On low-dimensional projections of high-dimensional distributions

Let P be a probability distribution on q -dimensional space. The so-called Diaconis-Freedman effect means that for a fixed dimension d<symmetric Gaussian distributions. The present paper provides necessary and sufficient conditions for this phenomenon in a suitable asymptotic framework with increasing dimension q . It turns out, that the conditions formulated by Diaconis and Freedman (1984) are not only sufficient but necessary as well. Moreover, letting P ^ be the empirical distribution of n independent random vectors with distribution P , we investigate the behavior of the empirical process n √ (P ^ −P) under random projections, conditional on P ^ .

Investigating interocclusal perception in tactile teeth sensibility using symmetric and asymmetric analysis

The purpose of this clinical trial was to determine the active tactile sensibility of natural teeth and to obtain a statistical analysis method fitting a psychometric function through the observed data points. On 68 complete dentulous test persons (34 males, 34 females, mean age 45.9 ± 16.1 years), one pair of healthy natural teeth each was tested: n = 24 anterior teeth and n = 44 posterior teeth. The computer-assisted, randomized measurement was done by having the subjects bite on thin copper foils of different thickness (5-200 µm) inserted between the teeth. The threshold of active tactile sensibility was defined by the 50% value of correct answers. Additionally, the gradient of the sensibility curve and the support area (90-10% value) as a description of the shape of the sensibility curve were calculated. For modeling the sensibility curve, symmetric and asymmetric functions were used. The mean sensibility threshold was 14.2 ± 12.1 µm. The older the subject, the higher the tactile threshold (r = 0.42, p = 0.0006). The support area was 41.8 ± 43.3 µm. The higher the 50% threshold, the smaller the gradient of the curve and the larger the support area. The curves showing the active tactile sensibility of natural teeth demonstrate a tendency towards asymmetry, so that the active tactile sensibility of natural teeth can mathematically best be described by using the asymmetric Weibull function.