Fan et al. (1997) considered two kinds of nonparametric estimators of the effects of the covariates in proportional hazards models. One of them has no parametric assumption on the baseline hazard function and is based on the integration of the estimated first order derivative of the regression function. We study the asymptotic properties of the estimator and consider another nonparametric estimator of the effects of the covariates in proportional hazards models. We show both of the estimators have very similar asymptotic properties. The latter is closely related to estimation in two-sample problems and is much easier to calculate.

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This paper is concerned with Dempster trace criterion for multivariate linear hypothesis which was proposed for high dimensional situation. First we derive asymptotic null and nonnull distributions of Dempster trace criterion when both the sample size and the dimension tend to infinity. Our approximations are examined through some numerical experiments. Next we compare the power of Dempster trace criterion with the ones of three classical criteria; likelihood ratio criterion, Lawley-Hotelling trace criterion, and Bartlett-Nanda-Pillai trace criterion when the dimension is large compared to the sample size.

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This paper considers the problem of estimating the optimal portfolio weight to the mean-variance model in finance when parameters are unknown. For this purpose, we consider the following two classes of estimators. One is the class of proportional type estimators and the other is the class of Stein type estimators. First, we derive an unbiased estimator of the optimal portfolio weight, which belongs to the class of proportional type estimators. Second, we obtain dominance results within each class. From this, we showed that the unbiased proportional estimator and the maximum likelihood estimator are inadmissible.

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In this paper we have proposed a class of estimators for the variance of the ratio estimator which includes two standard estimators V₀, V₂ and the estimators V_H and V₃ suggested by Royall and Eberhardt (1975) and Wu (1982) respectively. Under large sample approximation, the bias and mean-squared error of the proposed class of estimators are obtained. Based on this we obtain optimal variance estimator in the class and compare the relative merits with a number of estimators. For illustration an empirical study is provided. Here we confine ourselves to sampling scheme SRSWOR ignoring finite population correction.

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In this paper, the Vincze inequality for the Bayes risk of an estimator with the unbiasedness at any two specific values of the parameter is derived using the Lagrange method. The lower bound for the Bayes risk is also shown to be attained. The Cramér-Rao inequality is derived from the information inequality. Some examples on non-regular distributions are also given.

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This paper is concerned with parameter estimation in the presence of nuisance parameters. Usually, an estimator with known nuisance parameters is better than that with unknown nuisance parameters in reference to the asymptotic variance. However, it has been noted that the opposite can occur in some situations. In this paper we elucidate when and how this phenomenon occurs using the orthogonal decomposition of estimating functions. Most of the examples of this phenomenon are found in the case of semiparametric models, but this phenomenon can also occur in parametric models. As an example, we consider the estimation of the dispersion parameter in a generalized linear model.

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Asymptotic confidence intervals of location parameters are proposed in one- and two-sample models. These are robust procedures based on scale-invariant M-statistics. The one-sample procedures have the same robustness as Huber's M-estimators. Furthermore although the symmetry of the underlying distribution is needed in the asymptotic theory of Huber's M-estimators, the proposed procedures do not demand the symmetry in the two-sample model. The asymptotic efficiency of the proposed confidence intervals is given by a numerical integration.

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In this paper we consider a linear regression model when error terms obey a multivariate t distribution, and examine the effects of departure from normality of error terms on the exact distributions of the coefficient of determination (say, R²) and adjusted R² (say, R²). We derive the exact formulas for the density function, distribution function and m-th moment, and perform numerical analysis based on the exact formulas. It is shown that the upward bias of R² gets serious and the standard error of R² gets large as the degrees of freedom of the multivariate t error distribution (say, ν₀) get small. The confidence intervals of R² and R² are examined, and it is shown that when the values of ν₀ and the parent coefficient of determination (say, Φ) are small, the upper confidence limits are very large, relative to the value of Φ.

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