The "$S$-method" of multiple comparison ([5]; [6], Section 3.5) was intended for multiple estimation, possibly combined with multiple testing. It is shown that if only multiple testing is desired a certain "modified $S$-method" is more powerful. While this result is of some theoretical interest, it is recommended after a discussion of the relative advantages of the two methods, that the new one generally not be used in applications. The multiple testing problems considered are related to estimating the direction of a vector or its unoriented direction-estimation problems which also have an inherent interest. A confidence set for a parameter point is called improper if the probability that it gives a trivially true statement is positive. The problems of estimating the direction and unoriented direction of a vector are reformulated to permit solution by proper confidence sets. In the case of the unoriented direction of a $q$-dimensional vector the confidence sets yield solutions of the problem of joint estimation of $q - 1$ ratios and the problem of multiple estimation of all ratios in a certain infinite set. Specializing to the case $q = 2$ yields a proper confidence set as a substitute for Fieller's improper confidence set for a ratio.

Let $f(\omega), -\pi \leqq \omega \leqq \pi$ be the spectral density function of a discrete coordinate real-valued time series, stationary to order four. Assume that the covariance function $r(k)$ is such that $-\log r^2(k) \sim Ck^\gamma$, and $-\log(r^2(k + 1)/r^2(k)) \sim C_\gamma k^{\gamma-1}$, as $k \rightarrow \infty$, for some $C, \gamma, 0 < C, \gamma < \infty$. Then there exists a non-random sequence $t(n)$ which is such that the estimator $f^\ast(\omega) \equiv (2\pi)^{-1} \sum^{t(n)}_{k=-t(n)} (1 - n^{-1}|k|)\hat{r}(k) e^{ik\omega}$ is efficient where $\hat{r}(k) \equiv (n - |k|)^{-1} \sum^{n-|k|}_{j=1}(X(j) - \overline{X})(X(j + |k|) - \overline{X}), \overline{X} = n^{-1} \sum^n_{j=1} X(j)$, and an estimator $\hat{f}(\omega)$ is said to be efficient if $\lim_{n\rightarrow\infty}2\pi E\int^\pi_{-\pi} (\hat{f}(\omega) - f(\omega))^2 d\omega/I^2_{\min}(n) = 1$, where $I^2_{\min}(n)$ is the smallest integrated mean squared error which can be achieved using an estimator of the form $\tilde{f}(\omega) = (2\pi)^{-1} \sum^{n-1}_{k=-(n-1)} a(k,n)\hat{r}(k)e^{ik\omega}$, where $a(k,n)$ is nonrandom. In general a sequence $t(n)$ which is efficient for one covariance function is inefficient for another. A class of estimators $\hat{f}(\omega)$ is presented which are of the form $\hat{f}(\omega) = (2\pi)^{-1} \sum^{\hat{t}(n)}_{k=-\hat{t}(n)}(1 - n^{-1}|k|)\hat{r}(k)e^{ik\omega}$, where $\hat{t}(n)$ is a function of the observations. In an appropriate sense $\hat{t}(n)$ "estimates" $t(n)$. For any covariance function satisfying the above conditions $\sup_{-\pi\leqq\omega\leqq\pi}|\hat{f}(\omega) - \hat{f}^\ast(\omega)|/I_{\min}(n) \rightarrow 0$, in probability, where $f^\ast(\omega)$ is the unattainable efficient truncation estimator.

When proving results on the asymptotic behavior of estimates of the spectrum of a stationary time-series, it is invariably assumed that as the sample size $T$ tends to infinity, so does the truncation point $M_T$, but at a slower rate, so that $M_TT^{-1}$ tends to zero. This is a convenient assumption mathematically in that, in particular, it ensures consistency of the estimates, but it is unrealistic when such results are used as approximations to the finite case where the value of $M_TT^{-1}$ cannot be zero. We derive a formula for the asymptotic variance on the assumption that $M_TT^{-1}$ tends to a constant $\gamma$; a more accurate approximation to the variance in the finite case is then obtained by using this formula with $\gamma$ equal to the actual value of $M_TT^{-1}$. Numerical comparisons are made in the white noise case.

