Let $X$ denote a differential process, i.e., a stochastic process with independent increments for which the distribution of $X(t + h) - X(t)$ depends only on $h$. The parameter $t$ runs through the interval $\lbrack 0, + \infty)$, and the usual initial condition $P\lbrack X(0) = 0\rbrack = 1$ is assumed. Then it is known that the distribution of $X(t)$ is infinitely divisible, i.e., the logarithm of its characteristic function can be written as \begin{equation*}\tag{1.1} \log f_{x(t)} (u) = i\gamma tu + t \int^{+\infty}_{-\infty} \bigg(e^{iux} - 1 - \frac{iux}{1 + x^2}\bigg) \frac{1 + x^2}{x^2} dG(x).\end{equation*} In this canonical representation, $\gamma$ is a real constant, and $G$ is a bounded, nondecreasing function, it being permissible always to consider $G(- \infty) = 0$. In the usual probabilistic terminology, the probability law of $X(t)$ is a convolution of a normal law and a (possibly infinite) number of Poisson laws, or a limit of such laws. The function $G$ is called the jump function; its saltus at $x = 0, \sigma^2 = G(+0) - G(-0)$, is the variance of the normal component, and its set of points of increase for $x \neq 0$ gives information as to the nature of the Poisson components, viz., the "relative density" of the magnitudes of the discontinuities of the sample function. The purpose of this paper is to derive estimates for this jump function $G$ and for the "trend term", $\gamma$. Two estimates of $G$ are obtained, and one estimate is obtained for $\gamma$. In the first method of estimating $G$, considered in $\S 2$, it is assumed that the experimenter can observe a sample function of $X$ at any finite number of values of $t$ that he chooses. Accordingly, for any integer $n$, let $$X_{nk} = X \big(\frac{k}{n}\big) - X \big(\frac{k - 1}{n}\big).$$ Then the estimate $G^{\ast}_{N,n}(u)$ of $G(u)$ is defined as \begin{equation*}\tag{1.2}G^{\ast}_{N,n}(u) = \frac{1}{N} \sum^{nN}_{k = 1} \frac{X^{2_{nk}}}{1 + X^{2_{n^k}}} I_{\lbrack X_{nk \leqq u\rbrack}},\end{equation*} where \begin{equation*}I_{\lbrack X_{nk \leqq u}\rbrack} = \begin{cases}\begin{align*} & 1 & \text{if} & X_{nk} \leqq u \\ & 0 & \text{if} & X_{nk} > u,\end{align*}\end{cases}\end{equation*} and $N = \lbrack Tn\rbrack, T$ being the largest value of $t$ observed. It is proved that this estimate is strongly consistent in the following sense: \begin{equation*}\tag{1.3}P\{\lim_{n \rightarrow \infty} \lim_{N \rightarrow \infty} G^{\ast}_{N,n}(u) = G(u) \text{for all} u \varepsilon C(G)\} = 1,\end{equation*} where $C(G)$ denotes the set of all values of $u$ at which $G$ is continuous. This estimate is not necessarily an unbiased estimate of $G(u)$ for all $u \varepsilon C(G)$. The second method for estimating $G$, developed in $\S 3$, requires that the experimenter be able to observe all the discontinuities of a sample function of $X$ on a finite interval in addition to being able to observe $X(t)$ at any finite set of values of $t$. Not only is a consistent estimate obtained for $G$, but also a consistent estimate is obtained for the variance of the normal component, $\sigma^2 = G(+0) - G(-0)$. Let $\{k_n\}$ denote a sequence of positive integers such that \begin{equation*}\tag{1.4}\Sigma_nk^{-\frac{1}{3}}_n < \infty.\end{equation*} Further, let $T_0$ be a fixed value of time $T$, let \begin{equation*}\tag{1.5}S^2_n = \sum^{k_n}_{k = 1} \{X(kT_0/k_n) - X((k - 1)T_0/k_n)\}^2,\end{equation*} and \begin{equation*}\tag{1.6}D^2 = \text{the sum of the squares of the jumps of the sample function during the time interval} \lbrack 0, T_0\rbrack.\end{equation*} Finally, let \begin{equation*}\tag{1.7}J_T(x) = \int^x_{-\infty} \frac{b^2}{1 + b^2} dN_T(b),\end{equation*} where, for every Borel set $B, N_T(B)$ denotes the number of discontinuities observed for $X$ during $\lbrack 0, T\rbrack$ whose magnitudes lie in $B$. The estimate $\hat G_{n, T}(u)$ of $G(u)$ is then constructed as follows: \begin{equation*}\tag{1.8}\hat G_{n, T}(u) = \begin{cases}T^{-1} J_T(u) \text{if} u < 0 \\ T^{-1} J_T(u) + T_0^{-1} (S^2_n - D^2) \text{if} u > 0.\end{cases}\end{equation*} The estimate $\hat G_{n, T}(u)$ is an unbiased estimated of $G(u)$ if $\sigma^2 = 0$ or if $u < 0$, but in any case, $\hat G_{n, T}(b) - \hat G_{n, T}(a)$ is an unbiased estimate of $G(b) - G(a)$, provided $0 \not\in \lbrack a, b\rbrack$. Also, this estimate is consistent in the following sense: \begin{equation*}\tag{1.9}\operatornamewithlimits{\lim}_{\substack{T \rightarrow \infty\\n \rightarrow \infty}} \hat G_{n, T}(u) = G(u)\end{equation*} with probability one for every $u$. In addition, $(1/T_0) \lbrack S^2_n - D^2\rbrack$ is a consistent estimate of $\sigma^2 = G(+0) - G(-0)$, the variance of the normal component, in the sense that $\lim_{n \rightarrow \infty} (1/T_0) \lbrack S^2_n - D^2\rbrack = \sigma^2$ with probability 1. In $\S 4$ a comparison is made of the two estimates for $G(x)$ obtained in $\S\S 2$ and 3. It is found that both estimates do agree in a special limiting case. In particular it is proved that \begin{equation*}\tag{1.10}p \lim_{n \rightarrow \infty} G^{\ast}_{N, r}(u) = \lim_{n \rightarrow \infty} \hat G_{n, N}(u)\quad \text{for all} u \varepsilon C(\hat G_{n, N}).\end{equation*} Finally in $\S 5$ a consistent and unbiased estimate is derived from the "trend term", $\gamma$. This estimate is \begin{equation*}\tag{1.11}\hat \gamma(t) = \frac{1}{t} \big\{X(t) - \int^{+\infty}_{-\infty} \frac{b^3}{1 + b^2} dN_t(b)\big\}.\end{equation*} It is consistent in the sense that \begin{equation*}\tag{1.12}P\{\lim_{t\rightarrow\infty} \hat \gamma(t) = \gamma\} = 1.\end{equation*} Another way of writing this estimate is as follows. Let $J_1, J_2, \cdots, J_n, \cdots$ denote the discontinuities of the sample curve up to time $t$ (not necessarily in order); then $$\gamma(t) = \frac{1}{t}\big\{X(t) - \sum_n \frac{J^3_n}{1 + J^2_n}\big\}$$

