In recent years the study of sequential procedures which are asymptotically optimum in an appropriate sense as the cost $c$ per observation goes to zero has received considerable attention. On the one hand, Schwarz (1962) has recently given an interesting theory of the asymptotic shape, as $c \rightarrow 0$, of the Bayes stopping region relative to an a priori distribution $F$, for testing sequentially between two composite hypotheses $\theta \leqq \theta_1$ and $\theta \geqq \theta_2$ concerning the real parameter $\theta$ of a distribution of exponential (Koopman-Darmois) type, with indifference region the open interval $(\theta_1, \theta_2)$. (An example of Schwarz's considerations is described in connection with Figure 4.) One aim of the present paper is to generalize Schwarz's results to the case where (with or without indifference regions) the distributions have arbitrary form and there can be more than two decisions (Sections 2, 3, 4). In this general setting we obtain, under mild assumptions, a family $\{\delta_c\}$ of procedures whose integrated risk is asymptotically the same as the Bayes risk. (In fact, extending Schwarz's result, a family $\{\delta'_c\}$ can be constructed so as to possess this asymptotic Bayes property relative to all a priori distributions with the same support as $F$, or even with smaller indifference region support than $F$.) Procedures like our $\{\delta_c\}$ have already been suggested by Wald (1947) for use in tests of composite hypotheses (e.g., the sequential $t$-test), but his concern was differently inspired. At the same time, we show how such multiple decision problems can be treated by using simultaneously a number of sequential tests for associated two-decision problems. A second aim is to extend, strengthen, and somewhat simplify the asymptotic sequential design considerations originated by Chernoff (1959) and further developed by Albert (1961) and Bessler (1960) (Section 5). Our point of departure here is a device utilized by Wald (1951) in a simpler estimation setting, and which in the present setting amounts to taking a preliminary sample with predesignated choice of designs and such that, as $c \rightarrow 0$, the size of this preliminary sample tends to infinity, while its ratio to the total expected sample size tends to zero. The preliminary sample can then be used to "guess the true state of nature" and thus to choose the future design pattern once and for all rather than to have to reexamine the choice of design after subsequent observations. (In Wald's setting the only "design" problem was to pick the size of the second sample of his two-sample procedure.) The properties of the resulting procedure can then be inferred from the considerations of Sections 2, 3, and 4, where there is no design problem but where most of the work in this paper is done; using Wald's idea, we thereby obtain procedures for the design problem fairly easily, once we have the (non-design) sequential inference structure to build upon. The family $\{\delta^\ast_c\}$ so obtained has the same asymptotic Bayes property as that described above for the family $\{\delta_c\}$ of the non-design problem. Furthermore, a family $\{\delta^{\ast\ast}_c\}$ can be constructed so that, like $\{\delta'_c\}$ in the non-design problem, it is asymptotically Bayes for all a priori distributions with the same support. The value of the asymptotic Bayes risk of such a family is closely related to the lower bound which was obtained by Chernoff et al for the risk function of certain procedures, and which gives another form for the optimality statement. The role of the sequential procedures considered by Donnelly (1957) and Anderson (1960) for hypothesis testing with an indifference region is indicated at the end of Section 1. Asymptotic solutions to the problem of Kiefer and Weiss (1957) are given. An Appendix contains proofs of certain results on fluctuations of partial sums of independent random variables, which are used in the body of the paper.

In estimation problems where the parameter space is not compact the class of Bayes solutions $(\mathscr{B})$ is usually not a complete class and it is necessary to take the closure (in a suitable sense) of $(\mathscr{B})$ to obtain a complete class. When the parameter to be estimated is that of an exponential density the limits of Bayes solutions can be characterized as generalized Bayes solutions in the sense that they minimize a posteriori risk where the a priori distribution may have infinite variation (theorem and corollaries in Section 2). The extent to which exponential densities are necessary for this characterization and some consequences of this characterization are contained in a series of remarks at the end of Section 2. In Section 1 we motivate the ideas by obtaining the above characterization for the problem of estimating the mean of a normal distribution with the parameter space restricted to $\lbrack0, \infty)$ and with squared-error loss function.

