Let $I$ be a countable set with elements $i, j, k, \cdots$. Elements of $I$ are called "types." Let $T$ be the collection of all sequences $z = \{z(i):i \varepsilon I\}$ of non negative integers of which all, but at most finitely many, are 0. We shall identify elements of $T$ as real valued functions defined on $I$. The sum of two elements of $T$ is again an element of $T$. The 0 function shall be denoted by 0. The function which takes the value 1 at $i$ and 0 elsewhere shall be designated by $e_i$. Our branching process is a Markov chain $Z_0, Z_1, Z_2, \cdots$ with state space $T$ and stationary transition probabilities described as follows: The conditional distribution of $Z_{n + 1}$ given $Z_n = e_i$ is $p_i$ where $p_i$ is a probability distribution on $T$. Let $z \varepsilon T \cdot z(i_1) = n_1, \cdots, z(i_k) = n_k; z(i) = 0$ if $i$ is not one of $i_1, \cdots, i_k$. Then the conditional distribution of $Z_{n + 1}$ given $Z_n = z$ is the distribution of a sum of $n_1 + \cdots + n_k$ independent random variables, taking values in $T$, of which $n_1$ have distribution $p_{i_1}, \cdots$, and $n_k$ have distribution $p_{i_k}$. Finally, $\mathrm{Pr}\{Z_{n + 1} = 0 \mid Z_n = 0\} = 1$ completes the description of the transition probabilities of the process. Let $Z_n(i)$ be the $i$th component of $Z_n$, i.e., $Z_n(i) = z(i)$ if $Z_n = z. Z_n(i)$ represents the size of the population of type $i$ in the $n$th generation, and $\sum_i Z_n(i)$ represents the size of the total population of the $n$th generation. Let $(m_{ij})$ be the expectation matrix of the process, i.e., $m_{ij} = E\{Z_{n + 1}(j) \mid Z_n = e_i\}$. Let $m^{(n)}_{ij}$ be defined inductively by $m^{(n)}_{ij} = m_{ij},\quad m^{(n + 1)}_{ij} = \sum_k m^{(n)}_{ik} m_{kj}.$ We shall assume that all $m^{(n)}_{ij}$ are finite and the matrix $(m_{ij})$ is irreducible. For an extended real valued function $f$ defined on $I$, we define functions $Mf, fM$ by $Mf(i) = \sum_j m_{ij}f(j);\quad fM(j) = \sum_if(i)m_{ij},$ whenever the right sides of the equalities are well defined. Note that $Mf, fM$ are always well defined if $f$ is non negative since $m_{ij}$ are non negative. (We shall always adopt the following conventions concerning addition and multiplication involving $\infty$: \begin{align*}a + \infty = \infty + a = \infty\quad\text{if}\quad a > - \infty, \notag \\ a \cdot \infty = \infty \cdot a = \infty\quad\text{if}\quad a > 0, \notag \\ = 0\quad\text{if} \quad a = 0.\end{align*} Therefore, we have \begin{equation*}\tag{1}E\{Z_{k + n}(j) \mid Z_k = e_i\} = m^{(n)}_{ij}\end{equation*} and \begin{equation*}\tag{2}E\{Z_{k + n}(j) \mid Z_k = z\} = zM^n(j).\end{equation*} If $f$ is a non negative function, then \begin{equation*}\tag{3}E\{\sum_j f(j)Z_{k + n}(j) \mid Z_k = z\} = \sum_j zM^n(j)f(j) = \sum_i z(i)M^nf(i).