The theory of Part I is extended to problems in which it is permitted not to come to a definite conclusion regarding one or more of the questions under consideration. Some problems are also investigated in which, from a single set of observations, one wishes to answer a number of questions in sequence. Here the nature of the question at a later stage will depend on the answers obtained at the earlier stages.

The main purpose of this paper is to prove, by the method of invariance, that in certain sequential decision problems (discrete and continuous time) there exists a minimax procedure $\delta^\ast$ among the class of all sequential decision functions such that $\delta^\ast$ observes the process for a constant length of time. In the course of proving these results a general invariance theorem will be proved (Sec. 3) under conditions which are easy to verify in many important examples (Sec. 2). A brief history of the invariance theory will be recounted in the next paragraph. The theorem of Sec. 3 is to be viewed only as a generalization of one due to Peisakoff [1]; the more general setting here (see Sec. 2; the assumptions of [1] are discussed under Condition 2b) is convenient for many applications, and some of the conditions of Sec. 2 (and the proofs that they imply the assumptions) are new; but the method of proof used in Sec. 3 is only a slight modification of that of [1]. The form of this extension of [1] in Secs. 2 and 3, and the results of Secs. 4 and 5, are new as far as the author knows. In 1939 Pitman [2] suggested on intuitive grounds the use of best invariant procedures in certain problems of estimation and testing hypotheses concerning scale and location parameters. In the same year Wald [3] had the idea that the theorem of Sec. 3 should be valid for certain nonsequential problems of estimating a location parameter; unfortunately, as Peisakoff points out, there seems to be a lacuna in Wald's proof. During the war Hunt and Stein [4] proved the theorem for certain problems in testing hypotheses in their famous unpublished paper whose results have been described by Lehmann in [5a], [5b]. Peisakoff's previously cited work [1] of 1950 contains a comprehensive and fairly general development of the theory and includes many topics such as questions of admissibility and consideration of vector-valued risk functions which will not be considered in the present paper (the latter could be included by using the devise of taking linear combinations of the components of the risk vector). Girshick and Savage [6] at about the same time gave a proof of the theorem for the location parameter case with squared error or bounded loss function. In their book [7], Blackwell and Girshick in the discrete case prove the theorem for location (or scale) parameters. The referee has called the author's attention to a paper by H. Kudo in the Nat. Sci. Report of the Ochanomizu University (1955), in which certain nonsequential invariant estimation problems are treated by extending the method of [7]. All of the results mentioned above are nonsequential. Peisakoff [1] mentions that sequential analysis can be considered in his development, but (see Sec. 4) his considerations would not yield the results of the present paper. A word should be said about the possible methods of proof. (The notation used here is that of Sec. 2 but will be familiar to readers of decision theory.) The method of Hunt and Stein, extended to problems other than testing hypotheses, is to consider for any decision function $\delta$ a sequence of decision functions $\{\delta_i\}$ defined by $$\delta_i(x,\Delta) = \int_{G_n} \delta_i(gx,g\Delta)\mu(dg)/\mu(G_n)$$ where $\mu$ is left Haar measure on a group $G$ of transformations leaving the problem invariant and $\{G_n\}$ is a sequence of subsets of $G$ of finite $\mu$-measure and such that $G_n \rightarrow G$ in some suitable sense. If $G$ were compact, we could take $\mu(G) = 1$ and let $G_1 = G$; it would then be clear that $\delta_1$ is invariant and that $\sup_Fr_{\delta_1} (F) \leqq \sup_Fr_\delta(F),$ yielding the conclusion of the theorem of Sec. 3. If $G$ is not compact, an invariant procedure $\delta_0$ which is the limit in some sense of the sequence $\{\delta_i\}$ must be obtained (this includes proving that, in Lehmann's terminology, suitable conditions imply that any almost invariant procedure is equivalent to an invariant one) and $\sup_Fr_{\delta_0} (F) \leqq \sup_Fr_\delta(F)$ must be proved. Peisakoff's method differs somewhat from this, in that for each $\delta$ one considered a family $\{\delta_g\}$ of procedures obtained in a natural way from $\delta$, and shows that an average over $G_n$ of the supremum risks of the $\delta_g$ does not exceed that of $\delta$ as $n \rightarrow \infty$; there is an obvious relationship between the two methods. Similarly, in [7] the average of $r_\delta(gF_0)$ for $g$ in $G_n$ and some $F_0$ is compared with that of an optimum invariant procedure (the latter can thus be seen