For a wide class of stationary random sequences possessing a spectral density function, the variance of the best linear unbiased estimator for the mean is seen to depend asymptotically only on the behavior of the spectral density near the origin. Asymptotically efficient estimators based only on this behavior may be chosen. For spectral densities behaving like $\lambda^\nu$ at the origin, $\nu > -1$ a constant, the minimum variance decreases like $n^{-\nu-1}$, where $n$ is the sample size. Asymptotically efficient estimators depending on $\nu$ are given. Finally, the consequences of over- or under-estimating the value of $\nu$ in choosing an estimator are considered.

The theory of periodic smoothing splines is presented, with application to the estimation of periodic functions. Several theorems relating the order of the differential operator defining the spline to the saturation (order of bias) of the estimator are proven. The linear operator which maps a function to its periodic continuous smoothing spline approximation is represented as a convolution operator with a given convolution kernel. This operator is shown to be the limit of a sequence of operators which map a function into the periodic version of the usual lattice smoothing spline. The convolution kernel above appears as the kernel in a kernel type estimate of the spectral density. Thus, it is shown that, a smoothing spline spectral density estimate, is also asymptotically a kernel type spectral density estimate. Some numerical results are presented.

Consider the problem of estimating the location parameter $\theta \in R^d$ based on a sample of size $n$ from $(\theta + X, Y)$, where $X$ is a $d$-dimensional random vector, $Y$ is a random element of some measure space, and $(X, Y)$ has a known distribution. Let $\mathscr{J}^-$ denote the corresponding inverse Fisher information matrix. We show that there is always an invariant estimator $\hat{\theta}_n$ such that $\mathscr{L}(n^{\frac{1}{2}}(\hat{\theta}_n - \theta)) \rightarrow N(0, \mathscr{J}^-)$ as $n \rightarrow \infty$. Let $\rho$ be a fixed probability density on $R^d$, let $\tilde{\theta}_n$ be any estimator of $\theta$ and set $R_n(c) = \int \rho(\theta)d\theta E_\theta \min (c, n|\tilde{\theta}_n - \theta|^2)$. We show that $\lim_{c\rightarrow\infty} \lim\inf_{n\rightarrow\infty} R_n(c) \geqq$ trace $\mathscr{J}^-$ and that if $\lim_{c\rightarrow\infty} \lim\sup_{n\rightarrow\infty} R_n(c) =$ trace $\mathscr{J}^-$, then $\lim_{n\rightarrow\infty} \int \rho(\theta) d\theta P_\theta(n^{\frac{1}{2}}|\tilde{\theta}_n - \hat{\theta}_n| \geqq c) = 0$ for all $c > 0$. These results are obtained with no regularity conditions imposed on the distribution of $(X, Y)$.

We consider procedures for statistical inference based on the smallest $r$ observations from a random sample. This method of sampling is of importance in life testing. Under weak regularity conditions which include the existence of a q.m. derivative for the square root of the ratio of densities, we obtain an approximation to the likelihood and establish the asymptotic normality of the approximation. This enables us to reach several important conclusions concerning the asymptotic properties of point estimators and of tests of hypotheses which follow directly from recent developments in large sample theory. We also give a result for expected values which has importance in the theory of rank tests for censored data.

A random process called the Dirichlet process whose sample functions are almost surely probability measures has been proposed by Ferguson as an approach to analyzing nonparametric problems from a Bayesian viewpoint. An important result obtained by Ferguson in this approach is that if observations are made on a random variable whose distribution is a random sample function of a Dirichlet process, then the conditional distribution of the random measure can be easily calculated, and is again a Dirichlet process. This paper extends Ferguson's result to cases where the random measure is a mixing distribution for a parameter which determines the distribution from which observations are made. The conditional distribution of the random measure, given the observations, is no longer that of a simple Dirichlet process, but can be described as being a mixture of Dirichlet processes. This paper gives a formal definition for these mixtures and develops several theorems about their properties, the most important of which is a closure property for such mixtures. Formulas for computing the conditional distribution are derived and applications to problems in bio-assay, discrimination, regression, and mixing distributions are given.

Let $\{X_n\}$ be a sequence of i.i.d. random variables, each with probability density function $p(x \mid \theta, \xi)$ subject to certain regularity conditions. Here, $\theta$ is an $s$-dimensional vector of nuisance parameters, and $\xi \in (-r, r)$ is the parameter under test. The first $N$ members of the sequence $\{X_n\}$ are to be used for testing the hypothesis, $H_0: \xi = 0$, against the alternative, $H_1: \xi\neq 0$, while $\theta$ remains unspecified. The particular case considered is that in which the left-hand and right-hand derivatives, with respect to $\xi$, of the logarithm of the density function are unequal. It is shown that the class of $C(\alpha)$ tests based on linear combinations of the left and right derivatives, is an essentially complete class of these tests. The asymptotic power functions of these tests depend upon the coefficients of the linear combination. The maxmin test is deduced and compared with strongly symmetric and weakly symmetric tests. The motivation for the study is the vague notion of "fair" tests which do not arbitrarily favor detection of "positive" or "negative" alternatives.

