In this paper, a complete overview is given of the theoretical development of various estimators generated by the jackknife statistic. In particular, the jackknife method is extended to stochastic processes by means of two estimators referred to as the $J_\infty$-estimator and the $J_\infty^{(2)}$-estimator. These estimators are studied in some detail and shown to have the same properties as the jackknife when one considers the length of the process record as the sample size. Finally, it is shown that the entire development of the jackknife procedures discussed in this paper can be considered as a direct parallel of earlier developments in numerical analysis surrounding the study of a transformation referred to as the $e_n$-transformation.

For partial cumulative sums of independent and identically distributed random variables (i.i.d.r.v.) with a finite (positive) variance, weak convergence to Brownian motion processes has been established by Donsker (1951, 1952). The result is extended here to differentiable statistical functions of von Mises (1947) and $U$-statistics of Hoeffding (1948). Along with the extension to generalized $U$-statistics, a few applications are briefly sketched.

Linearized estimates, as functions of the ranks, are proposed for the general linear hypothesis. These estimates can be computed after a single ranking of the "centered" observations. The asymptotic distribution of the estimates is shown to be the same as the maximum likelihood estimates for fairly general sequences of design matrices.

Let $X_1^n \leqq X_2^n \leqq \cdots \leqq X_n^n$ be the order statistics of a size $n$ sample from any distribution function $F$ not necessarily continuous. Let $\alpha_j, \beta_j, (j = 1,2, \cdots, n)$ be any numbers. Let $P_n = P(\alpha_j < X_j^n \leqq \beta_j, j = 1,2, \cdots, n)$. A recursion is given which calculates $P_n$ for any $F$ and any $\alpha_j, \beta_j$. Suppose now that $F$ is continuous. A two-sided statistic of Kolmogorov-Smirnov type has the distribution function $P_{\mathrm{KS}} = P\lbrack\sup n^{\frac{1}{2}}\psi(F) \cdot |F^n - F| \leqq \lambda\rbrack$, where $F^n$ is the empirical distribution function of the sample and $\psi(x)$ is any nonnegative weight function. As $P_{\mathrm{KS}}$ has the form $P_n$, its calculation as a function of $\lambda$ can be carried out by means of the recursion. This has been done for the case $\psi(x) = \lbrack x(1 - x)\rbrack^{-\frac{1}{2}}$. Curves are given which represent $\lambda$ versus $1 - P_{\mathrm{KS}}$ for $n = 1,2, 10, 100$. From additional computations, the precision of a truncated development of $1 - P_{\mathrm{KS}}$ in powers of $\lambda^{-2}$ has been determined.

Consider the 2-sample problem where the null cdf $F(x)$ satisfies $F(0) = 0$ and the alternative is $F_\theta(x) = F(x/(1 + \theta))$ with $\theta > 0$. An asymptotically optimum statistic $z$ is obtained for a parametric model where $F(x)$ is a gamma distribution. The Mann-Whitney $U$ and Savage $T$ statistics are compared to $z$ for several null densities. It is shown that the Pitman asymptotic relative efficiency, ARE $(U/z)$, can approach zero if $\mu/\sigma\rightarrow 0$, where $\mu$ is the mean and $\sigma^2$ the variance of the null distribution. However, a lower bond on ARE $(U/z)$ is obtained as a function of $\mu/\sigma$ for general $F(x)$. Using the bound a composite test is constructed which has a specified minimum ARE of any desired value between 0 and .864. Densities exist for the composite test which result in arbitrarily large values of efficiency.

It is the purpose of this paper to explore the efficiency of a modified definition for maximum likelihood estimates which depends on the whole equivalence class of densities only and not--as in the classical case--on the particular choice of versions. We prove the existence of measurable maximum likelihood estimates in the new sense for compact metrizable families of probability measures without any continuity assumption for the densities. For appropriate families of probability measures the modified asymptotic maximum likelihood estimates are exactly the strongly consistent estimates. The paper uses Huber's concept of minimum contrast estimates which covers maximum likelihood estimates as a special case.