Let $x_t(t = 1, 2, \cdots)$ be defined recursively by \begin{equation*}\tag{1.1}x_t = ax_{t-1} + u_t,\quad t = 1, 2, \cdots,\end{equation*} where $x_0$ is a constant, $\varepsilon u_t = 0, \varepsilon u^2_t = \sigma^2$ and $\varepsilon u_tu_s = 0, t \neq s$. ($\varepsilon$ denotes mathematical expectation.) An estimate of $\alpha$ based on $x_1, \cdots, x_T$ (which is the maximum likelihood estimate of $\alpha$ if the $u$'s are normally distributed) is \begin{equation*}\tag{1.2}\hat \alpha = \bigg(\sum^T_{t=1} x_tx_{t-1}\bigg)\bigg/\big(\sum^T_{t=1} x^2_{t-1}\bigg).\end{equation*} If $|\alpha| < 1, \sqrt T (\hat \alpha - \alpha)$ has a limiting normal distribution with mean 0 under fairly general conditions such as independence of the $u$'s and uniformly bounded moments of the $u$'s of order $4 + \epsilon$, for some $\epsilon > 0$. (See [2], Chapter II, for example.) If $|\alpha| > 1$, White [3] has shown $(\hat \alpha - \alpha)|\alpha|^T/(\alpha^2 - 1)$ has a limiting Cauchy distribution under the assumption that $x_0 = 0$ and the $u$'s are normally distributed; he has also found the distribution when $x_0 \neq 0$. His results can be easily modified and restated in the following form $(\Sigma^T_{t=1} x^2_{t-1})^{\frac{1}{2}}(\hat \alpha - \alpha)$ has a limiting normal distribution if the $u$'s are normally distributed and if $|\alpha| \neq 1$. Peculiarly, for $|\alpha| = 1$ this statistic has a limiting distribution which is not normal (and is not even symmetric for $x_0 = 0)$. One purpose of this paper is to characterize the limiting distributions for $|\alpha| > 1$ when the $u$'s are not necessarily normally distributed; it will be shown that for $|\alpha| > 1$ the results depend on the distribution of the $u$'s. Central limit theorems are not applicable. Secondly, the limiting distribution for $|\alpha| < 1$ will be shown to hold under the assumption that the $u$'s are independently, identically distributed with finite variance. This was conjectured by White.

