It may fairly be said that modern time series analysis is a subject which embraces three fields which while closely related have tended to develop somewhat independently. These fields are (i) statistical communication and control theory, (ii) the probabilistic (and Hilbert space) theory of stochastic processes processing finite second moments, and (iii) the statistical theory of regression analysis, correlation analysis, and spectral (or harmonic) analysis of time series. In this paper it is my aim to show the close relation between these fields and to summarize some recent developments. The topics discussed are (i) stationary time series and their statistical analysis, (ii) prediction theory and the Hilbert space spanned by a time series, and (iii) regression analysis of time series with known covariance function. In particular, I describe a new approach to prediction and regression problems using reproducing kernel Hilbert spaces.

There are given a populations $\Pi_1, \cdots, \Pi_a$, of which we wish to select a subset. The quality of the $i$th population is characterized by a real-valued parameter $\theta_i$, and a population is said to be \begin{align*}\tag{1} positive \quad \text{(or} \quad good) \quad \text{if} \quad \theta_i &\geqq \theta_0 + \Delta, \\ \tag{2} negative \quad {(or} \quad bad) \quad \text{if} \quad \theta_i &\leqq \theta_0,\end{align*} where $\Delta$ is a given positive constant and $\theta_0$ is either a given number or a parameter that may be estimated. A number of optimum properties of selection procedures are defined (Section 3) and it is shown that for some of these, the optimum procedure selects $\Pi_i$ when \begin{equation*}\tag{3}T_i \leqq C_i,\end{equation*} where $T_i$ is a suitable statistic, the distribution of which depends only on $\theta_i$, and where $C$ is a suitable constant. (Sections 4 and 6.) Applications are given to distributions with monotone likelihood ratio in the case that $\theta_0$ is known (Sections 5 and 6), and to normal distributions when instead observations on $\theta_0$ are included in the experiment (Sections 10 and 11).

The paper is mainly concerned with the following multiple comparisons problem in the analysis of variance setting. In a balanced experiment $n$ treatments are to be compared. Each of the $\frac{1}{2}n(n - 1)$ pairwise comparisons is to be made, adjudging each difference as "positive", "negative", or "not significant"; overall decisions involving intransitivities are barred. The loss for each difference is proportional to the error; if a difference is asserted incorrectly the loss has proportionality constant $c_1$, if "not-significant" is the incorrect conclusion the proportionality constant is $c_0$; where $c_1 = k_1 + k_0, c_0 = k_0$ and $k_1 > k_0 > 0$. Total loss for the experiment is taken as the sum of the $\frac{1}{2}n(n - 1)$ component losses. The Bayes rule for any prior distribution is shown as a result to consist in the simultaneous application of Bayes rules to the $\frac{1}{2}n(n - 1)$ component problems. Each of these in turn is shown similarly to consist in the simultaneous application of Bayes rules to two subcomponent problems. The subcomponent Bayes rule for a normal prior density of treatment means is explicitly derived. The dependencies of the solution on the variance of the prior density, the degrees of freedom and the loss ratio $k_1/k_0$ are discussed. A principal finding is that the Bayes solution for the multiple comparisons problem corresponds to a tolerated error probability "of the first kind" for each single difference, that is independent of the number of treatments being compared.

The usual method of finding a most powerful size $\alpha$ test of a composite hypothesis against a simple alternative is the guessing of a Least Favorable Distribution (LFD)--introduced at various levels of generality by Neyman and Pearson [6], Wald [7], Lehmann [4], and Lehmann and Stein [5]--and testing the mixture of the distributions of the hypothesis over this LFD against the alternate using the Neyman-Pearson Fundamental Lemma. In guessing LFD's statisticians have looked for a mixture which is "like" the alternate. In this paper, the notion of Uniformly Least Favorable Mixture (ULFM) is introduced. In Section 2, we show that a ULFM is a point in the convex set of mixtures of the hypothesis which is closest (in the sense of the $\mathfrak{L}^1$ norm) to the alternate. The condition is not sufficient. More generally, any LFM corresponds to a point which is closest to the alternate in some expansion or contraction of this set of mixtures. A sufficient condition for ULFM's is, essentially, that the nuisance parameter can take on the same values in the alternate as in the hypothesis. In Section 3, we consider the case where no ULFM exists. We show, inter alia, that any distribution is least favorable for a closed set of $\alpha$'s. (A pathological example shows that this closed set need not be the union of a finite number of closed intervals.)