Consider a factorial system of order $N = p^m$, which consists of $m$ factors each at $p$ levels. The factorial model relates the expected yield to the various treatment combinations in terms of a linear function of $N = p^m$ parameters $(\beta_0, \beta_1, \cdots, \beta_{N-1})$, which represent the main effects and interactions. A subset of $S = p^s(s < m)$ preassigned parameters is specified for estimation and testing of hypotheses. The other $N - S$ parameters are considered as nuisance ones. Unbiased estimates of the $S = p^s$ preassigned parameters may be obtained by different procedures of balanced random allocation designs (see Dempster [2], [3]). In the present paper we consider unbiased estimators with two types of randomized fractional replication, studied previously by Ehrenfeld and Zacks [4]. These procedures, designated by R.P.I. and R.P.II, are based on orthogonal fractional replication designs. As will be shown in a subsequent paper, both procedures have some optimal properties in cases where the nuisance parameters may assume arbitrary values. R.P.I. consists of choosing at random, with or without replacement, $n$ blocks of treatment combinations from the set of $M = p^{m-s}$ blocks, constructed by confounding the nuisance parameters. R.P.II. consists of choosing at random $n$ treatment combinations independently from each one of the $p^s$ blocks, constructed by confounding the pre-assigned parameters. The estimator of the pre-assigned parameters studied in the previous paper is the "least squares estimator", commonly applied in fractional replication procedures (see Kempthorne [5], Cochran and Cox [1]). When all the nuisance parameters are zero then this estimator is the least-squares estimator, and thus the best unbiased linear estimator. However, if the nuisance parameters are not zero there is no uniformly best unbiased estimator. The first question raised is whether unbiased estimates of the pre-assigned parameters can be attained by choosing a block of treatment combinations, in a similar manner to R.P.I., but with unequal probabilities, (unbalanced designs). As proven in the present paper, unbiased estimates cannot be attained with a procedure that assigns unequal probabilities to different blocks. Thus, the class of linear unbiased estimators is studied with respect to R.P.I. The same estimators can be applied to R.P.II., or to any balanced allocation design that yields a factorial model with similar properties to those of R.P.I. The statistical model adopted in the present paper is the orthogonal factorial model. Accordingly, the main effects and interactions are represented by orthogonal contrasts of the expected values of the observed random variable. This is a common practice for measuring main-effects and interactions (see Kempthorne [5]). The resulting algebra and numerical procedures are relatively simple and free of the difficult problems associated with the approach of Dempster [2], [3]. Basic notions as well as the required algebra for the factorial model are represented in Section 2. Theorems concerning the specification of the class of linear unbiased estimators, and the characterization of the subclass of conditional least squares estimators are given in Section 3. In Section 4 we prove that the subclass of conditional least squares estimators is complete. For this purpose we first derive an explicit formula for the variance-covariance matrix of any linear unbiased estimator with R.P.I.; and then we show that to any linear unbiased estimator not in the subclass of the conditional least-squares estimators one can find a conditional least-squares estimator so that the difference between the corresponding variance-covariance matrices is positive definite. For simplifying the arguments and notation, the definitions and theorems in Sections 3 and 4 are given relative to a particular choice of the pre-assigned parameters for special defining parameters, by which the classification of the treatment combinations into blocks in R.P.I. is carried out. In Section 5 the results are generalized for arbitrary sets of pre-assigned and defining parameters. In Section 6 the results are extended for R.P.II.