In case the joint density $f$ of $X = (X_1, \cdots, X_n)$ is Schur-concave (is an order-reversing function for the partial ordering of majorization), it is shown that $P(X \in A + \theta)$ is a Schur-concave function of $\theta$ whenever $A$ has a Schur-concave indicator function. More generally, the convolution of Schur-concave functions is Schur-concave. The condition that $f$ is Schur-concave implies that $X_1, \cdots, X_n$ are exchangeable. With exchangeability, the multivariate normal and certain multivariate "$t$", beta, chi-square, "$F$" and gamma distributions have Schur-concave densities. These facts lead to a number of useful inequalities. In addition, the main result of this paper can also be used to show that various non-central distributions (chi-square, "$t$", "$F$") are Schur-concave in the noncentrality parameter.

The distribution $\Psi(\mathbf{x}; Z, M) = \operatorname{const}. \exp(\mathrm{tr} (ZM^T \mathbf{xx}^T M))$ on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix $Z$ and the orthogonal orientation matrix $M. \Psi$ is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for $Z$ and $M$ are found and their asymptotic properties elucidated. Inference procedures, including tests for isotropy and circular symmetry, are proposed. The application of $\Psi$ is illustrated by a numerical example.

The notion of $m, n$-pattern is introduced--namely, a division of the unit interval into at most $n$ cells (intervals or points), each having one of $m$ colors. Given an unknown $m, n$ pattern, it is desired to produce a reconstruction of the pattern using $r \geqq 1$ sample points (fixed or chosen at random) where the color is determined. The problem is studied from a decision-theoretic point of view. A way to obtain all the probability measures on the set of $m, n$-patterns is given. The notion of a Bayesian reconstruction rule (B.R.R.) is introduced. It is proved that when B.R.R.'s are considered, it is sufficient to use certain fixed sample points. A complete class of reconstruction rules is obtained. Finally an example of a B.R.R. is given for 2,2-patterns.

Connectedness is an important property which every block design must possess if it is to provide an unbiased estimator for all elementary treatment contrasts under the usual linear additive model. We have classified the family of connected designs into three subclasses: locally connected, globally connected and pseudo-globally connected designs. Basically, a locally connected design is one in which not all the observations participate in the estimation. A globally connected design is one in which all observations participate in the estimation. Finally, a pseudo-globally connected design is a compromise between locally and globally connected designs. Theorems and corollaries are given which characterize the different classes of connected designs. In our discussion on the optimality of connected designs we show that there is much to be gained by partitioning the family of connected designs in the above fashion. Our optimality criteria are $S$-optimality suggested by Shah, which selects the design with minimum trace of the information matrix squared and $(M, S)$-optimality which selects the $S$ optimal design from the class of designs with maximum trace of the information matrix. Using these optimality criteria, we have been able to derive some new results which we hope to be of interest to the users and researchers in the field of optimum design theory. To be specific, let BD $\{v, b, (r_i), (k_u)\}$ denote a block design on a set of $v$ treatments with $b$ blocks of size $k_u, u = 1,2, \cdots, b$ and treatment $i$ is replicated $r_i$ times. Then we have shown that for the family of connected block designs BD $\{v, b, (r_i), k\}$ with (i) less than $k - 1$ treatments having replication equal to one and binary (0, 1) the $S$-optimum design is pseudo-globally connected; (ii) the $S$-optimum design is globally connected if $r_i > 1$ and the designs are binary; and (iii) at least one treatment with replication greater than $b$, then the $(M, S)$-optimum design is pseudo-globally connected. In the final part of this paper we mention some unsolved problems in this area.

Balanced arrays were introduced and first studied by I. M. Chakravarti and called partially balanced arrays. J. N. Srivastava and D. V. Chopra have recently made major contributions to the theory and construction of these arrays. They suggested dropping the adjective "partially" and we have followed their lead. Whereas their work has been concerned mainly with arrays of strength four with two symbols, we are here concerned primarily with arrays of strength two with two symbols. However, when the proofs can be carried out as easily for any strength and any number of symbols, we do state the theorems in full generality. We are concerned here with finding bounds on the maximum possible number of rows and with the problem of constructing balanced arrays for given sets of parameters. Analogous to the problem of constructing other combinatorial configurations, we investigated whether some schemes, i.e. some subsets of columns of balanced arrays, could be extended to full balanced arrays. It is shown, analogously to orthogonal arrays, that balanced arrays of even strength, say $2u$ are extendable to arrays of strength $2u + 1$. A new technique of construction of balanced arrays with the maximum number of constraints is also described. It is shown that BIB designs with $\lambda = 1$ can be utilized for constructing balanced arrays with the number of symbols equal to the block size of the BIB design. For completeness we include an example of the analysis of a partially balanced array of strength two with two symbols when it is used as a fractional factorial design. We exhibit in this way an explicit method of estimating main effects when higher ordered interactions are assumed to be negligible.