to be Bayes in the wide sense); the method of [6] is in part similar. In some problems it is convenient (see Example iii and Remark 7 in Sec. 2) to apply the method of Hunt and Stein to a compact group as indicated above in conjunction with the use of Peisakoff's method for a group which is not compact. The possibility of having an unbounded weight function does not arise in the Hunt-Stein work. Peisakoff handles it by two methods, only one of which is used in the present paper, namely, to truncate the loss function. The other method (which also uses a different assumption from Assumption 5) is to truncate the region of integration in obtaining the risk function. Peisakoff gives several conditions (usually of symmetry or convexity) which imply Assumption 4 of Sec. 2 or the corresponding assumption for his second method of proof in the cases treated by him, but does not include Condition 4b or 4c of Sec. 2. Blackwell and Girshick use Condition 4b for a location parameter in the discrete case with $W$ continuous and not depending on $x$, using a method of proof wherein it is the region of integration rather than the loss function which is truncated. (The proof in [6] is similar, using also the special form of $W$ there.) It is Condition 4c which is pertinent for many common weight functions used in estimating a scale parameter, e.g., any positive power of relative error in the problem of estimating the standard deviation of a normal d.f. The overlap of the results of Secs. 4 and 5 of the present paper with previous publications will now be described. There are now three known methods for proving the minimax character of decision functions. Wolfowitz [8] used the Bayes method for a great variety of weight functions for the case of sequential estimation of a normal distribution with unknown mean (see also [9]). Hodges and Lehmann [10] used their Cramer-Rao inequality method for a particular weight function in the case of the normal distribution with unknown mean and gamma distribution with unknown scale (as well as in some other cases not pertinent here) to obtain a slightly weaker minimax result (see the discussion in Sec. 6.1 of [12]) than that obtainable by the Bayes method. The Bayes method was used in the sequential case by Kiefer [11] in the case of a rectangular distribution with unknown scale or exponential distribution with unknown location, for a particular weight function. This method was used by Dvoretzky, Kiefer and Wolfowitz in [12] for discrete and continuous time sequential problems involving the Wiener, gamma, Poisson, and negative binomial processes, for particular classes of weight functions. The disadvantage of using the Cramer-Rao method is in the limitation of its applicability in weight function and in regularity conditions which must be satisfied, as well as in the weaker result it yields. The Bayes method has the disadvantage that, when a least favorable a priori distribution does not exist, computations become unpleasant in proving the existence (if there is one) of a constant-time minimax procedure unless an appropriate sequence of a priori distributions can be chosen in such a way that the a posteriori expected loss at each stage does not depend on the observations (this is also true in problems where we are restricted to a fixed experimentation time or size, but it is less of a complication there); thus, the weight functions considered in [12] for the gamma distribution were only those relative to which such sequences could be easily guessed, while the proof in [11] is made messy by the author's inability to guess such a sequence, and even in [8] the computations become more involved in the case where an unsymmetric weight function is treated. (If, e.g., $\mathscr{F}$ is isomorphic to $G$, the sequence of a priori distributions obtained by truncating $\mu$ to $G_n$ in the previous paragraph would often be convenient for proving the minimax character by the Bayes method if it were not for the complication just noted.) The third method, that of invariance, has the obvious shortcoming of yielding little unless the group $G$ is large enough and/or there exists a simple sequence of sufficient statistics; however, when it applies to the extent that it does in the examples of Secs. 4 and 5, it reduces the minimax problem to a trivial problem of minimization. Several other sequential problems treated in Section 4 seem never to have been treated previously by any method or for any weight function; some of these involve both an unknown scale and unknown location parameter. A multivariate example is also treated in Sec. 4. In example xv of Sec. 4 will be found some remarks which indicate when the method used there can or cannot be applied successfully. In Sec. 5, in addition to treating continuous time sequential problems in a manner similar to that of Sec. 4, we consider another type of problem where the group $G$ acts on the time parameter of the process rather than on the values of the sample function.

