In the theory of analytic characteristic functions (c.f.'s), it is well-known that: (1) the order of an entire c.f. cannot be less than unity, unless the function is identically equal to one (P. Levy); and (2) a necessary and sufficient condition (NASC) for a distribution function (d.f.) $F(x) \neq \epsilon (x)$ to be a "finite" d.f. is that its c.f. be an entire function of order one and of exponential type (G. Polya). (Throughout this paper, $F(x)$ will invariably denote a d.f., and $f(t)$ the corresponding c.f.) These two results lead us naturally to the investigation of conditions under which a d.f. will have an entire c.f. (i) of order one and of maximal type, or (ii) of given finite order greater than one, or (iii) of infinite order. Sufficient conditions for (ii) were obtained for absolutely continuous d.f.'s by D. Dugue [2], and for general d.f.'s by E. Lukacs (see [4], p. 142). The scope of the present paper extends beyond the problems (i), (ii) and (iii) above. In Section 6 below, we obtain NASC's for a d.f. to have an entire c.f. of given finite order greater than one. Section 7 deals with NASC's for a d.f. to have an entire c.f. of given finite order greater than one and given type (maximal, intermediate, or minimal). In Section 8, we consider entire c.f.'s of order one: we first obtain NASC's for a d.f. to have an entire c.f. of order one and maximal type; next, we obtain a relation between the extremities of a "finite" d.f. $\neq \epsilon (x)$, and the type of the corresponding entire c.f. (which is of order one); and finally, we obtain as a corollary the fact that there cannot exist an entire c.f. of order one and of minimal type. These constitute the main results of this paper. In addition to the above, we obtain in Section 3 a theorem on the moments of a d.f., and in Section 4 a theorem on the interval of existence of the integral defining the moment-generating function of a d.f., these theorems being closely related in form and in content to the main results mentioned above. Allied results on analytic c.f.'s which are not entire, and on entire c.f.'s of infinite order are given in Section 9.

It is known that if $X$ and $Y$ are independent random variables having a Gamma distribution with parameters $(\theta, n)$ and $(\theta, m)$, i.e., with density function $p(x, \theta, n) = \frac{\theta^{n/2}x^{n/2 - 1}e^{-(\frac{1}{2})\theta x}}{2^{n/2}\Gamma(n/2)},\quad 0 < x, \theta; 1 \leqq n,$ then $X + Y$ and $X/(X + Y)$, or equivalently $X/Y$, are statistically independent. Lukacs [1] proved that this independence property characterizes the Gamma distribution, namely, if $X$ and $Y$ are two nondegenerate positive random variables, and if $X + Y$ is independent of $X/(X + Y)$, or equivalently of $X/U$, then $X$ and $Y$ have a Gamma distribution with the same scale parameter. In the present paper we present an extension to the case where $U$ and $V$ are symmetric positive definite matrices having a Wishart distribution. A number of difficulties are encountered in the generalization. First, there is no natural extension of a ratio, and we consider $Z = W^{-1}UW'$ $^{-1}$, where the "square root" $W = (U + V)^{\frac{1}{2}}$ is any factorization $WW' = (U + V)$. In the matrix case $Z$ is not a function of $V^{-\frac{1}{2}}UV^{-\frac{1}{2}}$ as was true in one dimension, and indeed if $U$ and $V$ are independent random matrices having a Wishart distribution, $U + V$ and $V^{-\frac{1}{2}}UV^{-\frac{1}{2}}$ need not be statistically independent, depending on which square root is chosen. This aspect will be treated in another paper. In the univariate case it is relatively straightforward to generate differential equations by differentiating under the expectation sign, but this is no longer true since the elements of $(U + V)^{\frac{1}{2}}$ do not bear a simple relation to the elements of $(U + V)$, and it is this point which leads to the difficulties in the proof. The characterization theorem is stated in Section 2. In Section 3 the differential equation is set up, and is solved in Section 4. The authors are indebted to Martin Fox for his comments and suggestions.

The work of Bush and Mosteller [1] on stochastic learning models leads to certain limiting distributions of response probabilities which are of a somewhat different character from those commonly encountered in mathematical statistics. These authors have derived many useful relations between the moments of the distributions but, in deducing the general nature of the distributions themselves, they have relied chiefly on Monte Carlo methods. In the present paper, we consider the two experimenter-controlled events model and, in the "equal alpha" case, derive limiting distributions for certain special cases by inverting the solution of a moment generating functional equation.