Asymptotic formulas are obtained for the generalized variance of the least squares estimates in polynomial regression under the assumption that the basic random variables are those of a stationary stochastic process, or a slight generalization of such a process. These formulas are used to study the information obtained by increasing the number of observational points in an interval and by increasing the length of the interval.

We shall consider the problem of estimating a linear functional relationship \begin{equation*}\tag{1.1}\alpha + \beta_1\tau^1 + \cdots + \beta_p\tau^p = 0\end{equation*} among $p$ variables $\tau^1, \cdots, \tau^p$ when the observed values do not satisfy it because all of them are subject to errors or fluctuations (superscripts will, in general, be indexing symbols, not powers, in this paper). Geometrically, the problem is equivalent to fitting straight lines or planes to a series of $q$ observed points when all the coordinates are subject to error. This problem has a long history. R. J. Adcock, in two papers written in 1877 [2] and 1878 [3], solves it by minimizing the sum of squares of the orthogonal distances from the points to the hyperplane (1.1). Adcock and many other authors used the model \begin{equation*}\tag{1.2}y_i = \tau_i + \epsilon_i\quad (i = 1, \cdots, q),\end{equation*} where $y_i$ and $\tau_i$ are column vectors representing the observed and true points, and the errors $\epsilon_i$ are independent random vectors with mean value zero. Since the $\tau_i$ are points lying on the hyperplane (1.1), we have in matrix notation \begin{equation*}\tag{1.3}\alpha + \beta\tau_i = 0\quad (i = 1, \cdots, q)\end{equation*} where $\beta$ is a row vector with components $\beta_1, \cdots, \beta_p$. If we assume that the $\tau_i$ are independently drawn from a probability distribution, then the estimate of $\beta$ obtained by Adcock is not consistent. In fact, in 1937, J. Neyman [21] pointed out that if the distribution of the true vectors $\tau_i$ and the errors $\epsilon_i$ is normal, then the distribution of the observed vectors $y_i$ is also normal and, being determined by moments of the first two orders, it is not sufficient to determine the parameters $\alpha$ and $\beta$; the functional relationship (1.1) is, therefore, nonidentifiable. Several methods have been proposed to overcome this difficulty, for which the reader is referred to a recent general survey of the literature by A. Madansky [18], which also contains an extensive bibliography. One approach is to assume that we know the covariance matrix of the errors up to a numerical factor. As was shown, in general, by C. F. Gauss [8], [9], in the case of independently and normally distributed observations whose variances are known up to a numerical factor, the maximum likelihood estimate is simply the weighted least-squares estimate. This estimate of the linear functional relationship was obtained as early as 1879 by C. H. Kummell [15], for the case in which the components $\epsilon^h_i$ of the vectors $\epsilon_i$ are independently distributed with variances $\kappa^{ii}\sigma_{hh}$, where the $\kappa^{ii}$ are known constants and the $\sigma_{hh}$ are known only up to a numerical factor. Kummell found that his estimate coincides with the estimate proposed by Adcock only in the case in which all the variances are equal. M. J. van Uven [24] considered the case in which the errors $\epsilon_i$ are independent and have the same multivariate normal distribution with a covariance matrix $\Sigma$ which is known only up to a numerical factor. His method is essentially the following. He considers $\tau^1, \cdots, \tau^p$ as skew coordinates in a new, "isotropic" space in which the rectilinear orthogonal coordinates are independent and have the same variance. In the new space he then uses Adcock's principle of adjustment, namely, he chooses as the estimate the hyperplane which minimizes the sum of orthogonal distances. Later, T. Koopmans [14] showed that van Uven's estimate is the maximum likelihood estimate for the case considered. If the $\tau_i$ are assumed independently drawn from a