On Mill's Ratio for the Type III Population

Abstract

Mills [1], Gordon [2], Birnbaum [3], and the author [4] have studied the ratio of the area of the standardized normal curve from $x$ to $\infty$ and the ordinate at $x$. The object of this note is to establish the monotonic character of, and to obtain lower and upper bounds for, the ratio of the ordinate of the standardized Type III curve at $x$ and the area of the curve from $x$ to $\infty$. This ratio, as shown by Cohen [5] and the author [6], has to be calculated for several values of $x$ when solving approximately the equations involved in the problem of estimating the parameters of Type III populations from truncated samples. It was found by the author that, for large values of $x$, when the ordinates and areas are small, either this ratio cannot be obtained from existing tables prepared by Salvosa [7] or that very few significant digits are available for its calculation. It was thus found desirable to obtain lower and upper bounds which could satisfactorily locate this ratio. The monotonic behavior of this ratio and the inequalities obtained may also prove useful in checking the accuracy of the tables in [7], and in studying the nature of the tail of the chi square distribution.