Let $Z_1, Z_2, \cdots, Z_n$ be jointly normally distributed random variables with $EZ_i = 0, EZ^2_i = 1, EZ_iZ_j = \rho, i \neq j, -1/(n - 1) \leqq \rho \leqq 1$. Let the collection of random variables $\{Z_i\}$ be ordered so that $Z^{(1)} \geqq Z^{(2)} \geqq \cdots \geqq Z^{(n)}$. It is the purpose of this note to show how the moments and product moments of the $\{Z^{(i)}\}$ for any $\rho$ can be obtained from the corresponding moments and product moments of the $\{Z^{(i)}\}$ for $\rho = 0$.

Numerous contributions have been made to the problem of order statistics in samples from normal and exponential populations. For the problem of location with symmetry Fraser [1] derived a locally most powerful rank test against normal alternatives. It is the Wilcoxon test statistic with the ranks replaced by the corresponding expected values of order statistics in a sample from the chi-distribution with one degree of freedom. Gupta [5] considered the order statistics from the standardized gamma distribution with the parameter $r$ defined on the positive integers (that is, from the chi-distribution with even degrees of freedom) and derived expressions for the $k$th moments of an order statistic and the covariance between two order statistics. He also presented a table of numerical values of the $k$th moments of an order statistic accurate to six magnificant digits for $k = 1 (1) N, N \leqq 15$ and $r = 1 (1) 5$, where $N$ is the sample size. It might be of interest to consider the problem of order statistics in samples from chi-populations with odd degrees of freedom. However, this problem seems to be more difficult than the one considered by Gupta [5]. In the present paper, the expected values for samples to size four and the mixed and second moments (about the origin) for samples to size five, drawn from the chi-population (1 d.f.) have been evaluated. Numerical values of these to eight decimal places are computed. Section 2 contains general formulae and some definite integrals used in the computation. The results in Section 3 have theoretical interest in showing the relationships between moments of order statistics from chi (1 d.f.) and the standard normal distributions. In Section 4, there is a discussion about the number of integrals required to evaluate the first, second and mixed moments of order statistics for each $N$, given these moments to $N - 1$ and the existence of the tables for the normal distribution. There is also a discussion about the cumulative rounding error involved in using the formulae recurrently.

A treatment is given of the M/G/1 queue with interruptions of Poisson incidence occasioned either by server breakdown or the arrival of customers with higher priority. Interruption times and priority service times have arbitrary distribution. After pre-emptive interruption, ordinary service is either repeated or resumed. The time dependent behavior of the system is discussed in a complete state space and the joint density in all system variables of this space is constructed systematically from the densities associated with a set of simpler first-passage problems. Equilibrium distributions are available as limiting forms and server busy period distributions obtained.

