Asymptotic Pitman efficiencies of multivariate spatial sign and rank methods are considered in the one-sample location case. Limiting distributions of the spatial sign and signed-rank tests under the null hypothesis as well as under contiguous sequences of alternatives are given. Formulae for asymptotic relative efficiencies are found and, under multivariate t distributions, relative efficiencies with respect to Hotelling's T 2 test are calculated.
Brown and Hettmansperger introduced affine-invariant bivariate analogs of the sign, rank, and signed-rank tests based on the Oja median. In this article affine-invariant k-variate extensions of the one-sample signed-rank test and the Hodges-Lehmann estimate are considered. The necessary distribution theory is developed, and asymptotic Pitman efficiencies with respect to Hotelling's T 2 test under multivariate t distributions are tabulated. An application of the signed-rank tests to a repeated-measurement setting is presented.
The association of perinatal events, childhood epilepsy, and central nervous system trauma with juvenile delinquency was studied prospectively in a geographically defined population of 5966 males in northern Finland. Those who had obtained a criminal record up to the age of 22 years, totalling 355, or 6-0%, were defined as delinquents. The incidence of delinquency was not increased in males with a birth weight less than 2500 g or greater than 4000 g, preterm births <37 weeks' gestation, or those with perinatal brain damage or having epileptic seizures before 14 years of age. The incidence was increased by 6-8% in the group of males with birth weights less than 3500 g, but not significantly increased after standardisation for a number of social and demographic background variables. The incidence was increased by 10-3% among the males who had had a central nervous system trauma by the age of 14 years, however, and this factor remained significant when social and demographic factors were standardised by regression analysis, with an odds ratio of 1.9 for all males with a criminal record and an odds ratio of 3-15 for those who had committed a violent crime. Previous central nervous system trauma may be a cause of delinquency, or another possibility is that the type ofbehaviour pursued by males who are likely to commit a violent crime will expose them more often to accidents which can result in central nervous system trauma.
For a set of p-variate data points y 1 , . . . , y n , there are several versions of multivariate median and related multivariate sign test proposed and studied in the literature. In this paper we consider the asymptotic properties of the multivariate extension of the Hodges-Lehmann (HL) estimator, the spatial HL-estimator, and the related test statistic. The asymptotic behavior of the spatial HL-estimator and the related test statistic when n tends to infinity are collected, reviewed, and proved, some for the first time though being used already for a longer time. We also derive the limiting behavior of the HL-estimator when both the sample size n and the dimension p tend to infinity.
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