The attention of statisticians has usually been focussed on single transformations, rather than on families of transformations. With a growing appreciation of the advantages of examining the behavior of data or approximations over whole families of transformations (Moore and Tukey [2], Anscombe and Tukey [1]), there arises a need for rationally planned charts for representing families of transformations. The contributions which (i) the topology of the family and (ii) a definition of the strength of a transformation can make to charting are studied in general and applied to the charting of the simple family of transformations. This family is defined to include all transformations of the form $$y \text{is replaced by} z = (y + c)^p$$ and all their limits. It thus includes $z = \log (y + c), z = e^{my}$ and the special case \begin{equation*}z = \begin{cases}0, & y = y_\min,\\1, & \text{otherwise},\end{cases}\end{equation*} where $y_\min$ is the least value of $y$ either (i) present in the data or (ii) possible, as well as all linear transformations of these transformations. Experience having shown that transformations with $p \leqq 1$ are much more frequently useful than any others, the charts developed, presented, and exemplified here are restricted to the part of the simple family--its central region--for which $p \leqq 1$. Separate charts are presented for two cases which should cover most cases which arise in practice: (1) Where, as with counted data and small counts, the least reasonable value for $y + c = 0$, and this value is likely to occur; (2) Where $y + c$ is always safely $>0$, and the range of $y$ is through not many powers of 10.

In the usual formulation of problems in statistical decision theory the probability distribution of the observations is assumed to be a member of some specified class of distribution functions. No a priori information is ordinarily assumed to exist concerning which member of this class is the true distribution of the observations although a priori probability measures defined over this class may be introduced as a technical device for generating complete classes of decision functions, minimax decision rules, etc. However, in some experimental situations it may be reasonable to suppose that such an a priori probability measure actually exists in the sense that the distributions of observations occurring in different experiments made under similar circumstances may be thought of as having been selected from a specified class of distribution functions according to some probability law. Such an assumption seems particularly apt in the case where measurements are made on an individual selected according to some probability law (e.g., "at random") from a population and where it is desired to make inferences about some characteristic of the individual on the basis of these measurements. If the class of probability distributions of the measurements for all individuals in the population and the law of selection are known, an optimum Bayes decision procedure can then be found. In general, however, such information will not be available to the experimenter, but there may be observations available on individuals previously selected in the same way from the same population and, under certain circumstances, these prior observations may be used to obtain approximations to the optimum Bayes decision procedure. The possibility of using prior observations to approximate Bayes procedures was first established for certain estimation problems in [1] by H. Robbins who coined the term "empirical Bayes procedures" to describe such approximations. Robbins in [1] discusses the estimation, using a squared error loss function, of the value $\lambda$ of a random variable $\Lambda$ associated with a discrete valued observation $X$ whose conditional probability function, given $\lambda$, is $p(x \mid \lambda),$ where $p(x \mid \lambda)$ is known for each $\lambda$ but where the (a priori) distribution of $\Lambda$ is unknown. For several specific parametric families of discrete probability functions $p(x \mid \lambda)$ Robbins shows that if prior independent observations $X_1, X_2, \cdots, X_n$, each having the same unconditional distribution as $X$ are available, then an empirical Bayes estimator Bayes estimator using $X_1, X_2, \cdots, X_n$ can be found which converges with probability 1 to the Bayes estimator as $n$ increases, for any a priori distribution of $\Lambda$. In Sec. 2 below a similar estimation problem is considered for the "non-parametric" case where the class of (conditional) probability distributions of $X$ is not restricted to a particular parametric family, but is instead the class of all probability functions assigning probability 1 to some specified denumerable set of numbers. The quantity to be estimated is the value of a functional defined on this class of probability functions and it is assumed that there exists an unknown a priori probability measure defined on a suitable $\sigma$-algebra of subsets of this class. For this case it is shown that under certain circumstances prior observations may be used to construct empirical Bayes estimators having the property that, as the number of prior observations increases, the risks of the empirical Bayes estimators converge to the risk of the Bayes estimator for any a priori probability measure provided that certain moments exist. The rate of convergence of these risks is also investigated for two special cases. In Sec. 3 the techniques of Sec. 2 are modified to apply to the case where the class of (conditional) distributions of $X$ is the class of all absolutely continuous distribution functions, and similar results are obtained. In Sec. 4 the results of the previous sections are used to obtain empirical Bayes solutions for certain two-decision problems of the hypothesis-testing type. Throughout this paper certain elementary properties of conditional expectations are used which are immediate consequences of results contained, for example, in Chapter VII of [2].

