This chapter discusses the distribution of vector random variables and its properties. Most of th... more This chapter discusses the distribution of vector random variables and its properties. Most of the commonly used test criteria in multivariate analysis are invariant test procedures in regard to a certain group of transformations. It is a generally accepted principle that if a problem with a unique solution is invariant under a certain transformation, then the solution should be invariant under that transformation because there should be or exists a unique best way of analyzing a collection of statistical information. Given a set of transformations, each leaving Ω invariant, the theorem asserts that one can always extend this set to a group G of transformations whose members also leave Ω invariant. Invariance by reducing the dimension of the sample space to that of the space of the maximal invariant shrinks the parametric space. Moreover, given any test function on the sample space one can always replace it by a test that depends only on the sufficient statistic such that both have the same power function.
This chapter discusses the multivariate normal distribution, its properties and characterization.... more This chapter discusses the multivariate normal distribution, its properties and characterization. A random vector X = ( X 1 , …, X p )′ taking values x = ( x 1 , …, x p )′ in E p (Euclidean space of dimension p ) is said to have a p -variate. If the co-variance matrix Σ of a normal random vector X = ( X t , …, X p )′ is a diagonal matrix, then the components of X are independently distributed normal random variables. The characteristic function determines uniquely the distribution function it follows that the p -variate normal distribution is completely specified by its mean vector μ and covariance matrix Σ . A standard notation for a p -variate normal distribution with mean μ and covariance matrix Σ is N p ( μ, Σ ). A p -variate random vector X with values in E p is said to have a normal distribution only if every linear combination of the components of X has a univariate normal distribution. In complex multivariate normal distribution, the real and imaginary parts of different components are pairwise independent, and the real and the imaginary parts of the same component are independent with the same variance.
This chapter discusses three interrelated concepts concerning multivariate covariance models: pri... more This chapter discusses three interrelated concepts concerning multivariate covariance models: principal components, factor models, and canonical correlations. All these concepts deal with the covariance structure of the multivariate normal distribution and aim at reducing the dimension of the observable vector variables. Principal components of a random vector X are normalized linear combinations of the components of X that have special properties in terms of variances, where X = ( X 1 , …, X p )′. The second principal component is a linear combination that has maximum variance among all normalized linear combinations. Factor analysis is a multivariate technique that attempts to account for the correlation pattern present in the distribution of an observable random vector X in terms of a minimal number of unobservable random variables, called factors. The canonical model selects linear combinations of variables from each of the two sets, so that the correlations among the new variables in different sets are maximized subject to the restriction that the new variables in each set are uncorrelated with mean 0 and variance 1.
This chapter describes the basic concepts and some basic results of group theory. It also present... more This chapter describes the basic concepts and some basic results of group theory. It also presents the results on the Jacobian of some specific transformations which are very useful in deriving the distributions of multivariate test statistics. In Jacobian of some transformations, it is assumed X 1 … X n to be a sequence of n continuous random variables with a joint probability density function f X1 … X n (x 1 , …., x n ) and Y i = g i ( X 1 , …, X n ) is a set of continuous one-to-one transformations of the random variables X 1 , …, X n . It is also assumed that the functions g 1 , …, g n have continuous partial derivatives in regard to x 1 , …, x n . The inverse function is denoted by X i = h i ( Y 1 , …, Y n ), i = 1, …, n . The determinant of the n × n square matrix is denoted by J is called the Jacobian of the inverse transformation.
This chapter discusses the distribution of vector random variables and its properties. Most of th... more This chapter discusses the distribution of vector random variables and its properties. Most of the commonly used test criteria in multivariate analysis are invariant test procedures in regard to a certain group of transformations. It is a generally accepted principle that if a problem with a unique solution is invariant under a certain transformation, then the solution should be invariant under that transformation because there should be or exists a unique best way of analyzing a collection of statistical information. Given a set of transformations, each leaving Ω invariant, the theorem asserts that one can always extend this set to a group G of transformations whose members also leave Ω invariant. Invariance by reducing the dimension of the sample space to that of the space of the maximal invariant shrinks the parametric space. Moreover, given any test function on the sample space one can always replace it by a test that depends only on the sufficient statistic such that both have the same power function.
This chapter discusses the multivariate normal distribution, its properties and characterization.... more This chapter discusses the multivariate normal distribution, its properties and characterization. A random vector X = ( X 1 , …, X p )′ taking values x = ( x 1 , …, x p )′ in E p (Euclidean space of dimension p ) is said to have a p -variate. If the co-variance matrix Σ of a normal random vector X = ( X t , …, X p )′ is a diagonal matrix, then the components of X are independently distributed normal random variables. The characteristic function determines uniquely the distribution function it follows that the p -variate normal distribution is completely specified by its mean vector μ and covariance matrix Σ . A standard notation for a p -variate normal distribution with mean μ and covariance matrix Σ is N p ( μ, Σ ). A p -variate random vector X with values in E p is said to have a normal distribution only if every linear combination of the components of X has a univariate normal distribution. In complex multivariate normal distribution, the real and imaginary parts of different components are pairwise independent, and the real and the imaginary parts of the same component are independent with the same variance.
This chapter discusses three interrelated concepts concerning multivariate covariance models: pri... more This chapter discusses three interrelated concepts concerning multivariate covariance models: principal components, factor models, and canonical correlations. All these concepts deal with the covariance structure of the multivariate normal distribution and aim at reducing the dimension of the observable vector variables. Principal components of a random vector X are normalized linear combinations of the components of X that have special properties in terms of variances, where X = ( X 1 , …, X p )′. The second principal component is a linear combination that has maximum variance among all normalized linear combinations. Factor analysis is a multivariate technique that attempts to account for the correlation pattern present in the distribution of an observable random vector X in terms of a minimal number of unobservable random variables, called factors. The canonical model selects linear combinations of variables from each of the two sets, so that the correlations among the new variables in different sets are maximized subject to the restriction that the new variables in each set are uncorrelated with mean 0 and variance 1.
This chapter describes the basic concepts and some basic results of group theory. It also present... more This chapter describes the basic concepts and some basic results of group theory. It also presents the results on the Jacobian of some specific transformations which are very useful in deriving the distributions of multivariate test statistics. In Jacobian of some transformations, it is assumed X 1 … X n to be a sequence of n continuous random variables with a joint probability density function f X1 … X n (x 1 , …., x n ) and Y i = g i ( X 1 , …, X n ) is a set of continuous one-to-one transformations of the random variables X 1 , …, X n . It is also assumed that the functions g 1 , …, g n have continuous partial derivatives in regard to x 1 , …, x n . The inverse function is denoted by X i = h i ( Y 1 , …, Y n ), i = 1, …, n . The determinant of the n × n square matrix is denoted by J is called the Jacobian of the inverse transformation.
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