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The well-known connection between the Poisson process and empirical c.d.f.'s is exploited from a new point of view. Distributions of functions of empirical c.d.f.'s for finite sample size n are explicitly described in some new examples, and new qualitative information is obtained for some classical examples.

random variables, each having the same continuous cdf. Let the 2n random variables be arranged in increasing order, and the values of the X's replaced by (-1 ) 's and the values of Y's replaced by l's. This sequence of (-l1)'s and l's is called the sequence of rank order indicators. There are ( 2, ) equally likely sequences of rank order indicators. A random variable which is a function of the X's and Y's only through these rank order indicators is called a rank order statistic. Often rank order statistics are defined in terms of the empirical cdf's Fn and G": F,(t) = (number of X's < t)/n, Gn(t) = (number of Y's < t)/n. Some examples of rank order statistics are Dn+ = sup_o

This paper is a continuation of [1]. The techniques of [1] are used to get specific information about the distribution of the total progeny in a branching process. This distribution is also related to one which arises in a random walk problem.

Hoeffding and Simons (Ann. Math. Statist. (1970) 341-352) show how to convert a sequence of i.i.d. indicator random variables with parameter $p$ into such a sequence with parameter $p = \frac{1}{2}$. In the present paper an extension is made to generating sequences of i.i.d. indicator random variables with parameter $p = \frac{1}{2}$ from arbitrary sequences of i.i.d. discrete random variables. Also considered is the generating of sequences of i.i.d. $r$-valued, equiprobable random variables. Some optimality criteria are established.

Let $(R_1, \cdots, R_n)$ be a random vector which takes on each of the $N!$ permutations of $(1, 2, \cdots, N)$ with equal probability, $1/N!$. Let $(a_{N1}, \cdots, a_{NN})$ and $(b_{N1}, \cdots, b_{NN})$ be two sets of real numbers given for every $N$. We will assume throughout that for no $N$ are the $a_{Ni}$ all equal or the $b_{Ni}$ all equal. We also assume that the $a_{Ni}$ and $b_{Ni}$ have been so normalized that $\sum a_{Ni} = \sum b_{Ni} = 0;\quad \sum a^2_{Ni} = N^{-1} \sum b^2_{Ni} = 1$. Unless otherwise stated, $\sum$ will mean $\sum^N_{i=1}$. Define \begin{equation*}\tag{1.1} S_N = \sum a_{Ni}b_{NR_i}.\end{equation*} Let $\Phi(x)$ be the unit normal c.d.f. In Section 2 sufficient conditions are given for $\mathrm{Pr}\{S_N < x\}$ to approach $\Phi(x)$ as $N \rightarrow \infty$. (The first two moments of $S_N$ are 0 and $N/(N - 1)$, respectively.) For every $N$, let $Y = (Y_{11}, \cdots, Y_{1N_1}, \cdots, Y_{m1}, \cdots, Y_{mN_m})$ be $N = N_1 + \cdots + N_m$ random variables which are mutually independent and independent of the $R_i$. We assume that all $Y_{ij}$ with the same first subscript are identically distributed. Define $\bar{Y} = N^{-1} \sum_{i,j}Y_{ij},\quad S^2 = N^{-1} \sum_{i,j} (Y_{ij} - \bar{Y})^2,\quad SY'_{ij} = Y_{ij} - \bar{Y}, Y'_{ij} = 0 \text{if} S = 0$. Let $Y'$ denote the vector of the $Y'_{ij}$. Let $y = (y_1, \cdots, y_N)$ denote a point in $N$-space. By $F_N(x, y)$ we mean the c.d.f. of the random variable $S_N = \sum a_{Ni}y_{NR_i}$. In Section 3 are considered sufficient conditions for convergence with probability one of the random c.d.f. $F_N(x, Y')$ to $\Phi(x)$.

Let $(R_1, \cdots, R_N)$ be a random vector which takes on each of the $N!$ permutations of the numbers $(1, \cdots, N)$ with equal probability, $1/N!$. Sufficient conditions are given for the asymptotic normality of $S_N = \sum^N_{i=1}a_{Ni}b_{NR_i}$, where $(a_{N1}, \cdots, a_{NN}), (b_{N1}, \cdots, b_{NN})$ are two sets of real numbers given for every $N$. These sufficient conditions are apparently quite different from those given by Wald and Wolfowitz [9] and extended by various writers [4, 7]. In some situations the conditions given here may be easier to apply than those given previously. The most general conditions available to date appear to be those of Hoeffding [4]. In the examples below, however, is given a case of an $S_N$ which does not satisfy the conditions required by Hoeffding's theorem but which is asymptotically normal by our results.

Let Y1, Y 2, … be a sequence of independent and identically distributed Poisson random variables with parameter λ. Let Sn = Y 1 + … + Yn, n = 1,2,…, S 0 = 0. The event Sn = n is a recurrent event in the sense that successive waiting times between recurrences form a sequence of independent and identically distributed random variables. Specifically, the waiting time probabilities are (Alternately, the fn can be described as the probabilities for first return to the origin of the random walk whose successive steps are Y1 − 1, Y2 − 1, ….)

