An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the last three decades. These matrices can be transformed to produce incomplete block designs, $t$-designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III designs, optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects. In this paper we survey the existence of Hadamard matrices and many of their applications.

The problem of finding an optimal design for the elimination of one-way heterogeneity when a balanced block design does not exist is studied. A general result on the optimality of certain asymmetrical designs is proved and applied to the block design setting. It follows that if there is a group divisible partially balanced block design (GD PBBD) with 2 groups and $\lambda_2 = \lambda_1 + 1$, then it is optimal w.r.t. a very general class of criteria including all the commonly used ones. On the other hand, if there is a GD PBBD with 2 groups and $\lambda_1 = \lambda_2 + 1$, then it is optimal w.r.t. another class of criteria. Uniqueness of optimal designs and some other miscellaneous results are also obtained.

The purpose of this paper is to study optimal designs for the elimination of multi-way heterogeneity. The $C$-matrix for the $n$-way heterogeneity setting when $n > 2$ is derived. It turns out to be a natural extension of the known formulas in the lower dimensional case. It is shown that under some regularity, the search for optimal designs can be reduced to that in a lower-way setting. Youden hyperrectangles are defined as higher dimensional generalizations of balanced block designs and generalized Youden designs. When all the sides are equal, they are called Youden hypercubes. It is shown that a Youden hyperrectangle is $E$-optimal and a Youden hypercube is $A$- and $D$-optimal. The latter is quite interesting since it is not always true in two-way settings.

For a regular optimality criterion function $\Phi$, a sequence of design measures $\{\xi_n\}$ is generated using the iteration $\xi_{n+1} = (1 - \alpha_n)\xi_n + \alpha_n\xi_n$, where $\xi_n$ is chosen to minimize $\nabla \Phi(M(\xi_n), M(\xi))$ over all $\xi$ and $\{\alpha_n\}$ is a prescribed sequence of numbers from (0, 1). This is called a general step-length algorithm for $\Phi$. Typical conditions on $\{\alpha_n\}$ are $\alpha_n \rightarrow 0$ and $\Sigma_n\alpha_n = \infty$. In this paper, a dichotomous behavior of $\{\xi_n\}$ is proved under the above conditions on $\{\alpha_n\}$ for $\Phi$ satisfying some mild regularity conditions. Sufficient conditions for convergence to optimal designs are also established. This can be applied to show that the $\{\xi_n\}$ as constructed above do converge to an optimal design for most of the trace-related and determinant-related design criteria.

The approximate optimal design problem is treated as a constrained convex programming problem. A general class of optimal design algorithms is proposed from this point of view. Asymptotic convergence to optimal designs is also proved. Related problems like the implementability problem for the infinite support case and the general step-length algorithms are discussed.

Properties of designs which are $(M, S)$ optimal within various classes of proper block designs are studied. The classes of designs considered are not restricted to connected designs. Connectedness is shown to be a property generally possessed by designs which are $(M, S)$ optimal within these more general classes of designs. In addition, we show that the complement of any proper binary $(M, S)$ optimal design is $(M, S)$ optimal within an appropriate class of complementary designs and that the dual of any proper equireplicated $(M, S)$ optimal design is $(M, S)$ optimal within an appropriate class of dual designs.

A methodology is presented for analyzing periodic autoregressions which is also applicable when inferring the second order properties of periodically correlated processes. In addition, capitalizing on the connection between periodic and multiple autoregressions, a method is set forth for analyzing the latter, which overcomes the usual requirements of a large number of both parameters and computer storage locations. This is achieved by introducing an orthogonal parameterization for multiple autoregressions.

As a natural, intuitive model for inferences about certain characteristics of finite populations, Bruce Hill has proposed a sequence of exchangeable variables $X_1, \cdots, X_{n + 1}$ which have distinct values with probability one and have the property that, conditional on $X_1, \cdots, X_n$, the next observation $X_{n + 1}$ is equally likely to fall in any of the $n + 1$ intervals determined by $X_1, \cdots, X_n$. Harold Jeffreys had previously assumed such a model (in the case $n = 2$) for normal measurements with unknown mean and variance. Hill has shown that, for $n \geqslant 1$, there exist no countably additive distributions with the prescribed properties. It is shown here that finitely additive distributions with these properties do exist for all $n$ and have a number of interesting properties.

Modifications of the $t$-test are considered which are robust under certain violations of the independence assumption. The additional number of observations these modified tests require in order to obtain under independence the same power as the $t$-test, is obtained asymptotically.

Let $x_1, \cdots, x_n$ be i.i.d. random variables with a distribution depending on the real parameter. Under what conditions is a generalized Bayes estimator independent of the choice of the even loss function? The known answer to this question is that this independence holds if the posterior density is symmetric and unimodal. The description of distributions and corresponding generalized prior densities on the real line, for which the posterior density is symmetric and unimodal, is presented. These families form an important subclass of all exponential laws with two-dimensional sufficient statistics.

A general method for approximating tail areas is developed through an extension of the methodology of Andrews. This extension is applied to both continuous and discrete distributions. Examples of the approximations are given for the standard normal, $t$, and chi-square distributions in the continuous case and for the Poisson and binomial distributions in the discrete case. Errors of the approximations are considered. The generality of the method shown indicates that extension is possible to other distributions.

It is shown that the bivariate density of the absolute normal distribution is totally positive of order 2. Necessary and sufficient conditions are given for the trivariate density of the absolute normal distribution to be totally positive of order 2 in pairs of arguments. These results are then used to show that certain generalized bivariate and trivariate $t, \chi^2$ and $F$ random variables are associated.

We consider the estimation of parameters in the time series model $X(t) = \sum^q_{j = 1}a_jX(t - j) + \varepsilon(t) - \varepsilon(t - j) - \sum^p_{j = 1} c_j\{\varepsilon(t - j) - \varepsilon(t - j - 1)\}$ where the $\varepsilon(t)$ are independently identically distributed random variables with zero mean and variance $\sigma^2$. We compute the exact $\log$ likelihood of the model, propose and justify an asymptotic approximation of it. The latter will be used to derive estimates of the parameters which are shown to be asymptotically normal and efficient.

Haberman's bound for a norm of the difference between the least squares and the best linear unbiased estimators in a linear model with nonsingular covariance structure is examined in the particular case when a vector norm involved is taken as the Euclidean one. In this frequently occurring case, a new substantially improved bound is developed which, furthermore, is applicable regardless of any additional condition.