As an approximation to the regression function $m$ of $Y$ on $X$ based upon empirical data, E. A. Nadaraya and G. S. Watson have studied estimates of $m$ of the form $m_n(x) = \sum Y_ik((x - X_i)/a_n)/\sum k((x - X_i)/a_n)$. For distinct points $x_1, \cdots, x_k$, we establish conditions under which $(na_n)^{\frac{1}{2}}(m_n(x_1) - m(x_1), \cdots, m_n(x_k) - m(x_k))$ is asymptotically multivariate normal.

Given a sample of size $n$, a continuous estimator for a distribution $F$ (based on Pyke's modified sample distribution) is shown to have the property that its expected squared error, for almost all $x$ in the positive sample space of $F$, is no larger than that of the sample distribution function given $F$ and $n$ sufficiently large. Letting risk be given by the expected squared error integrated with respect to $F$, it is shown that this estimator dominates both the sample distribution and the other best invariant estimator found by Aggarwal, given $F$ and $n$ sufficiently large. Other common estimators cannot serve in this dominating role. Explicit calculation of risk is made when $F$ is the uniform distribution. In this case the estimator strictly dominates the sample distribution for all $n \geqq 1$.

This paper considers some empirical Bayes procedures which have been discussed by H. Robbins, E. Samuel, and M. V. Johns. These procedures are shown to be inadmissible relative to a class of priors and by using some of the results of Rolph [6] admissible procedures are found for two examples. For an introduction to the empirical Bayes approach see Robbins [5].

Let $(\mathscr{X}, \mathscr{B}, P)$ be a probability measure space for each $P \in \mathscr{P} = \{F_0, \cdots, F_m\}, \mathscr{A}$ be an action space and $L$ be a loss function defined on $\mathscr{X} \times \mathscr{P} \times \mathscr{A}$ such that for each $i$, $c_i = \int_{\mathbf{V}_a} L(x, F_i, a) dF_i(x) < \infty$. In the compound problem, consisting of $N$ components each with the above structure, we consider procedures equivariant under the permutation group. With $\rho_{ij} = \mathbf{V}_{B\in\mathscr{B}}|F_i(B) - F_j(B)| \text{and} K(\rho) = .5012\ldots\rho(1 - \rho)^{-\frac{3}{2}},$ we show that the difference between the simple and the equivariant envelopes is bounded by \begin{equation*}\tag{T1} \{2K(\rho) \sum_i c_i^2\}^{\frac{1}{2}}N^{-\frac{1}{2}}\quad \text{where} \rho = \mathbf{V}_{i,j}\rho_{ij},\end{equation*} and by \begin{equation*}\tag{T2} 2^m\{2K(\rho') \sum_i c_i^2\}^{\frac{1}{2}}N^{-\frac{1}{2}} \text{where} \rho' = \mathbf{V}\{\rho_{ij}\mid\rho_{ij} < 1\}\end{equation*} The bound (T1) is finite iff the $F_i$ are pairwise non-orthogonal and (T2) is designed to replace it otherwise.

Let $f_\theta(x) = f(x - \theta), \theta, x\in R$, where $f(x) = 0$ for $x \leqq 0$ and let $\hat{\theta}_n$ be the maximum likelihood estimate (MLE) of $\theta$ based on a sample of size $n$. If $\alpha = \lim f'(x)$ exists as $x \rightarrow 0$, and $0 < \alpha < \infty$, then under some regularity conditions, it is shown that $\alpha_n(\hat{\theta}_n - \theta)$ has an asymptotic standard normal distribution where $2\alpha_n^2 = \alpha n \log n$ and that if $\theta$ is regarded as a random variable with a prior density, then the posterior distribution of $\alpha_n(\theta - \hat{\theta}_n)$ converges to normality in probability.