The system to be studied consists of a service unit and a queue of customers waiting to be served. Service-times of customers are independent, nonnegative variates with the common distribution $B(v)$ having a finite first moment $b_1$. Customers arrive in a Poisson process (see Feller [4], p. 364) of intensity $\lambda$; they form a queue and are served in order of arrival, with no defections from the queue. For previous work on this queueing system see for instance Pollaczek [11], Khintchine [9], Lindley [10], Kendall [7], [8], Smith [12], Bailey [1], and Takacs [14]. Let $W(t)$ be the time a customer would have to wait if he arrived at $t$. The forward Kolmogorov equation for the distribution of $W(t)$ is solved in principle by the use of Laplace integrals, and $E\{\exp\{ - sW(t)\}\}$ is determined in terms of $W(0)$ and the root of a possibly transcendental equation. It is shown that any analytic function of the root can be expanded in Lagrange's series, which provides a way of actually computing the transition probabilities of the process. Let $z$ be the first zero of $W(t)$. A series for $E\{\exp\{ - \tau z\}\}$ is obtained, and it is proved that $\mathrm{pr}\{z < \infty\} = 1$ if and only if $\lambda b_1 \leqq 1$. From a functional relation between $E\{W(t)\}$ and $\mathrm{pr}\{W(t) = 0\}$ the covariance function $R$ of $W(t)$ is determined. If the service-time distribution $B(v)$ has a finite third moment, then $R$ is absolutely integrable, and the spectral distribution of $W(t)$ is absolutely continuous.

It has been believed by some ([17], p. 204, and perhaps, by implication [21], p. 2, near line 10) that--and R. A. Fisher [e.g. at the Lake Junaluska conference in 1946] has urged the desirability of determining whether--the distribution induced by a pivotal, sufficient and smoothly invertible set of quantities is unique (that is to say, the induced distribution is independent of the choice of a particular set of pivotal quantities among those sets with these properties). If true, such uniqueness would be important in connection with the theory of fiducial probability. It is the purpose of this paper to present certain examples of particular interest showing that these conditions do not provide uniqueness. The first example applies to any family of two-dimensional normal distributions with fixed and known variances and covariances. A one-parameter family of pivotal pairs of quantities are provided, such that no two of the induced distributions are the same. Each pair is sufficient, and consists of two independent quantities, each distributed according to a unit normal distribution. Each pair is shown to be smoothly invertible of every finite order. This example can be extended to the Behrens-Fisher situation. The second example is due to L. J. Savage, and exhibits a two-parameter situation where the two alternative pairs of pivotal quantities constructed according to the prescription of Segal [24] give rise to different distributions. Mauldon [19] has recently published a quite different example of nonuniqueness which is also based on the bivariate normal distribution. In his example, the means are known and the second moments are to be estimated, so that there are 3 essential parameters. The paper concludes with a reasonably complete bibliography of papers on fiducial probability.

Given a fixed number $n$ of observations on a variate $x$ which has the Inverse Gaussian probability density function $$\exp\big\{-\frac{\phi^2x}{2\lambda} + \phi - \frac{\lambda}{2x}\big\} \sqrt{\frac{\lambda}{2\pi x^3}},\quad 0 < x < \infty$$, for which $E(x) = \lambda/\phi = \mu$, it is shown how to find functions of the sample mean $m$ whose expectations can be expressed suitably in terms of the parameter $\phi$ (or $\mu$). In particular, it is shown that the conditional expectation of any unbiased estimator $\tilde{\kappa}_r$ of the $r$th cumulant $\kappa_r$ is $$E(\tilde{\kappa}_r \mid m) = 2m(\frac{1}{2}\lambda n^2)^{r - 1}e^{\frac{1}{2}g} \int^\infty_1 (u - 1)^{2r - 3}e^{-\frac{1}{2}gu^2} du/(r - 2)!$$ where $g = \lambda n/m$. This expectation may be evaluated either by series given in the paper or by using published tables of numerical values of certain functions to which it can be related. The conditional variance of the usual mean square estimator $s^2$ of $\kappa_2$ is also found. These results give an asymptotic series for the conditional variance of a generalization $\chi^2_s = (n - 1)s^2/E(s^2 \mid m)$ of a statistic discussed by Cochran. Exact formulae for the expectation of the statistic $s^2/m^3$ and its mean square error as an estimator of $\lambda^{-1}$ are given or described. This statistic is a consistent estimator of $\lambda^{-1}$ and has asymptotically an efficiency of $\phi/(\phi + 3)$.