\end{equation*} In [6] it is shown that there is a unique non negative number $r$ which is the common radius of convergence of the power series $\sum^\infty_{n = 1} m^{(n)}_{ij} s^n$. We shall always assume $r > 0$. In this paper we shall extend the far reaching theorem of Everett, Ulam and Harris for a branching process with finitely many types to a branching process with countably many types. Let us first describe the above mentioned theorem. For a branching process with $k$ types, the expectation matrix $(m_{ij})$ is a $k \times k$ matrix. The matrix is assumed to be positive regular. It has a largest positive eigenvalue $\rho$ with corresponding right and left eigenvectors $u, v$. If we normalize $Z_n$ by dividing by $\rho^n$, the sequence $\{Z_n\rho^{-n}\}$ of vector valued random variables converges to a vector valued random variable $W$ with probability 1, if $\rho > 1$. Furthermore, the direction of $W$, if $W \neq 0$, coincides with that of $v$. The theorem has been proven under some conditions on the existence of moments of $Z_1$ in [1], [2], [3]. And recently Keston and Stigum have proven the theorem under less stringent conditions [4]. In fact, the theorem has attained its best possible form in their work. All these works are based on the theory of ergodic behavior of $(m^{(n)}_{ij})$ which is described completely in the classical Perron Frobenius theory of positive matrices. For an infinite, irreducible, non negative $(m_{ij})$ the number $r^{-1}$ shall take the place of $\rho$ of the finite case, in view of the fact that the radii of convergence of series $\sum^\infty_{n = 1} m^{(n)}_{ij}s^n$ are all equal to $\rho^{-1}$ for the finite case. The ergodic behavior of $(m^{(n)}_{ij})$ for an infinite matrix is far more complicated than its counterpart for a finite matrix. However, it has been worked out in great detail in [5]. There are two different but exhaustive cases: CASE I, $\sum^\infty_{n = 1} m^{(n)}_{ij} r^n < \infty\quad\text{for all}\quad i, j$; CASE II. $\sum^\infty_{n = 1} m^{(n)} r^n = \infty\quad\text{for all}\quad i, j$. In Case II there are two strictly positive functions $u$ and $v$ such that \begin{equation*}\tag{4}rvM(i) = v(i)\quad\text{for all}\quad i;\end{equation*} \begin{equation*}\tag{5}rMu(i) = u(i)\quad\text{for all}\quad i.\end{equation*} They are unique (up to constant multiples), non negative functions satisfying (4) and (5) respectively. The sum $\sum_i u(i)v(i)$ may be finite or infinite. In either Case I or Case II with $\sum_i u(i)v(i) = \infty$ we have $\lim_{n \rightarrow \infty} m^{(n)}_{ij} r^n = 0$. This being the case, if the process is initiated by finitely many particles $(\mathrm{Pr} \{Z_0 = z\} = 1 \text{for some} z \varepsilon T),\\E\{r^nZ_n(j)\} = r^nzM^n(j) = r^n\sum_i z(i)m^{(n)}_{ij} \rightarrow 0$ as $n \rightarrow \infty$. Hence all sequences $\{r^nZ_n(j)\}$ converge in mean to 0. One might say that the conclusion of the theorem of E. U. H. remains valid with $W = 0$ and mean convergence instead of convergence with probability 1. The most interesting case we shall analyze in detail is Case II with $\sum_i u(i)v(i) < \infty$. In this case the infinite matrix behaves strikingly like a finite matrix. We are able to use an approach similar to that of Harris, which may be described as the mean square approach. A condition is imposed on distributions $p_i$ so that all $Z_n(i)$ have finite second moments. We remark that there are other varieties of conditions which may serve the same purpose, but the one we have chosen has the appeal of being simple in appearance. The main result is Theorem 1 in which we prove the mean square convergence of $\{r^nZ_n(j)\}$.

