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    An overall test for multivariate normality

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    There are a number of methods in the statistical literature for testing whether observed data came from a multivariate normal(MVN) distribution with an unknown mean vector and covariance matrix. Let X1, ...,Xn be an iid sample of size n from a p-variate normal distribution. Denote the sample mean and sample variance-covariance matrix by X̅ and S respectively. Most of the tests of multivariate normality are based on the results that Yi=S-½(Xi-X̅), i=1,..., n, are asymptotically iid as p-variate normal than zero mean vector and identity covariance matrix. Tests developed by Andrews et al., Mardina and others are direct functions of Yi. We note that the N=np components of the Yi's put together can be considered as an asymptotically iid sample of size N from a univariate normal any well known test based on N independent observations for univariate normality. In Particular we can use univariate skewness and kurtosis tests, which are sensitive to deviations from normality

    A new four parameter q-series identity and its partition implications

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    We prove a new four parameter q-hypergeometric series identity from which the three parameter key identity for the Goellnitz theorem due to Alladi, Andrews, and Gordon, follows as a special case by setting one of the parameters equal to 0. The new identity is equivalent to a four parameter partition theorem which extends the deep theorem of Goellnitz and thereby settles a problem raised by Andrews thirty years ago. Some consequences including a quadruple product extension of Jacobi's triple product identity, and prospects of future research are briefly discussed.Comment: 25 pages, in Sec. 3 Table 1 is added, discussion is added at the end of Sec. 5, minor stylistic changes, typos eliminated. To appear in Inventiones Mathematica
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