The purpose of this paper is two-fold: (i) to develop the asymptotic distribution theory of the normal theory likelihood ratio test statistic for the (multivariate) general linear hypothesis problem when the parent distribution is not necessarily normal and (ii) to develop the theory of the multivariate analysis of covariance based on general rank scores. The problem (i) extends the distribution theory of the likelihood ratio statistic developed by the authors in [9] for the multivariate general linear hypothesis problem (for a class of simple alternatives) to the more general case where one has also to deal with a set of concomittant variables, and the problem (ii) extends the results of the authors' earlier paper [8] on the rank order theory of the univariate analysis of covariance to the corresponding multivariate case. Let $\mathbf{Z}_{k\alpha} = (\mathbf{Y}_{k\alpha}, \mathbf{X}_{k\alpha})$; \lbrack where $\mathbf{Y}_{k\alpha} = (Y^{(1)}_{k\alpha}, \cdots, Y^{(p)}_{k\alpha}) \text{and} \mathbf{X}_{k\alpha} = (X^{(1)}_{k\alpha}, \cdots, X^{(q)}_{k\alpha}), p, q \geqq 1 \rbrack, \alpha = 1, \cdots, n_k$ be $n_k$ independent and identically distributed random vectors (i.i.d.r.v.) having a $(p + q)$-variate continuous cumulative distribution function (cdf) $G_k(\mathbf{z}), \mathbf{z} \in R^{p+q}$, for $k = 1, \cdots, c$. It is assumed that $\mathbf{Z}_{11}, \cdots, \mathbf{Z}_{cn_c}$ are mutually independent. Let us denote by $F_k^{(1)}(\mathbf{x})$ the (marginal) joint cdf of $\mathbf{X}_{k\alpha}$, and let $F_k^{(2)}(\mathbf{y}\mid\mathbf{x})$ be the conditional cdf of $\mathbf{Y}_{k\alpha}$, given $\mathbf{X}_{k\alpha} = \mathbf{x}, k = 1, \cdots, c$. As in the univariate theory (cf. [8, 10]), we assume that \begin{equation*} \tag{1.1} F_1^{(1)}(\mathbf{x}) = \cdots = F_c^{(1)}(\mathbf{x}),\quad\mathbf{x} \in R^q\end{equation*} and frame the null hypothesis as \begin{equation*} \tag{1.2} H_0:F_1^{(2)}(\mathbf{y}|\mathbf{x}) = \cdots = F_c^{(2)}(\mathbf{y}|\mathbf{x}).\end{equation*} We may note that under the usual additive model, viz., \begin{equation*} \tag{1.3} F_k^{(2)}(\mathbf{y}|\mathbf{x}) = F^{(2)}(\mathbf{y} - \mathbf{\tau}\_k|\mathbf{x}), \tau_k = (\tau_k^{(1)}, \cdots, \tau_k^{(p)}),\end{equation*} $k = 1, \cdots, c$, the null hypothesis $H_0$ in (1.2) implies that $\tau_1 = \cdots = \tau_c$. We are interested in the set of alternatives that (1.2) does not hold, which under the model (1.3) implies that not all $\tau_k, k = 1, \cdots, c$ are identical. The problem of multivariate analysis of covariance (MANOCA) can be viewed as a special case of the general linear hypothesis problem, considered in Anderson (1958, chapter 8). Two problems arise in this context: (i) how the likelihood ratio (1.r.) test behaves when the parent distribution is not necessarily normal, and (ii) how the multivariate generalizations of the tests considered in [8, 10] compare with the normal theory l.r. test. The purpose of the present investigation is to study these problems thoroughly.