This paper introduces the concept of epsilon-delta entropy for "probabilistic metric spaces." The concept arises in the study of efficient data transmission, in other words, in "Data Compression." In a case of particular interest, the space is the space of paths of a stochastic process, for example $L_2\lbrack 0, 1\rbrack$ under the probability distribution induced by a mean-continuous process on the unit interval. For any epsilon and delta both greater than zero, the epsilon-delta entropy of any probabilistic metric space is finite. However, when delta is zero, the resulting entropy, called simply the epsilon entropy of the space, can be infinite. We give a simple condition on the eigenvalues of a process on $L_2\lbrack 0, 1\rbrack$ such that any process satisfying that condition has finite epsilon entropy for any epsilon greater than zero. And, for any set of eigenvalues not satisfying the given condition, we produce a mean-continuous process on the unit interval having infinite epsilon entropy for every epsilon greater than zero. The condition is merely that $\sum n\sigma_n^2$ be finite, where $\sigma_1^2 \geqq \sigma_2^2 \geqq \cdots$ are the eigenvalues of the process.

It is shown that the Kiefer-Wolfowitz procedure for estimating the maximum of a regression surface converges almost surely to the maximum if this is the only stationary point of the surface and some other conditions are satisfied. The result is in a sense stronger than those existing presently.

Brillinger [2] gives necessary and sufficient conditions for a model to be invariant under a Lie group of transformations. The problems that can be handled by his conditions are surveyed, and found effectively to be restricted to one-dimensional problems amendable to Lindley's [8] method and to problems connected with conflicts between Bayes' and fiducial theory. The problem of finding the general model invariant under a given group is proposed. Brillinger's theorem produces differential equations for the model. A general solution can be obtained by direct methods.

In recent years considerable research has been devoted to a class of problems which is concerned with ranking and/or selecting a subset of $k$ given populations where the ranking or the selection is defined in terms of a (scalar) parameter of the populations. In these problems the interest is often centered on the populations having large (small) values of the ranking parameter. One usually refers to the $t(< k)$ populations with largest (smallest) values of the ranking parameter as the $t$ best populations. In this paper we consider a problem of selecting a subset of specified size, from a given set of $k$ populations, which contains a subset of the $t$ best populations. Suppose that $\prod_1, \prod_2, \cdots, \prod_k$ is a given set of $k$ populations, where the distribution function of each observation from $\prod_i$ is $F(\cdot \mid \theta_i)$. The parameter $\theta_i$ is unknown, but it belongs to the interval, $\Theta$, of the real line $(1 \leqq i \leqq k)$. We assume that the functional form of $F$ is known. Let $\theta_{\lbrack 1\rbrack} \leqq \theta_{\lbrack 2\rbrack} \leqq \cdots \leqq \theta_{lbrack k\rbrack}$ be the ranked $\theta_i$; we assume that it is not known with which population $\theta_{\lbrack i\rbrack}$ is associated $(1 \leqq i \leqq k)$. The $t$ populations with largest $\theta$-values are defined as the $t$ best populations and we refer to $\theta$ as the ranking parameter. The problem of selecting the $t$ best populations in an unordered manner has been studied extensively from the sampling point of view, in relation to several distributions. The usual formulation of the problem is the following: the experimenter's goal is to select the $t$ best populations in an unordered manner. He specifies two positive constants $d^\ast$ and $P^\ast$, where $\binom{k}{t}^{-1} < P^\ast < 1$. He desires to have a fixed-sample procedure which has a probability of at least $P^\ast$ of selecting the $t$ best populations whenever $\theta_{\lbrack k - t + 1\rbrack}$ is at a distance not less than $d^\ast$ from $\theta_{\lbrack k - t\rbrack}$. Bechofer [3] developed a procedure based on predetermined number of observations from each population when $F$ is the normal distribution function with unknown mean $\theta$ and known variance. Bechhofer and Sobel [4] considered a similar problem in relation to normal populations where the ranking parameter is the variance. Sobel and Huyett [9], Sobel [8], Rizvi [7], Baar and Rizvi [2] considered similar problems for other distributions. Here we consider a generalized version of the above selection problem. We solve the problem in broad generality by not considering a specific family of distributions $F(\cdot \mid \theta)$, but by assuming certain properties of $F$. Thus many of the available results (concerning the above selection problem) can be obtained as special cases of our results. Let $c, s, t$ be integers such that $\max (1, s + t + 1 - k) \leqq c \leqq \min (s, t)$, which implies that $\max (s, t) \leqq k - 1$. The experimenter's goal, which is referred as Goal I, is to select a subset of size $s$ which contains at least $c$ of the $t$ best populations. The experimenter specifies two positive constants $d^\ast$ and $P^\ast(<1)$. He desires to have a fixed-sample procedure for which the probability of selecting the desired type of subset of populations is not less than $P^\ast$ whenever the distance between $\theta_{\lbrack k - t\rbrack}$ and $\theta_{\lbrack k - t + 1\rbrack}$ is at least $d^\ast$. Two particular cases of this goal are of special interest: (1) Selection of a subset of size $s(\geqq t)$ which contains the $t$ best populations, (2) Selection of a subset of size $s(\leqq t)$ which includes any $s$ of the $t$ best populations. Sobel pointed (see the footnote on page 22 of [3]) out that sometimes Case 2 is of interest. These two cases correspond to $c = t, s \geqq t$ and $c = s, s \leqq t$. Further when $c = s = t$, Goal I reduces to the goal of selecting the $t$ best in an unordered manner. The proposed selection procedure, $R_s$, is based on suitable statistics $T_1, T_2, \cdots, T_k$, where $T_i$ is computed from a random sample of size $n$ from $\prod_i(1 \leqq i \leqq k)$. The procedure $R_s$ selects the subset of populations which corresponds to the $s$ largest $T$-values. Having specified the procedure as above, the problem is to determine the common sample size $n$ so that the probability requirement imposed on the procedure is satisfied. This problem has been solved under the assumption that $T_i$ is an absolutely continuous random variable and its distribution function is stochastically increasing in $\theta_i$ for each value of $n$. This has been done in Section 5. It should be noted (by considering the subset not selected) that the above selection problem is logically equivalent to similar selection problem where the experimenter's goal is to select a subset of size $(k - s)$ which contains at least $(k - t) - (s - c)$ of the $(k - t)$ populations with smallest $\theta$-values. Thus solutions to the above problem for all admissible values of $c, s$ and $t$ (with fixed $k$) will provide solutions to the selection problem where the goal is the selection of a subset of size $s$, which contains at least $c$ of those $t$ populations with smallest values for the ranking parameter. Section 6 gives results for the two above mentioned particular cases of Goal I. A theorem, which relates the sample sizes necessary to achieve Goal I, and its particular cases, is given in that section. Section 7 deals with an easily verifiable sufficient condition for the existence of the required sample size. In Section 8, the general results have been applied to normal distributions with unknown mean and known common variance.

