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 Volume 7, Issue 2, 1973
South African Statistical Journal  Volume 7, Issue 2, January 1973
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Volume 7, Issue 2, January 1973

Nonparametric sequential procedures for selecting the best of k populations
Source: South African Statistical Journal 7, pp 85 –94 (1973)More LessLet ?1, ?2,....,?k denote k populations (k ? 2) with distribution functions F(X?1), F(x?2),...,F(x?k) where F is symmetric about 0, so that ?1,...,?k are the centres of symmetry of the respective distributions. The function F and the parameters ?i (i = 1, ... ,k) are considered to be unknown and we assume without loss of generality that ?i??2? ... ??k. If ?k > ?k1, ?k is referred to as the best population.

Symmetrlsed multivariate distributions
Author M.J. GreenacreSource: South African Statistical Journal 7, pp 95 –101 (1973)More LessIn multivariate analysis there exists a large class of important hypothesis testing problems all of which may be tested by a set of criteria that depend functionally on a matrix variate with a Beta distribution. Examples of such hypotheses are the following. (i) Hypothesis of independence of two sets of variates, considering the second set fixed. (We shall deal later with the case when both sets are comprised of random variables.) (ii) Linear hypotheses about regression coefficients. (iii) General linear hypothesis in MANOVA.

On the joint distributions of the largest and smallest latent roots of two random matrices (noncentral case)
Author V.B. WaikarSource: South African Statistical Journal 7, pp 103 –108 (1973)More LessThe joint distributions of the largest and smallest roots of certain random matrices are useful in the application of some test procedures in multivariate analysis. In the null case, Sugiyama (1970) derived the joint density of the largest and smallest roots of the central Wishart matrix. Waikar (1971) derived an alternate expression (which is better from the point of view of computing than that given by Sugiyama) and also derived the joint density of the largest and smallest roots of the MANOVA matrix in the null case. In the present paper, the author has derived the exact expressions for the joint nonnull density of the largest and smallest roots of (i) S1 S21 matrix, where S1 and S2 are independently distributed Wishart matrices and (ii) Wishart matrix. The expressions for the above densities are in terms of zonal polynomials.

Nonparametric estimation of the mode of a multivariate density
Author M. SamantaSource: South African Statistical Journal 7, pp 109 –117 (1973)More LessThe problem of estimating the mode of a probability density function is a matter of both theoretical and practical interest. This problem was first considered by Parzen (1962) in the univariate situation. He has shown that under certain regularity conditions the estimate of the population mode obtained by maximizing an estimate of the true probability density function is consistent and asymptotically normal. The strongest result in this direction is due to Nadaraya (1965) who has proved that under certain regularity conditions the estimate con verges to the theoretical mode with probability one. In this paper we consider the problem of estimating the mode of an unknown multivariate probability density function. We establish conditions under which our estimate of the . population mode is strongly consistent and asymptotically normal.

On the distribution of a generalised positive semidefinite quadratic form of normal vectors
Source: South African Statistical Journal 7, pp 119 –127 (1973)More LessLet the elements of X:pxn be distributed normal with E(X) = 0 and the covariance matrix of x'0 = (x'1,x'2,...,x'p), where x'i is the ith row of X, is the Kronecker product ?1 ? ?2 (?1 and ?2 are both positive definite).

The distribution of linear and quadratic forms in complex normal vectors
Author N.A.S. CrowtherSource: South African Statistical Journal 7, pp 129 –141 (1973)More LessThe complex normal distribution has been studied by Wooding (1956), Turin (1960) and Goodman (1963) and distribution problems concerning complex normal variables have been dealt with by James (1964), Khatri (1965), De Waal (1968) and others. There are examples in which complex measurements are involved (see e.g. Turin (1960) for examples in physics). This paper deals with (i) certain conditional distributions of complex normal variables and (ii) the distribution of second degree polynomials in complex normal vectors.

Random splittings: a model for a masssize distribution
Author C.F. Schultz, D.M. & CrouseSource: South African Statistical Journal 7, pp 143 –152 (1973)More LessIn certain breakage processes when a particle is split into two, it may be assumed that the fraction of the initial mass received by each descendant particle is distributed uniformly on (0,1).