In the first part of this note, we study functions of characteristic functions which are themselves characteristic functions and discussed also a property of analytic characteristic functions. In the second part, an example is constructed to answer a question raised by D. Dugue [3].

Maximum likelihood estimators of the parameters of a $p$-dimensional multinormal population are derived in this paper which are applicable when sample selection and observation is restricted with respect to $x_1$ but otherwise unrestricted with respect to $x_2, \cdots, x_p$. Restrictions imposed may consist of truncation, censoring, or a selection which results in full observation of all sample specimens with respect to $x_1$, but eliminates certain sample specimens from subsequent observation with respect to $x_2, \cdots, x_p$.

With the exception of papers by G. W. Snedecor, G. M. Cox, and H. F. Smith ([8], [9], [10]), there seems to be little about proportional frequencies in the literature. In this paper we consider two-way crossed classifications and two-way nested classifications. The expected values of the sums of squares are obtained in a form which is applicable to a variety of components of variance models. The tests of several hypotheses are considered.

Let $(x_{nk}), (k = 1, 2, \cdots, k_n; n = 1, 2, \cdots)$ be a double sequence of infinitesimal (i.e. $\lim_{n \rightarrow \infty} \max_{1 \leqq k \leqq k_n} P\{|x_{nk}| > \epsilon\} = 0$ for every $\epsilon > 0$) random variables such that for each $n, x_{n1}, \cdots, x_{nk_n}$ are independent. Let $S_n = x_{n1} + \cdots + x_{nk_n}$ and let $F_n(x)$ be the distribution function of $S_n$. For any $a > 0$ let the random variables $x^a_{nk}$ be defined by \begin{equation*}x^a_{nk} = \begin{cases}x_{nk}, & \text{if} -a < x_{nk} \leqq a, \\ 0, & \text{\otherwise},\end{cases}\end{equation*} and let $F^a_n(x)$ be the distribution function of $S^a_n = x^a_{n1} + \cdots + x^a_{nk_n}$. In the next section certain necessary and sufficient conditions are given for $F^a_n(x)$ to converge $(n \rightarrow \infty)$ to a limiting distribution and in particular it is shown that if $F^a_n(x)$ converges to $F(x)$, then $F(x)$ has finite moments of all orders. In Sec. 3 it is shown that if $F^a_n(x)$ converges to $F(x)$ then for each positive integer $k$ the $k$th moment of $F^a_n(x)$ approaches the $k$th moment of $F(x)$ as $n \rightarrow \infty$. We shall call the random variables $(x_{nk})$ a truncated system if there exists a $b > 0$ independent of $k$ and $n$ such that $P\{|x_{nk}| > b\} = 0$. We note that if we start with a truncated system we can choose $a > 0$ such that $x^a_{nk} = x_{nk}$.

The paper considers $\frac{1}{2}_r$ fractional replication designs of the factorial series with $n$ factors each at two levels. The identity relationship for such designs is often written in terms of a symbol $I$ and collections of letters which denote interactions among the factors. These collections may conveniently be called "words," and the number of letters in a collection, the "length" of the word. The problem considered is that of the existence of an identity relationship which contains words of specified lengths. It is known that the words of an identity, together with the symbol $I$, form an Abelian group. The group contains sets of independent generators, and the products of such generators. Necessary and sufficient conditions are developed for the existence of an identity relationship for which the lengths of a set of independent generators and their products are specified. Further, it is shown how to construct such an identity relationship, and it is proved that the identity relationship is unique, apart from renaming the letters. For the more general case in which the lengths of the words are given--but are not associated with particular generators and products--a necessary condition is developed for the existence of the identity relationship. It is shown by example that this condition is not sufficient.