Aspects of best linear estimation are explored for the model $y = X\beta + e$ with arbitrary non-negative (possibly singular) covariance matrix $\sigma^2V$. Alternative necessary and sufficient conditions for all simple least squares estimators to be also best linear unbiased estimators (blue's) are presented. Further, it is shown that a linear function $w'y$ is blue for its expectation if and only if $Vw \epsilon \mathscr{C} (X)$, the column space of $X$. Conditions on the equality of subsets of blue's and simple least squares estimators are explored. Applications are made to the standard linear model with covariance matrix $\sigma^2I$ and with additional known and consistent equality constraints on the parameters. Formulae for blue's and their variances are presented in terms of adjustments to the corresponding expressions for the case of the unrestricted standard linear model with covariance matrix $\sigma^2I$.

If a $2^{k - p}_R$ design, of fixed resolution $R$ and specified number of runs $2^q$, accommodates the maximum possible number of variables, we say that it is saturated. In this paper, we develop a method for constructing saturated designs and apply it to an example. We first show that when $R$ is odd, the set of all distinct $2^{k - p}_R$ designs (where $q = k - p$ is specified) can be obtained easily from a particular class of $2^{(k + 1) - p}_{R + 1}$ designs. We then develop a stage by stage method for constructing this "parent" class of designs of (even) resolution $R + 1$. This class is shown, incidentally, to contain a saturated design. The complete set of $2^{k - p}_R$ designs, which naturally includes all saturated $2^{k - p}_R$ designs, can then be obtained at once. The problem of arranging the designs constructed into blocks of runs, so that the blocked designs have certain desirable confounding properties, is also investigated, and a method for obtaining optimal blocking arrangements is given. As an important part of our method, a "sequential conjecture" procedure is developed and utilized to test the equivalence of any two designs. These procedures have been programmed for the computer, and are illustrated by the example $R = 5, q = 7$.