A stationary sequence $\{Y_n:n = 1, 2, \cdots\}$ of random variables with $D$ values (states) is said to be a function of a finite Markov chain if there is an integer $N \geqq D$, an $N \times N$ irreducible aperiodic Markov matrix $M$, a stationary Markov chain $\{X_n\}$ with transition matrix $M$, and a function $f$ such that $Y_n = f(X_n)$. For any finite sequence $s$ of states of $\{Y_n\}$, let $p(s) = P\{(Y_1, \cdots, Y_n) = s\}$. For any state $\epsilon$, let $s\epsilon t$ be the sequence $s$ followed by $\epsilon$ followed by the sequence $t$. For every state $\epsilon$, let $n(\epsilon)$ be the largest integer $n$ such that there are finite sequences $s_1, \cdots, s_n, t_1, \cdots, t_n$ such that the matrix $\| p(s_i\epsilon t_j):1 \leqq i, j \leqq n \|$ is nonsingular. If $\{Y_n\}$ is a function of a finite Markov chain, then $\Sigma n(\epsilon) \leqq N$. There is a finite set $\{s_1, \cdots, s_N, t_1, \cdots, t_N\}$ of finite sequences such that $p(s)$ satisfies the recurrence relations \begin{equation*}\tag{1}p(s\epsilon t) = \sum_{f(i)=\epsilon} a_i(s)p(s_i\epsilon t),\end{equation*} where $a_i(s)$ either is zero for all $s$ or else is a ratio of determinants involving only $p(s\epsilon t_k)$ and $p(s_j\epsilon t_k)$ for $f(j) = f(k) = f(i)$. If $\{Y_n\}$ has $D$ states and is a function of a Markov chain having $N$ states, then the entire distribution of $\{Y_n\}$ is determined by the distribution of sequences of length $\leqq 2(N - D + 1)$. For each $N$ and $D$, a function of a Markov chain is exhibited which attains this bound. If there is a Markov chain $\{X_n\}$ with $N = \Sigma n(\epsilon)$ states such that $\{Y_n\}$ is a function of $\{X_n\}$, then $\{Y_n\}$ is said to be a regular function of a Markov chain. If $\{Y_n\}$ is a regular function of a Markov chain having transition matrix $M$, then $M = X^{-1} AX$, where $A$ is an $N \times N$ matrix with elements $a_{ij} = a_j(s_if(i))$--defined by (1) above. $X = \|x_{ij}\|$ is a nonsingular $N \times N$ matrix such that $x_{ij} = 0$ unless $f(i) = f(j)$, the first row of each nonzero submatrix along the diagonal consists of positive numbers, and $\Sigma_jx_{ij} = p(s_if(i))$. Any $N \times N$ Markov matrix giving the same distribution for $\{Y_n\}$ can be written in this form, with the same $A$ and with an $X$ having the above properties. Any matrix of this form which has all elements nonnegative is a Markov matrix giving the same distribution for $\{Y_n\}$. There are $\Sigma\{n(\epsilon)\}^2 - N$ "unidentifiable" parameters in the matrix $X$, and at most $N^2 - \Sigma\{n(\epsilon)\}^2$ "identifiable" parameters, determined by the distribution of $\{Y_n\}$, in the matrix $A$.

