Maximum likelihood type robust estimates of regression are defined and their asymptotic properties are investigated both theoretically and empirically. Perhaps the most important new feature is that the number $p$ of parameters is allowed to increase with the number $n$ of observations. The initial terms of a formal power series expansion (essentially in powers of $p/n$) show an excellent agreement with Monte Carlo results, in most cases down to 4 observations per parameter.

The purpose of this paper is to find asymptotically optimal Bayes sequential procedures for estimating a function $g(\theta_1, \theta_2,\cdots, \theta_k)$ when there are $k$ experiments $E_1, E_2,\cdots, E_k$ and the performance of the experiment $E_i$ conducts to the observation of a random variable whose distribution depends on the vector parameter $\theta_i$. The term asymptotical refers here to the cost of experimentation tending to zero. The methods used are a generalization of those introduced by Bickel and Yahav.

Let $X_1, X_2,\cdots$ be a sequence of independent and identically distributed random variables governed by an unknown member of a countable family $\mathscr{P} = \{P_\theta: \theta \in \Omega\}$ of probability measures. The family $\mathscr{P}$ is said to be sequentially distinguishable if for any $\varepsilon (0 < \varepsilon < 1)$ there exist a stopping time $t$ and a terminal decision function $\delta(X_1,\cdots, X_t)$ such that $P_\theta\{t < \infty\} = 1 \forall\theta\in\Omega$ and $\sup_{\theta\in\Omega} P_\theta(\delta(X_1,\cdots, X_t) \neq \theta) \leqq \varepsilon$. Robbins [12] defined a general stopping time (see Section 2) as an approach to this problem. This paper is a study of this stopping time with applications to some exponential distributions.

Kesten proposed a method for adjusting the coefficients of a scalar stochastic approximation process, and proved w.p. 1 convergence. A family of multidimensional processes for function minimization are treated here. Each method consists of a sequence of truncated one-dimensional procedures of the Kesten type. The methods seem to offer a number of advantages over the usual Kiefer-Wolfowitz procedures, and are more natural analogs of the schemes in common use in deterministic optimization theory.

A complexity approach is used to decide whether or not the mean of a sequence of independent identically distributed random variables lies in an arbitrary specified countable subset of the real line. A procedure is described that makes only a finite number of mistakes with probability one. This leads to some speculations on inference of the laws of physics and the computability of the physical constants.

In this paper Laws of the Iterated Logarithm for maximums of absolute values of partial sums of permuted random variables are derived under conditions that are the same as or similar to conditions used by Kolmogorov, Hartman and Wintner, Petrov and Csaki in deriving Laws of the Iterated Logarithm for sums of random variables or semimartingales. These results are then applied to obtain logarithmic convergence rates for estimators of non-decreasing regression functions and integral regression functions.

This paper is concerned with maximum likelihood estimates for a large class of families of unimodal densities. The existence of measurable maximum likelihood estimates and the consistency of asymptotic maximum likelihood estimates are proved. By counterexamples it is shown that the conditions which are sufficient for consistency cannot be removed without compensation.

Let $f: X \times Y \rightarrow R$. We prove two theorems concerning the existence of a measurable function $\varphi$ such that $f(x, \varphi(x)) = \inf_y f(x,y)$. The first concerns Borel measurability and the second concerns absolute (or universal) measurability. These results are related to the existence of measurable projections of sets $S \subset X \times Y$. Among other applications these theorems can be applied to the problem of finding measurable Bayes procedures according to the usual procedure of minimizing the a posteriori risk. This application is described here and a counterexample is given in which a Borel measurable Bayes procedure fails to exist.

Methods for multiplication of polykays have been given by various authors. Carney used ordered partitions in considering double products. In this paper, relations between polykays, symmetric means, augmented monomial functions and unrestricted power sums are studied. The method of ordered partitions is extended to obtain multiple products of polykays. Kronecker products of ordered partitions are introduced and certain rules are suggested by whose application the size of the problem is reduced considerably. The method is illustrated on triple products of weight 7.

This paper characterizes balanced saturated main effect plans of the $2^n$ factorial in terms of $D'D$ rather than $X'X$, where $D$ is the $(n + 1) \times n$ treatment combination matrix and $X$ is the $(n + 1) \times (n + 1)$ design matrix. Besides this result, balanced optimal (in the sense of maximum determinant of $X'X$) saturated main effect plans of the $2^{4m-1}$ factorial are discussed for various classes of designs, each class consisting of designs having (0,0,$\cdots$ 0) and $n$ treatment combinations with exactly $t$ 1's among them. The optimality results are achieved by applying theorems associated with incidence matrices of $(\nu, k, \lambda)$ configurations. In addition results are given for designs associated with the permuted $(\nu, k, \lambda)$ configurations. Finally, the approach taken in the paper can be applied to $2^n$ factorials with $n \neq 4m - 1$.

Gontcharov obtained the joint probability generating function for the numbers of runs of all lengths, both of successes and failures, in a Bernoulli sequence. This is here generalized to a class of regenerative binary Markov processes. For an allied class of Markov processes, the probability generating function is obtained for a "total score" defined in terms of runs of successes only, and asymptotic formulas are derived for the expectation and variance of the score.