Let $\{X_n\}$ be a sequence of independent random variables and let $Y_n = B_n^{-1} \sum^n_{i=1} X_i - A_n$ be a sequence of normed, centered sums such that, for appropriately chosen normalizing coefficients $(B_n \rightarrow \infty)$ and centering constants, $\{Y_n\}$ converges in law to a nondegenerate limit distribution G. B. V. Gnedenko asked the following question: What characterizes the class of limit distributions $\{G\}$ when there are $r$ different distribution functions among those of the random variables $\{X_n\}$? Let $\mathscr{P}_r$ denote this class of distribution functions. As is well-known, $\mathscr{P}_1$ is the class of stable distributions. V. M. Zolotarev and V. S. Korolyuk [8] have shown that $\mathscr{P}_2$ consists solely of stable distributions and convolutions of two stable distributions. It was thought that O. K. Lebedintseva [4] had shown that this was true for $r > 2$ (with two replaced by less than or equal to $r$) with the added hypothesis that one of the $r$ possible distribution functions of the summands $X_n$ belongs either to the domain of attraction of a stable distribution, or to a domain of partial attraction of only one type. However, V. M. Zolotarev and V. S. Korolyuk [8] gave an example that showed that O. K. Lebedintseva's theorem did not completely settle the matter. However, A. A. Zinger [7] gave a necessary and sufficient condition on the Levy spectral function of $G$ in order that $G$ be in $\mathscr{P}_r$. His theorem shows that Lebedintseva's result is incorrect. In this same paper, A. A. Zinger proved a theorem that gives a necessary condition on the distribution functions of the summands $X_n$ in order that $G$ be a convolution of $r$ distinct stable distributions. Here we expand Zinger's theorem to obtain a necessary and sufficient condition that $G$ be a convolution of $r$ distinct stable distributions. Some related results are also obtained.

Motivated by ecological and genetic phenomena, Karlin and McGregor [3] introduced the following model to describe the continued formation and growth of mutant biological populations. It is assumed that a new mutant population arises at each event time of a stochastic process (referred to as the input process) $\{v(t),t > 0\}$ whose state space is the non-negative integers. Each new mutant population begins its evolution with a fixed number of elements and evolves according to the laws of a continuous time Markov chain $\mathscr{P}$ with stationary transition probability function $P_{i,j}(t)\quad i,j = 0, 1, 2, \cdots; t \geqq 0.$ We assume that all populations evolve according to the same Markov Chain and independent of one another. In terms of this structure, the basic question which we consider in this work can be formulated in the following manner: (A) Given an input process $\{v(t),t > 0\}$ and the individual growing process $\mathscr{P}$, determine the asymptotic behavior as $t \rightarrow \infty$ of the mean and variance of $S(t) = \{$number of different sizes of mutant populations at time $t\}$ and determine the limit distribution as $t \rightarrow \infty$ of $S(t)$ appropriately normalized. $S(t)$ is a special functional of the vector process $N(t) = \{N_0(t), N_1(t), N_2(t), \cdots\}\quad t > 0$ where $N_k(t) = \text{number of mutant populations with exactly} k \text{members at time} t$ and may be interpreted as a measure of the fluctuations in population size. We have restricted our considerations to this special case because it serves as a model problem for more general situations and possesses all the subtle difficulties of the general case. The random variable $S(t)$ can also be identified as the number of distinct occupied states at time $t$ among all Markov Chains which have begun their evolution up to that time. In the subsequent discussion we will refer to $\{S(t), t > 0\}$ as the "occupied states" process generated by the input process $\{v(t),t > 0\}$ and the Markov Chain $\mathscr{P}$. Without loss of generality we identify the state 0 as the initial state of all evolving Markov Chains and $-1$ as an absorbing state if absorption is possible. In this paper we introduce "occupied states" processes generated by a class of null recurrent, transient, and absorbing barrier Birth and Death processes and a Poisson input process. The special feature of this class is that with the normalization $Y(t;u) = t^{-1} X(t^\alpha u),\quad\alpha > 0$ $(\{X(t), t > 0\}$ is the growing process $\mathscr{P}$), the process $Y(t;u)$ converges weakly in the Markov sense as $t \rightarrow \infty$ to a Bessel diffusion (see C. Stone [8]). The main idea (also applicable to more general growing processes $\mathscr{P}$ is that one requires local limit theorems, and under some circumstances, specification of the rate of convergence of the transition density of $Y(t;u)$ to the density of the limiting diffusion in order to prescribe exact asymptotic formulas for $ES(t)$ and $\operatorname{Var} S(t)$ and to prove a central limit theorem for $S(t)$. The results of this paper are in sharp contrast with the asymptotic formulas for $ES(t)$ and $\operatorname{Var} S(t)$ which appear in the companion paper [6], where the growing process $\mathscr{P}$ is a general positive recurrent Markov chain and the input process remains Poisson. Section 2 contains basic definitions, some intuitive discussion, and precise statements of the main results on asymptotic behavior of $ES(t)$ and $\operatorname{Var} S(t)$. In Section 3, we present detailed proofs of the theorems of Section 2, and we conclude with a central limit theorem for $S(t)$ in Section 4. The appendix contains some technical lemmas which are essential for asymptotic formulas that incorporate speed of convergence theorems.