Suppose $X_1, \cdots, X_m, Y_1, \cdots, Y_n$ are $m + n = N$ independent random variables, the $X$'s identically distributed and the $Y$'s identically distributed, each with a continuous cdf. Let $$z = (z_1, \cdots, z_m, z_{m + 1}, \cdots, z_N) = (x_1, \cdots, x_m, y_1, \cdots, y_n)$$ represent an observation on the $N$ random variables and let $$u(z) = (1/m) \sum^m_{i = 1} z_i - (1/n) \sum^N_{i = m + 1} z_i = \bar x - \bar y$$. Consider the $r = N! N$-tuples obtained from $(z_1, \cdots, z_N)$ by making all permutations of the indices $(1, \cdots, N)$. Since we assume continuous cdf's, then with probability one, these $r N$-tuples will be distinct. Denote them by $z^{(1)}, \cdots, z^{(r)}$, and suppose that they have been ordered so that $$u(z^{(1)} \geqq \cdots \geqq u(z^{(r)})$$. Notice that since $$\bar x - \bar y = (1/m) \sum^N_{i = 1} z_i - (N/m)\bar y = (N/n)\bar x - (1/n) \sum^N_{i = 1} z_i,$$ the same ordering can be induced by choosing $u(z) = c\bar x$ or $u(z) = - c\bar y$ for any $c > 0$. Assuming that the cdf's of $X_1, Y_1$ are of the form $F(x), F(x - \Delta)$ respectively, Pitman [2] suggested essentially the following test of the hypothesis $H'$ that $\Delta = 0$. Select a set of $k (k > 0)$ integers $i_1, \cdots, i_k, (1 \leqq i_1 < \cdots < i_k \leqq r)$. If the observed $z$ is one of the points $z^{(i_1)}, \cdots, z^{(i_k)}$, reject $H'$, otherwise accept. When $H'$ is true, the type one error does not depend on the specific form of the distribution of the $X$'s and the $Y$'s and is in fact equal to $k/r$. The choice of the rejection set $i_1, \cdots, i_k$ should depend on the alternative hypothesis. For instance, if the experimenter wants protection against the alternative that the "$X$'s tend to be larger than the $Y$'s," then the labels $1, \cdots, k$ might be reasonable. For the alternative that the "$X$'s tend to be smaller than the $Y$'s" the analogous procedure is to use the other tail, $r - k + 1, \cdots, r$. Against both alternatives, a two-tail procedure could be used. Lehmann and Stein have shown in [1] that in the class of all tests (of size $\alpha = k/r$) of the hypothesis $$H: \text{the distribution of} X_1 \cdots, X_m, Y_1, \cdots, Y_n \text{is invariant under all permutations},$$ the single-tail test based on $1, \cdots, k$ is uniformly most powerful against the alternatives that $F_1$ is an $N(\theta, \sigma)$ cdf, $F_2$ is an $N(\theta + \Delta, \sigma)$ cdf, $\Delta < 0$; the test based on $r - k + 1, \cdots, r$ is uniformly most powerful for $\Delta > 0$. A practical shortcoming of this procedure is the great difficulty in enumerating the points $z^{(i)}$ and the evaluation of $u(z^{(i)})$ for each of them. For instance, even after eliminating those permutations which always give the same value of $u$, then for sample sizes $m = n = 5$, there are $\binom{10}{5} = 252$ permutations to examine, and for sample sizes $m = n = 10$, there are $\binom{20}{10} = 184,765$ permutations to examine. In the following section, we propose the almost obvious procedure of examining a "random sample" of permutations and making the decision to accept or reject $H$ on the basis of those permutations only. Bounds are determined for the ratio of the power of the original procedure to the modified one. Some numerical values of these bounds are given in Table 1. The bounds there listed correspond to tests which in both original and modified form have size $\alpha$, and for which the modified test is based on a random sample of $s$ permutations drawn with replacement. These have been computed for a certain class of alternatives which is described below. For simplicity, we have restricted the main exposition to the two-sample problem. In Section 5, we point out extensions to the more general hypotheses of invariance studied in [1].

Let N ( a ) N(a) be the number of overcrossings of height a a in a simple random walk. For p > 1 / 2 p > 1/2 , the process N ( 0 ) , N ( 1 ) , … N(0),N(1), \ldots is a branching process which eventually becomes extinct. For 1 / 2 > p , N ( 0 ) , N ( 1 ) , … 1/2 > p,N(0),N(1), \ldots is a stationary process which is a branching process with immigration.

We consider a simple sampling scheme where after each drawing there is a “replacement” whose magnitude is a fixed real number. This gives a unified approach to a family of discrete distributions that includes the hypergeometric, binomial, and Polya distributions as well as their multivariate versions.

What I will talk about relates to pedagogy the pedagogy of statistics in particular though some of my comments may have relevance to the teaching of other subjects as well. There is a growing involvement at universities these days in using the computer as an aid in teaching statistics. I want to describe our still limited experience at Northwestern University in these matters. I do not intend, nor am I able, to describe the 'state of the art' as a whole, but I will try to give some of my own philosophy, methods, guesses about costs and estimates of effectiveness and dangers. Whatever we are doing at Northwestern has no doubt been done in part or whole with varying points of view elsewhere. But I do believe it is worth passing on our own approach because there must be many teachers who have not become aware of or have not seriously considered computer-oriented methods in teaching. Moreover, there is still a minimal exchange of ideas and information on the subject. Up to this point, our experience at Northwestern has been limited to teaching several courses at the intermediate junior-senior level in statistics and a pre-calculus service course in statistics. In addition, we have taught some computer-oriented courses in finite mathematics for behavioral sciences, and several sections of calculus.