This paper deals with the problem of estimating the number of trials of a multinomial distribution, from an incomplete observation of the cell totals, under constraints on the cell probabilities. More specifically let $(n_1, \cdots, n_k)$ be distributed according to the multinomial law $M(N; p_1, \cdots, p_k)$ where $N$ is the number of trials and the $p_i$'s are the cell probabilities, $\sum^k_{i=1}p_i$ being equal to 1. Suppose that only a proper subset of $(n_1, \cdots, n_k)$ is observable, that $N, p_1, \cdots, p_k$ are unknown and that $N$ is to be estimated. Without loss of generality, $(n_1, \cdots, n_{l-1}), l \leqq k$ may be taken to be the observable random vector. For fixed $N, (n_1, \cdots, n_{l-1}, N - n)$ has the multinomial distribution $M(N; p_1, \cdots, p_l)$ where $n$ denotes $\sum^{l-1}_{i=1}n_i$ and $p_l$ denotes $1 - \sum^{l-1}_{i=1}p_i$. If the parameter space is such that $N$ can take any nonnegative integral value and each $p_i$ can take any value between 0 and 1, such that $\sum^{l-1}_{i=1}p_i < 1$ then, clearly, the only inference one can make about $N$ is that $N > n$. In specific situations, it might, however, be possible to postulate constraints of the type \begin{equation*}\tag{1.1} p_i = f_i(\theta),\quad i = 1, \cdots, l\end{equation*} where $\theta = (\theta_1, \cdots, \theta_r)$ is a vector of $r$ independent parameters and $f_i$ are known functions. This may lead to estimability of $N$. The problem of estimating $N$ in such a situation is studied here. The present investigation is motivated by the following problem. Experiments in particle physics often involve visual scanning of film containing photographs of particles (occurring, for instance, inside a bubble chamber). The scanning is done with a view to counting the number $N$ of particles of a predetermined type (these particles will be referred to as events). But owing to poor visibility caused by such characteristics as low momentum, the distribution and configuration of nearby track patterns, etc., some events are likely to be missed during the scanning process. The question, then, is: How does one get an estimate of $N$? The usual procedure of estimating $N$ is as follows. Film containing the $N$ (unknown) events is scanned separately by $w$ scanners (ordered in some specific way) using the same instructions. For each event $E$ let a $w$-vector $Z(E)$ be defined, such that the $j$th component $Z_j$ of $Z(E)$ is 1 if $E$ is detected by the $j$th scanner and is 0 otherwise. Let $\mathscr{J}$ be the set of $2^w w$-vectors of 1's and 0's and let $I_0$ by the vector of 0's. Let $x_I$ be the number of events $E$ whose $Z(E) = I$. For $I \in \mathscr{J} - \{I_0\}$, the $x_I$'s are observed. A probability model is assumed for the results of the scanning process. That is, it is assumed that there is a probability $p_I$ that $Z(E)$ assumes the value $I$ and that these $p_I$'s are constrained by equations of the type (1.1) (These constraints vary according to the assumptions made about the scanners and events, thus giving rise to different models. An example of $p_I(\theta)$ would be $E(\nu^{\Sigma^w_{j=1}I_j}(1 - \nu)^{w-\Sigma^w_{j=1}I_j})$ where $I_j$ is the $j$th component of $I$ and expectation is taken with respect to the two-parameter beta density for $v$. This is the result of assuming that all scanners are equally efficient in detecting events, that the probability $v$ that an event is seen by any scanner is a random variable and that the results of the different scans are locally independent. For a discussion of various models, see Sanathanan (1969), Chapter III. $N$ is then estimated using the observed $x_I$'s and the constraints on the $P_I$'s, provided certain conditions (e.g., the minimum number of scans required) are met. The following formulation of the problem of estimating $N$, however, leads to some systematic study including a development of the relevant asymptotic distribution theory for the estimators. The $Z(E)$'s may be regarded as realizations of $N$ independent identically distributed random variables whose common distribution is discrete with probabilities $p_I$ at $I$ (In particle counting problems, it is usually true that the particles of interest are sparsely distributed throughout the film on account of their Poisson distribution with low intensity. Thus in spite of the factors affecting their visibility outlined earlier, the events can be assumed to be independent.). The joint distribution of the $x_I$'s is, then, multinomial $M(N; p_I, I \in \mathscr{J})$. The problem of estimating $N$ is now in the form stated at the beginning of this section. Since the estimate depends on the constraints provided for the $p_I$'s, it is important to test the "fit" on the model selected. The conditional distribution of the $x_I$'s $(I \neq I_0)$ given $x$ is multinomial $M(x; p_I/p(I \neq I_0))$ where $x$ is defined as $\sum_{I\neq I_0} x_I$ and $p$ as $\sum_{I\neq I_0}P_I$. The corresponding $\chi^2$ goodness of fit test may therefore be used to test the adequacy of a model in question. Various estimators of $N$ are considered in this paper and among them is, of course, the maximum likelihood estimator of $N$. Asymptotic theory for maximum likelihood estimation of the parameters of a multinomial distribution has been developed before for the case where $N$ is known but not for the case where $N$ is unknown. Asymptotic theory related to the latter case is developed is Section 4. The result on the asymptotic joint distribution of the relevant maximum likelihood estimators is stated in Theorem 2. A second method of estimation considered is that of maximizing the likelihood based on the conditional probability of observing $(n_1,\cdots, n_{l-1})$, given $n$. This method is called the conditional maximum likelihood (C.M.L.) method. The C.M.L. estimator of $N$ is shown (Theorem 2) to be asymptotically equivalent to the maximum likelihood estimator. Section 5 contains an extension of these results to the situation involving several multinomial distributions. This situation arises in the particle scanning context when the detected events are classified into groups based on some factor like momentum which is related to visibility of an event, and a separate scanning record is available for each group. A third method of estimation considered is that of equating certain linear combinations of the cell totals (presumably chosen on the basis of some criterion) to their respective expected values. Asymptotic theory for this method is given in Section 6. This discussion is motivated by a particular case which is applicable to some models in the particle scanning problem, using a criterion based on the method of moments for the unobservable random variable, given by the number of scanners detecting an event (Discussion of the particular case can be found in Sanathanan (1969) Chapter III.). In the next section we give some definitions and a preliminary lemma.

