This paper discusses path analysis of categorical variables with logistic regression models. The total, direct and indirect effects in fully recursive causal systems are considered by using model parameters. These effects can be explained in terms of log odds ratios, uncertainty differences, and an inner product of explanatory variables and a response variable. A study on food choice of alligators as a numerical exampleis reanalysed to illustrate the present approach.
The RC(M) association model is designed for analyzing the association in two-way contingency tables. First, the RC(M) association model is discussed through entropy, and it is shown that the entropy in the model is decreased in the direction of the intrinsic association parameter vector, and that the Pearson product-moment correlation coefficients of row and column scores increase in the corresponding intrinsic association parameters. Second, a summary measure of association between two categorical variables in the RC(W) association model is proposed, and the relation-ship between the association measure and the intrinsic association parameter vector is investigated. Lastly, the present paper applies the discussion to the multivariate normal distribution.
This paper considers the chaotic time series with additive dynamic noise. Defining the embedding dimension and delay time, we discuss their mathematical properties. A method of estimating the embedding dimension and delay time is proposed, and the consistency of the estimators is proved.
In a nonlinear dynamic model, the consistency and asymptotic normality of the Nonlinear Least-Absolute Deviations (NLAD) estimator were proved by Weiss (1991), even though they are difficult to compute in practice. Overcoming this difficulty will be critical if the NLAD estimator is to become practical. We propose an approximated NLAD estimator with the same asymptotic properties as the original with the exception that ours is easier to use.
A crude odds ratio can differ from stratum-specific odds ratios conducted for controlling a potential confounder. In this paper, a contour method is proposed to prove conservative tendency of the crude odds ratio when the risk factor is independent of the exposure and the two stratum-specific odds ratios are common. The argument is also referred in the case when the risk factor is dependent on the exposure.
The HKB estimator of Hoerl, Kennard and Baldwin is known to be an ordinaryridge type shrinkage estimator and an adjustment of degrees of freedom in the ordinary ridge estimator of Farebrother. The HKB estimator has a smaller predictive mean squared error (MSE) than the positive-part Stein-rule (PP) estimator in the wide region of the noncentral parameter when the degrees of freedom q=3∼6, but does not satisfy the sufficient condition to dominate the OLS estimator of Baranchik or Efron and Morris when q=3. In this paper, a sufficient condition to dominate the OLS estimator is derived, and the modified HKB estimator is constructed to dominate the OLS estimator and have a smaller MSE than the PP estimator in the wide region of the noncentral parameter.
As estimators of estimable parameters, we consider three statistics which are U-statistic, V-statistic and limit of Bayes estimate. This limit of Bayes estimate, called LB-statistic in this paper, is obtained from Bayes estimate of estimable parameter based on Dirichlet process, by letting its parameter tend to zero. For the estimable parameter with non-degenerate kernel, the asymptotic relative efficiencies of LB-statistic with respect to U-statistic and V-statistic and that of V-statistic with respect to U-statistic are equal to one. We show asymptotic differences among LB-statistic, U-statistic and V-statistic by using the deficiency.
The inverse moment of the noncentral chi-squared variable is approximated in simple forms based on its asymptotic expansions. The inverse moment is expanded as the noncentrality parameter tends to infinity proportionally to degrees of freedom. Accuracies of our approximations can be examined through numerical evaluation. It is observed that our approximations perform well in a wide range of values of the noncentrality parameter or degrees of freedom.
This paper is concerned with an inequality for MSE in statistical prediction theory. Takeuchi (1975) provided the inequality for a risk of unbiased predictor under certain regularity conditions. We shall provide an inequality for MSE of an unbiased predictor from L2-differentiability of densities point of view. In addition, this inequality is simplified and corresponded to the above under slightly stronger conditions. We shall also state the criterion for L2-differentiability in the case that an observable random vector and a predictive random variable are not independent.