In recent years J. W. Tukey has introduced several algorithms for nonlinear smoothing of time series, but theoretical development has been slow owing to the extreme difficulty of obtaining analytical results even in the simplest cases. We present here some desiderata for smoothers generally, and give a framework within which some detailed comparisons between different smoothers can be made.

It is observed that in selecting an alternative to the usual maximum likelihood estimator, $\delta^0$, of a multivariate normal mean, it is important to take into account prior information. Prior information in the form of a prior mean and a prior covariance matrix is considered. A generalized Bayes estimator is developed which is significantly better than $\delta^0$ if this prior information is correct and yet is very robust with respect to misspecification of the prior information. An associated confidence region is also constructed, and is shown to have very attractive size and probability of coverage.

A recursive estimate of the stochastic structure of a stationary time series is constructed based on the assumption that the true structure is ARMA, i.e., has a rational spectrum. The estimate is recursive in the sense that each successive estimate is obtained from the previous one by a relatively simple adjustment, that could be effected in a "real time" situation. The procedure is basically that of updating a regression when all variates involved are constructed from previous estimates of the parameter vector. The strong convergence of the estimate to the true value is established as well as a result relating to the rate of convergence.

For the linear regression model $y = X \beta + u$ with stochastic regressor matrix, strong consistency of the least squares estimator of $\beta$ is proved in the case of martingale difference errors and predetermined regressors and for the case where errors and regressors are orthogonal up to the second order. The results obtained are applied to parameter estimation in autoregressive processes, leading to strong consistency if the errors are quasi-independent up to the fourth order.

Given a sequence of $p \times 1$ vectors $\mathbf{v} = \{\nu_i\}^\infty_{i=1}$ such that $M_n = \Sigma^n_{i=1} \nu_i\nu'_i$ is positive definite for some $n$, the linear space $\{u: u'M^{-1}_n u \rightarrow \infty\}$ is characterized in terms of the limiting properties of $\mathbf{v}$. This characterization result is applied to give a necessary and sufficient condition for the asymptotic consistency of any best linear unbiased estimator in terms of the limiting properties of the design sequence. For the polynomial regression model, it can be further related to the geometry of the polynomial system.

The problem of finding maximum likelihood estimates for patterned means and covariance matrices in multivariate analysis is considered. Necessary and sufficient conditions are presented for the existence of explicit solutions and the obtaining of these explicit solutions in one iteration of the scoring equations from any positive definite starting point. Cases in which averaging yields the explicit maximum likelihood estimates are discussed. These results can be applied to the problems of finding maximum likelihood estimates for the parameters in the complete, compound and circular symmetry patterns; mixed models in the analysis of variance; and for finding asymptotic distributions of likelihood ratio statistics when the parameters under the null hypothesis have explicit maximum likelihood estimates.

A result of Szatrowski, giving necessary and sufficient conditions for the existence of explicit maximum likelihood estimates for multivariate normal means and covariances with linear structure, can be applied to the problem of obtaining explicit maximum likelihood estimates in the mixed model of the analysis of variance. This application yields a simple procedure for checking whether or not explicit maximum likelihood estimates exist for the parameters in the balanced mixed model of the analysis of variance. Examples of this procedure as well as a discussion on finding the explicit maximum likelihood estimates are given.

A nonparametric probability density estimator is proposed that is optimal with respect to a discretized form of a continuous penalized-likelihood criterion functional. Approximation results relating the discrete estimator to the estimate obtained by solving the corresponding infinite-dimensional problem are presented. The discrete estimator is shown to be consistent. The numerical implementation of this discrete estimator is outlined and examples displayed. A simulation study compares the integrated mean square error of the discrete estimator with that of the well-known kernel estimators. Asymptotic rates of convergence of the discrete estimator are also investigated.

Extensions in two directions are described of the well-known stochastic ordering property of likelihood ratios. 1. It is shown that the stochastic ordering property holds in a very general conditional sense. 2. It is shown that, for the usual setting occurring in sequential analysis, sequences of likelihood ratios possess the same stochastic ordering property in a multivariate sense. Applications to sequential analysis, and elsewhere, are described.

This paper provides explicit solutions to the problem of estimating the arrival rate $\lambda$ of a Poisson process using a Bayes sequential approach. The loss associated with estimating $\lambda$ by $d$ is assumed to be of the form $(\lambda - d)^2\lambda^{-p}$ and the cost of observation includes both a time cost and an event cost. A discrete time approach is taken in which decisions are made at the end of time intervals having length $t$. Limits of the procedures as $t$ approaches zero are discussed and related to the continuous time Bayes sequential procedure.