probability distribution, the estimate of the linear functional relationship thus obtained is consistent, but the estimate of $\Sigma$ converges in probability to $p^{-1}\Sigma$ (see also [16]). B. M. Dent [7] solved the maximum likelihood equations in the case in which $\Sigma$ is not known, but, as was shown later by D. V. Lindley [16], her estimates are not consistent, and should, therefore, be rejected. More recently, J. Kiefer and J. Wolfowitz [13] showed that, under certain conditions of identifiability, when the $\tau_i$ have a probability distribution, the method of maximum likelihood, if properly applied, yields consistent estimates of both the linear functional relationship and the probability distribution of the $\tau_i$. However, Kiefer and Wolfowitz do not give explicit expressions for the maximum likelihood estimates. No difficulties with respect to the identifiability of the functional relationship or with respect to the consistency of the estimates arise if we can replicate the observations. The model is now, in matrix notation, \begin{equation*}\tag{1.4}y_{ij} = \tau_i + \epsilon_{ij}.\end{equation*} In general we have $n_i$ observed points for each of the true points $\tau_i$, and it is assumed that not all the $\tau_i$ lie on a translated subspace of dimension smaller than $p - 1$. Obviously this implies that $q \geqq p$. This model has been considered previously by G. W. Housner and J. F. Brennan [12], J. W. Tukey [23] and by F. S. Acton [1] (see also [11] and [25]). If we assume that the errors $\epsilon_{ij}$ are independently and normally distributed with a known covariance matrix $\Sigma$, we lose nothing if we consider only the averages $y_{i\cdot} = n^{-1}_i \sum_j y_{ij}$. We have then the same model (1.2), with the only difference that $y_i$ is replaced by $y_{i\cdots}$. If, however, the covariance matrix is not known, we can now obtain in the usual way an estimate $S$ of $\Sigma$. F. S. Acton [1] suggested the use of $S$ instead of $\Sigma$ in the estimate obtained by the method of maximum likelihood in the case of known $\Sigma$. In this paper it will be shown that the estimate thus obtained is the maximum likelihood estimate when $\Sigma$ is unknown. If the design is a (in general incomplete) block design, we have, if the treatment $i$ is applied on the block $j$, \begin{equation*}\tag{1.5}y_{ij} = \tau_i + b_j + \epsilon_{ij},\end{equation*} where $b_j$ is a column vector representing the block effect. Considering the block effects $b_j$ as unknown constants, we get in the usual way the intrablock estimates $t_i$ of the treatment effects $\tau_i$. Then, the same equation (1.2) still holds if $y_i$ is replaced by $t_i$, but in general the errors $\epsilon_i$ and consequently also the estimates $t_i$ will no longer be independent. If the design consists of $r$ replications of a basic design, then the covariance of two errors $\epsilon_i, \epsilon_i'$ will be given by \begin{equation*}\tag{1.6}\operatorname{cov} (\epsilon_i,\epsilon_i') = \frac{\kappa^{ii'}\Sigma}{r}\end{equation*} where $\kappa^{ii'}$ are known coefficients and the matrix $\Sigma$ is unknown. In this paper maximum likelihood estimates for $\Sigma$ and the parameters of the linear functional relationship will be found for the case in which (1.6) holds. It will be shown that the maximum likelihood estimates $\hat\alpha, \hat\beta$ in the case of unknown $\Sigma$ are obtained from the corresponding estimates in the case of known $\Sigma$ by simply replacing $\Sigma$ by the linear regression estimate $S$. In the last Section it will be shown that if the maximum likelihood method is applied directly to the variables $y_{ij}$ instead of the variables $t_i$ and $S$, then the same estimate $\hat\beta$ is obtained, but the estimate of $\Sigma$ is multiplied by $1 - k^{-1} + N^{-1}$, where $k$ is the number of experimental units in each block and $N$ is the total number of experimental units. All of the estimates obtained are consistent, with the exception only of the estimate of $\Sigma$ obtained by the direct approach in the last Section, which converges to $(1 - k^{-1})\Sigma$.