In a single-server first-come-first-served queue the waiting-times of successive customers are related by the equation \begin{equation*}\tag{1}w_{n + 1} = \lbrack w_n + U_n\rbrack^+\end{equation*} where \begin{equation*}\tag{2}U_n = S_n - T_n.\end{equation*} Here $S_n$ and $T_n$ are the service-time of the $n$th customer, and the time between the arrivals of the $n$th and the $(n + 1)$th customers, respectively. In the particular case of mutually independent identically distributed $U_n$ the basic investigation of this equation was carried out by Lindley [1], who found a simple necessary and sufficient condition for the existence of a stationary waiting-time distribution and derived this distribution in certain special cases. The theory was developed by Smith [2], who under fairly weak conditions gave a systematic treatment of the Wiener-Hopf equation obtained by Lindley. In the less restricted case when $U_n$ is a strictly stationary process it has been shown by the author (Loynes [3]) that under a simple condition the existence of a unique stationary distribution is again ensured: it is the purpose of this paper to show how (when this condition is satisfied) the stationary distribution may sometimes be found, and to obtain some qualitative results. The theory will be developed in Sections 3 and 4, and some examples discussed in Section 5. On the assumption that $U_n$ is a metrically transitive sequence the condition just referred to is \begin{equation*}\tag{3}E\lbrack U_n\rbrack < 0,\end{equation*} and we shall therefore suppose this satisfied throughout the paper. The assumption that $U_n$ is metrically transitive does not affect most of the arguments in this paper, but it will be made for convenience. For the existence of a unique stationary distribution (3) is both necessary and sufficient, but there is sometimes more than one stationary distribution when the inequality in (3) is replaced by equality. Such situations have been ignored here for several reasons: they are not very common, some of the arguments either break down or need adaptation, and in any case the problems are not then as difficult, for it follows from equation (7) of [3] that the function $\psi$ occurring in (5) below is then identically equal to one, so that the only unknown is $\phi$. There is never a stationary distribution when the inequality in (3) is reversed. Throughout the entire investigation we shall suppose the queue has a structure which satisfies the following condition (H). This is actually no restriction at all when the queue is in the stationary state, since the variables $z_n$ occurring in it may be taken as $(W_{n + 1}, T_{n + 1})$, where $\{W_n\}$ is the stationary waiting-time sequence. However, all the results will refer to the distribution of the waiting-time conditional on $z_n$, so that it will always be tacitly assumed that the distribution of $z_n$ is known in order to allow deductions about the unconditional distribution of the waiting-time. (H): There exists a sequence $\{z_n\}$ (defined on the same probability space as $w_n$) of random vectors--i.e., in finite-dimensional Euclidean space--with the following properties: (i) $\{z_n, S_n, T_n\}$ is a strictly stationary process. (ii) $S_n, T_n$, and $w_n$ are conditionally independent given $z_{n - 1}, z_n$. (iii) $w_n$ and $z_n$ are conditionally independent given $z_{n - 1}$. The assumption that (H) is satisfied allows us to write down the equation satisfied by the stationary (conditional) waiting-time distribution function. We cannot, however, in such a general situation actually find the solution to the equation, and to discover appropriate further restrictions which may make this possible we may consider further the case treated by Lindley and Smith. The general solution given by Smith was analytically complex and apparently only obtainable from the Wiener-Hopf technique or the equally specialized method due to Spitzer [4]. If, however, either the service-times or the inter-arrival times had a distribution with a rational characteristic function the solutions were much simpler in form, and in at least a few cases had been obtained previously by other methods. With this in mind we impose similar conditions here on the conditional characteristic functions; on that of the interarrival time in Section 3, and on that of the service-time in Section 4. The lack of success in treating the problem without these restrictions is probably no great loss in practice, in view of the complex nature of the Wiener-Hopf solution for the simple case. The results obtained under the conditions imposed so far are in the nature of aids to the solution of the problem, rather than the solution itself. A simple situation for which the required distribution may then be obtained straightforwardly occurs when the following "finite-matrix" condition is satisfied: (HF): Condition (H) holds, and $z_n$ has (with probability one) only a finite number $k$ of different possible values. An interesting result obtained by Smith showed that, roughly speaking, the characteristic functions of the waiting-time and the service-time had the same number of poles, and a somewhat similar result related the distribution of the inter-arrival times to that of the waiting-time. Under (HF) it will be found that similar results are also true. Smith in fact gave explicit formulae of a simple type for the two special cases, but we shall content ourselves here with pointing out how corresponding formulae could be obtained, since it appears that they are no longer simple when $k$ has a value other than one. As we have already remarked, all simple queues are included in (H) when in the stationary state. Even under the condition that the distribution of $z_n$ be known, a large class of queues satisfies (H), such as that in which $S_n$ and $T_n$ are independent stationary Markov processes. An example of a queue of considerable interest which apparently does not admit a specification of this type is that in which customers are due at regular intervals but are late independently with a lateness distribution which extends to $\infty$ (if the lateness distribution and service time distribution are both negative exponential, then (H) is satisfied, and a brief discussion of this example will be found in Section 5). It will be seen that by taking $z_n = 1$ the classical case with independence is included under (HF); two other types of queue satisfying (HF) are of some interest, and are discussed in Section 5--one being a simple queue whose input is a mixture of two streams of customers, and the other a queue whose input consists of customers who have already been served in a first queue. In addition to any other queues which may fall into this class, it is presumably true, that any queue may be approximated by those of this "soluble" class, although there are of course very great difficulties in deciding how such an approximation should be carried out. Finally we observe that since the content of a semi-infinite dam in discrete time is described by the same equation (1), the results obtained here may be applied in that context also, and in particular the finite-matrix case gives a class of dams with serially dependent inputs for which explicit solutions can be found.

