Let $x_1, x_2,\cdots$ be independent random variables which under $P_\theta$ have probability density function of the form $P_\theta\{x_k \in dx\} = \exp(\theta x - \Psi(\theta)) dH(x)$, where $\Psi$ is normalized so that $\Psi(0) = \Psi'(0) = 0.$ Let $a \leqq 0 < b, s_n = \sum^n_1 x_k$, and $T = \inf \{n: s_n \not\in (a, b)\}.$ For $u < 0$, an unbiased Monte Carlo estimate of $P_u(s_T \geqq b)$ is the average of independent $P_\theta$-realizations of $I_{\{s_T \geqq b\}} \exp\{(u - \theta)s_T - T(\Psi(u) - \Psi(\theta))\}$. It is shown that the choice $\theta = w$, where $w > 0$ is defined by $\Psi(w) = \Psi(u)$, is an asymptotically (as $b \rightarrow \infty)$ optimal choice of $\theta$ in a sense to be defined. Implications of this result for Monte Carlo studies in sequential analysis are discussed.

A theory of sequential Bayes interval estimation procedures for a single parameter is developed for the case where the loss for using an interval $I$ is a linear combination of the length of $I$, the indicator of noncoverage of $I$, and the number of observations taken. A class of stopping rules $\{t(c): c > 0\}$ is shown to be asymptotically pointwise optimal (A.P.O.) and asymptotically optimal (A.O.) for the confidence interval problem as the cost $c$ per observation tends to 0. The results require generalization of Bickel and Yahav's (1968) general conditions for the existence of A.P.O. and A.O. stopping rules to the case where the terminal risk $Y_n$ satisfies $f(n)Y_n \rightarrow V$ for $f(n)$ a regularly varying function on the integers.

Suppose a random variable has a density belonging to a one parameter family which has strict monotone likelihood ratio. For inference regarding the parameter (or a monotone function of the parameter) consider the loss function to be bowl shaped for each fixed parameter and also to have each action be a "point of increase" or a "point of decrease" for some value of the parameter. Under these conditions, given any nonmonotone decision procedure, a unique monotone procedure is constructed which is strictly better than the given procedure for all the above loss functions. This result has application to the following areas: combining data problems, sufficiency, a multivariate one-sided testing problem.

Consider a balanced array $T$ of strength 2$l,$ size $N, m$ constraints and index set $\{\mu_0, \mu_1,\cdots, \mu_{2l}\}$ with $\mu_l = 0$. Under some conditions $T$ yields a design of even resolution ($2l$, say) with $N$ assemblies such that all the effects involving up to $(l - 1)$-factor interactions are estimable provided $(l + 1)$-factor and higher order interactions are assumed negligible and that the covariance matrix of their estimates is invariant under any permutation of $m$ factors. The alias structure of the effects of $l$-factor interactions is explicitly given. Such an array $T$ is called an $S$-type balanced fractional $2^m$ factorial design of resolution $2l$. Necessary conditions for the existence of the design $T$ are given. For any given $N$, there are in general a large number of possible $S$-type balanced fractional $2^m$ factorial designs of resolution $2l$. Finally a criterion for comparing these designs is given.

This article discusses the sampling of stationary discrete-time stochastic processes at fixed but unequally spaced time points. The pattern of the sampling times is periodic with a cycle of $p$ time units. One of the major problems is to determine given $p$ the minimum number of sampling points required per cycle in order to estimate the covariances at all lags. The second problem is to find a pattern of distribution for the sampling points within the cycle which will allow the estimation of all covariances. A discussion of the references which describe the statistical properties of the estimates of covariances and spectra in this sampling situation is given.

The limiting distributions are obtained for two estimators of variance components: C. R. Rao's MINQUE, and an estimator produced by an iterative procedure, referred to as I-MINQUE. Limits are taken as the number of independent and identically distributed vector observations on the model assumed gets large. This approach provides the asymptotics of interest when an experiment with a large number of observations can be thought of as independent replications of a smaller experiment, a condition applying to some common experimental designs. The main result, from which the limiting distributions are obtained, is essentially an extension of a theorem due to T. W. Anderson (1973), who provides an application in time series. Both estimators considered here are consistent, and require only modest assumptions on the sampled distribution. The I-MINQUE has a limiting distribution which is functionally independent of the choice of norm; when it is further assumed that the sampled distribution is normal, the estimator is asymptotically equivalent to the m.1.e. and asymptotically efficient. The MINQUE itself is less robust in the sense that these two properties do not always apply, the conditions being dependent on the choice of design.

Conditions are given for the attainment of the Hammersley-Chapman-Robbins bound for the variance of an unbiased estimator, in both regular and nonregular cases. Comparisons are made between this bound and the Bhattacharyya system of bounds for a wide class of distributions and parametric functions. Sufficient conditions are provided to determine when one bound is sharper than the other one.

Sanov's theorem on the asymptotic behavior of probabilities of large deviations for an empirical probability distribution is established under different conditions than previously given by Sanov, Hoadley and Stone. The new conditions are based on likelihood ratio approximations rather than on multinomial approximations. It is shown that these conditions are strictly more general than those of Stone.

This paper gives rates of convergence for the $L_1$ distances between the distributions of standardized linear rank statistics and the standard normal random variable. These rates are $O(N^{-\frac{1}{2}})$ under various conditions on the score function and the distributions of the underlying observations.

A sum of squares can be partitioned into sums of quadratic forms whose kernels are projections. If these projections are mutually orthogonal and add to the identity, then, under the classical fixed effects linear model, the terms of the decomposition are mutually independent and are distributed as multiples of chi-square. In this paper we exhibit necessary and sufficient conditions for a specified sum of squares decomposition to have this property in the case of the mixed model.

In this note we discuss multivariate linear models from the coordinate-free point of view, as earlier done by Eaton (1970). We generalize the result of Eaton by allowing for missing observations. This leads to models of the kind $EY \in L, Cov Y \in\{P(I \otimes \sum)P'\}$ where $P$ is a diagonal mapping. The paper starts by deriving the conditions for existence of Gauss-Markov estimators (GME) of $EY$ in models where the covariance-mappings are not necessarily nonsingular. These conditions are then applied to the above models if $\Sigma$ runs either over all PSD-mappings or over all diagonal PSD-mappings. In the latter case $L$ must be of the form $L = L_1 \times \cdots \times L_p$ while in the general case some further conditions on the $L_i$ must be met. (If $P = I$, then $L_i = L_j$ must hold for all $i, j$; this is equivalent to the result obtained by Eaton). Examples show that these conditions are satisfied only under rather exceptional conditions.

In the usual linear regression model the sample regression coefficients converge with probability one to the population regression coefficients when the dependent variables are normally distributed and the inverse of the second-order moment matrix of the independent variables converges to the zero matrix.

Hypotheses generating groups are constructed for the class of multivariate symmetric distributions and for the corresponding multisample problems. Defining ranks as maximal invariants under these groups we get distribution free procedures. Applications to further multivariate test problems are indicated.

The noncentral distribution of latent roots arising in several situations in multivariate analysis involves the integration of a hypergeometric function of matrix variates over a group of orthogonal matrices in the real case and that of unitary matrices in the complex case. In this paper the subgroup of the orthogonal group (unitary group) for which the integrand is maximized has been found under mild restrictions. The results of earlier authors (Anderson, Chang, James, Li and Pillai) follow as special cases. Further, the maximization results concerning the integrand have been used to study asymptotic expansions of the distributions of the characteristic roots of matrices arising in canonical correlation analysis and MANOVA when the corresponding parameter matrices have several multiple roots.

Suppose an object is hidden in one of $N$ boxes. The initial prior probability that it is hidden in box $i$ is known to the searcher, who once a day must choose a box to be searched. The probability that the $j$th search of box $i$ will be unsuccessful, given that the object is in box $i$, is described by a random variable $\alpha_{ij}$. Assuming that each search of box $i$ costs $c_i > 0$, sufficient conditions on the joint distribution of the $\{\alpha_{ij}\}^\infty_{j=1}$ are found in order to guarantee that a particular search rule (analogous to earlier search rules found independently by R. Bellman, D. Blackwell, W. Black and J. Kadane) is optimal with respect to minimizing the total expected search cost of finding the object. An extension of a search problem studied by C. Sweat is also treated.