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Let $F(x)$ be the continuous distribution function of a random variable $X,$ and $F_n(x)$ the empirical distribution function determined by a sample $X_1, X_2, \cdots, X_n$. It is well known that the probability $P_n(\epsilon)$ of $F(x)$ being everywhere majorized by $F_n(x) + \epsilon$ is independent of $F(x)$. The present paper contains the derivation of an explicit expression for $P_n(\epsilon)$, and a tabulation of the 10%, 5%, 1%, and 0.1% points of $P_n(\epsilon)$ for $n =$ 5, 8, 10, 20, 40, 50. For $n =$ 50 these values agree closely with those obtained from an asymptotic expression due to N. Smirnov.

nential life distributions are accepted as characterizing the phenomenon of no wear. The problem of finding a class of life distributions which would similarly reflect the phenomenon of wear-out has been under investigation for some time. In answer to this problem we introduce in this paper the class of IHRA (Increasing Hazard Rate Average) distributions and show that it has, among others, the following optimal properties: (i) it contains the limiting case of no wear, i.e., all exponential distributions, (ii) whenever components which have IHRA life distributions are put together into a coherent system, this system again has an IHRA life distribution, i.e., a system wears out when its components wear out, and (iii) the IHRA class is the smallest class with properties (i) and (ii). 1. Introduction. The nature of many physical devices is to wear out in time. In this paper we endeavor to determine how the wear-out process is reflected as a property of the corresponding life distributions. Wear-out is an intuitively suggestive concept, but its precise stochastic meaning is not immediately evident. However, as a beginning, we review the long accepted stochastic characterization of the phenomenon of no wear. Let T ? 0 be the failure time of a device. Let F(t) = PI T > t} be the complement of the usual distribution function, which we will call the "survival probability." The conditional survival probability for remaining life, given that the device has survived to age x, is Fx(t) = F(x + t)/F(x) if F(x) > 0, Fx(t) = 0 if F(x) = O. A device does not wear if regardless of age an unfailed device is like new, i.e. Fx(t) = Fo(t) for all x, t > 0. It follows that the class of no wear survival probabilities is the class of solutions of the functional equation

Let $X, Y$ be independent random variables with continuous cumulative probability functions and let $$p = \mathrm{Pr}\{Y < X\}.$$ For the variance of the Mann-Whitney statistic $U,$ upper and lower bounds are obtained in terms of $p$, for the case of any $X$ and $Y$ as well as for the case of stochastically comparable $X, Y$. The results for the case of stochastic comparability are new, while the inequalities in the case of arbitrary $X, Y$ have either been obtained by van Dantzig or are a consequence of other inequalities due to van Dantzig.

The concepts of distribution-free statistics and statistics of structure (d) were defined relative to families Omega and Omega' of cumulative probability functions. A counter example was presented to disprove the conjecture that every distribution-free statistic, symmetric in X1, X2...., Xn (a sample of a 1-dimensional random variable X with a continuous cumulative probability function F) with Omega = Omega' = Omega 2, has structure (d). Strongly distribution-free statistics in a family Omega* with respect to Omega' were defined, and a proof was presented of the equivalence of statistics of structure (d) and strongly distribution-free statistics under the condition that Omega = Omega' = Omega*.