The history of fiducial probability dates back over thirty years and so is long by statistical standards; however thirty years have not proved long enough for agreement to be reached among statisticians as to the derivation, manipulation or interpretation of fiducial probability. The reason for this lack of agreement and the resulting controversy is possibly due to the fact that the fiducial method has been put forward as a general logical principle, but yet has been illustrated mainly by means of particular examples rather than broad requirements. This paper explores in two respects certain natural general requirements for the application of the fiducial argument in the bivariate case that have been proposed. An essential step in the process of fiducial inference is the derivation of the fiducial distribution. Properties that have been emphasized by Fisher in order that a genuine fiducial distribution may be obtained include sufficiency, the absence of any a priori information and the existence of a pivotal quantity. About ten years ago examples were discovered demonstrating that the distribution induced by a pivotal, sufficient and smoothly invertible set of quantities is not necessarily unique; that is to say the induced distribution depends on the particular set of pivotal quantities chosen (see [50]). the pivotal quantities used in these examples were denounced as having been "artificially constructed" (see [15]), and additional requirements were proposed. In the bivariate case requirements reduced to either $f(x, y \mid \alpha, \beta) = f(x, \alpha)f(y \mid x, \alpha, \beta)$, or $\alpha, \beta) = f(x, \alpha, \beta)f(y \mid x, \beta),$ $f(x, y \mid \alpha, \beta)$ being the given density. However as will be seen in this paper, neither of these requirements is sufficient to ensure the uniqueness of the resulting distribution. It may be argued (see [11]) that non-uniqueness of the induced distribution is no handicap. It is, however, more than intuitively appealing to feel that if one starts with a problem perfectly symmetrical in the parameters and the random variables, as the examples presented are, then the answer should be perfectly symmetrical as well. Recently Lindley proved a theorem in the one-dimensional case to the effect that a fiducial distribution is Bayes' posteriori if and only if the c.d.f. is invariant under a continuous one-parameter group of transformations. It is natural to seek an extension of this theorem to the multi-parameter case, for if it can be shown that there is a group of a certain form, then there exists an essentially unique pivotal, a frequency interpretation and certain consistency properties (see [21]). It is known in the multi-parameter case that if the c.d.f. is invariant under a particular type of group, then the fiducial distribution is Bayes' posteriori. However the final example presented in this paper demonstrates that a bivariate fiducial distribution may be Bayes' posteriori without possessing strong group invariance properties.