The present study is an extension of the previous work of the authors [3], [5], in which two randomization procedures in fractional factorial experiments were investigated. The general problem is to choose, in an optimal manner, a fractional replication of a full factorial system and an estimator, for the purpose of making inferences concerning a subset of pre-assigned parameters. We consider factorial systems of order $2^m$, which consist of $m$ factors each at 2 levels. The results can be generalized for factorial systems of order $p^m$, where $p > 2$. The factorial model adopted for the present study is the same as that adopted in the previous work, and the reader is referred to the preceeding papers [3], [5] for details and properties. The first two sections of [5] are essential for the present paper. The statistical properties of the conditional least-squares estimators (c.l.s.e.) were studied in [5] with respect to two specified randomization procedures (R.P.I. and R.P.II.), which are particular types of random allocation designs. It was shown that the c.l.s.e.'s constitute a complete class of linear unbiased estimators. The c.l.s.e. can be characterized as least-square estimators adjusted for the block of treatment combinations chosen and the information available on the nuisance parameters. In the present study we extend the investigation into the comparison of different randomization procedures. We consider a general class of procedures, characterized as follows: By some confounding method we construct $M = 2^{m-s}$ blocks, each one containing $S = 2^s$ treatment combinations. We choose one of the blocks with an arbitrary probability vector, $\xi$, and observe the associated random variables. R.P.I. is the special case where every block has the same probability of being chosen. A fixed fractional replication procedure is the special case where one of the blocks is chosen with probability one. Each randomization procedure is represented (uniquely) by a probability vector, $\xi$, and each c.l.s.e. is represented by a vector $\gamma$ in $(2^m - 2^s)$-dimensional Euclidean space. A strategy of the Statistician is thus represented by a pair of vectors $(\xi, \gamma)$. Minimax strategies are studied in the present paper for two states of information concerning the nuisance parameters: (i) all the nuisance parameters are bounded; (ii) all the nuisance parameters are bounded and all their signs are known. As proven in the present paper, the minimax strategy corresponding to Case (i) consists of R.P.I. with an unadjusted c.l.s.e. and is thus independent of the actual bounds of the nuisance parameters. On the other hand, the minimax strategy for Case (ii) consists of some fixed fractional replication with an adjusted c.l.s.e. which depends on the actual bounds of the nuisance parameters. These minimax theorems are proven with respect to a mean-square-error loss function, defined as the trace of the mean square-error matrix. A closeness loss function is considered too. The closeness of an estimator is defined as the probability that the values of the estimator will lie in a prescribed neighborhood of the true values of the parameters. In Section 2 the mean-square-error matrix and the closeness of a c.l.s.e., under an arbitrary randomization procedure, $\xi$, are derived. The mean-square-error and the closeness loss functions are defined in Section 3. Formulae for mean-square-error Bayes strategies, against any given a priori distribution of the nuisance parameters, are then derived. The closeness risk function is approximated by a similar function which has the same Bayes strategies as the mean-square-error loss function. It is shown that when the size of the experiment, $S = 2^s$, grows the closeness risk function and the approximative function converge. The minimax theorems are stated and proved in Section 4 only with respect to the mean-square-error loss function. It can be shown that the minimax closeness strategies, for the states of information studied, are the same as those for the mean-square-error loss function. This can be concluded also from the results of Section 3. In Section 5 we present a numerical example to illustrate the results and the computations involved.

Let $X$ be a generalized random variable taking values in an abstract set $\mathscr{X}$ on which is defined an appropriate $\sigma$-field of subsets. Suppose that the distribution of $X$ depends on a real parameter $\Theta$ and that it is desired to estimate the value of $\Theta$ from an observation on $X$. Let $W(\cdot)$ be a sufficiently smooth, non-negative, symmetric, convex function defined on the real line. Suppose that when the true value of $\Theta$ is $\theta$ and the estimated value is $\delta$, the loss incurred is $W(\theta - \delta)$. For a given prior distribution of $\Theta$ and any $x \epsilon \mathscr{X}$, let $F(\cdot \mid x)$ be the posterior distribution function of $\Theta$ when the observed value of $X$ is $x$. A Bayes estimate, for the given value of $x$, is a number $\delta^\ast$ such that \begin{equation*}\tag{1.1}\int_{-\infty < \theta < \infty} W(\theta - \delta^\ast) dF(\theta \mid x) = \inf{\tt\_}{\infty < \delta < \infty} \int_{-\infty < \theta < \infty} W(\theta - \delta) dF(\theta \mid x).\end{equation*} Thus, for each given $x$, the problem of finding a Bayes estimate reduces to the problem of minimizing the integral \begin{equation*}\tag{1.2}\int_{-\infty < \theta < \infty} W(\theta - \delta) dF(\theta)\end{equation*} where $F(\cdot)$ is a specified distribution function. In Section 2 the solution of this minimization problem is presented and some properties of the minimizing values of $\delta$ are discussed. In Section 3 it is shown that a Bayes estimator $\delta^\ast(\cdot)$ satisfying (1.1) for all $x \epsilon \mathscr{X}$ can be chosen so that it is a measurable function of $x$. In Section 4 the question of evaluating the expectation of $W\lbrack\Theta - \delta^\ast(X)\rbrack$ is considered and lower bounds for this quantity are presented. The special problem in which $W(\cdot)$ is of the form $W(t) = |t|^k, - \infty < t < \infty, k \geqq 1$, is considered in some detail. It is known (see e.g., [1], p. 302) that for $k = 1$ the integral (1.2) is minimized when $\delta$ is a median of the distribution function $F(\cdot)$, and for $k = 2$ it is minimized when $\delta$ is the mean of $F(\cdot)$. The solution of the minimization problem presented in Section 2 is a generalization of these familiar results.

A short exposition of the use of the generalized Jirina sequential tolerance limits procedure, and the necessary assumptions, is given. A method of obtaining the confidence level of the coverage for various values of the parameters from existing tables is included. Formulae for the expectation and variance of the random sample size are derived and usable approximations obtained. That the random sample size, appropriately scaled by one of the parameters, has an asymptotic distribution as the parameter increases is proved and the Laplace transform of this distribution is found. Also formulae for the asymptotic mean and variance are found and methods for their calculation exhibited. Some applications are given in the following paper.

