For multi-way tables, Bhapkar and Darroch (1990) gave the second-order marginal symmetry model. The present paper proposes a measure to represent the degree of departure from the second-order marginal symmetry model. The measure is expressed as the weighted sum of the Shannon entropy. The paper also gives the approximate confidence interval of the measure, and shows relationship between the measure and the trivariate normal distribution. Examples are given.

In this paper, we consider the one-sample problem of testing for the subvector of a mean vector with two-step monotone missing data. In the case that the data set consists of complete data with $p(=p_1+p_2+p_3)$ dimensions and incomplete data with $(p_1+p_2)$ dimensions, we derive the likelihood ratio criterion for testing the $(p_2+p_3)$ mean vector under the given mean vector of $p_1$ dimensions. Furthermore, we propose an approximation for the upper percentile of the likelihood ratio test (LRT) statistic. We investigate the accuracy and asymptotic behavior of this approximation using Monte Carlo simulation. An example is presented in order to illustrate the method.

It is well-known that the complement of a disconnected graph is connected, that is, the number of disconnected unlabeled graphs whose complement is also disconnected is zero. By this fact, we can easily express the number of connected unlabeled graphs whose complement is also connected, by the numbers of graphs and connected graphs. The generating functions of them are obtained by Harary [5]. In this paper, we express the number of unlabeled graphs such that both the graph and its complement are $2$-connected, by the numbers of graphs whose generating functions are known.

We study the large time asymptotics of solutions to the Cauchy problem for the modified Korteweg-de Vries equation

We develop the factorization technique to obtain the large time asymptotics of solutions in the neighborhood of the self-similar solution in the case of nonzero total mass initial data. Our result is an improvement of the previous work [18].