In [1], optimum $\beta$-expectation tolerance regions were found by reducing the problem to that of solving an equivalent hypothesis testing problem. The regions produced when sampling from a $k$-variate normal distribution were found to be of similar $\beta$-expectation and optimum in the sense of minimax and most stringency. It is the purpose of this paper to discuss the "Power" or "Merit" of such regions, when sampling from the $k$-variate normal distribution. Let $X = (X_1, \cdots, X_n)$ be a random sample point in $n$ dimensions, where each $X_i$, is an independent observation, distributed by $N(\mu, \sigma^2)$. It is often desirable to estimate on the basis of such a sample point a region which contains a given fraction $\beta$ of the parent distribution. We usually seek to estimate the center 100$\beta$% of the parent distribution and/or the 100$\beta$% left-hand tail of the parent distribution.

Suppose $g(u_1, \cdots, u_k)$ is a continuous function of its arguments, homogeneous of order $r$, monotonic nondecreasing in each of its arguments, which is positive whenever each of its arguments is positive, and is such that for any given $K(0 < K < \infty)$, there is a number $R(K)(0 < R(K) < \infty)$ such that $g(u_1, \cdots, u_k) < K$ and $u_1 \geqq 0, \cdots, u_k \geqq 0$ imply that $u_1 + \cdots + u_k < R(K)$. Let $U_1, \cdots, U_k$ be chance variables with joint density $e^{-(u_1 + \cdots + u_k)}$ for $u_1 \geqq 0, \cdots, u_k \geqq 0$, and zero elsewhere. For any $t$, define $U(t)$ as $P\lbrack g(U_1, \cdots, U_k) \geqq t\rbrack$. We note that $U(t)$ is a continuous distribution function, with $U(0) = 0$. Let $\rho(v)$ be a bounded nonnegative function with a finite number of discontinuities, defined for $0 \leqq v \leqq 1$. Suppose $X_1, X_2, \cdots, X_n$ are independently and identically distributed chance variables, each with density $f(x), f(x)$ being bounded, and having a finite number of discontinuities and oscillations. $F(x)$ denotes $\int^x_{-\infty} f(x) dx$. Define $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ as the ordered values of $X_1, \cdots, X_n$, and define $T_i$ as $Y_{i + 1} - Y_i(i = 1, \cdots, n - 1)$. Let $R_n(t)$ denote the proportion of the value $$\rho\big(\frac{1}{n}\big)g(T_1, \cdots, T_k), \quad\rho\big(\frac{2}{n}\big) g(T_2, \cdots, T_{k + 1}), \cdots,$$ $$\rho\big(\frac{n - k}{n}\big) g(T_{n - k}, \cdots, T_{n - 1})$$ which are less than or equal to $t/n^r$. Let $\overline{U}\lbrack\lbrack tf^r(x)\rbrack / \{\rho\lbrack F(x)\rbrack\}\rbrack$ be defined as follows. If $f(x) = 0$, $$\overline U\lbrack\lbrack tf^r(x)\rbrack / \{\rho\lbrack F(x)\rbrack\}\rbrack = 0$$ regardless of the value of $t$. If $x$ is such that $f(x) > 0$ and $\rho\lbrack F(x)\rbrack = 0$, then $\overline U\lbrack\lbrack tf^r(x) \rbrack / \{\rho\lbrack F(x)\rbrack\}\rbrack = 1$ regardless of the value of $t$. If $f(x) > 0$ and $\rho\lbrack F(x) \rbrack > 0$, then $\overline U\lbrack\lbrack tf^r(x) \rbrack / \{\rho\lbrack F(x) \rbrack\}\rbrack = U\lbrack\lbrack tf^r (x) \rbrack / \{\rho \lbrack F(x) \rbrack\}\rbrack$. Let $S(t)$ denote $$\int^\infty_{-\infty} \overline{U}\lbrack\lbrack t \cdot f^r(x)\rbrack / \{\rho\lbrack F(x)\rbrack\}\rbrack f(x) dx$$, and let $V(n)$ denote $\sup_{t \geqq 0}|R_n(t) = S(t)|$. Then $V(n)$ converges to zero stochastically as $n$ increases. This generalizes the result of [1], where $k = 1, g(u_1) = u_1, \rho(v) = 1$. The present result may be used to construct tests of fit in the presence of unknown location and scale parameters.