Assume that each random variable of a sequence had a density which is a Polya frequency function of order two. To this sequence we apply the Jirina sequential procedure to determine a tolerance interval. In this paper we find some sufficient conditions on the type of trend permissible for this sequence which enable us to show that when the Jirina procedure is used the sampling will stop sooner and the tolerance interval cover more of the population (in a stochastic sense) than would occur in the case without trend. Similar considerations are shown to hold when the sequences of observations have densities which have non-decreasing hazard rates.

This paper presents null hypothesis distribution theory for certain methods of significance testing applicable to multivariate data. This theory is derived in Sections 2 and 3 using simple geometrical reasoning. Section 3 derives theory related to stepwise methods in a form sufficiently general to include both the standard methods of Section 4 and certain new methods based on principal variables of Section 5. The generality in Section 3 is made possible by the independence results proved in Section 2.

Fisher wrote as though the fiducial argument were a well-defined form of reasoning: witness phrases like "the fiducial argument itself," (Fisher (1956), p. 120) and "a genuine fiducial argument" (Fisher (1956), p. 172). However, attempts to extend the set of examples of the fiducial argument beyond the set personally approved by Fisher have often run into the difficulty that several inconsistent fiducial arguments appeared to be available for a single situation, e.g., the pair of papers by Creasy (1954) and Fieller (1954), Mauldon (1955), Tukey (1957) and Brillinger (1962). Apparently in response to the Mauldon example, Fisher intimated that the joint fiducial distribution of several parameters should be built up "rigorously by a step by step process" (Fisher (1956), p. 172). Fisher's comment is rather confusing because the Mauldon approach does appear to be a step by step approach. Brillinger (1962) presented an artificial two-parameter example in which Fisher's type of step by step approach can be applied in inconsistent ways. In Section 2 I will show that a step by step approach leads to alternative inconsistent answers even in the basic two parameter situation of sampling from the normal distribution. In Section 3 I will demonstrate a difference between the fiducial distribution for the means $(\mu_1, \mu_2)$ of a bivariate normal distribution given by Fisher (1954) and the marginal distribution of $(\mu_1, \mu_2)$ under the joint fiducial distribution of all five parameters of the bivariate normal given by Fisher (1956).

For various but sound practical reasons it has become desirable to approximate to previously tabulated mathematical functions by polynomials or rational fractions. In this paper Chebyshev polynomials are used to approximate Mills' Ratio over two separate ranges [0, 1], [1, $\infty$] of the argument. Some new asymptotic expansions for this ratio are also obtained by an extended use of the symbolic operator method, revealing incidentally that Ruben's (1962) expansion is a special but not necessarily superior case.

The principle of maximum entropy, together with some generalizations, is interpreted as a heuristic principle for the generation of null hypotheses. The main application is to $m$-dimensional population contingency tables, with the marginal totals given down to dimension $m - r$ ("restraints of the $r$th order"). The principle then leads to the null hypothesis of no "$r$th-order interaction." Significance tests are given for testing the hypothesis of no $r$th-order or higher-order interaction within the wider hypothesis of no $s$th-order or higher-order interaction, some cases of which have been treated by Bartlett and by Roy and Kastenbaum. It is shown that, if a complete set of $r$th-order restraints are given, then the hypothesis of the vanishing of all $r$th-order and higher-order interactions leads to a unique set of cell probabilities, if the restraints are consistent, but not only just consistent. This confirms and generalizes a recent conjecture due to Darroch. A kind of duality between maximum entropy and maximum likelihood is proved. Some relationships between maximum entropy, interactions, and Markov chains are proved.