Let $X_1, \cdots$ be i.i.d. $\sim N(\mu_1,1)$ and $Y_1, \cdots$ be i.i.d. $\sim N(\mu_2,1)$. A symmetric sequential procedure for $H_0 : \mu_1 > \mu_2$ vs. $H_1 : \mu_1 < \mu_2$ is proposed in this paper. The expected number of observations taken from the inferior population is given in an asymptotic form, which is optimum in Farrell's sense.

Let $V_n$ be the standardized sum of squares of the means of $r + 1$ random samples of sizes $s_0, s_1, \cdots, s_r$, where $n = s_0 + s_1 + \cdots + s_r$, taken without replacement from $n$ numbers. Then using an approximation to the characteristic function of the means, an asymptotic expansion is obtained for the distribution of $V_n$ with first term being the distribution function of $\chi^2_r$ and with error of approximation generally of smaller order than $1/n$. When the numbers are the first $n$ integers, $V$ is the Kruskal-Wallis statistic and the approximation is compared with the exact distribution in some examples of this special case.

Let $\mathbf{X} = (X_1, \cdots, X_m)'$ and $\mathbf{Y} = (Y_1, \cdots, Y_n)'$ be two random vectors. Given any random vector $\mathbf{Z}$, let $\mathbf{Y}^\ast_Z$ be the best linear predictor of $\mathbf{Y}$ based on $\mathbf{Z}$. Let $p$ be any natural number smaller than $m$. We consider the problem of finding the $p$-dimensional random vector $\mathbf{Z} = (Z_1, \cdots, Z_p)'$ where each component $Z_i$ is a linear function of $\mathbf{X}$, which minimizes the determinant of $E(\mathbf{Y} - \mathbf{Y}^\ast_Z)(\mathbf{Y} - \mathbf{Y}^\ast_Z)'$. We show that $Z_1, \cdots, Z_p$ coincide with the first $p$ canonical variables (except for a nonsingular linear transformation). We also show that the square of the $(p + 1)$th canonical correlation coefficient measures the relative improvement in the prediction of $\mathbf{Y}$ when $p + 1 Z_i$'s are used instead of $p$.

Let $X_1, \cdots, X_n$ be independent observations with common density $f$. A kernel estimate of the mode is any value of $t$ which maximizes the kernel estimate of the density $f_n$. Conditions are given restricting the density, the kernel, and the bandwidth under which this estimate of the mode has an asymptotic normal distribution. By imposing sufficient restrictions, the rate at which the mean squared error of the estimator converges to zero can be decreased from $n^{-\frac{4}{7}}$ to $n^{-1+\varepsilon}$ for any positive $\varepsilon$. Also, by bounding the support of the kernel it is shown that for any particular bandwidth sequence the asymptotic mean squared error is minimized by a certain truncated polynomial kernel.

A common polynomial trend in one or more dimensions is assumed to exist over the plots in each block of a classical experimental design. An analysis of covariance model is assumed with trend components represented through use of orthogonal polynomials. The objective is to construct new designs through the assignment of treatments to plots within blocks in such a way that sums of squares for treatments and blocks are calculated as though there were no trend and sums of squares for trend components and error are calculated easily. Such designs are called trend-free and a necessary and sufficient condition for a trend-free design is developed. It is shown that these designs satisfy optimality criteria among the class of connected designs with the same incidence matrix. The analysis of variance for trend-free designs is developed. The paper concludes with two examples of trend-free designs.

Let an optimality criterion $\Phi$ satisfy regularity conditions, including convexity and possession of two continuous derivatives at nonsingular $\mathbf{M}$. $\Phi\{\mathbf{M})$ may be minimized at a singular $\mathbf{M}$. A class of design sequences $\{\xi_n\}$ is shown to make $\Phi\lbrack \mathbf{M}(\xi_n) \rbrack$ converge monotonically to the minimum value. An equivalence theorem for $\Phi$-optimality follows. Techniques which are applicable include vertex direction, conjugate gradient projection, quadratic and "diagonalized quadratic" methods for changing the design weights, and gradient-based methods for making small changes in the support points. Methods are also considered which approximate $\Phi$ by a criterion which is infinite for singular $\mathbf{M}$. The results are applied to an example with $D_s$-optimality.

Admissible designs for polynomial monospline regression are shown to be a proper subset of the admissible designs for spline regression. Sufficient conditions are given for designs for polynomial monospline regression to be admissible, and examples are used to show that simply counting points in intervals does not determine admissibility for monospline functions.

In this paper, we obtain main effect plans with $N = m + 1 + (m - h) (h + 1)$ treatments for $2^m$ factorial experiments, $m = 2^h - 1, h(\geqslant 2)$ an integer, which permit search and estimation of one possible nonzero effect among two factor and higher order interactions.

Taking the sample mean of a set of measurements need not be uniformly advantageous, even when $n = 2$ and all moments are finite. The examples presented include one given by Edgeworth in 1883.