Properties of the Median and Other Order Statistics of Logistic Variates

Abstract

Recently, increasing use has been made of the logistic distribution. In addition to the many investigations of the logistic as a regression curve, several authors have investigated the properties and suggested the use of the logistic as a density function. One reason for this is the property that the inverse of the logistic distribution function can be expressed as the logarithm of a rational function. A general background for the work in this field can be gained by reading references [1], [2], [3], [7], and [10]. In this paper several additional properties of a sample of logistic variates will be described. In Section 2, it will be shown that the logistic distribution and its moment generating function can be expressed as a Maclaurin series where the coefficients are simple functions of Bernoulli numbers. In Section 3, the moment generating function of the median is given. In Section 4, the variance of the median will be determined and the relative efficiency of the median of logistic variates to the mean for various sample sizes as well as the asymptotic efficiency is given. In Section 5, the cumulant generating function of the median of logistic variates is derived and the cumulants, themselves, given in terms of zeta values. In Section 6, the variance and covariance of any two order statistics are given. The order statistic variance-covarance matrix can be used to determine the BLUE estimators of the population location and scale parameters. These may be expected to have high efficiency for the estimation of logistic parameters [4]. Although the maximum likelihood estimate of the population location parameter $\hat\theta$ satisfies the relationship, $\frac{1}{2} = \lbrack\sum^n_{i = 1} F(x_i - \hat{\theta})\rbrack/n,$ iterative methods must be used to solve for $\hat{\theta}$. Therefore the computational advantage of the BLUE estimates as well as their probable efficiency would tend to make them useful in the case of the logistic distribution.