Let $\varepsilon(t,\mathbf{x})$ denote a stationary Gaussian process with $E_\varepsilon (t, \mathbf{x}) = 0, E_\varepsilon^2(t, \mathbf{x}) = \sigma^2\mathbf{\forall t} \in \mathscr{J}, \mathbf{x} \in \mathscr{X}$, and $E_\varepsilon(t_1, \mathbf{x}_1)_\varepsilon(t_2, \mathbf{x}_2) = 0 \mathbf{\forall t}_1 \neq t_2$ or $\mathbf{x}_1 \neq \mathbf{x}_2$. Let $\mathscr{J}$ be the set of integers and $\mathscr{X}$ a subset of the $r$-dimensional Euclidean space $R^r$. Given a coordinate system in $R^r$ and a time origin, observe $y(t, \mathbf{x}) = s(t, \mathbf{x}) + \varepsilon(t, \mathbf{x})$, where $s(t, \mathbf{x}) = \sum^{T-1}_{j=0}A(\omega j) \exp\{i\lbrack\omega jt - \mathbf{\kappa}(\omega j)'\mathbf{x}\rbrack\}, \omega j = 2\pi j/T, j = 0,1,\cdots, T - 1$, and $\mathbf{\kappa}(\omega j)$ is a vector of parameters in $R^r$. If $\mathbf{\kappa}(\omega) = (\omega/\nu)\mathbf{e}$, where $\mathbf{e'e} = 1, s(t, \mathbf{x})$ is the $r$-dimensional generalization of a (discrete-time) plane wave which is propagating with phase velocity $\nu$ in a direction parallel to $\mathbf{e}$. For a finite time let the process $y(t, \mathbf{x})$ be simultaneously observed at each $\mathbf{x} \in \mathscr{X} = S_1 \times S_2 \times \cdots \times S_r, S_j = \{1,2,\cdots, n\}$. The maximum likelihood estimators $\hat{A}(\omega j)$ and $\hat{\mathbf{\kappa}}(\omega j)$ of $A(\omega j)$ and $\mathbf{\kappa}(\omega j)$, respectively, have a joint limiting normal distribution in which appropriately normalized estimators of the $r$ components of $\mathbf{\kappa}(\omega j)$ are mutually independent, for each $j = 1,\cdots, T - 1$. The distributions of the estimators for different $\omega j$'s are mutually independent. The analysis is generalized to the case where $s(t, \mathbf{x})$ is a sum of plane waves with separation between the phase velocities.