In the classical linear estimation set-up, we have \begin{equation*}\tag{1}E\mathbf{y} = X\mathbf{\theta},\end{equation*} where $\mathbf{y}' = (y_1, \cdots, y_n)$ is a random vector whose components are uncorrelated and have equal variance, $\mathbf{\theta}' = (\theta_0, \theta_1, \cdots, \theta_p)$ is a vector whose elements are unknown constants and $X$ is a matrix of $n$ rows and $p + 1$ columns, $n \geqq p + 1$, which has full rank and whose elements are known constants. Plackett [2] gives a historical note on the least squares estimator $\hat{\mathbf{\theta}} = (X'X)^{-1}X'\mathbf{y}$ of $\mathbf{\theta}$ for which the following property is well-known, (I) Each component $\hat{\theta}_j$ of $\hat{\mathbf{\theta}}$ is the estimator with uniformly minimum variance among all unbiased linear estimators of the corresponding component $\theta_j$ of $\mathbf{\theta}$. It can also be easily seen that (II) For a quadratic loss function for the estimation of each component $\theta_j$ of $\mathbf{\theta}$, the least squares estimators have uniformly minimum risk among the class of all linear (in $y$'s) estimators with bounded risk. Properties (I) and (II) hold for model (1) with uncorrelated and homoscedastic $y_1, \cdots, y_n$. Hodges and Lehmann [1] have shown that if we do not restrict ourselves to estimators which are linear in $y$'s, then the least squares estimators have the following weaker property: (III) If the loss in estimating the true vector $\mathbf{\theta}$ by another $\mathcal{B}$ is $(\mathbf{\theta} - \mathcal{B})' (\mathbf{\theta} - \mathcal{B})$, then the least squares estimator is minimax among the class of all estimators of $\mathbf{\theta}$ if there exists a number $v$ such that $\operatorname{Var} y_i \leqq v, i = 1, \cdots, n$, and the family of distributions of $\mathscr{F}$ of $(y_1, \cdots, y_n)$ contains the sub-family $\mathscr{F}_0$ of all independent normal distributions of $(y_1, \cdots, y_n)$ which satisfy (1) for some $\mathbf{\theta}$, and have $\operatorname{Var} y_i = v, i = 1, \cdots, n$. In many situations, however, $(y, x_1, \cdots, x_p)$ follows a $(p + 1$-variate distribution on which observations are made and the method of least squares is applied to estimate the linear regression of $y$ on $x_1, \cdots, x_p$, regarding the $x$-observations to be non-stochastic. This problem differs from the classical problem of linear estimation because instead of (1) the model is $E\lbrack\mathbf{y} \mid X\rbrack = X\mathbf{\theta}$, where the elements of the $X$ matrix are stochastic. For reasons given in Section 2, the loss in estimating the true regression function $\phi(x_1, \cdots, x_p)$ by another function $\psi(x_1, \cdots, x_p)$ is considered to be of the form, $\int \lbrack\phi(x_1, \cdots, x_p) - \psi(x_1, \cdots, x_p)\rbrack^2 dF(x_1, \cdots, x_p),$ where $F$ is the distribution function of $(x_1, \cdots, x_p)$. For the above loss function, it is shown under certain conditions that if the class of estimates which are linear in $y$'s and have bounded risk is non-empty, then the estimate obtained by the method of least squares belongs to this class and has uniformly minimum risk in this class. A necessary and sufficient condition on $F(x_1, \cdots, x_p)$ is obtained for this class to be non-empty, which unfortunately is not easy to verify in particular cases. However, by a sequential modification of the sampling scheme, this condition may always be satisfied at the cost of an arbitrarily small increase in the expected sample size. It is also shown under certain further conditions on the family of admissible distributions that the least squares estimator is minimax in the class of all estimators. For the case of multivariate normal distribution of $(y, x_1, \cdots, x_p)$, Stein [3] has considered this problem under a loss function similar to the one given above. He has shown the minimax property of the least squares estimates (which also happen to be the maximum likelihood estimates in a multivariate normal model) for the regression coefficients, and has raised many interesting questions about the admissibility of these estimates.

In [2], the exact power of Mood's and Massey's two-sample tests for discriminating between two populations was derived. Two types of alternatives were considered--change in the location of an exponential distribution and change in the location and scale of a rectangular distribution. The asymptotic relative efficiency of Mood's test based on the median against an alternative of change in location of a normal distribution was shown ([1], [8]) to be $2/\pi$. A limited comparison of powers of the tests based on the median, on the first quartile and the median and on the likelihood-ratio against the exponential alternative is given in [3]. In this paper, the asymptotic relative efficiencies of Mood's test based on the median and Massey's test based on the first quartile and the median are shown to be zero, when these tests are compared against the likelihood-ratio test appropriate for detecting a shift in location of an exponential distribution. Massey's test is found to be about three times as efficient as Mood's test, for exponential distribution. But so also is the test based on the first quartile alone. If the order of the fractile is lowered, the efficiency of the test based on it is increased. Similar comparisons are also made for the normal distribution.

This paper derives and exhibits the optimum Bayes solution to the following problem: Given a continuous-time Poisson process with unknown mean occurrence rate $\lambda$; to decide whether $\lambda > k$ or $\lambda < k$. The prior distribution is taken to be of Gamma type, with positive mean and finite variance. The cost of observation is taken proportional to the length of time the process is observed, and the cost of a wrong decision proportional to $|\lambda - k|$. The decision rule derived is optimum (in the sense of minimum expected cost) among all non-randomized sequential rules. Some of the results hold true, of course, for other cost functions and/or prior distributions. A method for treating the same problem with the inclusion of a constant setup cost is also given.

