This paper studies the variance estimators for a class of U-statistics. We obtain asymptotic representations of jackknife, Hinkley's (1978) corrected jackknife, unbiased, Sen's (1960), and new variance estimators. The asymptotic mean square errors of these estimators are theoretically investigated. The Edgeworth expansions of the estimators with remainder term o(n^-1) are also established. We show that the normalized Hinkley's corrected estimator coincides with the normalized unbiased estimator up until the order n^-1/2o_p(n^-1).

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Many numerical examples have demonstrated that the saddlepoint approximation for the cumulative distribution function of a general normalised statistic behaves better in comparison with the third order Edgeworth expansion. This effect is especially pronounced in the tails. Here we are dealing with the inverse problem of quantile evaluation. The inversion of the Lugannani-Rice approximation is compared with the Cornish-Fisher expansion both theoretically and numerically. We show in a very general setting that the expansion of the inversion of the Lugannani-Rice approximation up to third order coincides with the Cornish-Fisher expansion. Based on this, an explanation of the superiority of the former in comparison with the latter in the tails and for small samples is given. An explicit approximation of the inversion of the Lugannani-Rice formula is suggested that utilizes the information in the cumulant generating function and improves upon the Cornish-Fisher formula.

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Generalized mixtures of exponential random variables (Everitt and Hand (1981) and Johnson et al. (1994)) X₁ and X₂ are identified in terms of their relations between the best predictor of X_2:2 given X_1:2 and the functions of the failure rate (or hazard function) of the distribution. Here X_1:2 and X_2:2 denote the corresponding order statistics. In addition, we also mention some related theorems in order to characterize the generalized mixtures of exponential distribution. Moreover, when the sample size is n, the above results are also valid.

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A new higher order approximation formula for a percentage point of the distribution of the sample correlation coefficient is given up to the order O(n^-1), using the Cornish-Fisher expansion for the statistic based on a linear combination of a normal random variable and chi-random variables. The numerical comparison of the formula with others shows that it dominates the others and gives almost precise values in various cases even for the size n=10 of sample.

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The problem of constructing a fixed-size confidence region for a linear function of mean vectors of k multinormal populations is considered when auxiliary information about covariance matrices exists. A two-stage procedure is proposed to derive such a confidence region by incorporating such information. The proposed two-stage procedure is asymptotically efficient and more economical than a previous attempt given by Aoshima, Takada and Srivastava (1997) in terms of the sample size.

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This paper investigates the robustness properties of linear discriminant analysis. We study the problem distorting the assumptions of the linear discriminant rule that the populations are normally distributed and they have equal variance and covariance matrices. Assuming that the populations are lognormally distributed and they have equal or unequal variance and covariance matrices, we investigate the robustness of validity and the robustness of efficiency of the rule. In the case of one dimension, we obtain the asymptotic expansion of the misclassification probability and the risk of classification rule, and we investigate the robustness properties using approximate probabilities. In the case of more than one dimension, we investigate them using Monte Carlo simulation.

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In this paper, we present some partially balanced fractional 2^m₁+m₂ factorial designs of resolution IV derived from partially balanced arrays of full strength such that (A) all the main effects and all the two-factor interactions within the m₁ factors and the m₂ ones each are estimable, (B) all the main effects and all the two-factor interactions within the m₁ ones and between the m₁ and m₂ ones are estimable, (C) all the main effects and all the two-factor interactions within the m₁ ones are estimable, (D) all the main effects and all the two-factor interactions between the m₁ and m₂ ones are estimable, and (E) all the main effects are estimable.

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Changes in asset return variance or volatility over time may be modeled using the GARCH class models or stochastic volatility (SV) models. The log-GARCH models are the logarithmic extension of the GARCH models. The GARCH models are popular and easily estimated. Compared to the GARCH models, the SV models are more general in several respects, but it is well recognized that they are not easy to estimate. In this paper, we derive a log-GARCH representation of a class of SV models, including the ARMA-SV model, and analyze the finite sample properties of a Quasi-Maximum Likelihood (QML) estimator based on the log-GARCH representation. Our Monte Carlo results indicate that their finite sample properties are superior to those of the Generalized Method of Moments estimator and those of the QML estimator based on the Kalman filter; and close to those of the nonlinear filtering maximum likelihood estimator which is a computationally intensive method. We present an empirical example of daily observations on the yen/dollar exchange rate.

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This paper deals with a regression model with covariates subject to misclassification. Simply ignoring the misclassification generally yields bias in parameter estimation. A corrected score function (Nakamura, 1990) is developed to obtain asymptotically unbiased estimates adjusting for the misclassification. It is proved that a corrected score function always exists for a misclassification model whose likelihood consists of independent contributions. This existence theorem for misclassification model is conspicuous since corrected score functions do not always exist for measurement error models with continuous covariates. For instance, a corrected score function does not exist for a logistic model with continuous covariates subject to normal random error (Stefanski, 1989). For illustration, the correction method is applied to generalized linear models and a simulation study based on the logistic regression model is performed. The misclassification are assumed to be nondifferential, which is normally assumed with prospective studies.

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