Estimation of the value $f(0)$ of a density function evaluated at 0 is studied, $f: \mathbb{R}_m \rightarrow \mathbb{R}, 0 \in \mathbb{R}_m$. Sequences of estimators $\{\gamma_n, n \geqq 1\}$, one estimator for each sample size, are studied. We are interested in the problem, given a set $C$ of density functions and a sequence of numbers $\{a_n, n \leqq 1\}$, how rapidly can $a_n$ tend to zero and yet have $\lim\inf_{n\rightarrow\infty} \inf_{f\in C}P_f(|\gamma_n(X_1,\cdots, X_n) - f(0)|\leqq a_n) > 0?$ In brief, by "rate of convergence" we will mean the rate which $a_n$ tends to zero. For a continuum of different choices of the set $C$ specified by various Lipschitz conditions on the $k$th partial derivatives of $f, k \geqq 0$, lower bounds for the possible rate of convergence are obtained. Combination of these lower bounds with known methods of estimation give best possible rates of convergence in a number of cases.

Conditions are given for the strong consistency and asymptotic normality of the MLE (maximum likelihood estimator) for multiparameter exponential models. Because of the special structure assumed, the conditions are less restrictive than required by general theorems in this area. The technique involves certain convex functions on Euclidean spaces that arise naturally in the present context. Some examples are considered; among them, the multinomial distribution. Some convexity and continuity properties of multivariate cumulant generating functions are also discussed.

Let $S_1, S_2,\cdots, S_n$ be integrable random variables (rv). Upper bounds of the Hajek-Renyi type are presented for $P(\max_{1\leqq k\leqq n} \phi_k S_k \geqq \varepsilon \mid \mathscr{G})$ where $\phi_1 \geqq \cdots \geqq \phi_n > 0$ are rv, $\varepsilon > 0$ and $\mathscr{G}$ is a $\sigma$-field. The theorems place no further assumptions on the $S_k$'s; some, in fact, do not even require the integrability. It is shown, however, that if the $S_k$'s are partial sums of independent rv or if $S_1, S_2,\cdots, S_n$ forms a submartingale, then some well-known inequalities follow as consequences of these theorems.

The finite sequence of $n$ random variables $U_1 U_2,\cdots, U_n$ is divided into two complementary groups of random variables in one of $2^n$ ways. The random variables in each group are summed and the two sums are compared. Let $|S_n|$ be the minimum of the difference of the sums out of all the $2^n$ possible divisions. A lower bound to all sequences $\{\epsilon_n\}$ such that $P\{|S_n| < \varepsilon_n\} \rightarrow 1$ as $n \rightarrow \infty$ is found in two cases: - $U_i = X_i i = 1,2,\cdots n$ and $U_i = X_i/\sum^n_{i=1} X_i, i = 1,2\cdots n$ where the $X_i$ are independent and identically distributed random variables which have densities and satisfy certain regularity conditions. The results lead to the solution of the particular problem of minimising the difference between two sums formed from segments of a fractured unit interval.

Certain families of probability distribution functions maintain their infinite divisibility under repeated mixing and convolution. Examples on the continuum and lattice are given. The main tools used are Polya's criteria and the properties of log-convexity and complete monotonicity. Some light is shed on the relationship between these two properties.

Conditions are given which imply that the partial sums of a sequence of independent integer-valued variables which satisfy the classical Lindeberg conditions for the central limit theorem also obey a strong version of the local limit theorem. Application is made to the problem of establishing the interval recurrence of the partial sums.

In Section 1 it is shown that $n$ interior points of the $(k - 1)$-dimensional simplex $S_k$ define a partition of $S_k$ into $\binom{n+k-1}{k-1}$ convex polytopes $R(\mathbf{n})$ which are in one-one correspondence with the partitions of $n$ into a sum of $k$ nonnegative integers. If the $n$ points are uniformly and independently distributed over $S_k$, then $R(\mathbf{n})$ becomes a random polytope. Basic properties of the random $R(\mathbf{n})$ are given in Section 2. Section 3 presents an algorithm which can be used to compute the distribution of any extremal vertex of $R(\mathbf{n})$.

