The Birnbaum-Saunders distribution has received some attention in the statistical literature since its inception. The univariate Birnbaum-Saunders distribution has been used quite effectively in analyzing positively skewed data. Recently, bivariate and multivariate Birnbaum-Saunders distributions have been introduced in the literature. In this paper we propose a new generalization of the multivariate (p-variate) Birnbaum-Saunders distribution based on the multivariate skew normal distribution. It is observed that the proposed distribution is more flexible than the multivariateBirnbaum-Saunders distribution, and the multivariate Birnbaum-Saunders distribution can be obtained as a special case of the proposed model. We obtain the marginal, reciprocal and conditional distributions, and also discuss some otherproperties. The proposed p-variate distribution has a total of 3p + parameters. We use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters. One data analysis has been performed forillustrative purposes.
In this paper, we clarify conditions for consistency of a log-likelihood-based information criterion in multivariate linear regression models with a normality assumption. Although normality is assumed for the distribution of the candidate model, we frame the situation so that the assumption of normality may be violated. The conditions for consistency are derived from two types of asymptotic theory; one is based on a large-sample asymptotic framework in which only the sample size approaches ∞, and the other is based on a high-dimensional asymptotic framework in which the sample size and the dimension of the vector of response variables simultaneously approach infinity. In both cases, our results are independent of any indicator measuring a discrepancy between the true distribution and the normal distribution, e.g., skewness, kurtosis and other higher-order cumulants of the true distribution.
We propose a method for estimating the parameters of contingency table models, which is motivated by a geometrical idea. Our method—bisector regression for contingency tables (BRCT)—is based on a nested structure of contingency table models. Our method estimates parameters corresponding to the interactions of lower orders after estimating or eliminating those of higher orders. BRCT generates a sequence of parameter estimates, each element of which represents a model and a parameter estimate. The length of the sequence is equal to the number of parameters, which is much smaller than the total number of models. We describe the BRCT algorithm and show an example. We provide explanations for two cases: (a) two factors and (b) K factors.
We consider φ-divergence statistics Cφ for the test of one factor independence from the other two in an r × s × t contingency table. Statistics Cφ include the statistics Ra based on the power divergence as a special case. Statistic R0 is the log likelihood ratio statistic and R1 is Pearson's X2 statistic. Statistic R2/3 corresponds to the statistic for the goodness-of-fit test recommended by Cressie and Read (1984). Statistics Cφ have the same chi-square limiting distribution under the hypothesis that one factor and the other two are independent. In this paper, when we assume that the distribution of Cφ is continuous, we show the derivation of an expression of approximation based on a multivariate Edgeworth expansion for the distribution of Cφ under the hypothesis that one factor and the other two are independent. Using the expression, we propose a new approximation of the distribution of Cφ. In addition, on the basis of the approximation, we obtain transformed statistics that improve the speed of convergence to a chi-square limiting distribution of Cφ. By numerical comparison in the case of Ra, we show that the transformed statistics perform well for a small sample.