Suppose $X_1, X_2, \cdots, X_n$ are independent and identically distributed chance variables, each with density $f(x)$, where $\int^1_0 f(x) dx = 1, f(x)$ has a finite number of discontinuities, and there are two constants $A, B(0 < A < B < \infty)$ such that $A \leqq f(x) \leqq B$ for all $x$ in $\lbrack 0, 1\rbrack$. Let $Y_0$ denote zero, $Y_{n + 1}$ denote unity, and let $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ be the ordered values of $X_1, X_2, \cdots, X_n$. Define $R_i$ as $Y_i - Y_{i - 1}$ for $i = 1, \cdots, n + 1$. Let $r$ be any positive number greater than unity, and let $V(n)$ denote $\sum^{n + 1}_{i = 1} T^r_i$. The following theorem was proved in [1]. THEOREM A. If $f(x) = 1$ for $x$ in $\lbrack 0, 1\rbrack$, then the distribution of $$\frac{n^{r-1/2}V(n) - \sqrt n\Gamma(r + 1)}{\sqrt{\Gamma(2r + 1) - (r^2 + 1)\lbrack\Gamma(r + 1)\rbrack^2}}$$ approaches the standard normal distribution as $n$ increases. In the present paper, we prove the following generalization of Theorem A: THEOREM 1: The distribution of $$\frac{n^{r-1/2}V(n) - \sqrt n\Gamma(r + 1) \int^1_0 f^{1 - r}(x) dx}{\sqrt{\lbrack\Gamma(2r + 1) - 2r\Gamma^2(r + 1)\rbrack \int^1_0 f^{1 - 2r}(x) dx - \big\lbrack(r - 1)\Gamma(r + 1) \int^1_0 f^{1-r}(x) dx \big\rbrack^2}}$$ approaches the standard normal distribution as $n$ increases. Theorem 1 can be used to compute the asymptotic power of certain tests of fit based on $V(n)$.

The distribution of the number of different locally maximal elements in a random sample is found, where the sampling is from a continuous population of real numbers. This distribution has application in certain nonparametric tests; the problem of finding the distribution may be regarded as identical with the enumeration of permutations according to the number of distinct locally maximal elements.

The first order Gaussian auto-regressive process $(x_t)$ may be defined by the stochastic difference equation \begin{equation*}\tag{1}x_t = \rho x_{t-1} + u_t,\end{equation*} where the $u$'s are NID(0, 1) and $\rho$ is an unknown parameter. The choice of a statistic as an estimator for $\rho$ depends on the initial conditions imposed on the difference equation (1). The so-called "circular" model is obtained by considering a sample of size $N$ and then assuming that $x_{N + 1} = x_1$. An appropriate estimator for $\rho$ in this case is the circular serial correlation coefficient \begin{equation*}\tag{2} r = \frac{\sum^N_{t = 1} x_tx_{t + 1}}{\sum^N_{t = 1} x^2_t}\quad (x_{N + 1} = x_1).\end{equation*} Leipnik [1] has derived an approximate density function \begin{equation*}\tag{3} f(t) = \frac{\Gamma\big(\frac{N + 2}{2}\big)}{\Gamma\big(\frac{N + 1}{2} \Gamma\big(\frac{1}{2}\big)} (1 - 2t\rho + \rho^2)^{-N/2}(1 - t^2)^{(N - 1)/2}\end{equation*} for the estimator $r$. Leipnik also evaluated the first two moments of this distribution. In this paper a formula is obtained which gives $E(r^k)$ as a polynomial of degree $k$ in $\rho$.

In this paper a decision procedure based on the Tukey Studentized range ([5], [6], [8]) has been shown to be an optimum procedure for a particular type of slippage of means of univariate normal populations based on a common but unknown variance. The method given here is similar to that used by Paulson [2] and Truax [7].

Several simple estimates of the mean and standard deviation of a normal population are discussed. The efficiencies of these estimates are compared to the sample mean and sample standard deviation and to the best linear unbiased estimates. Little efficiency is lost when simple rather than optimum weights are used. Since moments of the order statistics are now available for samples of sizes $N \leqq 20$ from normal populations [3] it is a simple matter to find the variances of linear combinations of order statistics. The sample values are denoted $X_1 \leqq X_2 \leqq X_3 \leqq \cdots \leqq X_N$.