural stability with nonparametric regression residuals. The results here allow for dynamic models and dependent errors. Although the test statistics in this paper are similar to those of Hidalgo (1995), they are more widely applicable and the proofs for the asymptotics are mathematically correct.

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The Bhattacharyya type bound for the variance of sequential estimation procedures is given. Some examples on the attainment of the bound are also given, and in one of them it is shown that although there exists no complete sufficient statistic, there is a uniformly minimum variance unbiased sequential estimation procedure of estimable parametric function.

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Stability of the Multidimensional Scaling (MDS) procedure is investigated under conditions where the dissimilarity data matrix involves small errors, in particular, in a somewhat ideal stochastic error model. The asymptotic performances of the solution of MDS and some related statistics are studied by using their asymptotic expansions under the model. Numerical examples which illustrate our theory are also given.

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This paper discusses the statistical inference problems seen in a canonical correlation analysis with linear constraints (CCALC), previously formulated by Yanai and Takane [15]. Linear constraints are imposed on the coefficient vectors of canonical variates, and canonical correlations with the constraints are not greater than the usual canonical correlations without the constraints. It is thus important to evaluate the effect of the linear constraints on the canonical correlations. We consider here in the test of the hypothesis that the canonical correlations are invariant by imposing the linear constraints. This testing problem can be treated as an additional information test in the usual (without any constraints) canonical correlation analysis (CCA) discussed by McKay [9] and Fujikoshi [4], and the likelihood ratio criterion and an asymptotic expansion formula of its null distribution are derived. An asymptotic distribution of the sample canonical correlation decrease by imposing the linear constraints is also obtained. Using this, an approximate confidence interval for the decrease can be constructed. Another problem discussed in this paper is the test of a hypothesis concerning a number of useful pairs canonical variates in CCALC, an important test in obtaining the number of useful canonical variates. The likelihood ratio criterion for this testing problem and an asymptotic expansion formula of its null distribution are obtained.

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This paper considers the utility of the RC (M) association model for analysing association between two sets of ordinal variables. It is shown that the RC (M) association model approximates the discretized multivariate normal distribution and that the association model relates to canonical correlation analysis. Numerical examples demonstrate the present discussion.

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