The confidence sets for linear functions $\mu + \lambda\sigma^2$ of the mean $\mu$ and variance $\sigma^2$ of a normal distribution, defined in terms of the uniformly most powerful unbiased level $\alpha$ tests of hypotheses of form $H_0(\lambda, m): \mu + \lambda\sigma^2 = m$ against the two-sided alternative $H_1(\lambda, m): \mu + \lambda\sigma^2 \neq m$ for $-\infty < m < \infty$, for fixed $\alpha$ and $\lambda$, are shown to be intervals if the number of degrees of freedom for estimating $\sigma^2$ is $\geqq 2$.

$f(x)$ is a uniformly continuous density which equals zero for negative values of $x$, has a right-hand derivative equal to $\alpha$ at $x = 0$, where $0 < \alpha < \infty$, and satisfies certain regularity conditions. $X_1,\cdots, X_n$ are independent random variables with the common density $f(x - \theta), \theta$ an unknown parameter. Let $\hat{\theta}_n$ denote the maximum likelihood estimator of $\theta$, and define $\alpha_n$ by the equation $2\alpha_n^2 = \alpha n \log n$. It was shown by Woodroofe that the asymptotic distribution of $\alpha_n(\hat{\theta}_n - \theta)$ is standard normal. It is shown in the present paper that $\hat{\theta}_n$ is an asymptotically efficient estimator of $\theta$.

The large-sample distributions of the maximum-likelihood estimates for the index, skewness, scale, and location parameters (respectively $\alpha, \beta, c$, and $\delta$) of a stable distribution are studied. It is shown that if both $\alpha$ and $\delta$ are unknown, then the likelihood function $L$ will have no maximum within $0 < \alpha \leqq 2, -\infty < \delta < \infty$, but that $L(\alpha, \delta) \rightarrow \infty$ as $(\alpha, \delta) \rightarrow (0, x_k)$ where $x_k$ is any one of the $n$ observed sample values. However, it is shown that the centroid of $L$ is little affected by this behavior and, if the estimate $\hat{\alpha}$ is restricted to $\hat{\alpha} \geqq \varepsilon > 0$, then the maximum-likelihood estimates are consistent and $n^{\frac{1}{2}}(\hat{\alpha} - \alpha, \hat{\beta} - \beta, \hat{c} - c, \hat{\delta} - \delta)$ has a limiting normal distribution with mean (0,0,0,0) and covariance matrix $\mathbf{I}^{-1}$, where $\mathbf{I}$ is the Fisher information matrix. There are some exceptional values of $\alpha$ and $\beta$ for which the argument presented does not hold. The argument consists in showing that the family of stable distributions satisfies conditions given in the literature and in doing so it is proven that certain asymptotic expansions for stable densities can be differentiated arbitrarily with respect to the parameters.

A class of statistics is considered, which are based on order-statistics and have properties analogous to Student's $t$. They can be used to estimate a required quantile of a random variable or to test hypotheses about a quantile; they are simple to compute, and can be calculated when not all sample values are available, e.g. for censored samples. The limiting distributions of these statistics are derived and shown to be independent of the distribution of the underlying random variable. A numerical tabulation of the limiting distribution is included, for the special case when the quantile considered is the median.

The classification statistic $W$ is used to classify an observasion as coming from one of two multivariate normal populations with common covariance matrix and different means when these parameters are estimated from two samples, one from each population. The distribution of $W$ depends on the Mahalanobis distance between the populations, $\alpha$. When the sample sizes approach infinity, the limiting distribution of $(W - \frac{1}{2}\alpha)/\alpha^{\frac{1}{2}}$ is the standard normal distribution if the observation is from the first population; the same is true of $(W - \frac{1}{2}a)/a^{\frac{1}{2}}$, where $a$ is an estimate of $\alpha$. This paper gives an asymptotic expansion of the distribution of $(W - \frac{1}{2}a)/a^{\frac{1}{2}}$ with an error of the order of the square of the number of observations. The correction to the standard normal distribution function is the standard normal density times a third-degree polynomial in the argument divided by the sum of the observations (less 2).

It is shown that under certain conditions, the maximum likelihood and the minimum variance unbiased estimators of a positive integral power of the natural parameter in an exponential family have the same asymptotic distribution.

Let $Z_n = \max_{1\leqq j\leqq n} X_j$ where $\{X_n: 1 \leqq n < \infty\}$ is a Gaussian process. Some known limit theorems for $Z_n$ when $\{X_n\}$ is stationary are extended to the nonstationary case.

A new type of $m$-class cyclic association scheme is defined for $v$ objects, where $v$ is either a prime or power of a prime. The two cases, when $v$ is even and when $v$ is odd are dealt with separately. A simple method of construction of PBIB designs based on this scheme is proposed. The PBIB designs presented in this paper have cyclic solutions and are obtainable through more than one initial block. The proposed scheme has been called an $NC_m$ association scheme and the number of independent $p_{jk}$'s for this scheme is obtained.