Let $\{x_1, x_2,\cdots\}$ be a sequence of i.i.d.r.v. with mean zero, variance one, and (1) $\mathbf{P}(|x_k| \geqq \lambda) \leqq C \exp(-\alpha\lambda^\varepsilon)$ for positive $\alpha, \varepsilon$. Let $f(t, x)$ (with its first partial derivatives) be of slow growth in $x$, let $F_n(x)$ be the distribution function of $(1/n) \sum^n_1 f(k/n, s_k/n^{\frac{1}{2}})$ where $s_k = x_1 + x_2 + \cdots + x_k$, and let $F(x)$ be the distribution function of $\int^1_0 f(t, w(t)) dt$ where $\{w(t)\}$ is Brownian motion. Then $\sup_x |F_n(x) - F(x)| = O((\log n)^\beta/n^{\frac{1}{2}})$ provided $F(x)$ has a bounded derivative. The proof uses the Skorokhod representation; also, a theorem is proven which would indicate that the Skorokhod representation cannot be used in general to obtain a rate of convergence better than $O(1/n^{\frac{1}{4}})$. A corresponding result is obtained if (1) is replaced by the existence of a finite $p$th moment, $p \geqq 4$.

Let $(S, d)$ be a compact metric space; let $(\Omega, \mathscr{F}, P)$ be a probability space, and for each $t \in S$ let $X_t: \Omega \rightarrow \mathbb{R}$ be a random variable, with $E(X_t) = 0$ and such that $\{X_t\}_{t\in S}$ forms a Gaussian process. In this paper we find sufficient conditions for the Gaussian process $\{X_t\}_{t\in S}$ to admit a separable and measurable model whose sample functions are continuous with probability one. The conditions involve the covariance, $E(X_s, X_t)$, of the process and also the $\varepsilon$-entropy of $S$.

Let $A_0(x), x \in Z$ be independent nonnegative integer-valued random variables with $\mu_j(x) = E(A_0(x)(A_0(x) - 1)\cdots(A_0(x) - j + 1))$. Assume that $\{\mu j(x)\}_x$ has limits for $j = 1,2$ and that it is bounded for $3 \leqq j \leqq 6$. Suppose at time zero there are $A_0(x)$ particles at $x \in Z$ and subsequently the particles move independently according to the transition function $P(x, y)$ of a recurrent random walk. For a finite nonempty subset $B$ of $Z$ denote by $A_n(B)$ the number of particles in $B$ at time $n$. Then $S_n(B) = \sum^n_{k=1} A_k(B)$ is the total occupation time of $B$ by time $n$ of all particles. Assuming that the $n$ step transition function $P_n(x, y)$ is such that there is an $\alpha$, with $1 < \alpha \leqq 2$, so that $P_n(0, x) \sim cn^{-1/\alpha}$ for all $x$, it is proved that the strong law of large numbers and the central limit theorem hold for the sequence $\{S_n(B)\}$.

Let $\mathscr{P} = \{F_0,\cdots, F_m\}$ be a class of probability measures on $(\mathscr{X}, \mathscr{B})$. For any signed measure $\tau$ on $\mathscr{B}^N$, let $\tau^\ast$ be the average of $\tau g$ over all $N$! permutations $g$ and let $\|\tau\| = \mathbf{V}\{|\tau(C)|: C \in \mathscr{B}^N\}$. Let $d_{ij} = \|F_i - F_j\|$ and $K(x) = .5012\cdots x(1 - x)^{-\frac{3}{2}}$. For any nonnegative integral partitions $\mathbf{N} = (N_0,\cdots, N_m)$ and $\mathbf{N}' = (N_0',\cdots, N_m')$ of $N$, let $\delta_i = N'_i - N_i$ and $\wedge_i = (N'_i \wedge N_i) + 1$. With $\tau = \times F_i^{N_i} - \times F_i^{N_i'}$ and $n = {\tt\#}\{i\mid\delta_i \neq 0\} - 1$, we bound $\|\tau^\ast\|^2$ by \begin{equation*}\tag{T3} nK(d)\sum \delta_i^2\wedge_i^{-1}\quad \text{with} d = \vee\{d_{ij}\mid\delta_i \neq 0, \delta_j \neq 0\}\end{equation*} and, if $\mathscr{P}$ is internally connected by chains with non-orthogonal successive elements, by \begin{equation*}\tag{T4} \frac{1}{2}mK(\check{d})(\sum |\delta_i|)^2(\sum \wedge_i^{-1}) \quad\text{with} \check{d} = \vee\{d_{ij} \mid F_i ?? F_j\}.\end{equation*} The bound (T3) is finite iff the $F_i$ with $\delta_i \neq 0$ are pairwise non-orthogonal and (T4) is designed to replace it otherwise.

