Joint and the marginal densities of the latent roots of the bivariate Wishart matrix are shown for small degrees of freedom with different covariance matrices. Corresponding contours of the joint densities arc also given. Skewness, kurtosis and correlation are computed and compared with the asymptotic values. It is shown that the distribution of the minimum root is negatively skewed when the number of degrees of freedom is larger than 557 and the scale matrix is proportional to the unit matrix.

View full abstract

Let X_t be a stochastic process satisfying some conditions. We assume that EX_t=m_t is a periodic function, i.e., m_t=m_t-s for a positive integer s. There are many popular predictors for X_h (0<h_??_s) under the observation {X_t;-(T-1)_??_t_??_0}. In this paper, we show some statistical properties of one of these predictors, which we denote _??__h. The predictor _??__h is composed of the sum of the least squares estimator _??__t of m_t, and a linear predictor _??__h of Z_h=X_h-m_h, which is obtained by the least squares method. When Z_t is a stationary process, _??__h is a reasonable predictor. The main point of this paper is to discuss the robustness of _??__h when Z_t deviates from a stationary process. As a nonstationary process Z_t we consider the case when ∇_??_Z_t=Z_t-Z_t-s=Y_t is stationary. We show comparisons between the asymptotic prediction error of _??__h and that of another predictor when the sample size tends to infinity.

View full abstract

In some regression model, the minimum (asymptotic) variance estimator of a ratio is discussed for some class of linear combinations of ratio estimators, and the jackknife procedure is considered. It is seen that the grouped jackknife estimator is optimal in the sense that it has asymptotically the minimum variance in the class. Higher order bias reduction of the estimators is discussed, and some examples are given.

View full abstract

Several decision problems such as bandit problems can be considered as special sequential two-action Markov decision models as described in [2]. In this paper a uniform two-armed bandit problem with one arm known is studied by embedding it in the general framework developed in [2]. Two cases of this problem are examined. The first case assumes that one end point of the uniformity interval of the unknown arm is Pareto distributed. In the second case the joint distribution of the two end points of the uniformity interval of the unknown arm is bilateral Pareto. The results obtained extend and complete those obtained in [6, 7].

View full abstract

A graphical representation of ranked data and test statistics based on the representation for the nonparametric k-sample problem is considered. Asymptotic distributions of the rank statistics are derived; asymptotic comparisons of these relative to some standard nonparametric tests are performed; and illustrative examples are given.

View full abstract

B-splines with equally spaced knots (cardinal B-splines) provide a simple, locally flexible and computationally efficient basis for approximation problems in regression and density estimation. The fidelity to the data (small bias, large variance) and smoothness (large bias, small variance) of a B-spline fit depends on the number of knots employed.
This paper studies the problem of selecting the number of knots in cardinal B-splines in density estimation. Akaike's Information Criterion (AIC)is proposed and discussed as a data-based criterion for selecting the number of knots in density estimation. The EM algorithm is developed for a “mixture” of B-splines to obtain the maximum likelihood estimators of the parameters and to score AIC. Numerical examples are provided to illustrate the versatility and efficiency of the proposed approach by automating the curve fitting process.

View full abstract

For estimating matrix of means, we consider two classes of estimators and obtain conditions under which these estimators are minimax with respect to the quadratic loss function where the covariance matrix is unknown.

View full abstract

This paper derives the asymptotic expansions of the distribution function of the maximum likelihood estimator (MLE) and the log likelihood ratio (LR) test in a nonlinear regression model. It reports on an investigation of the effects of nonlinearity of a model on the asymptotic expansions by making use of two kinds of curvature measures: intrinsic curvature and parameter effect curvature defined by Bates and Watts (1980). It shows, after suitable transformation, that the distribution function of the MLE up to O(T^-1/2) is related to only the parameter effect curvature. The intrinsic curvature appears only in a term of O(T^-1) in the distribution of LR. Furthermore, this paper illustrates that the intrinsic curvature is essentially equivalent to Efron's statistical curvature.

View full abstract

The present paper gives a brief explanation of the Autoregressive Conditional Heteroscedasticity (ARCH) model and its extensions. It also discusses the estimation and hypothesis testing procedures of these models. The ARCH model is applied to estimate time-varying risk premium of a long term bond in the U. S.

View full abstract

We estimate the two-stage CES production function with capital and labor inputs in Japanese manufacturing sectors to measure the elasticities of substitution. The labor inputs are differentiated by educational attainment with Fallon-Layard's procedure. We statistically test the following three hypotheses: (1) capital and educated labor are complements, (2) capital is more complementary to educated labor than to less-educated labor, (3) unions are likely to be relatively weak in the sectors where the elasticity of substitution between capital and labor is large.

View full abstract

In 1986 the Japanese Ministry of Health and Welfare carried out the Comprehensive Survey of Living Condition of the People on Health and Welfare which included the Survey on Household Questionnaire and the Survey on Income Questionnaire. The former was done for 4966 survey units (about 240 thousands households) drawn from 750 thousands survey units in the National Census. Among these, 940 survey units (about 40 thousands households) were drawn and the latter survey was carried out. In the statistics the mean of household incomes was estimated only on the basis of the information obtained from the Survey on Incomes Questionnaire. However, some items in the questionnaire of the Survey on Households Questionnaire must have correlation with the mean of household incomes. Hence the precision of the estimation may be improved by using the information obtained from the Survey on Households Questionnaire, whose scale was five times larger than that of the Survey on Incomes Questionnaire.
In this study we propose four schemes which improve the precision of estimating the mean of household incomes classified by the areas and age-classes of householders. We then make some investigation and comparison of these schemes together with the traditionally used scheme. In the analysis we assume a regression model where the objective variable y is a household income and the explanatory variables' vector x is corresponding to the questionnaire of the Survey on Households.

View full abstract

[in Japanese]

View full abstract

Binary choice models, in which the dependent variable takes on the values of 0 or 1, are widely used in the various fields of social and natural sciences. These models are usually estimated by the probit or logit maximum likelihood method. However, unlike standard regression models, these estimators are inconsistent if the distribution of the error term is not correctly specified.
Recently, a great deal of effort has been spent by econometricians and statisticians to find out distribution-free or semiparametric estimation methods of binary choice models. This paper summarizes important developments of these methods.

View full abstract