Ahmad, Ohlson, and von Rosen (2011a) present asymptotic distribution of a one-sample test statistic under non-normality, when the data are high dimensional, i.e., when the dimension of the vector, p, may exceed the sample size, n. This paper extends the case to a two-sample statistic to test the difference of mean vectors of two independent multivariate distributions, again under high-dimensional set up. Using the asymptotic theory of U-statistics, and under mild assumptions on the traces of the unknown covariance matrices, the statistic is shown to follow an approximate normal distribution when n and p are large. However, no relationship between n and p is assumed. An extension to the paired case is given, which, being essentially a one-sample statistic, supplements the asymptotic results obtained in Ahmad, Ohlson, and von Rosen (2011a).

2.

Ahmad, M. Rauf

Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.

A test statistic is considered for testing a hypothesis for the mean vector for multivariate data, when the dimension of the vector, p, may exceed the number of vectors, n, and the underlying distribution need not necessarily be normal. With n, p large, and under mild assumptions, the statistic is shown to asymptotically follow a normal distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of well-known Box's approximation, under high-dimensional set up.

4.

Ahmad, M. Rauf

et al.

Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.

Ohlson, Martin

Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.

von Rosen, Dietrich

Department of Energy and Technology, Swedish Univerity of Agricultural Sciences, SE-750 07 Uppsala, Sweden.

Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p >> n. The statistics are derived under very general conditions, particularly avoiding any strict assumptions on the traces of the unknown covariance matrix. Neither any relationship between n and p is assumed. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. The validity of the commonly used assumptions for high-dimensional set up is also briefly discussed.

5.

Ahmad, M. Rauf

et al.

Swedish University of Agricultural Sciences, Uppsala, Sweden and Department of Statistics, Uppsala University, Sweden.

von Rosen, Dietrich

Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.

Singull, Martin

Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.

A test statistic is considered for testing a hypothesis for the mean vector for multivariate data, when the dimension of the vector, p, may exceed the number of vectors, n, and the underlying distribution need not necessarily be normal. With n,p→∞, and under mild assumptions, but without assuming any relationship between n and p, the statistic is shown to asymptotically follow a chi-square distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of the well-known Box's approximation, under high-dimensional set up. Using a classical limit theorem, the approximation is further extended to an asymptotic normal limit under the same high dimensional set up. The simulation results, generated under different parameter settings, are used to show the accuracy of the approximation for moderate n and large p.

Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.

Ahmad, M. Rauf

Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.

von Rosen, Dietrich

Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.

More on the Kronecker Structured Covariance Matrix2012In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 41, no 13-14, p. 2512-2523Article in journal (Refereed)

Abstract [en]

In this paper, the multivariate normal distribution with a Kronecker product structured covariance matrix is studied. Particularly focused is the estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The suggested estimation generalizes the procedure proposed by Srivastava et al. (2008) for a separable covariance matrix. The restrictions imposed by separability and double separability are also discussed.

Linköping University, Department of Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics .

Ahmad, M. Rauf

Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.

von Rosen, Dietrich

Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.

In this paper the multivariate normal distribution with a Kronecker product structured covariance matrix is studied. Particularly, estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The estimation procedure, suggested in this paper, is a generalization of the procedure derived by Srivastava et al. (2008), for a separable covariance matrix.

Furthermore, the restrictions imposed by separability and double separability are discussed.

In this paper, the multilinear normal distribution is introduced as an extension of the matrix-variate normal distribution. Basic properties such as marginal and conditional distributions, moments, and the characteristic function, are also presented.

The estimation of parameters using a flip-flop algorithm is also briefly discussed.

Linköping University, Department of Mathematics. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.

Ahmad, M. Rauf

Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.

von Rosen, Dietrich

Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.

In this paper, the multilinear normal distribution is introduced as an extension of the matrix-variate normal distribution. Basic properties such as marginal and conditional distributions, moments, and the characteristic function, are also presented. The estimation of parameters using a flip-flop algorithm is also briefy discussed.