Jackson and Bradley (1959, 1961a, 1961b) developed and studied a sequential (multivariate) $T^2$ test of hypotheses on a vector of means, and an analogous $\chi^2$ test for known covariance structure. The present paper presents the results of Monte Carlo sampling on the operating characteristics and average sample numbers (ASN) of these tests. Consideration is restricted to the behavior of these tests at specific null and alternate hypotheses $(H_0$ and $H_1)$ with nominal $\alpha$ and $\beta$ errors of .05. The empirical $\alpha$ and $\beta$ errors are, in general, less than .05 and appear to decrease as the number of variables increases. The empirical ASN are appreciably smaller than the corresponding fixed sample sizes, and approximate the ASN that Jackson and Bradley obtained using Bhate's conjecture. The estimation of the fixed sample sizes were based on the nominal $\alpha$ and $\beta$ errors of .05 while the sequential test ASN were, of course, associated with the resulting smaller error probabilities. Thus the true advantage of the sequential test is understated by the above sample size comparisons. This study investigates the behavior of the sequential test at $H_0$ and $H_1$ only. A similar study involving points between $H_0$ and $H_1$ would be of definite values for (1) ascertaining whether the advantages of the sequential procedure hold under situations other than $H_0$ and $H_1$, and (2) suggesting methods to overcome the conservatism of the test as it now stands.

Rotatable designs were introduced by Box and Hunter (1954, 1957) for the exploration of response surfaces. They constructed these designs through geometrical configurations and obtained several second order designs. Afterwards, Gardiner and others (1959) obtained some third order designs through the same technique for two and three factors and a third order design for four factors. Bose and Draper (1959) obtained some second order designs by using a different method. Draper (1960 a) gave a method of construction of an infinite series of second order designs in three and more factors. Recently, Box and Behnken (1960 a) have obtained a class of second order rotatable designs from those of first order. Draper (1960 b) has obtained some third order rotatable designs in three dimensions and a third order rotatable design in four dimensions. Das (1961) has obtained such designs, both second and third orders up to 8 factors as fractional replicates of factorial designs. The method of construction of the designs presented in this paper is essentially based on that presented by Das (1961). After the manuscript of this paper was submitted for publication the authors' attention was drawn to the work of Box and Behnken (1960 b). They have obtained some second order designs by following a procedure which uses balanced incomplete block designs in the same manner as described below. They did not, however, extend the method to include other complementary sets of points which, as will be shown, allow one to obtain rotatable second and third order designs based on any balanced incomplete block design. In the present paper a method has been given by using the properties of balanced incomplete block designs through which second order rotatable designs with any number of factors, with a reasonably small number of points, can be obtained. By extending the method, third order rotatable designs, both sequential and nonsequential, up to 15 factors have been obtained with the help of doubly balanced incomplete block designs and complementary B.I.B. designs.

The problem of constructing fractional replicates of the $s^m$ designs, where $s$ is a prime power is not new in literature. There are several papers which deal with this problem. However, so far as the subject matter of this paper is concerned, the contributions made by Banerjee [2], Rao [8], Dykstra [6] and very recently by Addelman [1] are of special interest. This is yet another attempt in the same direction. The basic concept is the same as in [8], where Rao gives a method of obtaining block designs for the fractional replicates of the $s^m$ designs so as to estimate the main effects and the two-factor interactions orthogonally assuming all other interactions to be absent. With the same assumptions, a method of construction is given in this paper which gives in many cases block designs for the fractional replicates of the $2^m$ designs with lesser number of treatment combinations than that of the corresponding fractional designs given earlier. This is achieved by allowing the estimates to be correlated. The scheme allows the estimates of treatment-effects and block effects to be mutually orthogonal. An additional example is given at the end to indicate the possibility of improving the construction.