Doob introduced the standard modifications or extensions of a stochastic process and proved that every stochastic process has a separable standard modification. In 1964 Elliott and Morse developed a general theory of product measures with implications in the theory of continuous parameter processes with mutually independent random variables. In particular, they gave a new method for obtaining extensions which considerably generalizes the notion of separability. For a separable process only certain events specified by restrictions of the random variables at a nondenumerable collection of time points are measurable. Under their generalization, the restriction to only certain events is virtually removed. The key to the new method for obtaining extensions is a modification by means of nilsets. The definition of nilsets has recently been adjusted to enable the application of this method to general stochastic processes.

This is an extension of D.A.S. Fraser's structural inference to statistical problems invariant under a group which is not necessarily exactly transitive on either the sample or parameter spaces. Explicit formulas are given for the extended structural distribution of the parameter given observations, in a class of cases for which a relatively invariant measure exists. The connection with Bayesian inference with invariant priors is discussed.

Bounds on the risks, under squared error loss, of a family of estimators of a multivariate location parameter are given for both fixed and random unknown location parameters when the covariance matrix of the observed random variable is unknown. The class of estimators considered in this paper contains Cogburn's [2].

A conjecture is made for the expected value of the $j$th elementary symmetric function (ESF) of the roots of a noncentral Wishart matrix with covariance matrix $\sigma^2I$. A conjecture is also made if $j = p$ for any covariance matrix $\Sigma$. The expected value of the noncentral Wishart matrix is derived for any covariance matrix $\Sigma$ and therefore also the expected value of the first ESF of the roots.

In The Second Berkeley Symposium Robbins raised the question whether or not his "bootstrap" procedure might be dominated by a Bayes procedure against the prior which puts equal mass on each "orbit" and is uniform within each orbit. We show that this is not true for the compound problem with two components.

Let $x_1, x_2,\cdots$ be independent and positive random variables with the common distribution function $F$. We show that if $\int^1_0|F(x) - x/b| \times x^{-2}dx < \infty$ for some $0 < b < \infty$, then $\sum^n_{k=1} \min(x_1,\cdots, x_k)$ is asymptotically normal with expectation $b \log n$ and variance $b^2 2 \log n$.

A law of the iterated logarithm is proved using an extension of the Borel-Cantelli lemma and the asymptotic expansion of the left tail of this stable law.

It is shown that if $F_n$ is a stable distribution function for each value of $n$ and if $F_n$ converges completely to $F$, then $F$ is a stable distribution function.

This note contains a simple proof of the following theorem of G. I. Kalmykov. Let $\{X_n\}$ and $\{Y_n\}$ be real-valued, discrete time Markov processes. Suppose $P(X_0 \leqq z) \leqq P(Y_0 \leqq z)$ for all real $z$ and $P(X_n \leqq z|X_{n-1} = x) \leqq P(Y_n \leqq z| Y_{n-1} = y)$ for $n = 1,2,\cdots$ and all $z$, whenever $y \leqq x$. Then $P(X_n \leqq z) \leqq P(Y_n \leqq z)$ for all $n$ and $z$. Some converse results are also given.

In this paper the following result is proved: THEOREM. If $\lambda$ is a measure on a product $\sigma$-algebra $\mathfrak{U} \times \mathscr{B}$ and $\lambda_1$ and $\lambda_2$ are the corresponding marginals on $\mathfrak{U}$ and $\mathscr{B}$ respectively, then $\lambda$ is nonatomic $\operatorname{iff}$ at least one of $\lambda_1$ and $\lambda_2$ is nonatomic.