The main problem encountered with any $s^m$ lattice design, where $s(= p^n)$ is a prime positive integer, is the construction of a balanced lattice so that after ordering the experimenter may select a best set of arrangements. When $s$ and $m$ are large the usual method of construction becomes quite laborious. The object of this paper is to develop a method of obtaining a balanced set of arrangements by means of cyclic collineations on the finite projective geometry $PG(k, s)$, where $k = m - 1$. Considerations are to be limited to collineations whose characteristic matrices $A (\rho) = A - \rho I$ have a single non-trivial invariant factor. The Smith canonical form of $A(\rho)$ is then diagonal $(1, 1, \cdots, 1, f_m(\rho))$ and thus we can limit consideration to the associated rational canonical form. Such matrices can be generated and the orders found by electronic computers for any $s^m$ lattice. It is shown that with proper choice of $GF(s)$ any balanced $l$-restrictional $s^m$ lattice is given by $\alpha = \sum^{m - 1}_{i = 0} s^i$ arrangements. Associating with the $\alpha$ arrangements the $\alpha$ powers of a cyclic collineation in rational canonical form of order $\alpha$ it is shown that the generators of the confounding scheme in each arrangement can immediately be taken from the columns of the respective powers of the matrix of the cyclic collineation. Balanced arrangements for lattices with $s^m < 1000$ are typified by presenting the associated cyclic collineations of order $\alpha$

The distribution of the largest latent root of the covariance matrix calculated from a sample from the normal normalitive multivariate population with population covariance matrix $\sigma^2 I$ are presented by author [10] in 1966. The purpose of this paper is to find the distribution of the largest latent root for arbitrary $\Sigma$.

It is shown that by drawing multiple samples from $N(\mu_i, \sigma^2_i) = 1, 2)$ it is possible to have confidence interval of preassigned length for the Behrens-Fisher problem.

Tsao (1954) deficumulative distribution function $S_n(x) = k/n$ if $X_k \leqq x < X_{k + 1}$, where $X_0 = -\infty$ and $X_{n + 1} = \infty$. Let $Y_1 < Y_2 < \cdots < Y_m$ represent an ordered random sample from the continuous distribution function $G(x)$, with the empirical cumulative distribution function $S'_m (x)$. As test statistics for testing $H_0:F(x) \equiv G(x)$ against $H_1:F(x) \not\equiv G(x)$, Tsao (1954) proposed $d_r = \max_{x \leqq X_\tau} |S_n(x) - S'_m(x)|,\quad r \leqq n,$ and $d'_r = \max_{x \leqq \max(X_r, Y_r)} |S_n(x) - S'_m(x)|,\quad r \leqq \min (m, n).$ It seems natural to consider also the test statistic $d"_r = \max_{x \leqq \min (X_r, Y_r)} |S_n(x) - S'_m(x)|,\quad r \leqq \min (m, n)$ Tsao described a counting procedure to obtain the probabilities associated with the distribution functions of $d_r$ and $d'_r$, and illustrated this procedure in the relatively simple case where $m = n$. Tables were compiled using the procedure for various values of $r$ and $m(= n)$. In this paper the asymptotic distributions of $N^{\frac{1}{2}} d_r, N^{\frac{1}{2}} d'_r$, and $N^{\frac{1}{2}} d''_r$ are given, where $N = mn/(m + n)$. Also, for $m = n$, the exact closed form of the distribution functions of $d_r, d'_r$, and $d''_r$ are derived under the null hypothesis. Also shown are the relationships \begin{align*}P(d_r \leqq x) = \frac{1}{2}P(d'_r \leqq x) + \frac{1}{2}P(d"_r \leqq x); \notag \\ P(d"_r \leqq x) = P(d'_{r - c} \leqq x),\quad\text{for} c < r, \text{where} c = \lbrack nx\rbrack, \notag \\ = 1, \text{for} c \geqq r,\end{align*} and therefore \begin{align*}P(d_r \leqq x) = \frac{1}{2}P(d'_r \leqq x) + \frac{1}{2}P(d'_{r - c} \leqq x), \quad\text{for} c < r, \notag \\ = \frac{1}{2}P(d'_r \leqq x) + \frac{1}{2}, \text{for} c \geqq r,\end{align*} illustrating that tables for $P(d_r \leqq x)$ and $P(d''_r \leqq x)$ are superfluous while tables for $P(d'_r \leqq x)$ exist. Epstein (1955) compared the power of Tsao's $d'_r$ with three other nonparametric statistics on the basis of 200 pairs of random samples of size 10 drawn from tables of normal random numbers. Rao, Savage, and Sobel (1960) considered $d'_r$ as a special case in the general scheme of censored rank order statistics.

