On Huberty's Coefficient of Irregularity of Yarns and its Estimators

Abstract

The coefticient K (Huberty) has been introduced to obtain a characterization of the irregularity of yarns, which depends only on the structure of the distribution of the fiber endpoints along the axis of the yarn and not on the length L and cross section Q of the fibers, and also not on the expected number of fiher endpoints per unit length. The coefficient K is a parameter which has to be estimated from a piece of yarn of length T. In this paper, bias and standard error of the customary ( K1) and an alternative ( K2) estimator are obtained. For ideal yarn, the mean and variance of K1 and K2 depend only on the sampling ratio s = L/T. The estimators have, therefore, properties which one would demand in analogy to the requirements which led to the introduction of k. Both, in particular K2, are very suitable for a test of the hypothesis that the yarn is ideal. It is conjectured that the estimators are approximately distributed as a constant times χn2 (chi-squared with n degrees of freedom), where the constant and n are functions of s only. Significant points of K are given. If the yarn is real, k has certain disadvantages: it depends not only on the structure of the distribution of endpoints (on the statistical anomaly), but also on L; it covers only "short-term irregularity." Bias and standard error of the estimator depend—besides on s—also on the anomaly and similar higher- order parameter functions. If the logic leading to k is applied to some of the usually used parameter functions of the yarn (covariance function, spectral density, variance length curve) one obtains— for ideal yarn—standardized parameter functions which depend only on L, and this dependence is such that it can be eliminated by a suitable choice of scale on the abscissa. The paper is based on the traditional model which has been considerably extended by Giesekus [1, 2] whose terminology is also used here.