a test of the gaussian ness of a data set using clustering the properties of the magnifying glass method of clustering are discussed these properties which include unbiased and consistent estimation of the mean for gaussian distributions and biased and inconsistent estimation of the mean for non gaussian distributions lead to the development of a technique for testing data to determine whether or not it is gaussian an example of a non gaussian distribution is given to show the sensitivity of the proposed gaussian test

entire solutions of nonhomogenous differential equations abstract in this paper entire solutions of nonhomogenous differential equations of second order with polynomial coefficients in complex domain are considered bounds on the type and maximum modulus are obtained conditions for a solution to be entire and univalent are also given

a test to determine the multivariate normality of a data set a test is described for multivariate normality that is useful in pattern recognition the test is based on the friedman rafsky 1979 multivariate extension of the wald wolfowitz runs test the test data are combined with a multivariate swarm of points following the normal distribution generated with mean vector and covariance matrix estimated from the test data the minimal spanning tree of this resultant ensemble of points is computed and the count of the interpopulation edges in the minimal spanning tree is used as a test statistic the simulation studied both the null case of the test and one simple deviation from normality two conclusions are made from this study first the test can be conservatively applied by using the asymptotic normality of the test statistic even for small sample sizes second the power of the test appears reasonable especially in high dimensions monte carlo experiments were performed to determine if the test is reliable in high dimensions with moderate sample size the method is compared to other such tests available in the literature

a new test for multivariate normality and homoscedasticity given data x ij j 1 to n i i 1 to g one may wish to test whether the x ij are normal with common covariance matrix letting x i denote the mean of the ith group and s the pooled covariance matrix the quantities v ij x ij x i s 1 x ij x i are used in the test the exact null distribution of v ij is noted and the anderson darling statistic is proposed for testing whether the data fit this null distribution the new test has acceptable power is flexible and is easily implemented

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

two statistics for testing for multivariate normality malkovich afifi 1973 proposed measures of multivariate skewness and kurtosis in the present paper distributional properties of the two statistics are examined a method is given for calculating approximate percentage points of their distributions for all sample sizes greater than 8 for dimensions 2 3 and 4 and in principle for all higher dimensions the percentage points are derived from appropriate transformations of the asymptotic distributions a simulation study shows the approximations to be good for sample sizes of 25 and above

the potentially stable tree sign patterns for dimensions less than five the problem of characterizing potentially stable sign pattern matrices remains unsolved even for tree patterns we present here in tabular form all irreducible n by n potentially stable sign patterns whose undirected graphs are trees for n 5 in hopes that this may be useful to others in formulating and evaluating conjectures on the subject some background and a descriptio of methodology which includes some new theoretical ideas are also included

further properties of the star left star right star and minus partial orderings certain classes of matrices are indicated for which the star left star right star and minus partial orderings or some of them are equivalent characterizations of the left star and right star orderings similar to those devised by hartwig and styan linear algebra appl 82 1986 145 for the star and minus orderings are established along with other auxiliary results which are of independent interest as well some inheritance type properties of matrices are also given the class of ep matrices appears to be essential in several points of our considerations

some further results on hermitian matrix inequalities abstract some new results on the lowner partial ordering between certain sums products and direct products of matrices are derived proofs of these results are based on a necessary and sufficient condition for the nonnegative definiteness of the difference of two hermitian matrices which follows from the criterion for the nonnegative definiteness of a partitioned hermitian matrix due to albert 1969

two local operators and the blue abstract for a given matrix a a matrix p such that pa a is said to be a local identity and such that p2a pa is said to be a local idempotent in the paper some simple properties of such operators are presented their relation to the best linear unbiased estimation in the general gauss markov model is also demonstrated

classification error for a very large number of classes classification error is analyzed for a situation where the number of possible classes may be on the order of a hundred or more the error associated with classifying to a single class is shown to depend mainly on average nearest neighbor distance between class means noise level and effective dimensionality of the class mean distribution and not much on other aspects of the distribution noise correlation or number of classes since single class error is large separation of classes into groups is also explored group classification error has the same properties as single class error but the size of the error is moderated by the bayes overlap between groups standard curves are provided to predict single class and group error also discussed are the effect of pattern blurring on classification error and the nearest neighbor distance statistics throughout a distribution

partial orders based on outer inverses classes of outer inverses are used to create new partial orders and unify the classification of some of the old partial orders these classes are then compared with those obtained by using g inverses inner inverses

symmetrizers of matrices abstract a symmetrizer of a given pair of matrices a and b is defined as a matrix x for which the product axb is symmetric right and left symmetrizers of a given matrix a are defined accordingly the main results of the paper are general representations of all three types of symmetrizers the problem considered arose in connection with certain questions pertaining to admissible linear estimation in a gauss markoff model