Bunches of individual customers approach a single servicing facility according to a stationary compound Poisson process. The resulting waiting line process is studied in continuous time by the method of the imbedded Markov chain, cf. Kendall [7], [8], and of renewal theory, cf. Blackwell [3], Feller [5], and Smith [9]. Busy period phenomena are discussed, cf. Theorem 1, in which the transform of the joint d.f. of busy period duration and the number of departures in that duration is expressed as the root $x_1(s, z)$ of a functional equation, a generalization of a result of Takacs [12]. In terms of $x_1(s, z)$ "zero-avoiding" transition probabilities are characterized. A simple model for "instantaneous defection" is analyzed. Using renewal theory, ergodic properties of waiting line lengths and waiting times are discussed for the "general" process, in which idle and busy periods recur.

For well over two centuries mathematicians have considered the conditions under which it is possible to obtain a good approximation of one probability distribution by another of a given type. However, the conditions which assure the existence of a best approximation of a given type seem to have been virtually neglected. Because of their intrinsic interest and because of their relevance for an estimation problem which is discussed later, such conditions are examined here with respect to the following example: Suppose that $F$ and $G$ are distribution functions and that an ordered real number pair $(a, b)$, with $a > 0$, is desired such that $F(ax + b)$ is close to $G(x)$ for all real $x$. Is there a best pair? For instance, is there a pair $(a_0, b_0)$ satisfying \begin{equation*}\tag{1}\sup_{-\infty < x < \infty} |F(a_0 x + b_0) - G(x)| = \inf_{\substack{0 < a < \infty\\-\infty < b < \infty}} \sup_{-\infty < x < \infty} |F(ax + b) - G(x)|?\end{equation*} In this note we give an example in which a pair $(a_0, b_0)$ satisfying (1) does not exist. We then prove two theorems each giving a simple sufficient condition for the existence of such a pair. One or the other condition is almost always satisfied in practice. For example, the first requires, merely, that both of the sets $\{x \mid \frac{1}{3} \leqq F(x) \leqq \frac{2}{3}\}$ and $\{x \mid \frac{1}{3} \leqq G(x) \leqq \frac{2}{3}\}$ be nondegenerate. Next, we show that in any case if the set of minimizing pairs is nonempty then it is convex. This fact is used to obtain a fairly precise description of the set of minimizing pairs for the case $F$ is increasing and continuous. In this case, simple conditions on $G$ imply the uniqueness of a minimizing pair. Applications, especially to an estimation problem involving an unknown scale and location parameter, are then discussed. Throughout the paper, the right hand side of (1) is denoted by $M$. Also, $F$ and $G$ are understood to be continuous on the right.

Response surface techniques are discussed as a generalization of factorial designs, emphasizing the concept of rotatability. It is shown that the necessary and sufficient conditions for a configuration of sample points to be a rotatable arrangement of a specified order are greatly simplified if, in the case of two factors, the factor space is considered as the complex plane. A theorem giving these conditions is proved, with an application to the conditions governing the combination of circular rotatable arrangements into configurations possessing a higher order of rotatability. This is done by showing that certain coefficients must vanish in the "design equation" whose roots are the (complex) values of the various sample points. A method is presented by which any configuration of sample points (for example, some configuration fixed by extra-statistical conditions) may be completed into a rotatable design of the first order by the addition of only two properly chosen further sample points.

The $L_2$ association scheme for a class of partially balanced incomplete block designs determines the parameters of the second kind. This paper considers the converse problem: do these parameters imply the $L_2$ association scheme? Necessary conditions for the existence of such designs are also obtained.

For tests, $\mathbf{\Phi} = \{\phi_k\}$, of composite hypotheses, $\omega_1$ and $\omega_2$, asymptotic efficiency is defined in terms of the behavior as $\alpha \rightarrow 0$ of the sample size $N_{\phi}$ required to reduce the maximum risk to $\alpha$. For problems where the $\omega_i$ contain elements $\theta_i$ whose relative densities satisfy $$\sup_{\omega_1} \inf_{t>0} E_{\theta}(f_2/f_1)^t = \inf_t E_i(f_2/f_1)^t = \sup_{\omega_2} \inf_{t<0} E_{\theta}(f_2/f_1)^t,$$ Chernoff's Theorem 1 [2] is applied to the non-randomized test $\mathbf{\Phi}^{\ast}$, with $\phi^{\ast}_k = 1$ or 0 according as $\Sigma \log (f_2/f_1) > 0$ or not, and proves $\mathbf{\Phi}^{\ast}$ asymptotically efficient (Theorem 2.1). The principal results of the paper are applications of Theorem 2.1 to tests of the difference $(\xi - \eta)$ of binomial probabilities with samples of relative size $m/n$. For $\omega_1 = \{\xi - \eta \leqq - \delta\}, \omega_2 = \{\xi - \eta \geqq \delta\}$, certain tests of the form $\phi^{\ast}_k = 1$ if and only if $\lambda(\hat \xi - \frac{1}{2}) > (\hat \eta - \frac{1}{2})$, with $\lambda$ increasing in $m/n$, turn out to be asymptotically efficient, while all tests of the form $\psi_k = 1, a_k, 0$ according as $(\hat \xi - \hat \eta)$ is greater than, equal to, or less than $c_k$ are asymptotically inefficient when $m \neq n$. For given relative sampling costs, the ratio $m/n$ may be chosen so that the asymptotic cost of observations is minimized.

In this paper, the known methods of linear estimation are extended to various cases of censored samples from multivariate normal populations. The two estimators considered correspond to the minimum variance and the `alternative' estimators treated by Gupta [3] and Sarhan and Greenberg [4], [5] for univariate samples. It is found that the `alternative' estimator has important applications in multivariate samples, being easy of computation and of low variance.

A standard linear regression model is \begin{equation*}\tag{1}x_t = \alpha y_t + u_t\quad (t = 1, 2, 3, \cdots T)\end{equation*} where $\alpha$ is an unknown parameter, the $y$'s are known parameters and the $u$'s are NID $(0, \sigma^2)$. The maximum likelihood estimators for $\alpha$ and $\sigma^2$ are \begin{equation*}\tag{2}\hat \alpha = \frac{\Sigma x_t y_t}{\Sigma y^2_t},\quad\hat \sigma^2 = \frac{\Sigma (x_t - \hat \alpha y_t)^2}{T}.\end{equation*} The statistic \begin{equation*}\tag{3}\frac{(\hat \alpha - \alpha)}{\hat \sigma} (\Sigma y^2_t)^{\frac{1}{2}} \big(\frac{T - 1}{T}\big)^{\frac{1}{2}}\end{equation*} then has a $t$ distribution with $T - 1$ d.f. and its limiting distribution is $N(0, 1)$. One approach to time-series analysis is to set $y_t = x_{t-1}, y_1 = x_0 = \text{a constant}$. The model (1) is then transformed into the stochastic difference equation. \begin{equation*}\tag{4}x_t = \alpha x_{t-1} + u_t.\quad (t = 1,2, \cdots, T)\end{equation*} The maximum likelihood estimators for $\alpha$ and $\sigma^2$ in (4) are \begin{equation*}\tag{5}\hat \alpha = \frac{\Sigma x_t x_{t-1}}{\Sigma x^2_{t-1}},\quad \hat \sigma^2 = \frac{\Sigma (x_t - \hat \alpha x_{t-1})^2}{T}\end{equation*} which are exactly the values one would obtain by substituting $y_t = x_{t-1}$ in (2). In this paper it is shown that the limiting distribution of \begin{equation*}\tag{6}W = \frac{(\hat \alpha - \alpha)}{\hat \sigma} (\Sigma x^2_{t-1})^{\frac{1}{2}},\end{equation*} which is the analogue of (3), has a limiting $N(0, 1)$ distribution, except perhaps when $|\alpha| = 1$. This result is well-known for $|\alpha| < 1$ and was proved by Mann and Wald [1] under much more general conditions. The feature of the proof presented here is that it also holds in the explosive case $(|\alpha| > 1).$