The aim of this paper is to give an extended front door criterion and a mixed door criterion as new identifiability criteria for causal effects of a joint intervention when causal diagrams for empirical research have some unobserved variables. The criteria enable us to recognize from the structure of the graph whether the causal effect can be expressed by a subset of observed variables. If the answer is affirmative, the criteria provide a closed-form expression for the effect of the external joint intervention.

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Threshold methods for estimating area rain-rate first product moment and covariance are proposed. Empirically, we observed a strong correlation between the first product moment and the probability of exceeding two threshold values for two geographically different locations, land and island, for Tokyo Metropolitan rainfall data. Theoretical optimal thresholds under the bivariate lognormal distibution as a model are chosen by minimizing the asymptotic normalized variance. A good agreement between the emprical and theoretical results is observed and this observation can be reproduced for other locations too.

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An algorithm for the construction and classification of two-symbol orthogonal arrays which is based on the extension of ideas given in yamamoto et al. (1996) is given. Computational results on the enumeration and classification of all possible two-symbol orthogonal arrays of strength t, m=t+4 constraints and index λ≤5 are given.

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This paper is concerned with the null distribution of the one-way ANOVA test statistic T for testing the equality of means of q nonnormal populations with common variance. It is known that the null distribution of T converges to X²_q-1 when all the sample sizes of the q populations are large. We extend this result by obtaining the Bartlett correction and an asymptotic expansion under general conditions. Numerical accuracies are studied for some approximations of the percentage points and actual test sizes of T based on the limiting distribution, Bartlett correction and an asymptotic expansion.

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Robinson (1987) proposed to use a nearest neighbor approach to estimate the regression coefficient in a heteroskedastic linear model. While this estimator is asymptotically efficient, it has been said to be inefficient in small samples compared with other semiparametric estimators such as those using a kernel. Like other semiparametric methods, his estimators of variances, which are used to get the weighted least squares estimator of the regression coefficient, are constructed as a weighted sum of the squared ols residuals. As Robinson (1987) indicated himself, however, there exists a sample splitting problem in his estimator and this may cause the small-sample inefficiency. Therefore a slight modification improves the small sample property of the k-NN estimator. In this paper we report Monte Carlo experiments and show that the modified Nearest Neighbor estimator has a sufficient level of efficiency in small samples.

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Many studies on economic time series focus much attention on unit root processes and there is some concern about time series data with missing observations. However most previous research on unit root processes was done only for complete observed data. This paper therefore considers unit root processes with missing observations, proposes two estimators for them and investigates their asymptotic properties (limiting distributions and order of convergence). The properties depend on the pattern of occurrence of missing observations. As an application of the main results, we also consider unit root tests for data with missing observations.

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Recently, for a one-parameter exponential family of discrete distributions, the higher order approximation of randomized confidence intervals was derived from the optimum test by Akahira, Takahashi and Takeuchi[1]. In this paper, a numerical comparison of approximations of randomized confidence intervals is given and the higher order approximation is shown to be useful to construct confidence intervals for the negative binomial and logarithmic series distributions.

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A test of homogeneity of independent populations is introduced in terms of the φ-divergence. The asymptotic power of the test is determined for local and nonlocal alternative hypotheses. A power approximation is also proposed, based on a normalizing transformation of the test statistics for testing homogeneity. The behaviour of the proposed approximations of the power function is numerically compared. A new statistic is obtained and the respective test is more powerful, in some cases, than the existing tests for testing homogeneity of multinomial populations.

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This paper points out that Gott III's delta t argument is a practical example that a uniformly most powerful test exists for his argument but it creates a confidence interval with a trivial infimum. In addition, investigation into this problem from several standpoints is given.

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