A k-sample modified Baumgartner statistic is proposed. For k=3, the limiting distribution of a k-sample Baumgartner statistic is derived with a procedure similar to Anderson-Darling (1952). For the case of k ≥ 4, a saddlepoint approximation is used to approximate the limiting distribution of the k-sample Baumgartner statistic. The critical values are given for k=3 to 10, 25, 50 and 100.

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The approximation for the distribution function of a test statistic is extremely important in statistics. On testing the hypothesis in a multisample problem, the Jonckheere-Terpstra test is often used for testing the ordered location parameters. Herein, we performed a saddlepoint approximation with continuity correction in the upper tails for the Jonckheere-Terpstra statistic under finite sample sizes. We then compared the saddlepoint approximation with Odeh's approximation to obtain the exact critical value. The table of critical values was extended by using the saddlepoint approximation. Additionally, the orders of errors of a saddlepoint approximation were derived.

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In the estimation of a multivariate normal mean, it is shown that the problem of deriving shrinkage estimators improving on the maximum likelihood estimator can be reduced to that of solving an integral inequality. The integral inequality not only provides a more general condition than a conventional differential inequality studied in the literature, but also handles non-differentiable or discontinuous estimators. The paper also gives general conditions on prior distributions such that the resulting generalized Bayes estimators are minimax. Finally, a simple proof for constructing a class of estimators improving on the James-Stein estimator is given based on the integral expression of the risk.

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Box and Pierce (1970) proposed a test statistic T_BP which is the squared sum of m sample autocorrelations of the estimated residual process of an autoregressive-moving average model of order (p,q). T_BP is called the classical portmanteau test. Under the null hypothesis that the autoregressive-moving average model of order (p,q) is adequate, they suggested that the distribution of T_BP is approximated by a chi-square distribution with (m-p-q) degrees of freedom, ``if m is moderately large". This paper shows that T_BP is understood to be a special form of the Whittle likelihood ratio test T_PW for autoregressive-moving average spectral density with m-dependent residual processes. Then, it is shown that, for any finite m, T_PW does not converge to a chi-square distribution with (m-p-q) degrees of freedom in distribution, and that if we assume Bloomfield's exponential spectral density, T_PW is asymptotically chi-square distributed for any finite m. From this observation we propose a modified T^†_PW which is asymptotically chi-square distributed. In view of the likelihood ratio, we also mention the asymptotics of a natural Whittle likelihood ratio test T_WLR which is always asymptotically chi-square distributed. Its local power is also evaluated. Numerical studies illuminate interesting features of T_PW, T^†_PW, and T_WLR. Because many versions of the portmanteau test have been proposed and been used in a variety of fields, our systematic approach for portmanteau tests and proposal of tests will give another view and useful applications.

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In this paper, we construct a confidence region for the efficient frontier assuming the asset returns to be matrix elliptically contoured distributed. Our results extend the findings of Bodnar and Schmid (2009) to the non-normal distributed asset returns. In order to correct the overoptimism of the sample efficient frontier documented in Siegel and Woodgate (2007), the unbiased estimator of the efficient frontier is suggested. Moreover, we derive an exact overall F-test for the efficient frontier in elliptical models.

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The conditional least squares (CLS) estimator proposed by Tj\o stheim (1986) is convenient and important for nonlinear time series models. However this convenient estimator is not generally asymptotically efficient. Hence Chandra and Taniguchi (2001) proposed a G estimator based on Godambe's asymptotically optimal estimating function. For important nonlinear time series models, e.g., RCA, GARCH, nonlinear AR models, we show the asymptotic variance of the G estimator is smaller than that of the CLS estimator, and the G estimator is asymptotically efficient if the innovation is Gaussian. Numerical studies for the comparison of the asymptotic variance of the G estimator, that of the CLS estimator and the Fisher information are also given. They elucidate some interesting features of the G estimator.

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This paper deals with the two sample problem for rounded data in the i.i.d. model. It is well known that under the null hypothesis the two sample Kolmogorov-Smirnov statistic without rounding converges in distribution to the supremum of a standard Brownian bridge. We establish that a natural statistic of the Kolmogorov-Smirnov type based on the rounded data converges in distribution to the same limit as the full observation case. Our result is based on ``Donsker's theorem for discretized data'' given by Nishiyama (2008, J. Japan Statist. Soc.).

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This paper proposes the shrinkage generalized method of moments estimator to address the ``many moment conditions'' problem in the estimation of conditional moment restriction models. This estimator is obtained as the minimizer of the function constructed by modifying the GMM objective function, such that we shrink the effect of a subset of moment conditions that are less important and used only for efficiency. We provide the closed form of the shrinkage parameter that minimizes the asymptotic mean squared error. A simulation study shows encouraging results.

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