Consider the problem of estimating the common mean of two normal distributions. Two new unbiased estimators of the common mean are offered for the equal sample size case. Both are better than the sample mean based on one population for sample sizes of 5 or more. A slight modification of one of the estimators is better than either sample mean simultaneously for sample sizes of 10 or more. This same estimator has desirable large sample properties and an explicit simple upper bound is given for its variance. A final result is concerned with confidence estimation. Suppose the variance of the first population, say, is known. Then if the sample mean of that population, plus and minus a constant, is used as a confidence interval, it is shown that an improved confidence interval can be found provided the sample sizes are at least 3.

Using a locally Lipschitz function $T$ of $n > 1$ variables one can reduce data consisting of a sample of size $n$ to one real number. If we are given a family of probability measures on the real line which are equivalent to Lebesgue measure then $T$ yields a sufficient data reduction only if the given family is exponential. This result is compared with the results of Brown (1964) and Denny (1970).

It is shown that $(k, n)$-outlier-proneness of a family of distributions is equivalent to complete outlier-proneness.

The conditions $|\phi_k| \leqq 1$ for all $k = 1,2, \cdots$ and $|\phi_k| = 1$ implies $\phi_{k+1} = \phi_k$ are both necessary and sufficient for a sequence of real numbers $\{\phi_k; k = 1,2, \cdots\}$ to be the partial autocorrelation function for a real, discrete parameter, stationary time series. If all partial autocorrelations beyond the $p$th are zero, the series is an autoregression. If all beyond the $p$th have magnitude unity, the series satisfies a homogeneous stochastic difference equation. A stationary series is singular if and only if $\sum^N_1 \phi_k^2$ diverges with $N$. The likelihood function for the partial autocorrelation function is produced, assuming normality.

We consider functions $\alpha(\bullet)$ and $\hat{\alpha}(\bullet)$ on a finite set $S$ which correspond to a function $M(\bullet)$ on the nonempty subsets of $S$ which has the Cauchy mean value property (i.e., $M(A + B)$ is between $M(A)$ and $M(B)$ whenever $A$ and $B$ are nonempty disjoint subsets of $S$). $\hat{\alpha}(\bullet)$ is isotone with respect to a partial ordering on $S$ and is equal to $\alpha(\bullet)$ when $\alpha(\bullet)$ is isotone. It is shown that $\hat{\alpha}(\bullet)$ has the following norm reducing property: $\max_{s\in S} |\hat{\alpha}(s) - \theta(s)| \leqq \max_{s\in S} |\alpha(s) - \theta(s)|$ for all isotone $\theta(\bullet)$. Computation algorithms for $\hat{\alpha}(\bullet)$ are discussed and the norm reducing property is shown to give consistency results in several isotonic regression problems.

In the estimation of a location parameter the Hodges-Lehmann estimator is known to have some "robust" properties, but it is very "expensive" for large sample sizes. By using the linearity of a special rank statistic we can find a linearized version which requires only $O(n \log n)$ operations.

Two models for modified, one-sample, sequential probability ratio tests based on Lehmann alternatives are considered, one developed by Weed and Bradley and one by Govindarajulu. It is shown how they are related. Sure termination of the SPRT's is established under very general conditions.

For a given sufficiently regular distribution $F$ two efficient location estimators are given. One is a linear combination of order statistics, called $L(F)$, and the other is an estimator derived from a rank test, called $R(F)$. The asymptotic variance of both estimators is then compared for various underlying distributions $H$ and it is shown that the asymptotic variance of $R(F)$ is never larger than the one of $L(F)$.

If a set of states is given in a problem of dynamic programming in which each state can be observed only partially, the given model is generally transformed into a new model with completely observed states. In this article a method is introduced with which Markov models of dynamic programming can be transformed and which preserves the Markov property. The method applies to relatively general sets of states.

It is shown in this paper that the product family of countably many families of perfect probability measures defined on countably generated $\sigma$-fields admits a minimal sufficient statistic if and only if each component family admits a minimal sufficient statistic. Moreover, the minimal sufficient statistic of the product family is the "product" of the minimal sufficient statistics for the component families. Examples show that the assumptions on the component families cannot be omitted.

A formulation is given and a procedure is proposed for constructing a confidence interval for a certain ordered location or scale parameter and for simultaneously selecting all populations having parameters equal or larger than this ordered parameter with a preassigned minimal probability. The well-known indifference-zone formulation of the ranking problem is obtained as a special case as is the problem of interval estimation of an ordered parameter.

For BIB designs $N_i$ and their complements $N_i^\ast (i = 1,2, \cdots, n)$, Kageyama (1972) gave necessary and sufficient conditions for a PBIB design $N = N_1 \otimes N_2 + N_1^\ast \otimes N_2^\ast$ with at most three associate classes having the rectangular association scheme to be reducible to a PBIB design with only two distinct associate classes having the $L_2$ association scheme. In this paper similar results for the PBIB design $N_1 \otimes N_2 \otimes \cdots \otimes N_n + N_1^\ast \otimes N_2^\ast \otimes \cdots \otimes N_n^\ast$, which is in a sense a generalization of the Kronecker products of the above type, are described.

In this paper it is shown that a new series of group divisible designs can be obtained by using a geometrical method.