In the multivariate one sample location problem, the theory of permutation distribution under sign-invariant transformations is extended to a class of rank order statistics, and this is utilized in the formulation of a genuinely distribution free class of rank order tests for location (based on Chernoff-Savage (1958) type of test-statistics). Asymptotic properties of these permutationally distribution free rank order tests are studied, and certain stochastic equivalence relations with a similar class of multivariate extensions of one sample Chernoff-Savage type of tests are derived. The power properties of these tests are studied.

The object of the present investigation is to extend the findings of Hajek [7] on asymptotically most powerful rank order tests (AMPROT) to grouped data where the underlying distributions are essentially continuous but the observable random variables correspond to a finite or countable set of contiguous class intervals. In this context, the two sample problem for grouped data is considered and various efficiency results are also studied.

Suppose that the members of a population fall into an unknown number of classes of various sizes. A random sample of $N$ observations is taken and from the information in the sample must be estimated the parameters of the population; roughly speaking these are the number of classes of a given size. This description is deliberately vague because there are two models available in the literature for this problem. Good (1953), Good and Toulmin (1956), Harris (1959), and Trybula (1959) use an infinite population and investigate estimators for useful functions of the population parameters. Examples of applications are given. Goodman (1949) uses a finite population and obtains rather more restricted results. Des Raj (1961) also treats a very special case of this model. Such problems have been called cataloguing problems, see Harris (1959), which is the reason for the title. It is the object of this paper to show that the second model, that with finite population, is more suitable for these problems, and to extend the results of Goodman to match those available for the model with infinite population. The unbiased estimators obtained would be modified for practical use, and the main contribution of the paper is thought to lie in the simplification of the theory connected with the problem.

Let $X_1 \cdots X_n$ represent a sequence of independent random variables with a common (unknown) density function $f(x)$. In this paper, an estimate of $f(x)$ of the form $\hat{f}_n(x) = \sum^{q(n)}_{j = 0} \hat{a}_{jn\varphi_j}(x)$ is considered, where $\hat{a}_{jn} = (1/n) \sum^n_{i = 1\varphi_j}(X_i), \varphi_j( )$ is the $j$th Hermite function and $q(n)$ is an integer dependent on $n$. Assuming $f(x)$ is $L_2$, it is shown that the sequence of estimates is consistent in the sense of mean integrated square error, $\lim_{n \rightarrow \infty} E \cdot \int (f(x) - \hat f_n(x))^2 dx = 0$ and, under additional conditions on $f(x)$, the sequence of estimates is also consistent in mean square error, $\lim_{n \rightarrow \infty} E(f(x) - \hat{f}_n(x))^2 = 0$, uniformly in $x$. For both error criteria, bounds on the rate of convergence of the estimate are obtained. The rate of convergence is seen to depend on the smoothness and integrability properties of $f(x)$--the maximum rate being bounded by $1/n$. In order for the series method to achieve the same rate of convergence as an estimate which uses the "kernel" technique [4], [6], more assumptions on $f(x)$ are required. However, in estimating a multivariate density, with the same type of conditions as in the univariate case, the rate of convergence remains the same for the multivariate series estimate. With the "kernel" method, the rate depends on the dimension of the density being estimated; the rate of convergence of the estimate decreases as the dimension increases. In the next section, we introduce notation and give some preliminary results. Conditions for consistency and rates of convergence are established in Section 3. These results are then compared in Section 4 to previous work in the area.