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

on some linear combinations of hypergeneralized projectors abstract the concept of a hypergeneralized projector as a matrix h satisfying h2 h where h denotes the moore penrose inverse of h was introduced by gros and trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 in the present paper the problem of when a linear combination c1h1 c2h2 of two hypergeneralized projectors h1 h2 is also a hypergeneralized projector is considered although a complete solution to this problem remains unknown this article provides characterizations of situations in which c1h1 c2h2 2 c1h1 c2h2 derived under certain commutativity property imposed on matrices h1 and h2 the results obtained substantially generalize those given in the above mentioned paper by gros and trenkler

characterizations of the best linear unbiased estimator in the general gauss markov model with the use of matrix partial orderings abstract under the general gauss markov model y x 2 v two new characterizations of blue x are derived involving the lowner and rank subtractivity partial orderings between the dispersion matrix of blue x and the dispersion matrix of y as particular cases of these characterizations three new criteria for the equality between olse x and blue x are given

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

automorphisms of mn partially ordered by rank subtractivity ordering let mn be the space of all n n matrices with coefficients in r or c where n 3 the rank subtractivity or minus partial order on mn is defined by a rsb iff rankb ranka rank b a we characterize bijective mappings on mn such that a rsb iff a rs b a b mn

characterizations and dispersion matrix robustness of efficiently estimable parametric functionals in linear models with nuisance parameters abstract algebraic representations dimensional expressions and characterizations are given for the subspace of functionals of the main parameters which can be estimated with full efficiency under a linear model y w z 2 v containing nuisance parameters subspaces of functionals of for which the ordinary least squares estimator is robust against an alternative dispersion matrix v are obtained and a particular subspace of such functionals is found wherein the ordinary least squares estimator is both dispersion matrix robust and robust against the presence of nuisance parameters

partial orderings of matrices referring to singular values or eigenvalues abstract a new partial ordering in the set of complex matrices is defined which is weaker than the star ordering introduced by drazin in 1978 and stronger than the minus ordering introduced by hartwig in 1980 this ordering refers to singular values of matrices and the interest in it was generated by canonical interpretations of the minus and star orderings given by hartwig and styan in 1986 for hermitian matrices a similar ordering referring to eigenvalues is also considered and its connection with a problem concerning distributions of quadratic forms in normal variables is pointed out

univalence of a function f and its successive derivatives when f satisfies a differential equation ii abstract solutions of second order differential equations with quadratic polynomials as coefficients are studied by a suitable choice of constants it is shown that there exists a solution which is an entire function of exponential type one and such that this function and all its derivatives are close to convex in the unit disc by another choice of constants we get an entire solution of exponential type 2 with gap power series such that all odd or all even derivatives are close to convex in the unit disc

a relationship between the star and minus orderings abstract the result of hartwig and styan 1986 stating that matrices a and b are star ordered if and only if they are minus ordered and b a b a 1 3 or b a b a 1 4 is extended by considering generalized inverses of a linear combination b a with any 0

some properties of matrix partial orderings abstract the matrix partial orderings considered are 1 the star ordering and 2 the minus ordering or rank subtractivity both in the set of m n complex matrices and 3 the lowner ordering in the set of m m matrices the problems discussed are 1 inheriting certain properties under a given ordering 2 preserving an ordering under some matrix multiplications 3 relationships between an ordering among direct or kronecker and hadamard products and the corresponding orderings between the factors involved 4 orderings between generalized inverses of a given matrix and 5 preserving or reversing a given ordering under generalized inversions several generalizations of results known in the literature and a number of new results are derived

the matrix inequality m b mb abstract the loewner partial ordering between nonnegative definite matrices m and b mb is considered the main result obtained generalizes a number of results known in the literature the rank subtractivity partial ordering and the drazin partial ordering between m and b mb are also discussed

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses