It is shown that an $L$ function is unimodal if its Levy spectral function has support on $(-\infty, 0\rbrack$ or on $\lbrack 0, \infty)$, and that this implies that every $L$ function is the convolution of at most two unimodal $L$ functions. Other results concerning the unimodality of $L$ functions and other infinitely divisible distribution functions are also obtained.

We find an explicit formula for the first passage probability, $Q_a(T | x) = P_r(S(t) < a, 0 \leqq t \leqq T | S(0) = x)$, for all $T > 0$, where $S$ is the Gaussian process with mean zero and covariance $ES(\tau)S(t) = \max (1 - |t - \tau|, 0)$. Previously, $Q_a(T\mid x)$ was known only for $T \leqq 1$. In particular for $T = n$ an integer and $-\infty < x < a < \infty$, $Q_a(T \mid x) = \frac{1}{\varphi(x)} \int_D \cdots \int \det \varphi(y_i - y_{j+1} + a) dy_2 \cdots dy_{n+1},$ where the integral is an $n$-fold integral on $y_2, \cdots, y_{n+1}$ over the region $D$ given by $D = \{a - x < y_2 < y_1 < \cdots < y_{n+1}\}$ and the determinant is of size $(n + 1) \times (n + 1), 0 < i, j \leqq n$, with $y_0 \equiv 0, y_1 \equiv a - x$.

In this paper it is shown that the optimal design for estimating any specific parameter in a polynomial spline regression with a single multiple knot at the center is supported by one of two sets of points. Familiarity with the notation and terminology used in the paper by Studden (1968) is assumed.

Since the asymptotic expression for the variance of estimators of the spectrum of a stationary time series was derived, it has often been used as an approximation to the variance of estimators using finite samples. Little attempt seems to have been made to investigate the nature of the convergence to the asymptotic form. In this paper an exact expression for the variance is derived on the additional assumption that the time series is a normal process, and is used to study estimators of various different spectra. A philosophy for choosing spectrum estimators is proposed which attempts to place the two forms of error, bias and variance, in their true perspective.

In the recent literature a system of inference which leads to upper and lower posterior distributions based on sample data has been proposed by Dempster (1966, 1967, 1968). The purpose of this paper is to apply this method to the problem of finding the upper and lower probabilities that the mean of a distribution falls within a given interval, the only prior information being that the distribution is continuous. Included in the class of all continuous distributions are those having the property that $0 < F(y) < 1$ for all real $y$. For distributions of this type the approach used here leads to trivial results. The reason for this is discussed in Section 2. To circumvent this difficulty the mean of the truncated distribution is used. That is, for given $\varepsilon_1$ and $\varepsilon_2$ such that $0 < \varepsilon_1 < 1 - \varepsilon_2 < 1$, \begin{equation*}\tag{1.1} \mu(\varepsilon_1, \varepsilon_2) = \frac{1}{1 - \varepsilon_1 - \varepsilon_2} \int^{\xi 2}_{\xi 1} td F(t)\end{equation*} where $F$ is a distribution function and $\xi_1 < \xi_2$ are real constants such that $F(\xi_1) = \varepsilon_1$ while $F(\xi_2) = 1 - \varepsilon_2$. The device of truncating the mean is to some extent artificial and is unnecessary when the continuous distribution function is constant except on an interval of finite length. However, it does offer an approach to the more general (and in the author's opinion more interesting) problem where no restriction is placed on the interval on which the distribution function is neither zero nor one. Section 2 contains a brief restatement of some of Dempster's basic definitions in the context of this particular problem along with an exposition of the type of reasoning required whenever this method is applied to problems involving truncated moments. Exact expressions for the upper and lower probabilities are derived in Section 3, and some asymptotic results are presented in Section 4. In particular, it is shown that if the upper and lower probabilities converge, they converge to the same limit. All relevant distribution theory is presented in an appendix.

Conditions are given under which a semi-Markov process $Z(t)$ can be obtained from a Markov process $Y(t)$ by a time change (i.e. $Z(t) = Y(\gamma(t))$). Estimates are given for $P\{\sup_{s\leqq t} |s - \gamma(s)| > \varepsilon\}$ and the construction is used to give conditions under which a sequence of semi-Markov processes will have the same convergence properties as the corresponding sequence of Markov processes.

In this paper the exact distribution of Wilks' likelihood ratio criterion for testing linear hypothesis about regression coefficients is discussed. The exact distribution, in the most general case, is given in simple algebraic functions which can be computed without much difficulty. Explicit expressions for the density function as well as for the cumulative distribution function are given under the null hypothesis.

The problem of finding location estimators which are "robust" against deviations from normality has received increasing attention in the last several years. See, for example, Tukey (1960), Huber (1968), and papers cited therein. In the theoretical work done on the estimation of a location parameter, the underlying distribution is usually assumed to be symmetric, and the estimand is taken to be the center of symmetry, a natural quantity to estimate in this situation. Since the finite sample size properties of many proposed estimators are difficult to study analytically, most research has focussed on their more easily ascertainable asymptotic properties, which, it is hoped, will provide useful approximations to the finite sample size case. Most of the estimators commonly studied are, under suitable regularity conditions, asymptotically normal about the center of symmetry, with asymptotic variance depending on the underlying distribution. We thus have a simple criterion, the asymptotic variance, for comparing the performance of different estimators for a given underlying distribution, and of a given estimator for different underlying distributions. Huber (1964) has formulated and solved some minimax problems, in which the estimators are judged by their asymptotic variance. In Section 2 we define and state the asymptotic variances which have been found for the three most commonly studied types of location estimators. In Section 3 we demonstrate some relationships among the three types of estimators, and in Section 4 we show that Huber's minimax result applies to all three types. Then, in Section 5 we consider an aspect of the more general estimation problem in which the distributions are not assumed symmetric. A model of asymmetric contamination of a symmetric distribution is formulated, in which the amount of asymmetry tends to zero as the sample size increases. The estimators here are thought of as estimating the center of the symmetric component of the distribution. The maximum likelihood type estimators are shown to be asymptotically normal under this model, but with a bias that tends to zero as the sample size increases. The estimators may be judged by their asymptotic mean squared error, a concept which is made meaningful by the model. We conclude in Section 6 with a minimax result analogous to Huber's, for which we allow both symmetric and asymmetric contamination of a given distribution and judge the estimators by their asymptotic mean squared error.

We consider the experimental design problem of Sacks and Ylvisaker. We consider only the case of the (noise) stochastic process $X$ satisfying a stochastic differential equation of the form \begin{equation*}\tag{0.1} L_mX(t) = dW(t)/dt\quad 0 \leqq t \leqq 1\end{equation*} where $L_m$ is an $m$th order differential operator whose null space is spanned by an ECT system and $W(t)$ is a Wiener process. We show that the non-degeneracy of the covariance matrix of $\{X^{(\nu)} (t_i), \nu = 0, 1, 2, \cdots, m - 1, t_i \in \lbrack 0, 1 \rbrack, i = 1,2, \cdots, n\}$ is equivalent to the total positivity properties of the Green's function for $L_m^\ast L_m$ with appropriate boundary conditions. An asymptotically optimal sequence of designs is found for this case and its dependence on the characteristic discontinuity of the above mentioned Green's function is exhibited. Finally we show that a special case of the problem is equivalent to the problem of the optimal approximation of a monomial by a Spline function in the $L_2$ norm. Some recent results are available on this latter problem which provide some information concerning existence and uniqueness of optimal designs with distinct points.

The Radon-Nikodym derivative (RND) with respect to Wiener measure of a measure determined by the sum of a differentiable (random) signal process and a Wiener process is shown, under rather general conditions, to have the same form as the RND for the case of a known (nonrandom) signal plus a Wiener process. The role of the known signal is played by the causal least-squares estimate of the signal process given the sum process. This formula can be shown to be equivalent to all previously known explicit formulas for RND's relative to Wiener measure. Moreover, and more important, the formula suggests a general structure for engineering approximation and implementation of signal detection schemes. Secondly, an explicit necessary and sufficient characterization, in signal plus noise form, is given of all processes absolutely continuous with respect to a Wiener process. Finally, the results are extended to some reference measures related to Wiener measure, in particular to measures induced by martingales of a Wiener process. We also note that the case where both measures are Gaussian permits some stronger results. The proofs are based on several recent results in martingale theory.

It is known that deterministic automata are generally not optimal in the problem of learning with finite memory. It is natural to ask how much memory is saved by randomization. In this note it is shown that the memory saving is arbitrarily large in the sense that for any memory size $m < \infty$, and $\delta > 0$, there exist problems such that all $m$-state deterministic algorithms have probability of error $P(e) \geqq \frac{1}{2} - \delta$, while the optimal two-state randomized algorithm has $P(e) \leqq \delta$.

Variance estimation in systematic sampling by splitting the sample into equal halves can lead to very serious bias. The expression for this bias relative to the true variance is given in terms of intraclass correlation coefficients. The danger of serious bias is still present when successive pairs of units are treated as "independent" replicates; an expression for this relative bias is also given.

Let $X_1, X_2, \cdots, X_n$ be a random sample of size $n$ from a continuous distribution with cdf $P(x)$ and pdf $p(x)$. Let $X_{1:n} \leqq X_{2:n} \leqq \cdots \leqq X_{n:n}$ be the corresponding order statistics. Denote the first moment $E(X_{r:n})$ by $\mu_{r:n} (1 \leqq r \leqq n)$ and the mixed moment $E(X_{r:n}, X_{s:n})$ by $\mu_{r,s:n} (1 \leqq r \leqq s \leqq n)$. We assume that all these moments exist. Several recurrence relations between these moments are summarized by Govindarajulu [1]. In this note, we give a simple argument which generalizes some of the results given in [1]. These generalizations then lead to some modifications in the theorems given by Govindarajulu.

Let $S_0 = 0, S_n = \sum^n_1 X_i, n = 1,2, \cdots$ be the sequence of partial sums of independent, identically distributed real valued random variables $X_1, X_2, \cdots$ The sum $S_k$ is a (strict) ladder sum if $S_j < S_k$ for $0 \leqq j < k$. Contrary to usual practice, we always count $S_0$ as a ladder sum. A run of ladder sums of length $i$ starts at $S_k$ if $S_k, S_{k+1}, \cdots, S_{k+i-1}$ are ladder sums but $S_{k-1}$ and $S_{k+i}$ are not. Thus $S_0$ is always the beginning of a run, of length one if $S_1 \leqq 0$. Except at the beginning of Section 3, we assume that $X_1$ has continuous law, symmetric with respect to 0. Thus ties $S_i = S_j$ can be disregarded. Let \begin{align*} L_n &= "\text{index of} \max \{S_i, 0 \leqq i \leqq n\}," \\ \tag{1} G_n &= "\text{number of ladder sums among} 0, S_1, \cdots, S_n," \\ R_n &= "\text{number of runs they form}." \\ \end{align*} Two sets of probabilities concerning those variables are obtained, namely the joint law \begin{equation*}\tag{2} p_n(k + 1, m + 1) = P(G_n = k + 1, R_n = m + 1) = 2^{-2n+k}\binom{2n - 2k}{n - k - m}\binom{k}{m}\end{equation*} where $0 \leqq m \leqq k \leqq n$ and $k + m \leqq n$, and the probabilities \begin{equation*}\tag{3} p_n^\ast(k + 1, m + 1) = P(G_n = k + 1, R_n = m + 1, L_n = n) = \frac{m}{n - k}p_n(k + 1, m + 1),\end{equation*} valid for $1 \leqq m \leqq k \leqq n$ and $k + m \leqq n$. If $m = 0$ and $0 \leqq k < n$, then clearly $p_n^\ast(k + 1, 1) = 0$, while on the other hand $p_n^\ast(n + 1, 1) = P(0 < S_1 < \cdots < S_n) = 2^{-n}$. Considering similarly the behavior beyond the maximum, let $X_i' = -X_{n-i+1}, S'_i = \sum^i_{k=1} X'_k = S_{n-1} - S_n, i = 1, \cdots, n$, and \begin{align*}\tag{4} G'_n &= "\text{number of ladder sums among} 0, S'_1, \cdots, S'_n," \\ R'_n &= "\text{number of runs they form}."\end{align*} so that $G'_n$ counts descending ladders from $S_{L_n}$ on. Various limit laws are obtained, in particular it is shown that $(2L_n)^{-\frac{1}{2}} G_n, (2n - 2L_n)^{-\frac{1}{2}}G'_n, n^{-1} L_n, G^{-\frac{1}{2}}_n (2R_n - G_n)$ and $G_n^{'- \frac{1}{2}}(2R'_n - G'_n)$ are asymptotically independent, the first two having a limiting $\chi_2$ law and the latter two a standard normal one.

Consider $n$ replications of an incomplete block design $D$ consisting of $b$ blocks of constant size $k(\geqq 2)$ to which $v(\geqq k)$ treatments are applied in such a way that (i) no treatment occurs more than once in any block, (ii) the $j$th treatment occurs in $r_j(\leqq b)$ blocks, and (iii) the $(j, j')$th treatments occur together in $r_{jj}'(> 0)$ blocks, for $j \neq j' = 1, \cdots, v$. Let then $S_i$ stand for the set of treatments occurring in the $i$th block, $i = 1, \cdots, b$. For the $\alpha$th replicate, the response of the plot in the $i$th block and receiving the $j$th treatment is a stochastic $p$-vector $\mathbf{X}_{\alpha ij}$ and is expressed as \begin{equation*}\tag{1.1} \mathbf{X}_{\alpha ij} = \mathbf{\mu}_\alpha + \mathbf{\beta}_{\alpha i} + \mathbf{\tau}_j + \mathbf{\varepsilon}_{\alpha ij},\quad j \in S_i, i = 1, \cdots, b, \alpha = 1, \cdots, n; \quad \sum \mathbf{\tau}_j = 0,\end{equation*} where the $\mathbf{\mu}_\alpha$ and the $\mathbf{\beta}_{\alpha i}$ are respectively the replicate and block effects (nuisance parameters in "fixed effect" model or spurious random variables in "mixed effect" model), $\mathbf{\tau}_1, \cdots, \mathbf{\tau}_v$ are the treatment effects (parameters of interest) and the $\mathbf{\varepsilon}_{\alpha ij}$ are the error vectors. It is assumed that $\{\mathbf{\varepsilon}_{\alpha ij}, j\in S_i\}$ have jointly a continuous cumulative distribution function ($\operatorname{cdf}$) $G(\mathbf{x}_1,\cdots, \mathbf{x}_k)$ which is symmetric in its $k$ vectors. This includes the conventional assumption of independence and identity of the $\operatorname{cdf's}$ of all the $N( = nbk)$ error vectors as a special case. We want to test the null hypothesis \begin{equation*}\tag{1.2} H_0:\mathbf{\tau}_1 = \cdots = \mathbf{\tau}_\nu = \mathbf{0} \text{vs} H_1:\mathbf{\tau}_j \neq \mathbf{0},\quad \text{for at least one} j(=1,\cdots, \nu)\end{equation*} In the univariate case (i.e., $p = 1$), intra-block rank tests for this problem are due to Durbin (1951), Benard and Elteren (1953), and Bhapkar (1961), among others. For some special balanced designs, the studies made by Elteren and Noether (1959) and Bhapkar (1963) reveal the low (Pitman-) efficiency of these tests, particularly when $k$ is small. In complete block designs, it is known [cf. Hodges and Lehmann (1962) and Sen (1968)] that the use of ranking after alignment increases the efficiency of the rank tests. The purpose of the present paper is to show that this merit of the ranking after alignment is preserved for a broad class of incomplete block designs. In fact, certain bounds for the efficiency are derived which do not depend on the design $D$, i.e., on the particular values of $b, \nu, r_j, r_{jj'}$ and $k$.

One of the strongest features of conventional sample survey theory is that very little needs to be assumed about the form of the population distribution. Hartley and Rao (1968) and Ericson (1969) have recently developed a Bayesian approach to sampling from finite populations that shares this feature. In this note, the resulting posterior distribution of the population elements is shown to approach normality for a broad class of prior distributions when the population size $N$ and sample size $n$ increase so that $n \rightarrow \infty$ and $N - n \rightarrow \infty$. In most cases this leads to the same large-sample interval estimates for population moments as the usual approach invoking the Central Limit Theorem for random sampling from a finite population (Madow (1948), Erdos and Renyi (1958), Hajek (1960)).

Block designs with parameters $(p^n, \Delta p^n, \Delta((p^n - 1)/2), (p^n - 1)/2, \Delta((p^n - 3)/4))$ are shown to exist whenever $p^n$, a prime power, can be expressed as $2^mt + 1$ with $m$ a positive integer and $t$ an odd integer $>1$, that is, whenever $p^n$ is not one greater than a power of 2; $\Delta$ is equal to $2^{m-1}$.

In a generalized age-dependent branching process, immigrants are allowed to arrive according to a non-homogeneous Poisson process. A limit theorem is given for the critical case.

If $\{Z_n\}$ is a Galton-Watson process with mean one, and $\tau$ is the extinction time, it is shown that $EZ^{1+\alpha} < \infty$ implies $E\tau^\beta = \infty$ for $\beta > 1/\alpha, 0 < \alpha < 1$. Conditions which imply $EZ^{1+\alpha} = \infty$ and $E\tau^\beta < \infty$ for $\beta < 1/\alpha, 0 < \alpha < 1$ are given. Necessary and sufficient conditions for $EX^{m+\alpha} < \infty$ or $EX^m \log X < \infty$ are given in terms of the Laplace transform of a general nonnegative random variable $X, 0 < \alpha < 1, m = 0, 1,\cdots$.

Let $x_n(t)$ be the position of a particle in one dimension that switches between uniform velocities $+n$ and $-n$ at the jump times of a Poisson process with intensity $n^2$. In this note are constructed realizations of the processes $x_n(t)$ that converge almost surely to Brownian motion, uniformly on the unit time interval.

For a sequence of stochastic vectors forming either a forward or a reverse martingale, a Hajek-Renyi type inequality is derived, and its applications in some problems of simultaneous confidence regions are stressed.