Consider a multivariate normal population with mean $\mathbf{u} = (\mu_1, \cdots, \mu_p)$ and covariance matrix $\Sigma$. Let $\mathbf{u}_0$ be a vector of constants, $_n\mathbf{\bar x}$ a vector of sample means based on $n$ observations, and $\mathbf{S}_n$ the corresponding sample covariance matrix. The statistics considered are \begin{equation*}\tag{1.1}\chi^2_n = n(_n\mathbf{\bar x} - \mathbf{u}_0)\Sigma^{-1}(_n\mathbf{\bar x} - \mathbf{u}_0)'\end{equation*} and \begin{equation*}\tag{1.2}T^2_n = n(_n\bar\mathbf{x} - \mathbf{u}_0)S^{-1}_n(_n\bar\mathbf{x} - \mathbf{u}_0)'.\end{equation*} It is shown that probability-ratio tests for a sequential test of the composite hypothesis, \begin{equation*}\tag{1.3}H_0: (\mathbf{u} - \mathbf{u}_0)\Sigma^{-1}(\mathbf{u} - \mathbf{u}_0)' = \lambda^2_0\end{equation*} against the alternative \begin{equation*}\tag{1.4}H_1: (\mathbf{u} - \mathbf{u}_0)\Sigma^{-1}(\mathbf{u} -\mathbf{u}_0)' = \lambda^2_1\end{equation*} may be based on \begin{equation*}\tag{1.5}p_{1n}/p_{0n} = \lbrack\exp - n(\lambda^2_1 - \lambda^2_0)/2\rbrack _0F_1(p/2; n\lambda^2_1\chi^2_n/4)/_0F_1(p/2; n\lambda^2_0\chi^2_n/4)\end{equation*} when $\Sigma$ is known and \begin{equation*}\tag{1.6}p_{1n}/p_{0n} = \lbrack\exp - n(\lambda^2_1 - \lambda^2_0)/2\rbrack _1F_1\lbrack n/2, p/2; n\lambda^2_1T^2_n/2(n - 1 + T^2_n)\rbrack/ \end{equation*} $_1F_1\lbrack n/2, p/2; n\lambda^2_0T^2_n/2(n - 1 + T^2_n)\rbrack$ when $\Sigma$ is unknown and must be estimated from the sample. The sequential $\chi^2$-test is associated with (1.5) and the sequential $T^2$-test with (1.6). $_0F_1$ and $_1F_1$ are respectively forms of the generalized hypergeometric function $_pF_q(a_1, \cdots, a_p; c_1, \cdots, c_q; x)$, the second being the confluent hypergeometric function. It is shown that the use of these probability ratios in sequential tests results in Type I and Type II errors of approximately $\alpha$ and $\beta$ when these values are used to obtain bounds on the probability ratios in the traditional way. It is also shown that the sequential tests terminate with probability unity. Bounds on the probability ratios are translated into bounds on $\chi^2_n$ and $T^2_n$ themselves and tables have been prepared with more tables in preparation. Procedures are also given to test sequentially whether or not two samples come from populations with the same means. The $\chi^2$-test is generalized to give simultaneous sequential tests on both the means and the covariance matrix. The average sample number functions (ASN functions) are considered and approximations to them suggested. The operating characteristic functions (OC functions) are difficult to investigate and essentially are only known approximately at $\lambda^2_0$ and $\lambda^2_1$.

