An overview is given of an approach to the definition of descriptive characteristics of populations and to their estimation. The emphasis is on the robustness and efficiency of the estimators. Detailed summaries will be found in successive papers of the series dealing with the problems of location, scale and kurtosis.

Measures of location (without assumption of symmetry) are defined as functionals satisfying certain equivariance and order conditions. Three classes of such measures are discussed whose estimators are respectively linear functions of order statistics, $R$-estimators and $M$-estimators. It is argued that such measures can be compared in terms of the (asymptotic) efficiencies of their estimators. Of the three classes considered, it is found that trimmed expectations (and certain other weighted quantiles) are the only ones which are both robust and whose estimators have guaranteed high efficiency relative to the mean $\bar{X}$ for all underlying distributions.

This paper investigates the effect of serial dependence in the data on the efficiency of some robust estimators. When the observations are from a stationary process satisfying certain mixing conditions, linear combinations of order statistics and the Hodges-Lehmann estimator are shown to be asymptotically normally distributed. Gaussian processes are studied in detail and it is shown that when all the serial correlations $(\rho_n)$ are $\geqq 0$, the efficiency of the robust estimators relative to the mean is greater than in the case of independent observations.

Conditions are given for the weak convergence of weighted empirical cumulative processes of three types of mixing random variables (rv's) on $\lbrack 0, 1\rbrack$.

Estimation of linear functions of a vector parameter $\theta$ when an observed discrete- or continuous-time vector-valued stationary process has mean value $H\theta, H$ a known matrix, is considered. Large-sample comparisons of best linear unbiased estimators and estimators based on the sample mean are made. Limits and rates of convergence of the variances of these estimators are obtained. It is shown that under general conditions there are asymptotically efficient estimators based on the sample mean, their form determined by the spectrum at the origin. Conditions under which all least squares estimators are asymptotically efficient are also given.

Spjotvoll [1967] has obtained a test associated with an unbalanced oneway layout for Model II ANOVA. Under the assumption of normality, his test possesses several optimum properties. Without the normality assumption, the significance level is (in general) highly nonrobust. An attempt to remedy this situation, using a test based on the jackknife technique, appears in Arvesen [1969]. The present paper proposes as an alternative a jackknifed version of Spjotvoll's test. The new test is not sensitive to departures from normality, and Monte Carlo sampling and asymptotic efficiency results suggest that it is more powerful than Arvesen's test. The paper also includes some general results for use of the jackknife technique with nonidentically distributed random variables.

The statistical analysis which is carried out in conducting a $p_1^{n_1}p_2^{n_1} \cdots p_k^{n_k}$ factorial experiment in blocked designs requires that the treatment combinations be randomly arranged for each treatment run. When the nature of the process under investigation restricts the number of factor levels which can be changed from treatment combination to treatment combination, the usual technique of full randomization cannot be carried out. This paper presents methods of constrained randomization for $p_1^{n_1}p_2^{n_2} \cdots p_k^{n_k}$ factorial experiments in blocked designs when the requirement on adjacent treatment combinations is that the number, $\Delta$, of factor levels which can be changed is less than $t$, where $t = \sum^k_{i = 1} n_i(p_i - 1)$. If $\Delta = t$ this is ordinary full randomization. The method of constrained randomization contained in this paper requires the construction of an operational sequence in which $\Delta = 1$ for the first and last treatment combinations in the sequence as well as for all adjacent treatment combinations. The existence and construction of such operational sequences present interesting graph theory problems whose formulations and solutions are found in this paper. This method of constrained randomization provides a basis for a statistical analysis utilizing the randomization model, which results in unbiased estimates of treatment effects and an unbiased estimate of experimental error.

Hedayat and John (1974) studied the case of resistance of a balanced incomplete block design (BIBD) to the removal of one treatment. They showed that a necessary and sufficient condition for a BIBD to be resistant (that is, retain its variance balance) upon removal of a single treatment is that the two proper subdesigns created by this removal be BIBDs. Herein their results on the structure of resistant BIBDs are generalized and the relationship of such designs to $t$-designs (which are of interest in combinatorial theory) is shown. The main result is a structure theorem which states that a BIBD in $v$ treatments and $k$ plots per block is resistant to the removal of any subset of a specific set of $n$ treatments, $n < k < v$, if and only if all of the $2_n$ proper subdesigns formed when all $n$ treatments are removed are BIBDs in all $v - n$ remaining treatments. This leads to a lower bound (of $2^n(v - n))$ on the number of blocks required for fully locally resistant BIBDs which are not essentially trivial. Examples of resistant designs are given.

A simple general approach to inference about the tail behavior of a distribution is proposed. It is not required to assume any global form for the distribution function, but merely the form of behavior in the tail where it is desired to draw inference. Results are particularly simple for distributions of the Zipf type, i.e., where $G(y) = 1 - Cy^{-\alpha}$ for large $y$. The methods of inference are based upon an evaluation of the conditional likelihood for the parameters describing the tail behavior, given the values of the extreme order statistics, and can be implemented from both Bayesian and frequentist viewpoints.

The following proposition is sometimes used in distribution theory: for each fixed $z$ suppose that $T(X, z)$ has the distribution $Q$ and is independent of $Y$; then $T(X, Z(Y))$ has the distribution $Q$ and is independent of $Y$. An example is presented to show this result is false in general. Additional conditions under which the proposition becomes valid are presented.

We prove in a discrete setting that if for all test functions, $t$, there is a $\mathbf{B}$ measurable test function, $s$, such that $E_p(t) = E_p(s)$ for all $p \in P$ then some subfield of $\mathbf{B}$ is sufficient for $P$.

Assume given the $(n + 1)$-first generation sizes of a supercritical branching process. An estimator is proposed for the variance $\sigma^2$ of this process when the mean is known. It is shown to be unbiased, consistent and asymptotically normal. From that one deduces a consistent and asymptotically normal estimator for $\sigma^2$ in the case of an unknown mean. Finally, the maximum likelihood estimator of $\sigma^2$, based on a richer sample, is found and asymptotic properties are studied.