It is well known, in testing an hypothesis concerning the means of several independent normal distributions with common but unknown variance $\sigma^2$, that the likelihood ratio $\lambda$, raised to an appropriate positive power, is equal to the ratio of two quadratic forms. That is, there exists a positive constant $c$ so that $\lambda^c = \mathbf{X'AX/X'BX}$, where $\mathbf{A}$ and $\mathbf{B}$ are real symmetric matrices and $\mathbf{X}$ is a column matrix whose elements have independent normal distributions. Since $\lambda \leqq 1$, we see that $\mathbf{X' (B - A)X}$ is a non-negative quadratic form. In addition, we usually find that $\mathbf{X'AX}/\sigma^2$ and $\mathbf{X'BX}/\sigma^2$ have chi-square distributions. That is, $\mathbf{A}^2 = \mathbf{A}$ and $\mathbf{B}^2 = \mathbf{B}$. Consequently, in accordance with a theorem of Hogg and Craig [5], $\mathbf{A(B - A)} = \mathbf{0}$ and $(\mathbf{B - A})^2 = (\mathbf{B - A)}$. Hence $\mathbf{X'AX}$ and $\mathbf{X'(B - A)X}$ are stochastically independent and $\mathbf{X'(B - A)X}/\sigma^2$ is chi-square. Hence $\lambda^c$ has a beta distribution provided each chi-square is central; this is usually the case under the null hypothesis. The analogous situation in multivariate statistical analysis introduces a rather intriguing theorem which almost seems obvious upon first inspection. We will describe this situation after we present some notation and certain preliminary results. Let the $(n \times p)$ matrix $\mathbf{X} = (x_{ij})$ have the p.d.f. $\exp\{-\frac{1}{2}\operatorname{tr}\lbrack \mathbf{K}^{-1}(\mathbf X - \mathcal{u})'\mathbf{V}^{-1} (\mathbf X - \mathcal{u})\rbrack\ /(2\pi)^{np/2} |\mathbf{K}|^{n/2} |\mathbf{V}|^{p/2}, - \infty < x_{ij} < \infty,$ where the $(n \times n)$ matrix $\mathbf{V}$ and the $(p \times p)$ matrix $\mathbf{K}$ are real symmetric positive definite matrices and $\mathbf{\mathcal{u}}$ is a real $(n \times p)$ matrix. If $\mathbf{V = I}$, the $(p \times 1)$ column matrices $\mathbf{Y}_1, \cdots, \mathbf{Y}_n$ of $\mathbf{X'}$ are independent and have, respectively, the $p$-variate normal distributions $N(\mathcal{\mathbf{u}}_k, \mathbf{K}), k = 1, 2, \cdots, n$, where $\mathbf{\mathcal{u}}_1, \cdots,\mathbf{\mathcal{u}}_n$ are the $(p \times 1)$ column matrices of $\mathbf{\mathcal{u}}'$. In a recent article [7], Roy and Gnanadesikan proved the following results which are generalizations of two theorems on quadratic forms [3], [4]. Let $\mathbf{A, A}_1$, and $\mathbf{A}_2$ be real symmetric $(n \times n)$ matrices. Then $\mathbf{X'AX}$ has the Wishart distribution $W(\mathbf{K}, r, \mathbf{\mathcal{u}}'A\mathbf{\mathcal{u}})$ if and only if $\mathbf{AVA} = \mathbf{A}$ (or $\mathbf{A}^2 = \mathbf{A}$ provided $\mathbf{V = I}$). Here $r$ is the rank of $\mathbf{A}$ and $\mathbf{\mathcal{u}}'A\mathcal{u}$ is the matrix of the non-centrality parameters. The forms $\mathbf{X'A}_1\mathbf{X}$ and $\mathbf{X'A}_2\mathbf{X}$ are stochastically independent if and only if $\mathbf{A}_1\mathbf{VA}_2 = 0$ (or $\mathbf{A}_1\mathbf{A}_2 = 0$ provided $\mathbf{V = I}$). These results permit us to state, without proof, the chi-square decomposition theorem of Hogg and Craig [5] in terms of Wishart variables. THEOREM 1. Let $\mathbf{A, A}_1, \cdots, \mathbf{A}_{k-1}, \mathbf{A}_k$ be real symmetric $(n \times n)$ matrices so that $\mathbf{A = A}_1 + \cdots + \mathbf{A}_{k-1} + \mathbf{A}_k$. Let $\mathbf{X'AX, X'A}_1\mathbf{X}, \cdots, \mathbf{X'A}_{k-1}\mathbf{X}$ have Wishart distributions and let $\mathbf{A}_k$ be positive semidefinite. Then $\mathbf{X'A}_1\mathbf{X}, \cdots, \mathbf{X'A}_{k-1}\mathbf{X, X'A}_k\mathbf{X}$ are mutually stochastically independent and $\mathbf{X'A}_k\mathbf{X}$ has a Wishart distribution. Moreover, if $\mathbf{A} = \mathbf{V}^{-1}$ and if $k = 2$, the above conclusion is still valid even though the hypothesis that $\mathbf{A}_2$ is positive semidefinite is omitted. Now, in most tests of an hypothesis concerning the means of a multivariate normal distribution with unknown matrix $\mathbf{K}$, the likelihood ratio, raised to an appropriate positive power, is equal to the ratio of two determinants, say $U = |\mathbf{X'AX}| / |\mathbf{X'BX}|$, where $\mathbf{A}$ and $\mathbf{B}$ are real symmetric matrices with ranks greater than or equal to $p$. Usually $\mathbf{V = I}$ and hence, for simplicity, we assume that this is the case. In addition, we frequently know, or it can be easily shown, that both $\mathbf{X'AX}$ and $\mathbf{X'BX}$ have Wishart distributions. Accordingly, if the fact that the likelihood ratio is less than or equal to one (or $|mathbf{X'AX}| \leqq| \mathbf{X'BX}|$ for all $\mathbf{X}$) implies that $\mathbf{B - A}$ is positive semidefinite, then Theorem 1 requires that $\mathbf{X'AX}$ and $\mathbf{X'(B - A)X}$ be stochastically independent and that $\mathbf{X'(B - A)X}$ have a Wishart distribution. Thus, if this is true, $U$ has a well known distribution, for it is distributed like $|\mathbf{W}_1|/|\mathbf{W}_1 + \mathbf{W}_2|$, where $\mathbf{W}_1$ and $\mathbf{W}_2$ are independent Wishart variables. In the next section, we show that this is, in fact, the situation. Finally, in the last section of this paper, we consider certain other theorems on independence that involve Wishart variables.