Let $\mathfrak{F}$ be the collection of cumulative distribution functions on $(- \infty, \infty)$ and $\mathfrak{F}_{\lbrack a, b\rbrack}$ that subset of $\mathfrak{F}$ all of whose elements have $F(a - 0) = 0$ and $F(b) = 1$. Let $\mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)} (\mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack a, b\rbrack})$ be the class of cumulative distribution functions on $(- \infty, \infty) (\lbrack a, b\rbrack)$ whose first $k$ moments are $\mu_1, \mu_2, \cdots, \mu_k$ respectively. We will suppose throughout that $\mu_1, \mu_2, \cdots, \mu_k$ is a legitimate moment sequence, i.e., that there exists a cumulative distribution function $F(x) \varepsilon \mathfrak{F} (\mathfrak{F}_{\lbrack a, b\rbrack})$ whose first $k$ moments are $\mu_1, \mu_2, \cdots, \mu_k$. Let $g(x)$ be a continuous and bounded function on $\lbrack a, b\rbrack$. Then, we wish to determine $F^\ast (x) \varepsilon \mathscr{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack a, b\rbrack}$ with \begin{equation*}\tag{1}\int^b_a g(x) dF^\ast (x) = \min (\max)_{F\varepsilon \mathfrak{F}^{{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack a, b\rbrack}}} \int^b_a g(x) dF(x).\end{equation*} Any $F^\ast (x)$ satisfying (1) will be called an extremal distribution with respect to $g(x)$. Let $\mathscr{G}^{(k)}_{\lbrack a, b\rbrack}$ be the set of continuous, bounded, and monotonic functions on $\lbrack a, b\rbrack$, whose first $k$ derivatives exist and are monotonic in $(a, b)$. In addition, we further require that $\mathscr{G}^{(k)}_{\lbrack a, b\rbrack}$ contain only functions not linearly dependent on the monomials $1, x, x^2, \cdots, x^k$. This paper characterizes the extremal distributions for $g(x) \varepsilon \mathscr{G}^{(k)}_{\lbrack a, b\rbrack}$. The results are extended to $\mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack 0, \infty)}$ and $\mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}$, in that we investigate determining $\inf (\sup)_F \varepsilon\mathscr{F}^{(\mu_1, \mu_2, \cdots, \mu_k)}_{\lbrack 0, \infty)} \int^\infty_0 g(x) dF(x) \text{and} \inf (\sup)_F \varepsilon \mathfrak{F}^{(\mu_1, \mu_2, \cdots, \mu_k)} \int^\infty_{-\infty} g(x) dF(x).$ These results are then applied to the computation of bounds on the moment generating function, knowing the first $k$ moments, in some specific cases. The methodology is largely a straightforward extension of results in an earlier paper by the author [1].

The role of serial correlations in time series analysis is well known. Considerable attention has been given to the derivation of their sampling properties when the sample size is both small and large. In all these discussions it has been tacitly assumed that these correlations are bounded between -1 and 1. At least, no literature exists which considers it otherwise. Whereas it is true that the serial correlations are bounded it is not true that the bounds are -1 and 1. In fact, in small samples these bounds may very well be lower than -1 and higher than 1. To the best of the author's knowledge, this fact has not been mentioned anywhere. The purpose of this note is to discuss this particular aspect.

The sums of squares and cross products of groups of random variables drawn from a multivariate normal population whose covariance matrix has a certain pattern indicating intra-class correlations are shown to follow the Wishart distribution. It follows immediately from this result that the usual distributions of the simple, partial, and multiple correlation coefficients obtain, although with non-centrality parameters that reflect the effect of the intra-class correlations.

The purpose of this paper is to establish the monotonicity property of some tests suggested by Roy and Gnanadesikan [2] for the problem of testing the null hypothesis of equality of two dispersion matrices against some specific alternatives. If $\Sigma_1$ and $\Sigma_2$ denote the dispersion matrices of two non-singular $p$-variate normals and $\gamma_1, \gamma_2, \cdots, \gamma_p$ denote the characteristic roots (all positive) of $\Sigma_1\Sigma^{-1}_2$, then the null hypothesis is $H_0$: all $\gamma_i$'s are equal to unity. The alternative hypotheses to be considered are: (i) $H_1 : \gamma_m > 1$; (ii) $H_2 : \gamma_M < 1$; (iii) $H_3 : \gamma_M > 1$; (iv) $H_4 : \gamma_m < 1$, where $\gamma_m$ and $\gamma_M$ denote, respectively, the smallest and the largest of the $\gamma_i$. Let us denote the largest and smallest characteristic roots of any square matrix $A$ by $\operatorname{ch}_{\max} (A)$ and $\operatorname{ch}_{\min} (A)$, respectively.