The Box-Cox transformation has been used as a simple method of transforming dependent variable in ordinary-linear regression circumstances for improving the Gaussian-likelihood fit and making the disturbance terms of a model reasonably homoscedastic. The paper introduces a new version of the Box-Cox transformation and investigates how it works in terms of asymptotic performance and application, focusing in particular on inference on stationary multivariate ARMA models. The paper proposes a computational estimation procedure which extends the three-step Hannan and Rissanen method so as to accommodate the transformation and, for the purpose of parameter testing, the paper proposes a Monte-Carlo Wald test. The allied algorithm is applied to a bivariate series of the Tokyo stock-price index (Topix) and the call rate.
Testing procedures can be considered decision procedures, and there exists the problem of how to deal with observed values of test statistics which fall in a neighborhood of the critical values. On the other hand, in the framework of the multiple decision problem, even if an incorrect decision is made, it may still be close to the correct decision, and may not exhibit the degree of error that would result from making decisions completely contrary to the truth. While testing procedures are conservative only in respect of rejection of the null hypothesis, multiple decision procedures are more conservative. In this paper we discuss the multiple decision problem on the signs of the components of the three-variate normal mean vector based on Takeuchi (1973), and study the behavior of multiple decision procedures by simulation.
In this article, we develop a multivariate theory for analyzing multivariate datasets that have fewer observations than dimensions. More specifically, we consider the problem of testing the hypothesis that the mean vector μ of a p-dimensional random vector x is a zero vector where N, the number of independent observations on x, is less than the dimension p. It is assumed that x is normally distributed with mean vector μ and unknown nonsingular covariance matrix ∑. We propose the test statistic F+ = n−2 (p − n + 1) N ¯x′S+¯x, where n = N − 1 < p, ¯x and S are the sample mean vector and the sample covariance matrix respectively, and S+ is the Moore-Penrose inverse of S. It is shown that a suitably normalized version of the F+ statistic is asymptotically normally distributed under the hypothesis. The asymptotic non-null distribution in one sample case is given. The case when the covariance matrix ∑ is singular of rank r but the sample size N is larger than r is also considered. The corresponding results for the case of two-samples and k samples, known as MANOVA, are given.
In this expository paper, we illustrate the generality of the game-theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe how discrete distributions with finite support and discrete pricing formulas, such as the Cox-Ross-Rubinstein formula, are naturally derived from game-theoretic probability protocols. Corresponding to any discrete distribution with finite support, we construct a finite-horizon discrete game, a replicating strategy of Skeptic, and a neutral forecasting strategy of Forecaster, such that the discrete distribution is derived from the game. Construction of a replicating strategy is the same as in the standard arbitrage arguments of pricing European options in binomial tree models. However the game-theoretic framework is advantageous because it eliminates the need for any a priori probabilistic assumption.
An algebraic method is suggested to search for the optimal solution that maximizes a correlation criterion under a quadratic constraint. First it is shown that the problem formulated in a sample space can be reformulated in a parameter space, and then some properties of a matrix which specifies the quadratic constraint are provided along with its geometrical interpretation; the solution can be obtained by solving a nonlinear equation derived from the singular value decomposition of the matrix. Numerical results based on artificial data and entrance examination data are given to examine how our solution differs from the least squares solution under a quadratic constraint.
In microarray experiments, the dimension p of the data is very large but there are only a few observations N on the subjects/patients. In this article, the problem of classifying a subject into one of two groups, when p is large, is considered. Three procedures based on the Moore-Penrose inverse of the sample covariance matrix, and an empirical Bayes estimate of the precision matrix are proposed and compared with the DLDA procedure.
This paper presents the conditions for robustness to the nonnormality on\\break three test statistics for a general multivariate linear hypothesis, which were proposed under the normal assumption in a generalized multivariate analysis of variance (GMANOVA) model. The proposed conditions require the cumulants of an unknown population's distribution to vanish in the second terms of the asymptotic expansions for both the mean and variance of the test statistics. With the proposed conditions, the test statistic can be investigated for robustness to nonnormality of the population's distribution. When the conditions are satisfied, the Bartlett correction and the modified Bartlett correction in the normal case improve the quality of the chi-square approximation even under nonnormality.