This paper discusses the use of order statistics in estimating the parameters of (negative) exponential populations. For the one-parameter exponential population, the best linear unbiased estimators, $\tilde \sigma_k = c_kx_k$ and $\tilde \sigma_{lm} = c_lm_l + c_mx_m$, of the parameter $\sigma$ are given, based on one order statistic $x_k$ and on two order statistics $x_l$ and $x_m$. For samples of any size up through $n = 100$, a table is given of $k, l$, and $m$ and of the coefficients $c_k, c_l$, and $c_m$, together with the coefficients of $\sigma^2$ in the variances $V_k$ and $V_{lm}$ of the estimators, and the corresponding efficiencies $E_k$ and $E_{lm}$ (relative to the best linear unbiased estimator based on all order statistics). For the two-parameter exponential population, the best linear unbiased estimators, $\tilde \alpha = c_{\alpha l}x_l + c_{\alpha m}x_m, \tilde \sigma = c_{\sigma l}x_l + c_{\sigma m}x_m$, and $\tilde \mu = c_{\mu l}x_l + c_{\mu m}x_m$, of the parameters $\alpha$ and $\sigma$ and the mean $\mu = \alpha + \sigma$ are given, based on two order statistics $x_l$ and $x_m$. For samples of any size up through $n = 100$, a table is given of $m(l$ is always 1 for the best estimator) and of factors $c_\alpha$ and $c_\alpha$ for computing the coefficients $c_{\alpha l} = 1 + c_\alpha, c_{\alpha m} = -c_\alpha, c_{\sigma l} = -c_\sigma, c_{\sigma m} = c_\sigma, c_{\mu l} = 1 + c_\alpha - c_\sigma$, and $C_{\mu m} = C_\sigma - C_\alpha$, together with the coefficients of $\sigma^2$ in the variances $V_{\tilde \alpha}, V_{\tilde \sigma}$, and $V_\mu$ of the estimators, and the corresponding efficiencies $E_{\tilde \alpha}, E_{\tilde \sigma}$, and $E_{\tilde \mu}$ (relative to the best linear unbiased estimators based on all order statistics).

In this paper, a new nonparametric test for the problem of $c$ samples is offered. It is based upon the numbers of $c$-plets that can be formed by choosing one observation from each sample such that the observation from the $i$th sample is the least, $i = 1, 2, \cdots, c$. The asymptotic distribution of the new test statistic is derived by an application of the extension of Hoeffding's theorem [4] on $U$-statistics to the case of $c$ samples. The asymptotic power and the asymptotic efficiencies of this test relative to the Kruskal-Wallis $H$-test [7] and the Mood-Brown $M$-test [10] are computed in standard fashion along the lines of Andrews' paper [1].

The probabilities of misclassification involved in the use of estimated discriminant functions are subject to chance variations. The author's purpose in this paper is to derive the distribution laws that the probabilities of misclassification follow and to obtain their expected values. The parent populations are assumed to be normal. The first part of the paper considers the univariate case and the second part the multivariate case. The discussion of the multivariate case proceeds in three stages of increasing complexity. When the exact results are complicated, asymptotic results or approximations are given. Finally, the problem of estimating the expected probabilities of misclassification is considered. Interval estimates as well as point estimates are given.

The largest characteristic root has been proposed in [2] as a test statistic in (i) the multivariate analysis of variance test, and (ii) testing that two sets of variates are independent. In this paper it is shown that, in each case, the power function is a monotonically increasing function of each non-centrality parameter, separately. This property was stated in [2] without proof. This provides a stronger result than would be obtained by any direct use of Anderson's Theorem [1] which implies that the power function increases when all the roots are simultaneously increased in the same ratio. The proof of the monotonocity property for the multivariate analysis of variance is given in Section 3, and in Section 4 it is shown how the proof is modified for testing independence between two sets of variates.

Pillai and Mijares [7] gave the exact expressions for the first four moments of the sum of $s$ non-zero roots of a matrix occurring in multivariate normal analysis as studied independently by R. A. Fisher [3], P. L. Hsu [4] and S. N. Roy [9]. In this paper some properties of completely homogeneous symmetric functions and certain determinantal results (Section 2) are used to give an inverse derivation of those moments (Section 4). The method is further extended to the moments in general of elementary symmetric functions (e.s.f.) of the roots of a matrix in multivariate analysis (Section 6) through the use of certain properties of compound matrices (Section 5).