Let $\mathbf{W}_{jj}$ and $\mathbf{\Sigma}_{jj}, 1 \leqq j \leqq q$, respectively denote the diagonal blocks of a partitioned Wishart matrix $\mathbf{W}$ and its matrix $\mathbf{\Sigma}$ of parameters. A Laguerrian expansion is given for the joint distribution of $v_j = \mathrm{tr} \mathbf{W}_{jj}\mathbf{\Sigma}^{-1}_{jj}, 1 \leqq j \leqq q$, which is a generalization of known multivariate chi-square distributions. Approximations to the joint distribution function are discussed, and probability inequalities are given for this and a related multivariate $F$-distribution. Applications are made to some simultaneous multivariate test procedures.

The interarrival times past a point at the intersection of cars that queued up during the stop phase were considered by the authors of [1]. Denoting by $\{T_i\}, i = 1,2, \cdots$, the process of these consecutive interarrival times or time headways, an examination of experimental data indicated that the random variables $\{T_{n_o+i}, i = 1,2, \cdots\}$ were independently identically distributed. For $n_0 = 2$ the $\chi^2$-test for homogeneity showed this hypothesis to be tenable. Therefore the data were pooled to estimate the parameters of a suitable density function. As a final step, each of the subpopulations was tested, using the $\chi^2$-statistic, against the population whose parameters were determined using the pooled data. Two methods of estimation, namely, the modified minimum $\chi^2$ and the method of maximum likelihood, were considered. In each case, this paper shows that these $\chi^2$-statistics do not have the usual limiting $\chi^2$-distribution, but are stochastically larger than would be expected under the $\chi^2$-theory. More generally and precisely, let $\Pi_i, i = 1,2, \cdots, k$ populations and let the mathematical form of $\Pi_i$ be known except for the unknowns $\mathbf{\alpha}_i = (\alpha_{1i}, \alpha_{2i}, \cdots, \alpha_{si})$. Assume that the hypothesis $H_0:\Pi_1 = \Pi_2 = \cdots = \Pi_k$ is true, and also that a random sample of size $n_i$ from the $i$th population $(i = 1,2, \cdots, k)$ is available. The authors propose to treat in a subsequent communication a more complicated case in which the samples are tested separately, initially, and the null hypothesis merely fails to be rejected (so that it is not known whether or not it is true). Let $\mathbf{\alpha}^{(N)}$ and $\mathbf{\alpha}^{\ast(N)}$ be respectively the modified minimum $\chi^2$ and maximum likelihood estimates of $\mathbf{\alpha} = \mathbf{\alpha}_1 = \cdots = \mathbf{\alpha}_k$ based on the pooled sample of size $N = n_1 + n_2 + \cdots + n_k$. Let $\hat{\Pi}$ and $\Pi^\ast$ denote respectively $\Pi(\hat{\alpha}^{(N)})$ and $\Pi(\mathbf{\alpha}^{\ast(N)})$. The hypothesis $H_{01}:\Pi_i = \hat{\Pi}$ and $H_{02}:\Pi_i = \Pi^\ast$ are considered for any specified $i$. The following two theorems for the case $k = 2$ are proven.

