Exact Moments and Percentage Points of the Order Statistics and the Distribution of the Range From the Logistic Distribution

Abstract

The logistic curve $y = k/(1 + \alpha e^{-\beta t})$ has been used in studies pertaining to population growth by Verhulst [17] and by Pearl and Reed [14] and by several later authors. The logistic function $P = 1/(1 + e^{-(\alpha+\beta x)})$ has been very widely used by Berkson [1], Berkson and Hodges [2] as a model for analyzing bioassay and other experiments involving quantal response. Gumbel [8] has shown that the asymptotic distribution of the midrange of exponential type initial distributions is logistic. In connection with problems involving censored data, Plackett [15], [16] has considered the use of the logistic distribution. In this paper a random variable $Y$ is said to follow a logistic distribution (denoted by $L(\mu, \sigma^2))$ if its cumulative distribution function (c.d.f.) is \begin{equation*}\tag{1.1}F(y; \mu, \sigma) = 1/\lbrack 1 + e^{-\lbrack(y-\mu)/\sigma\rbrack\cdot(\pi/3\frac{1}{2})}\rbrack. \end{equation*} The probability density function (p.d.f.) corresponding to (1.1) is \begin{equation*}\tag{1.2}f(y; \mu, \sigma) = (\pi/\sigma 3^{\frac{1}{2}})e^{-\pi(y-\mu)/3\frac{1}{2}\sigma}/\lbrack 1 + e^ {-\pi(y-\mu)/3\frac{1}{2}\sigma}\rbrack^2,\end{equation*} where $-\infty < y < \infty, -\infty < \mu < \infty$ and $\sigma > 0$. It should be noted that the distribution (1.2) is symmetrical with mean $\mu$ and variance $\sigma^2$. The moment generating function of $X = (Y - \mu)/\sigma$ is easy to derive (see for example, Gumbel [8]) and is \begin{equation*}\tag{1.3}M_X(t) = \Gamma(1 + t/g)\Gamma(1 - t/g),\quad g = \pi/3^{\frac{1}{2}}.\end{equation*} In this paper order statistics from the standard logistic distribution $L(0, 1)$ are studied. If $X_1, X_2, \cdots, X_n$ are $n$ independent and identically distributed logistic random variables with density function, \begin{equation*}\tag{1.4}f(x) = (\pi/3^{\frac{1}{2}})(e^{-x\pi/3\frac{1}{2}})/(1 + e^{-x\pi/3\frac{1}{2}})^2,\quad -\infty < x < \infty,\end{equation*} then we are concerned with the moments, the distribution and some estimation problems using the ordered random variables $X_{(1)}, X_{(2)}, \cdots, X_{(n)}$ where \begin{equation*}\tag{1.5}X_{(1)} \leqq X_{(2)} \leqq \cdots \leqq X_{(k)} \leqq \cdots \leqq X_{(n)}.\end{equation*} In the sequel, we shall call $X_{(k)}$, the $k$th order statistic in a sample of size $n$ from the logistic distribution $L(0, 1)$. In this paper the exact expressions for the moments of $X_{(k)}$ have been derived. The values of the first four exact moments for all sample sizes $n$ from 1 to 10 have been tabulated (Table I). More generally, the moments of $X_{(k)}$ have been expressed in terms of expressions involving Bernoulli and Stirling numbers of 1st kind. These derivations are obtained from the moment generating function which has been derived. The cumulants of $X_{(k)}$ are expressed in terms of polygamma functions, as was pointed out by Plackett [15]. Birnbaum and Dudman [3] have tabulated expected values and standard deviations from the logistic distribution using tabulated values of the digamma and trigamma functions. Table III of the present paper gives the percentage points of $X_{(k)}$ (i) for all $k(k \leqq n)$ and all $n$ from 1 to 10 (ii) for $k = 1, n$ and $\frac{1}{2}n$ and $\frac{1}{2}(n + 2)$ ($n$ even) or $\frac{1}{2}(n + 1)$ ($n$ odd) for $n = 11(1)25$. In Section 3, we obtain series expansions for the joint moment generating function and covariance of the two order statistics. In Section 5, the use of one and two order statistics for estimating $\mu$ and $\sigma$ in $L(\mu, \sigma^2)$ is shown. In Section 6, expressions (closed form) are derived for the cumulative distribution function and the density function of the sample range. Using the results of Section 6, a short table (Table II) of the sample range of the logistic is given for $n = 2$ and 3. Section 7 gives a description of the tables in this paper.