In this paper we present a method of constructing main-effect plans for symmetrical factorial experiments which can accommodate up to $\lbrack 2(s^n - 1)/(s - 1) - 1\rbrack$ factors, each at $s = p^m$ levels, where $p$ is a prime, with $2s^n$ treatment combinations. As main-effect plans are orthogonal arrays of strength two the method presented permits the construction of the orthogonal arrays $(2s^n, 2\lbrack s^n - 1\rbrack/\lbrack 2 - 1\rbrack - 1, s, 2)$.

Partially balanced arrays are generalizations of orthogonal arrays. Multifactorial designs derived from partially balanced arrays require a reduced number of assemblies in order to accommodate a given number of factors. For instance, an orthogonal array of strength two, six symbols and four constraints, would require at least $2.6^2 = 72$ assemblies. This is because there does not exist a pair of mutually orthogonal Latin Squares of order six. But for the same situation, a partially balanced array in 42 assemblies, is constructed in this paper. The method of construction is one of composition which utilizes the existence of a pairwise partially balanced incomplete block design and an orthogonal array.

A new method of deriving designs of type O:PP is described. The method gives rise to some designs previously obtained by other methods, and also to some entirely new designs. These new designs are described in detail and a worked example given.

In this paper the concept of statistical sufficiency is studied within a general probability setting. It is not assumed that the family of probability measures is dominated. That is, it is not assumed that there is a $\sigma$-finite measure $\mu$ such that each probability measure in the family is absolutely continuous with respect to $\mu$. In the dominated case, the theory of sufficiency has received a thorough-going and elegant treatment by Halmos and Savage [6], Bahadur [2], and others. Although many families of probability measures of importance for statistical work are dominated, many others are not. Nonparametric statistical work, especially, abounds with undominated families. It seems appropriate, therefore, to see what can be learned about sufficiency in the undominated case. Let $X$ be a set, $\mathbf{A}$ a $\sigma$-field of subsets of $X$, and $P$ a family of probability measures $p$ on $\mathbf{A}$. The probability structure $(X, \mathbf{A}, P)$ is to be kept in mind throughout the paper and is unrestricted except where specifically stated to the contrary. Any subfield (= sub-$\sigma$-field) entering the discussion is implicitly assumed to be a subfield of $\mathbf{A}$. If $H$ is a collection of subfields, let $\bigvee H$ denote the smallest $\sigma$-field containing each member of $H$. If $\mathbf{A}_1, \mathbf{A}_2, \cdots$ are subfields, write $\mathbf{A}_1 \bigvee \mathbf{A}_2$ for $\bigvee\{\mathbf{A}_1, \mathbf{A}_2\}, \mathbf{\bigvee}^\infty_{n = 1} \mathbf{A}_n$ for $\bigvee\{\mathbf{A}_1, \mathbf{A}_2, \cdots\}$, and so forth. A set $N$ is $P$-null if $N$ is $p$-null for each $p$ in $P$, that is, if $N$ is in $\mathbf{A}$ and $p(N) = 0, p \varepsilon P$. If $f$ and $g$ are $\mathbf{A}$-measurable functions, write $f = g\lbrack p\rbrack$ if the set $\{x\mid f(x) \neq g(x)\}$ is $p$-null and write $f = g\lbrack P\rbrack$ if this set is $P$-null. Let $\mathbf{N}$ be the smallest $\sigma$-field containing the $P$-null sets. If $\mathbf{A}_1$ and $\mathbf{A}_2$ are subfields, write $\mathbf{A}_1 \subset \mathbf{A}_2\lbrack P\rbrack$ if $\mathbf{A}_1 \subset \mathbf{A}_2 \bigvee \mathbf{N}$, and so forth. A subfield $\mathbf{B}$ is sufficient if, for each bounded $\mathbf{A}$-measurable function $f$, there is a $\mathbf{B}$-measurable function $g$ such that $\int_Bf dp = \int_Bg dp, B \varepsilon \mathbf{B}, p \varepsilon P$, that is, such that $g = E_p(f \mid\mathbf{B})\lbrack p\rbrack, p \varepsilon P$. Equivalent definitions are obtained if "bounded $\mathbf{A}$-measurable function" is replaced by $\mathbf{A}$-measurable characteristic function" or by "$P$-integrable function." Of course, $f$ is $P$-integrable if $f$ is $\mathbf{A}$-measurable and $\int _X|f| dp$ is finite for each $p$ in $P$. A subfield $\mathbf{B}$ is separable if it contains a countable subcollection such that $\mathbf{B}$ is the smallest $\sigma$-field containing the subcollection. In Section 2, we give an example of a nonsufficient subfield containing a sufficient subfield. This solves a problem posed by Bahadur (Problem 1 on page 441 of [2]). In fact, we show that often the collection of such nonsufficient subfields is much larger than the collection of sufficient subfields. Analogous results hold for statistics. Some of these and later results depend on Theorem 1 which gives a necessary condition for a subfield to be sufficient in the case that $\mathbf{A}$ is separable. Let $\mathbf{A}_1, \mathbf{A}_2, \cdots$ be a sequence of sufficient subfields. Are the subfields $\mathbf{A}_1 \bigcap \mathbf{A}_2, \mathbf{A}_1 \bigvee \mathbf{A}_2, \mathbf{\bigcap}^\infty_{n = 1}\mathbf{A}_n$, and $\mathbf{\bigvee}^\infty_{n = 1}\mathbf{A}_n$, necessarily sufficient? This question is investigated in Sections 3 and 4. Using martingale theory, we show that, if the sequence is decreasing (increasing), then $\mathbf{\bigcap}^\infty_{n = 1}\mathbf{A}_n(\mathbf{V}^\infty_{n = 1}\mathbf{A}_n)$ is sufficient. If the sequence is not necessarily monotone, it is still possible to show that $\mathbf{A}_1 \bigcap \mathbf{A}_2$ and $\bigcap^\infty_{n = 1}\mathbf{A}_n$ are sufficient under a small extra assumption involving $\mathbf{N}$. This result rests on a theorem proved in [3] regarding iterates of conditional expectation operators. One consequence of this result is of interest in connection with the theory of minimal sufficient subfields. It is not necessarily true that $\mathbf{A}_1 \bigvee \mathbf{A}_2$ is sufficient. This is shown in Example 4. Conditions under which $\mathbf{A}_1 \bigvee \mathbf{A}_2$ is sufficient are examined. The main result of Section 5 is related to Theorem 1 and indicates that if $\mathbf{A}$ is separable then each sufficient subfield is essentially equal to one of a very special type.