Let $\{\mathbf{Z}_n:n \geqq 0\}$ be a supercritical $p$-type $(p \geqq 2)$ Galton-Watson branching process with offspring probability generating functions (pgf) $h_i(\mathbf{s}) i = 1,2,\cdots, p$. Assume (i) $m_{ij} \equiv \partial h_i/\partial s_j\mid_{s=1} < \infty$ for all $i$ and $j$ where $\mathbf{s} = (s_1, \cdots, s_p)$ and $\mathbf{1} = (1, 1, \cdots, 1)$, (ii) $\exists n_0 > 0 \ni$ if $M \equiv ((m_{ij}))$ then $M^{n0} \simeq 0$ (i.e. each element of $M^{n0}$ is $> 0$) and (iii) the largest real eigenvalue $\rho$ of $M$ is $> 1$. Let $\mathbf{u} \simeq 0$ and $\mathbf{v} \simeq 0$ be column vectors such that $M\mathbf{v} = \rho\mathbf{v}, \mathbf{u}'M = \rho\mathbf{u}', \mathbf{u} \cdot \mathbf{1} = 1, \mathbf{u} \cdot \mathbf{v} = 1$ where $\mathbf{u}'$ denotes transpose of $\mathbf{u}$ and $\cdot$ refers to inner product. Kesten and Stigum [6] showed (i) there always exists a nonnegative random variable $W$ such that $\mathbf{Z}_n\rho^{-n}$ converges almost surely (a.s.) to $\mathbf{u}W$ and (ii) $P(W = 0) < 1$ if and only if $E(Z_1^j \log Z_1^j\mid\mathbf{Z}_0 = e_i) < \infty$ for all $i$ and $j$ where $e_i = (\delta_{i1}, \delta_{i2}, \cdots, \delta_{ip}), \delta_{ij} = 1$ if $i = j$ and 0 if $i \neq j, Z_1^j$ is the $j$th coordinate of $\mathbf{Z}_1$. We give here a simple proof of a modified result which is exactly the same as above except that convergence a.s. is replaced by convergence in probability. We do this by showing that without any extra assumption other than the existence of $M$ the vector $(\mathbf{v}\cdot \mathbf{Z}_n)^{-1}Z_n$ converges in probability to $\mathbf{u}$ on the set of non-extinction

For the $2^n$ factorial a treatment design to estimate the $n$ main effects and the mean with $(n + 1)$ treatment combinations is known in the literature as a saturated main effect plan. Let the $(n \times 1) \times n$ matrix $D$ consisting of the 0's and 1's making up the subscripts of the observations, denote such a plan and let the $(n + 1) \times (n + 1)$ matrix $X$ stand for the corresponding design matrix of -1's and 1's, then optimal (in the sense of maximum absolute value of the determinant of $X'X$) designs have been characterized in terms of the information matrix $X'X$ by many authors, such as Plackett and Burman [6] and Raghavarao [7]. Williamson [9], Mood [3], and Banerjee [2], among others, have used (0, 1)-matrices to construct optimal and weighing designs. If the elements of the first row of $D$ are set equal to zero, then the $n \times n$ (0, 1)-matrix used in weighing designs is obtained from the last rows of $D$. However, $D$ is not restricted to always include the combination having all zero levels in this paper. For a summary concerning several aspects of optimal saturated main effect plans the reader is referred to Addelman's [1] paper. The aim of this paper is to characterize the optimal saturated main effect plans in terms of D'D rather than $X'X$. A major consequence of this is that all theory available for semi-normalized $(-1, 1)$-matrices is applicable to semi-normalized (0, 1)-matrices and vice versa. A second major consequence is that the normal equations for saturated main effect plans need not be obtained as they are readily derivable from the $D$ matrix.