In a one-way analysis of variance situation in which the populations are ordered under the alternative hypothesis, one desires a test that, unlike the usual normal theory $F$ test, concentrates its power on the ordered alternatives, not on any alternatives. In this paper, two contributions are made. First, under the usual normal assumptions, work by Bartholomew [4], [7] on the likelihood ratio test, when the ordering is complete under the alternative hypothesis, is extended. By suitable characterization of the partition which the likelihood ratio induces on the sample space, the likelihood ratio test is shown to depend on incomplete Beta functions and certain probabilities of the above partitions of the sample space. The major contribution in this paper is for the case of equal sample sizes, where explicit expressions for these probabilities are obtained by indicating their relationship to Sparre Andersen's [1], [2] results. Second, under the analogous nonparametric assumptions and for equal sample sizes, a parallel test based on ranks is proposed and discussed for stochastic ordering of the populations. The asymptotic Pitman efficiency of the nonparametric test relative to the test in the normal case is derived.

In linear models with several observations per cell, estimates of all contrasts are given whose small and large sample behaviour is analogous to that of the estimate of a shift parameter proposed in [2]. In particular, the asymptotic efficiency of these estimates relative to the standard least squares estimates, as the number of observations in each cell gets large, is shown to be the same as the Pitman efficiency of the Wilcoxon test relative to the $t$-test.

Let $X_i, Y_i i = 1, 2, \cdots n$ be mutually independent binomial random variables with $P\{X_i = 1\} = p_1, P\{Y_i = 1\} = p_2$. Let $\bar X_n = (1/n) \sum^n_{i = 1} X_i$ and $\bar Y_n = (1/n) \sum^n_{i = 1} Y_i$. We wish to estimate the parameter $p = (p_1, p_2)$. If the parameter space is $\Omega = \{p \mid 0 \leqq p_1 \leqq 1, 0 \leqq p_2 \leqq 1\}$ then the usual non-sequential estimator $\delta = (\delta_1, \delta_2)$ is of the form $(f(\bar X_n)), f(\bar Y_n))$, that is, if no restriction is placed on the parameters, the estimator for the paired parameters is the pair of estimators for each parameter separately. In this paper, however, we are concerned with the parameter space $\Omega = \{p \mid 0 \leqq p_1 \leqq p_2 \leqq 1\}$. We show that estimators constructed as before are no longer admissible with respect to a class of reasonable loss functions. Square error loss is included in this class. In particular, we show that for such loss functions, estimators not ordered in the same way as the parameters are inadmissible. A class of estimators which retains the same ordering as the parameters, that is with $\delta_2 \geqq \delta_1$, is investigated and the asymptotic behavior of the minimax member is described. Finally, an asymptotic estimator based on a normal approximation is given. This estimator is minimax and admissible. The problem of estimating ordered probabilities arises in estimating a section of an unknown distribution function.