This paper contains the definition of and some preliminary results on Markov Renewal processes and Semi-Markov processes. The close relationship between these two types of processes is described. The concept of regularity is introduced and characterized. A classification of the states of a Markov Renewal process is described and studied.

In this paper, Markov Renewal processes having a finite number of states are studied. Explicit expressions are derived for the distribution functions of first passage times, as well as for the marginal distribution function of the corresponding Semi-Markov process. Double generating functions are obtained for the distribution functions of the $N_j$-processes. The limiting behavior of a Markov Renewal process is discussed, the stationary probabilities being derived completely. General Markov Renewal processes are introduced, and a related stationary process is determined. Several examples are given.

Let $P = (p_{jk})$ be the transition matrix of an ergodic, finite Markov chain with no cyclically moving sub-classes. For each possible transition $(j, k)$, let $H_{jk}(x)$ be a distribution function admitting a moment generating function $f_{jk}(t)$ in an interval surrounding $t = 0$. The matrix $P(t) = \{p_{jk}f_{jk}(t)\}$ is of interest in the study of the random variable $S_n = X_1 + \cdots + X_n$, where $X_m$ has the distribution $H_{jk}(x)$ if the $m$th transition takes the chain from state $j$ to state $k$. The matrix $P(t)$ is non-negative and therefore possesses a maximal positive eigenvalue $\alpha_1(t)$, which is shown to be a convex function of $t$. As an application of the convexity property, we obtain an asymptotic expression for the probability of tail values of the sum $S_n$, in the case where the $X_m$ are integral random variables. The results are related to those of Blackwell and Hodges [1], whose methods are followed closely in Section 5, and Volkov [4], [5], who treats in detail the case of integer-valued functions of the state of the chain, i.e., the case $f_{jk}(t) = \exp(\beta_kt) (\beta_k$ integral).