Let $J$ be a finite non-empty set and let $S(J)$ denote the set of all finite sequences of elements of $J$. If $s = (\delta_1, \cdots, \delta_m)\in S(J)$ and $t = (\mu_1, \cdots, \mu_n)\in S(J)$, then $st$ will denote the combined sequence $(\delta_1, \cdots, \delta_m, \mu_1, \cdots, \mu_n)$. The singleton sequence $(\delta)$ will be denoted by $\delta$. The symbol $s^2$ will mean the sequence $ss$ and the symbols $s^3,s^4$, etc., are defined similarly. Suppose $\{Y_n\}$ is a stationary process with state-space $J$. If $s \in S(J)$ and has length $n, p(s)$ denotes $P\lbrack (Y_1, \cdots, Y_n) = s \rbrack$. The rank $n(\delta)$ of a $\delta \in J$ is defined to be the largest integer $n$ such that we can find $2n$ sequences $s_1, \cdots, s_n, t_1, \cdots, t_n$ in $S(J)$ such that the $n \times n$ matrix $\|p(s_i\delta t_j)\|$ is non-singular. Suppose now that $\{Y_n\}$ is a function of a finite Markov chain (hereafter abbreviated ffMc). That is, let there exist a stationary Markov chain $\{X_n\}$ with a finite state-space $I$ and a function $f$ on $I$ onto $J$ such that $\{Y_n\}$ and $\{f(X_n)\}$ have the same distribution. Then Gilbert [5] has shown that $n(\delta) \leqq N(\delta)$ for all $\delta \in J$, where $N(\delta)$ is the number of elements in $f^{-1}\lbrack\{\delta\}\rbrack$. If we can find $\{X_n\}$ and $f$ in such a way that $n(\delta) = N(\delta)$ for all $\delta \in J$, then $\{Y_n\}$ is said to be a regular ffMc. The motivation for investigating the regularity property of a ffMc has been made clear by Gilbert in the first and the last paragraphs of Section 2 of [5]. Fox and Rubin [3] have given an example of a process $\{Y_n\}$ which has $n(\delta) < \infty$ for all $\delta \in J$ but which is not a ffMc. In the first part of this paper we expand their example into a class of examples and show that some of these examples yield nonregular ffMc. These examples are of a different nature than those given in [1], Section 4. Further our method of investigation is different from that employed by Fox and Rubin. The second part of this paper is devoted to proving that an exchangeable process which is a ffMc is a regular ffMc.

The Bayesian theory for testing a sharp hypothesis, defined by fixed values of parameters, is here presented in general terms. Arbitrary positive prior probability is attached to the hypothesis. The ratio of posterior to prior odds for the hypothesis is given by the weighted likelihood ratio, shown here to equal Leonard J. Savage's (1963) ratio of a posterior to a prior density (2.21). This Bayesian approach to hypothesis testing was suggested by Jeffreys (1948), Savage (1959), (1961), Lindley (1961), and Good (1950), (1965), but obscured some what by approximations and unique choices of prior distributions. This Bayesian theory is distinct from that of Lindley (1965) and that of Dickey (1967a). Applications are given to hypotheses about multinomial means, for example, equality of two binomial probabilities. A new test is presented for the order of a finite-state Markov chain.