In a recent paper [1] Ansari and Bradley have shown the equivalence of two rank tests for comparing dispersion, one test due to Barton and David [2], the other to Ansari and Freund, and have provided tables of the exact distribution. They observe that Siegel and Tukey have proposed [11] a similar test which permits use of existing tables. They also exhibit the mean of the limiting normal distribution under the alternative hypothesis. Later Klotz [7] established the equivalence of all these tests. In the present paper it is shown that (1) Any of these tests is consistent against differences in dispersion if the two distributions have a common median and differ in a scale parameter, and under some less restrictive circumstances. But without such restrictions bizarre asymptotic behavior can arise--including good sensitivity against translation for some non-symmetric densities. One (not very natural) example is offered in which the test constructed for rejection if one of two scale parameters is the larger, actually turns out to be consistent against that parameter's being the smaller of the two. (2) No rank test (i.e., a test invariant under strictly increasing transformation of the scale) can hope to be a satisfactory test against dispersion alternatives without some sort of strong restrictions (e.g., equal or known medians) being placed on the class of admissible distribution pairs. (3) Box [3] has proposed testing equality of variances by applying the $t$ test to the logarithms of variances computed within small subgroups. He indicates how such tests should be robust (though not of exact size). Distribution free tests of exact size can be constructed by applying a rank test in place of the $t$ test. Wilcoxon's test applied to variances-within-triads has asymptotic efficiency .5 against normal alternatives. If the two samples each have 9 observations then the exact power is readily calculated and "efficiency" is again about .5.

Assuming that the wide sense stationary process being sampled has an absolutely continuous spectrum, Parzen [5] has shown the consistency of a general class of estimators for estimating the spectral density at a given frequency. He has shown that the class of estimators he considers, contains as a particular case estimators considered earlier by Grenander and Rosenblatt [2], Barlett [1], Tukey [7], [8], Daniell, and Lomnicki and Zaremba [3]. In this paper we extend Parzen's results to the case of a Stationary Gaussian Vector process whose spectrum is not necessarily absolutely continuous and show that this general class of estimators consistently estimate the co- and quadrature spectral densities at all those frequencies where they exist. It has been earlier shown by the author [4], using an entirely different approach from the one presented in this paper, that for a normal Stationary process whose spectrum besides the absolutely continuous part contains a step function with a finite number of saltuses, the weighted periodogram estimator, which is a particular case of the general class of estimators considered by Parzen [5], is still a consistent estimate of the spectral density at any point of continuity of the spectrum. Thus this paper also substantially generalizes the earlier result of the author, where he limits himself to a finite number of saltuses.

This paper came out of an attempt to solve the following general problem: Suppose $\{Y_n, n \geqq 1\}$ is a stationary process with a finite state-space $J$. Under what conditions can we express it as a function of a finite Markov chain? More precisely, when can we find a stationary Markov chain $\{X_n, n \geqq 1\}$ with a finite state-space $I$ and a function $f$ on $I$ onto $J$ such that the process $\{f(X_n)\}$ has the same distribution as $\{Y_n\}$? We do not solve the general problem here but for mixing processes we obtain a theorem which is the best possible in a certain sense. Suppose $\epsilon$ denotes a state of $J$ and suppose $s, t$ denote finite sequences of states of $J$. If $s = \epsilon_1 \cdots \epsilon_n$, let $p(s) = P\lbrack (Y_1, \cdots, Y_n) = s\rbrack$. For each $\epsilon$, define $n(\epsilon)$ to be the largest integer $n$ such that we can find $s_i, t_i (i = 1, \cdots, n)$ such that the matrix $\|p(s_i\epsilon t_j)\|$ is nonsingular. Gilbert [4] has shown that if $\{Y_n\}$ is a function $f$ of a finite Markov chain $\{X_n\}$ and if $f$ takes $N(\epsilon)$ states of $I$ into the state $\epsilon$ of $J$, then $n(\epsilon) \leqq N(\epsilon)$. If $n(\epsilon) = N(\epsilon)$, then $\{Y_n\}$ is said to be a regular function of a Markov chain. Thus a necessary condition for $\{Y_n\}$ to be a function of a finite Markov chain is that $\sum n(\epsilon)$ is finite. It is proved here that if $\sum n(\epsilon) < \infty$ and if the process $\{Y_n\}$ is mixing, then there exists a positive integer $m^\ast$ such that for every $m \geqq m^\ast$ the process $\{Y_{nm + 1}, n \geqq 0\}$ is a function of a Markov chain with $\sum n(\epsilon)$ states. An example is constructed to show that $m^\ast$ cannot, in general, be brought down to 1. Thus the whole process $\{Y_n, n \geqq 1\}$ may still not be a function of a Markov chain with $\sum n(\epsilon)$ states.