Let $\{\mathbf{X}_n\}^\infty_{n = 1}$ be the sequence of random variables associated with the sequence of densities of the form \begin{equation*}\tag{1.1}C_n(x)f^n(x)\quad n = 1, 2, \cdots, \end{equation*} where $k$ is a positive function and $f$ has a unique mode $m$ at which it is sufficiently smooth. It is known that $n^{\frac{1}{2}} (X_n - m)b$ converges in law to the standard normal distribution when $b$ is a suitably chosen scaling constant (see Laplace (1847), pages 400-403, or von Mises (1964), page 269). In Section 2, it is shown that the cumulative distribution function $F_n$ possesses an asymptotic expansion in powers of $n^{\frac{1}{2}}$ where each coefficient is the product of a polynomial and the standard normal density. The polynomials have coefficients which are expressible in terms of $k$ and $\log f$, together with their derivatives evaluated at $m$. Section 3 shows that the normalizing transformation $\eta_n(\xi) = \Phi^{-1}(F_n(\xi))$, where $\Phi$ is the standard normal cdf, also has an asymptotic expansion in powers of $n^{-\frac{1}{2}}$ and Section 4 makes the same conclusion for the percentiles of $F_n$. The coefficients in each of these expansions are polynomials. Similar theorems are given by Bol'shev (1959), (1963), Dorogovcev (1962), Peiser (1949), and Wasow (1956). Examples of these expansions, namely the $t$-distribution and central order statistics, are given. 1.1. General assumptions. Consideration of the random variables $b(X_n - m)$ where $b^2 = -f''(m)/f(m)$ shows that it is possible, without loss of generality, to specialize to the case where $m = 0, f(0) = 1, f'(0) = 0$, and $f''(0) = -1$. The general assumptions are stated for this case. ASSUMPTION (i). $f(x)$ and $k(x)$, considered as functions of a complex variable, are analytic for $|x| \leqq \delta_1$ where $\delta_1$ is a positive constant. ASSUMPTION (ii). $f(x)$ has an absolute maximum at $x = 0$ and $f(x) \leqq \rho_1 < 1$ for all real $x$ with $|x| \geqq \delta_1$. We further assume that $k(0) \neq 0$ and also that $f(x) \neq 0$ whenever $|x| \leqq \delta_1$. These assumptions are chosen for simplicity and certainly are not the most general under which Laplace's approximation provides an asymptotic expansion. 1.2. Notation. Let $F_n$ denote the cdf of $n^{\frac{1}{2}}X_n$ where $X_n$ has a density satisfying the general assumptions. Denote by $\Phi$ and $\varphi$ the standard normal cdf and density respectively. For any $0 < \alpha < 1$, let $\xi_n$ denote the upper $\alpha$th percentile of $F_n$. It follows from the general assumptions that $\xi_n$ is uniquely determined if $n$ is sufficiently large. When considering percentiles, we shall assume that this is the case. The notation $\thicksim$ is used rather than = for relating a function to its asymptotic series since the series may or may not converge.

In certain multivariate problems involving several populations, the covariance structure of the populations is such that all covariance matrices can be diagonalized simultaneously by a fixed orthogonal transformation. In the transformed problem one has a number of independent univariate populations. Consequently certain hypotheses in the original problem become equivalent to simultaneous hypotheses on these univariate populations in the transformed model. Using this approach we propose a test procedure for testing the hypothesis of equality of covariance matrices against a certain alternative under the intraclass correlation model. The relative advantages of our procedure over that of Srivastava's procedure [6] are also discussed. Finally we indicate how the problem of testing for the equality of covariance matrices under a more general set up can be reduced to a univariate problem.

Raktoe [3] has recently developed a procedure for obtaining a balanced confounding scheme for any $l$-restrictional lattice design of $s^m$ treatments where $s$ is a prime or a power of a prime and $m$ is a positive integer. He has shown that the generators of the confounding scheme in each arrangement can be taken from the columns of different powers of the rational canonical form of a matrix of cyclic collineation of a particular order. However, he did not indicate how to construct the generator matrices analytically except for the case $s = p = 2$. In all other cases, he obtained these matrices empirically. The present paper gives an analytic method for constructing the generator matrices of collineations for all values of $s$, by the application of a particular theorem in projective geometry and another one from group theory.