For the analysis of square contingency tables with ordered categories, Tomizawa, Miyamoto and Ashihara (2003) considered the measure which represents the degree of departure from the marginal homogeneity (MH) model and does not depend on the diagonal probabilities in the table. This paper proposes another measure which represents the degree of departure from MH and depends on the diagonal probabilities. The measure proposed is expressed by using the Cressie-Read power-divergence or Patil-Taillie diversity index, which is applied for the cumulative marginal probabilities that an observation will fall in row (column) category $i$ or below [or in row (column) category $i+1$ or above]. The measure is useful for seeing how far the cumulative marginal probabilities are distant from those with a MH structure, and for comparing the degree of departure from MH in several tables. Examples are given.

For statistical models including continuous time stochastic processes, two types of information criteria based on the expected Kullback-Leibler information are proposed. The information criteria are applied to the evaluation of various types of statistical models and they are generally different from the results proposed in Uchida and Yoshida [33], which are based on the estimated Kullback-Leibler information. As an example, we present two information criteria for ergodic diffusion processes.

In the modified second approximation method for multiple comparisons of mean vectors that Professor Minoru Siotani (1959, 1960, 1964) originally proposed, it is crucial to consider the bivariate distribution of two correlated Hotelling’s $T^2$ statistics. We focus on an application of the differential operator developed by Kakizawa and Iwashita (2005a) and then derive asymptotic expansion of the bivariate distribution of $(T_{ab}^2,T_{cd}^2)$ as well as its marginal $T_{ab}^2$.

Suppose that a statistic $S$ is asymptotically distributed as a distribution function $G(x)$ as some parameter $ϵ\to 0$. We consider monotone transformations of $S$ in order to improve the asymptotic approximation. The transformations proposed here preserve monotonicity and give transformed statistics $T(S)$ whose distribution function is coincident with $G(x)$ up to the order $O({ϵ}^{r-1})$. It may be observed that the proposed transformations give a significant improvement to approximations. Further, we shall also consider error bounds for the remainder term of an asymptotic expansion for the distribution of $T(S)$. Finally, some applications of the findings are demonstrated for some test statistics.

In repeated measures, $p$ dimensional measurements are observed, then testing a hypothesis for equality of means of $p$ components is one of the simplest inference. If the hypothesis were accepted, it is interesting to estimate of the mean. In this paper, we consider the problem of constructing a fixed width confidence interval of equal normal means, when the covariance matrix is intraclass correlation model.

Simultaneous confidence intervals for all contrasts of the mean components in repeated measures with the intraclass correlation model are considered when the observations are missing at random. An exact test statistic for the equality of means and Scheffé, Bonferroni and Tukey types of simultaneous confidence intervals are given under the general case of missing observations. Finally, numerical examples by simulation are given to illustrate the method.

In this paper, we introduce a comprehensive family of harmonic univalent functions which contains various well-known classes of harmonic univalent functions as well as many new ones. Coefficient bounds, distortion bounds, extreme points, convolution conditions, and convex combination are determined for functions in this family. Consequently, many of our results are either extensions or new approaches to those corresponding previously known results.