This paper considers two procedures for estimating the center of a symmetric distribution, which use the observations themselves to choose the form of the estimator. Both procedures begin with a family of possible estimators. We use the observations to estimate the asymptotic variance of each member of the family of estimators. We then choose the estimator in the family with smallest estimated asymptotic variance and use the value given by that estimator as the location estimate. These procedures are shown to be asymptotically as good as knowing beforehand which estimator in the family is best for the given distribution, and using that estimator.

Random variables $X, Y_1, Y_2, \cdots$ are available for observation with $X$ real valued and $Y_1, Y_2, \cdots$ taking values in arbitrary spaces. The distribution of $Y = (Y_1, Y_2, \cdots)$ is given by $\mu_j (j = 1, \cdots, r)$ and the conditional density with respect to Lebesgue measure given $Y_i = y_i(i = 1, \cdots, n - 1)$ is $p_{jn}(x - \theta, y)$ where $y = (y_1, y_2, \cdots)$. The parameters $j$ and $\theta$ are unknown. A decision $k \in \{1, \cdots, m\}$ is to be made with loss $W(j, k, n, y)$ when $n$ observations are taken. Following Brown's (1966) methods admissibility is proved for the decision procedure which is Bayes in the class of invariant procedures. The result contains that of Lehmann and Stein (1953).

Many well-known procedures in multivariate data analysis are invariant under the group, $L(p)$, of translations and nonsingular linear transformations. New maximal $L(p)$ invariant statistics are derived and are shown to have the geometrical interpretation of a scatter of points in Euclidean space. The distribution of maximal $L(p)$ invariants for the case of a single multivariate normal population is shown to follow from a result of James (1954). Finally we consider tests of the null hypothesis that $k > 1$ populations are identical and show that optimal $L(p)$ invariant tests are similar tests of randomness.

A new criterion for determining the estimability of linear combinations of the parameters of a linear model is established. The result consists of evaluating the trace of a matrix and thus only one number must be checked to determine estimability. The sums of squares necessary to test hypotheses about estimable linear combinations are also derived. Finally, a stepwise computational procedure to compute generalized inverses and matrix products involving generalized inverses is presented. Using the theory and computational techniques, a computer program can be developed to provide a complete analysis of the linear model using generalized inverses.

A comparison, by means of Pitman asymptotic efficiency, is made between the Kolmogorov-Smirnov test and the locally most powerful rank and the locally asymptotically most powerful (Neyman) test for testing two-sided shifts in the two-sample problem under the assumption that the true distribution is different from the one assumed. It is shown that the behavior of the bounds for the Pitman asymptotic efficiencies are the same as those for testing the one-sided shift using the Smirnov test in place of the Kolmogorov-Smirnov test.

This paper investigates the effectiveness of the first and second order jackknife estimators, as tools for bias reduction. The biases of the estimators are characterized in terms of the bias of the original estimator. The biases of the two estimators are compared asymptotically with each other and with the bias of the original estimator.

This paper provides sufficient conditions for the weak convergence in the Skorohod space $D^d\lbrack a, b\rbrack$ of the processes $\{(Y_{1,\lbrack nt\rbrack} - b_n)/a_n, (Y_{2,\lbrack nt\rbrack} - b_n)/a_n, \cdots, (Y_{d,\lbrack_{nt\rbrack}} - b_n)/a_n\}, 0 < a \leqq t \leqq b$, where $Y_{i,n}$ is the $i$th largest among $\{X_1, X_2, \cdots, X_n\}, a_n$ and $b_n$ are normalizing constants, and $\langle X_n: n \geqq 1\rangle$ is a stationary strong-mixing sequence of random variables. Under the conditions given, the weak limits of these processes coincide with those obtained when $\langle X_n: n \geqq 1\rangle$ is a sequence of independent identically distributed random variables.

Let $(X, Y)$ be an ordered pair of real-valued random variables. Say that $(X, Y)$ is fair if $E(Y \mid X) = X$ a.s. It is shown, for example, that if $X$ has a finite mean and the pair $(X, Y)$ is fair, then $X$ and $Y$ cannot be stochastically ordered unless $X = Y$ a.s. The conclusion is in general false, if $X$ does not have a mean. On the other hand, if $X$ is independent of the increment $Y - X$, the preceding statement remains in force without any moment restrictions on $X$. The last assertion, combined with a gambling idea of Dubins and Savage, yields a simple proof of a theorem of Chung and Fuchs on the upper limit of a random walk with mean zero.

Chentsov-Billingsley type fluctuation inequalities for stochastic processes whose time parameter ranges over the $q$-dimensional unit cube are derived and used to establish weak convergence results for such processes.

Consider a situation in which balls are falling into $N$ cells with arbitrary probabilities. A limiting distribution for the number of occupied cells after $n$ falls is obtained, when $n$ and $N \rightarrow \infty$, so that $n^2/N \rightarrow \infty$ and $n/N \rightarrow 0$. This result completes some theorems given by Chistyakov (1964), (1967). Limiting distributions of the number of falls to achieve $a_N + 1$ occupied cells are obtained when $\lim \sup a_N/N < 1$. These theorems generalize theorems given by Baum and Billingsley (1965), and David and Barton (1962), when the balls fall into cells with the same probability for every cell.

For a probability distribution function on the real line, a necessary condition on its infinite divisibility is given which deals with its one-sided asymptotic behavior; the proof is based on properties of characteristic functions which are analytic in the upper (lower) half plane.