Let $x_{i1}, x_{i2}, \cdots, x_{ini}$ be a random sample of real observations from the $i$th population with cumulative distribution function (cdf) $F_i(x), i = 1,2, \cdots, c$. Let the $c$ samples be independent and the $F$'s continuous. In this paper we shall consider tests for the null hypothesis $H_0:F_1(x) = F_2(x) = \cdots = F_c(x) = F(x), \text{say}$. The statistics and tests, proposed in this paper, are based upon $c$-plets of observations which are formed by selecting one observation from each of the $c$ samples. The total number of distinct $c$-plets that can be formed in this way is $\prod^c_{i=1}n_i$. In each $c$-plet we compare and rank observations appearing therein. Let $v_{ij}$ be the number of $c$-plets in which the observation selected from the $i$th sample is larger than exactly $(j - 1)$ observations and smaller than the other $(c - j)$ observations. Since the distributions are assumed to be continuous the probability of the existence of ties is zero. Let us define $u_{ij} = \nu_{ij}/\prod^c_{i=1}n_i$; it is the proportion of $c$-plets which give rank $j$ to the observation from the $i$th sample. Let us have $N = \sum^c_{i=1}n_i, p_i = n_i/N, L_i = \sum^c_{j=1} a_ju_{ij}$, where the $a$'s are real constants such that they are not all equal and \begin{equation*} \tag{1.1} A = \sum^c_{j=1} \sum^c_{l=1} a_ja_l \big\{\frac{\binom{c-1}{j-1}\binom{c-1}{l-1}}{(2c - 1)\binom{2c-2}{j+l-2}} - \frac{1}{c^2}\big\}.\end{equation*} Then we define a class of statistics $\mathscr{L}$ as \begin{equation*} \tag{1.2} \mathscr{L} = \frac{N(c - 1)^2}{Ac^2} \big\lbrack \sum^c_{i=1} p_iL_i^2 - \big(\sum^c_{i=1} p_i L_i\big)^2 \big\rbrack.\end{equation*} A particular member of the class is found by specifying the real constants $a$'s. With each member of this class we associate a test of $H_0$: Reject $H_0$ at a significance level $\alpha$ if $\mathscr{L}$ exceeds some predetermined constant $\mathscr{L}_\alpha$. We, later in this paper, show that under $H_0, \mathscr{L}$ is distributed as a $\chi^2$ variate with $c - 1$ degrees of freedom, in the limit as $N \rightarrow \infty$. Hence for sufficiently large $N, \mathscr{L}_\alpha$ may be approximated by the corresponding significance point of the $\chi^2$ distribution with requisite degrees of freedom. Tests proposed by Bhapkar [2], [3], Sugiura [13], and the author [5], [6] may be seen to belong to this class. In this paper it is attempted to provide a unified treatment of statistics and tests based on $c$-plets--particularly those based on linear combinations of the $u$'s. The detailed properties of statistics belonging to this class are discussed under the null hypothesis and the following two alternative hypotheses. (I) the alternative of different locations or shift, the distributions being equal in all other respects and, (II) the alternative of different scales, the distributions again being equal in all other respects. Haller [7] has discussed the use and the properties of some statistics belonging to this class for testing $H_0$ against an alternative of stochastically ordered variables and for selection and ranking procedures. In the fourth section we give a condition on the distributions under which these tests are consistent against specified alternatives. In the fifth section $\mathscr{L}$ is shown to have a limiting noncentral $\chi^2$ distribution with $c - 1$ degrees of freedom under the pertinent alternative hypotheses. The noncentrality parameter is seen to be a quadratic form in the constants $a$'s, involving $F$. The earlier test statistics, mentioned above, were constructed taking into account the relative magnitudes of the $u$'s under the null and under the alternative hypotheses. The idea was to emphasize the difference between the two magnitudes. This "difference" is, in some sense, maximized if we are able to obtain the statistics, from the class, which has the largest noncentrality parameter under the alternative hypothesis of interest. This statistic would then be recommended to test $H_0$ whenever the particular alternative is suspected as likely. Also, for this particular alternative hypothesis, this test shall have maximum asymptotic relative efficiency (in the Pitman sense) among the class of statistics proposed. In the sixth section we show how to obtain the statistics with the above property and do so for certain specified alternatives. In the same section we compute the ARE of these tests with respect to certain of their competitors.