Let $\{Y_n, n \geqq 1\}$ be a stationary process with a finite state-space $J$. We will use the definitions of a function and of a regular function of a finite Markov chain given in [1]. In Section 1 of this paper we define, for each state $\epsilon$ of $J$, a convex cone $\mathscr{C}(\pi_\epsilon)$. The main theorem (Section 2) asserts that if each $\mathscr{C}(\pi_\epsilon)$ is polyhedral, then $\{Y_n\}$ is a function of a finite Markov chain. The hypothesis that each $\mathscr{C}(\pi_\epsilon)$ is polyhedral is not quite necessary and some results are given in Section 3 under weaker assumptions. It is also shown that these weaker conditions are necessary for $\{Y_n\}$ to be a regular function of a finite Markov chain. The final section presents an example which shows that not every function of a finite Markov chain is a regular function of a Markov chain. The results of Gilbert [3] are at the root of our investigation. Gilbert always assumed that the given stationary process $\{Y_n\}$ and the underlying Markov chain were irreducible and aperiodic. However, his results continue to hold even when these assumptions are dropped. In particular, the results of Section 1 of [3] depend only on the stationary and the Markov character of the underlying chain. Theorem 2 of [3] also holds in a more general set-up (see Lemma 3.1 of [1]). Our stationary process $\{Y_n\}$ need not be irreducible or aperiodic. Our sufficient conditions also do not necessarily yield such a chain.

Let $T$ be a subset of the set $I$ of non-negative integers. Define $f(\theta) = \sum a(x)\theta^x$ where the summation extends over $T$ and $a(x) > 0, \theta \geqq 0$ with $\theta \epsilon \Theta$, the parameter space, such that $f(\theta)$ is finite and differentiable. One has $\Theta = \{\theta:0 \leqq \theta < R\}$ where $R$ is the radius of convergence of the power series of $f(\theta)$. Then a random variable $X$ with probability function \begin{equation*}\tag{1}\operatorname{Prob}\{X = x\} = p(x, \theta) = a(x)\theta^x/f(\theta) \quad x \epsilon T\end{equation*} is said to have the generalized power series distribution (GPSD) with range $T$ and the series function $f(\theta)$. It may be observed that the range $T$ could be a countable subset of the real numbers. The GPSD is thus an exponential-type discrete distribution; whereas the power series distribution (PSD) as defined by Noack [5], is a special case of a GPSD. The author [7], [8], [9], [10], [11] has discussed some problems of statistical inference associated with the GPSD and some of its particular forms. Roy and Mitra [14] and Guttman [2] have studied the problem of the minimum variance unbiased (MUV) estimation for the PSD's. In this paper, we investigate the problem of the existence of the MVU estimator for the parameter $\theta$ of the GPSD in terms of the number theoretic structure of its range $T$. We further provide the MVU estimators for the probability and distribution functions of the GPSD and consider a few special cases of some practical significance.

Known PBIB designs with two associate classes were classified according to their association schemes by Bose and Shimamoto [3]. Association schemes for three or more associate classes have been little developed except the rectangular association scheme of Vartak [15] and GD $m$-associate designs of Raghavaroa [8] and Hierarchical group divisible designs with $m$-associate classes of Roy [11]. In this paper we define a new association scheme known as right angular scheme which defines four associate class designs. For convenience we call PBIB designs with right angular scheme, right angular designs. In Section 2 we define the association scheme. Sections 4 and 5 deal with the analysis and construction of right angular designs. In Section 6 we study some combinatorial properties and Sections 3 and 7 give the non-existence of some right angular designs.

A procedure for adjusting the treatment design matrix to furnish estimates as orthogonal linear functions of stochastic variates has been given for fractional replicates of factorial experiments composed of $ks^{-m}s^n$ treatments for $m < n$, for $s$ a prime number, and $k$ not a multiple of $s$. These are irregular fractional replicates. Ordinarily irregular fractional replicates do not lead to estimates of effects which are orthogonal linear combinations of the observations. In this paper a new relationship and some generalizations on the theory of irregular fractional replicates of complete factorials have been developed. En route to these developments two theorems in matrix transformations were proved. In addition, the relationship between the method utilized here and ordinary missing plot techniques is pointed out.

This paper considers the sequential compound decision problem for the case where the component decisions are of the simple versus simple hypothesis testing type, and thus can be stated in terms of testing whether $\theta = 0$ or $\theta = 1$. From this point of view it may be considered to be the sequential analogue of [3]. The risk for the compound decision is taken to be the average of the risks in the component problems. We prove that whenever the Bayes envelope power function of the component has a derivative, the decision function which at each stage plays Bayes against an estimate of the average of the previous parameter values, has a risk which for all $n$ sufficiently large lies uniformly close to the envelope risk function. We also exhibit a randomized version of the above decision function which has this property for all possible envelope functions.

This note describes explicitly a minimal complete class of decision rules for the problem of sequentially classifying the individuals of a group which is known to come from one of two completely specified populations.