The problem of estimation of a single regression parameter for a process with fixed known regression function and unknown covariance is attacked using a Hilbert space representation of the process. Some general results are obtained which characterize efficiency classes of covariances--that is, classes for each of which there exists a single estimate that is efficient for all members. These results are applied to both the discrete parameter and the continuous parameter stationary process with rational spectral density. Some special results are also obtained concerning the efficiency of the least square estimate.

When a group of $k$ individuals is required to make a joint decision, it occasionally happens that there is agreement on a utility function for the problem but that opinions differ on the probabilities of the relevant states of nature. When the latter are indexed by a parameter $\theta$, to which probability density functions on some measure $\mu(\theta)$ may be attributed, suppose the $k$ opinions are given by probability density functions $p_{s1}(\theta), \cdots, p_{sk}(\theta)$. Suppose that $D$ is the set of available decisions $d$ and that the utility of $d$, when the state of nature is $\theta$, is $u(d, \theta)$. For a probability density function $p(\theta)$, write $u\lbrack d\mid p(\theta)\rbrack = \int u(d, \theta)p(\theta) d\mu(\theta)$. The Group Minimax Rule of Savage [1] would have the group select that $d$ minimising $\max_{i = 1, \cdots, k}\{\max_{d'\epsilon D} u\lbrack d' \mid p_{si}(\theta)\rbrack - u\lbrack d \mid p_{si}(\theta)\rbrack\}$. As Savage remarks ([1], p. 175), this rule is undemocratic in that it depends only on the different distributions for $\theta$ represented in those put forward by the group and not on the number of members of the group supporting each different representative. An alternative rule for choosing $d$ may be stated as follows: "Choose weights $\lambda_1, \cdots, \lambda_k (\lambda_i \geqq 0, i = 1, \cdots, k$ and $\sum^k_1 \lambda_i = 1)$; construct the pooled density function $p_{s\lambda}(\theta) = \sum^k_1 \lambda_ip_{si}(\theta);$ choose the $d$, say $d_{s\lambda}$, maximising $u\lbrack d \mid p_{s\lambda}(\theta)\rbrack$." This rule, which may be called the Opinion Pool, can be made democratic by setting $\lambda_1 = \cdots = \lambda_k = 1/k$. Where it is reasonable to suppose that there is an actual, operative probability distribution, represented by an `unknown' density function $p_a(\theta)$, it is clear that the group is then acting as if $p_a(\theta)$ were known to be $p_{s\lambda}(\theta)$. If $p_a(\theta)$ were known, it would be possible to calculate $u\lbrack d_{s\lambda} \mid p_a(\theta)\rbrack$ and $u\lbrack d_{si} \mid p_a(\theta)\rbrack$, where $d_{si}$ is the $d$ maximising $u\lbrack d \mid p_{si}(\theta)\rbrack, i = 1, \cdots, k$ and then to use these quantities to assess the effect of adopting the Opinion Pool for any given choice of $\lambda_1, \cdots, \lambda_k$. It is of general theoretical interest to examine the conditions under which \begin{equation*}\tag{1.1}u\lbrack d_{s\lambda} \mid p_a(\theta)\rbrack \geqq \min_{i = 1, \cdots, k} u\lbrack d_{si} \mid p_a(\theta)\rbrack.\end{equation*} Theorems 2.1 and 3.1 provide different sets of sufficient conditions for (1.1) to hold. Theorem 2.1 requires $k = 2$ and places a restriction on $p_a(\theta)$ (or, equivalently, on $p_{s1}(\theta)$ and $p_{s2}(\theta)$); Theorem 3.1 puts conditions on $D$ and $u(d, \theta)$ instead.