Let $\{F_n\}$ be a sequence of distribution functions defined on the real line, and suppose $\{F_n(x)\}$ converges to some limiting distribution function $F(x)$. It is of interest to investigate the error involved in using $F(x)$ as an approximation to $F_n(x)$, that is to investigate the rate of convergence of $\{F_n\}$ to $F$. This leads to the problem of finding bounds on $M_n = \sup_{-\infty<x<\infty}|F_n(x) - F(x)|$. In particular, this problem has been studied by several authors for cases where $F_n(x)$ represents the distribution function of a certain sum of independent random variables. For cases involving the classical forms of the central limit theorem Berry [1] and Esseen [3] have obtained certain bounds on $M_n$ which have been reinvestigated and improved by many authors (c.f. [4] Chapter XVI). Let $(X_{nk}), k = 1,2, \cdots, k_n; n = 1,2, \cdots$ be a system of random variables such that for each $n, X_{n1}, \cdots, X_{nk}$ are independent (we say the system is independent within each row). In [6], under suitable conditions, bounds have been obtained on $M_n$ for the case where $F_n(x)$ is the distribution function of $S_n = X_{n1}, + \cdots + X_{nk}$ and $F(x)$ is an infinitely divisible distribution. A basic assumption made in [6] was that both $X_{nk}$ and $F(x)$ have finite variances. The purpose of this study is to extend the results of [6] to include the case where neither $F(x)$ nor $X_{nk}$ need have finite variance. Our main theorem (Theorem 1) gives a bound on $M_n$ under a mild assumption on $X_{nk}$ and a certain assumption on the derivative of the infinity divisible distribution $F(x)$. It is shown in Section 4, that if $F(x)$ satisfies an additional condition which is considerably weaker than that having finite variance, then the bound obtained tends to zero as $n$ becomes infinite under necessary and sufficient conditions that $\{F_n(x)\}$ converge to $F(x)$. In Section 5 our general results are applied to the case of convergence of distribution functions of normed sums of independent identically distributed random variables to an arbitrary stable law with exponent $\alpha, 0 < \alpha < 2$.

Given a collection of analysis of variance mean squares, not all of which necessarily have the same degrees of freedom, the present paper describes a method of "mapping" them so as to facilitate the statistical structuring of the mean squares. Even under a null model of no real effects, the mean squares do not have the same distribution because their degrees of freedom may differ, and the ordered mean squares cannot be regarded as the usual order statistics of a sample from a single common distribution. If the ordered mean squares in a general orthogonal analysis of variance are $0 < S_1 \leqq S_2 \leqq \cdots \leqq S_K$ with corresponding degrees of freedom, $\nu_1,\nu_2, \cdots, \nu_K$, then the inferential reference set in the present approach is one obtained by so-called complete conditioning, i.e., repeated sampling from a set of $K$ populations such that the $i$th ordered mean square will be considered to have come from the population associated with $v_i$ degrees of freedom, for $i = 1,2, \cdots, K$. The approach consists of obtaining from each of the ordered mean squares, in turn, a maximum likelihood estimate of a presumed common error variance based on an order statistics formulation which employs complete conditioning of the mean squares. Methods of obtaining the sequence of maximum likelihood estimates as well as two graphical modes of displaying them are described. Illustrative examples are included.

Let $X = (X_1, \cdots, X_k)$ have the multinomial distribution, given by \begin{equation*} \tag{0.1} \operatorname{Pr}\{X = x\} = n!\prod^k_{i=1} (p^{xi}_i/(x_i!))\end{equation*} where $x = (x_1, \cdots, x_k), \sum^k_{i=1} x_i = n$ and $\sum^k_{i=1}p_i = 1$, and let \begin{equation*} \tag{0.2} C(p_1, \cdots, p_m) = \operatorname{Pr}\{X_i \geqq s_i; i = 1, \cdots, m\}\end{equation*} where $\sum^m_{i=1}s_i\leqq n$ and $m \leqq \min (k - 1, n)$. We show that $C(p_1, \cdots, p_m)$ is nondecreasing in $p_i$ for $i = 1, \cdots, m$ and that for $s_i = s_j$, \begin{equation*} \tag{0.3} C(p_1, \cdots, p_m) \leqq C_{ij}(p_1, \cdots, p_m)\quad\text{and}\end{equation*} \begin{equation*} \tag{0.4} C(p_1, \cdots, p_m) \geqq C_{ijt}(p_1, \cdots, p_m)\end{equation*} where $C_{ij}(p_1, \cdots, p_m)$ is obtained from $C(p_1, \cdots, p_m)$ by substituting $p = \frac{1}{2}(p_i + p_j)$ for $p_i$ and $p_j$ and $C_{ijt}(p_1, \cdots, p_m)$ is obtained from $C(p_1, \cdots, p_m)$ by substituting $t$ for $p_i$ and $p_i + p_j - t$ for $p_j$ where $0 \leqq t \leqq \min (p_i, p_j)$. These and similar results are shown. An application of these results to a multiple decision problem is indicated.