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Ahmad, M. Rauf
Publications (9 of 9) Show all publications
Ahmad, M. R., von Rosen, D. & Singull, M. (2013). A note on mean testing for high dimensional multivariate data under non-normality. Statistica neerlandica (Print), 67(1), 81-99
Open this publication in new window or tab >>A note on mean testing for high dimensional multivariate data under non-normality
2013 (English)In: Statistica neerlandica (Print), ISSN 0039-0402, E-ISSN 1467-9574, Vol. 67, no 1, p. 81-99Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
non-normality;high dimensionality;Box's approximation
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-87379 (URN)10.1111/j.1467-9574.2012.00533.x (DOI)000313270000005 ()
Available from: 2013-04-03 Created: 2013-01-16 Last updated: 2017-12-06Bibliographically approved
Ohlson, M., Ahmad, M. R. & von Rosen, D. (2013). The Multilinear Normal Distribution: Introduction and Some Basic Properties. Journal of Multivariate Analysis, 113(S1), 37-47
Open this publication in new window or tab >>The Multilinear Normal Distribution: Introduction and Some Basic Properties
2013 (English)In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 113, no S1, p. 37-47Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Maryland Heights, MO, United States: Academic Press, 2013
Keywords
Flip-flop algorithm; matrix normal distribution; marginal and conditional distributions; maximum likelihood estimators; moments; tensor product
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-67577 (URN)10.1016/j.jmva.2011.05.015 (DOI)000310865300005 ()
Available from: 2011-04-18 Created: 2011-04-18 Last updated: 2017-12-11Bibliographically approved
Ohlson, M., Ahmad, M. R. & von Rosen, D. (2012). More on the Kronecker Structured Covariance Matrix. Communications in Statistics - Theory and Methods, 41(13-14), 2512-2523
Open this publication in new window or tab >>More on the Kronecker Structured Covariance Matrix
2012 (English)In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 41, no 13-14, p. 2512-2523Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Taylor & Francis, 2012
Keywords
Kronecker product structure, Separable covariance, Double separable covariance, Maximum likelihood estimators
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-67576 (URN)10.1080/03610926.2011.615971 (DOI)000305208600019 ()
Available from: 2011-04-18 Created: 2011-04-18 Last updated: 2017-12-11
Ahmad, M. R. (2011). A two-sample test statistic for high-dimensional multivariate data under non-normality. Linköping
Open this publication in new window or tab >>A two-sample test statistic for high-dimensional multivariate data under non-normality
2011 (English)Report (Other academic)
Abstract [en]

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).

Place, publisher, year, edition, pages
Linköping: , 2011. p. 17
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2011:12
Keywords
two-sample test, high-dimensionality, U-statistics
National Category
Algebra and Logic
Identifiers
urn:nbn:se:liu:diva-70060 (URN)LiTH-MAT-R-2011-12 (Local ID)LiTH-MAT-R-2011-12 (Archive number)LiTH-MAT-R-2011-12 (OAI)
Available from: 2011-08-17 Created: 2011-08-17 Last updated: 2013-11-26
Ahmad, M. R., Ohlson, M. & von Rosen, D. (2011). A U-statistics Based Approach to Mean Testing for High Dimensional Multivariate Data Under Non-normality. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>A U-statistics Based Approach to Mean Testing for High Dimensional Multivariate Data Under Non-normality
2011 (English)Report (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2011. p. 16
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2011:16
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-67237 (URN)LiTH-MAT-R-2011-06 (Local ID)LiTH-MAT-R-2011-06 (Archive number)LiTH-MAT-R-2011-06 (OAI)
Available from: 2011-04-06 Created: 2011-04-05 Last updated: 2014-09-29Bibliographically approved
Ohlson, M., Ahmad, M. R. & von Rosen, D. (2011). More on the Kronecker Structured Covariance Matrix. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>More on the Kronecker Structured Covariance Matrix
2011 (English)Report (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2011. p. 15
Series
LiTH-MAT-R, ISSN 0348-2960 ; 11:1
Keywords
Kronecker product structure, Separable covariance, Double separable covariance, Maximum likelihood estimators
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-64428 (URN)LiTH-MAT-R--2011/01--SE (Local ID)LiTH-MAT-R--2011/01--SE (Archive number)LiTH-MAT-R--2011/01--SE (OAI)
Available from: 2011-01-24 Created: 2011-01-24 Last updated: 2018-10-08Bibliographically approved
Ahmad, M. R., Ohlson, M. & von Rosen, D. (2011). Some Tests of Covariance Matrices for High Dimensional Multivariate Data. Linköping
Open this publication in new window or tab >>Some Tests of Covariance Matrices for High Dimensional Multivariate Data
2011 (English)Report (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Linköping: , 2011. p. 28
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2011:13
Keywords
covariance testing; high dimensional data; sphericity
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-70062 (URN)LiTH-MAT-R--2011/13--SE (Local ID)LiTH-MAT-R--2011/13--SE (Archive number)LiTH-MAT-R--2011/13--SE (OAI)
Available from: 2011-08-17 Created: 2011-08-17 Last updated: 2014-09-29
Ohlson, M., Ahmad, M. R. & von Rosen, D. (2011). The Multilinear Normal Distribution:Introduction and Some Basic Properties. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>The Multilinear Normal Distribution:Introduction and Some Basic Properties
2011 (English)Report (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2011. p. 20
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2011:2
Keywords
Flip-flop algorithm, matrix normal distribution, marginal and conditional distributions, maximum likelihood estimators, moments, tensor product
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-64483 (URN)LiTH-MAT-R--2011/02--SE (ISRN)
Available from: 2011-01-26 Created: 2011-01-26 Last updated: 2014-09-29Bibliographically approved
Ahmad, M. R. (2010). Robustness of a one-sample statistic for mean testing of high dimensional longitudinal data. Linköping
Open this publication in new window or tab >>Robustness of a one-sample statistic for mean testing of high dimensional longitudinal data
2010 (English)Report (Other academic)
Place, publisher, year, edition, pages
Linköping: , 2010. p. 13
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2010:11
Keywords
longitudinal data; high dimensionality; robustness
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-63640 (URN)LiTH-MAT-R--2010/11--SE (Local ID)LiTH-MAT-R--2010/11--SE (Archive number)LiTH-MAT-R--2010/11--SE (OAI)
Available from: 2010-12-29 Created: 2010-12-29 Last updated: 2013-11-26Bibliographically approved
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