The principal result of this paper is the introduction of a readily verifiable sufficient condition for the existence of an invariant measure on transient Markov chains. Specifically, subject to mild additional regularity conditions, it is enough to check that for every compact set $A$ in the state space $X$, there exists a compact $C \supset A$ such that \begin{equation*}\tag{0.1} \lim_{y\in F, y\rightarrow\infty} \frac{P(y, A)}{P(y, C)} = 0\quad\text{if} \bar{F} \text{is not compact and} F = \{y: P(y, A) > 0\},\end{equation*} $P(y, A)$ denote the 1-step transition probabilities of the Markov chain. (0.1) is readily implied by simple conditions on the generating functional of the offspring distribution (Theorems 2.1 and 2.2) for discrete time multi-type Markov processes. These conditions together with some additional regularity-hypotheses are verified for (i) A 1-dimensional neutron branching model of Harris, (ii) Discrete-time age dependent branching processes, (iii) Galton-Watson processes with immigration. A generalization (to discrete-time temporally homogeneous Markov processes with $\sigma$-compact metric state space) of a condition of T. Harris (1957) for existence of invariant measures on transient Markov chains is also given. This condition is, unfortunately, difficult to check in specific examples. The sufficient condition (0.1) involves only 1-step transition probabilities, as opposed to the $n$-step transitions incorporated in the Harris condition, which is also necessary. Throughout this paper, all subsets of a topological space considered are assumed Borel measurable. The closure, interior, boundary and complement of a set $A$ are denoted by $\bar{A}, A^0, \partial A$ and $A^c$ respectively, and $I_A$ denotes the indicator function of $A$. The sets of positive integers, nonnegative integers, are denoted by $I$ and $I_0$ respectively. For any $\sigma$-compact metric space $X$, the following notations are used: $M(X) =$ set of regular measures (Borel measures finite on compact sets) on $X$. $B(X) =$ set of bounded measurable functions on $X$ with sup norm. $B'(X) = \{s \in B(X): \|s\| < 1\}$. $C(X) =$ set of bounded continuous functions on $X$. $C_{00}(X) = \{s\in C(X): s$ has compact support$\}$. $C_0(X) = \{s \in C(X):\lim_{x\rightarrow\infty}s(x) = 0\}.$ $C'(X) = C(X) \cap B'(X)$, similarly for $C_0'(X)$ and $C'_{00}(X)$. For any $\mathscr{L} \subset B(X), \mathscr{L}^+ = \{s \in \mathscr{L}: s \geqq 0\}$. If $\mu \in M(X)$ and $\in B(X)$, then $\mu f = \int_xf(x)\mu(dx)$ when defined.

A proof is given for a gambling theorem which was stated by Dubins and Savage. Connections are made with optimal stopping theory and the usual abstract stopping problem is generalized to a situation where stopping is allowed only at certain times along a given path.

We consider the problem of maximizing the long-run average (also the long-run average expected) reward per unit time in a semi-Markov decision processes with arbitrary state and action space. Our main result states that we need only consider the set of stationary policies in that for each $\varepsilon > 0$ there is a stationary policy which is $\varepsilon$-optimal. This result is derived under the assumptions that (roughly) (i) expected rewards and expected transition times are uniformly bounded over all states and actions, and that (ii) there is a state such that the expected length of time until the system returns to this state is uniformly bounded over all policies. The existence of an optimal stationary policy is established under the additional assumption of countable state and finite action space. Applications to queueing reward systems are given.

Following the work of Poisson (1837), there has been considerable practical and theoretical interest in how well the Poisson distribution approximates the binomial distribution. The approximation, which was initially suggested by a limit theorem (see (1) below), has been shown in numerical examples to be very good for certain binomial parameters within a useful range. (See Feller (1950), page 143.) Subsequently, (nonasymptotic) theoretical results have confirmed the approximation's accuracy. (See (2.1) and (2.2) below.) The purpose of this note is to demonstrate, with an elementary argument, that the binomial distributions converge very strongly to the Poisson distributions.

Let $X_1, X_2,\cdots, X_m, Y_1, Y_2,\cdots, Y_n$ be independent random samples from absolutely continuous distributions $F$ and $G$ respectively. Several standard tests of the hypothesis $H:F = G$ against the one-sided shift alternative $A: G(v) = F(v - \theta); (\theta > 0)$, are defined in terms of $F$. If, however, the true distributions of $X$'s and $Y$'s are $\Psi(v)$ and $\Psi(v - \theta)$ respectively, with $\Psi$ not necessarily equal to $F$, these tests are no longer optimal. It will be shown that there exist continuous distributions $\Psi$ (with density $\psi$), which are quite similar to $F$ but for which the Smirnov test--in terms of generalized Pitman efficiency (defined below) is considerably superior.

The problem of finding a Chebyshev type inequality for random variables with unknown or non-existent variance was considered by Z. W. Birnbaum (1970). In this present paper, a statistic, $T$, similar to, but simpler than Birnbaum's, is considered. The statistic is independent of location and scale parameters for families of bell-shaped distributions and so may be considered to be a competitor to Student's $t$. An inequality establishing an upper bound for $P(|T| > \lambda)$ is proved. This bound is considerably smaller than the corresponding bound found by Birnbaum. Finally, an improvement of the latter is offered.

The purpose of this note is to give a counterexample to the following statement. Let $Y_1, Y_2, \cdots$ be i.i.d rv with distribution function $F$ and $P\lbrack Y_1 \geqq 0\rbrack = 1$. For any set $A \subset \lbrack 0, \infty)$ let $U(A) = \sum^\infty_{k=0} F^{\ast k}(A)$ be the usual renewal measure. If $A \subset \lbrack 0, \infty)$ and $U(A) = +\infty$ then there is a renewal in $A$ almost surely.