This paper is concerned with testing the hypothesis that the parameter in a multivariate exponential distribution lies in a linear subspace of the natural parameter space. Our main result characterizes a complete class of tests which is independent of the particular exponential distribution. This class is, in fact, complete relative to the stronger ordering among tests which compares conditional power, given a certain statistic, pointwise. The conclusion holds without any restriction on the exponential distribution. Many of the tests are admissible, but examples show that although the class is essentially the smallest class complete relative to all exponential distributions, it is not in general minimally complete. Some special cases where the class is minimally complete are discussed.

In a recent paper A. N. Shiryaev [1] discusses the problem of detecting the arrival of a "disorder" in an observed stochastic process, as quickly as possible subject to a limitation on the number of false alarms. He considers two versions of a simple model. In the first, the disorder arrives at a discrete instant $\theta$ according to a geometric distribution. The process disturbed by this event consists of a sequence of independent observations $\{\xi_t\}$, such that $\xi_1, \xi_2, \cdots, \xi_{\theta - 1}$ arise from a certain distribution $F_0$, whereas $\xi_\theta, \xi_{\theta + 1}, \cdots$ come from a different distribution $F_1$. In the continuous time version of the model, the a-priori distribution of $\theta$ is exponential: $P(\theta > t) = e^{-\lambda t} \quad (t \geqq 0)$ and the disorder is represented by a change in the mean drift of an observed Wiener process $\{\eta(t)\}$. More precisely, for any given value of $\theta$, this process has independent normal increments $\delta\eta = \eta(t + \delta t) - \eta(t)$, with \begin{align*}E (\delta\eta) = 0 \quad (0 \leqq t < \theta), \notag \\ E(\delta\eta) = \delta t \qquad (t \geqq \theta), \notag \\ \operatorname{Var} (\delta\eta) = \delta t \quad (t \geqq 0)\end{align*}. In both versions it may be decided at any instant $t$ to carry out a detailed inspection in order to ascertain whether or not the disturbance has occurred. Then, if it is found that $\theta < t$ the process terminates but observation must be resumed immediately after a false alarm. Within these rules it is required to find a decision procedure which determines the instants at which a thorough inspection is worthwhile. Assuming that $N$, the expected number of false alarms, is specified in advance Shiryaev establishes the general form of policy which minimizes $\tau$, the expected delay in verifying the arrival of the disorder. The a posteriori distribution of $\theta$ at any time, does not depend on anything which took place before the last false alarm. For example, in continuous time $p(t) = P(\theta \leqq t \mid \eta(t'), 0 \leqq t' \leqq t) = P(\theta \leqq t \mid \eta(t'), s \leqq t' \leqq t),$ where $s$ is the instant of the most recent false alarm. The geometric and exponential distributions have the useful property that $P(\theta > t + s \mid \theta > s) = P(\theta > t \mid \theta > 0).$ He deduces that the optimal policy for the period following any false alarm must correspond exactly with the procedure applied initially, before the first inspection. In addition, he proves the existence of a critical level $p^\ast = p^\ast(N)$ such that, in general, observations should continue so long as $0 \leqq p(t) < p^\ast$ with an immediate inspection as soon as $p(t) \geqq p^\ast$. These and other similar results are established first for the discrete time model and then extended to the continuous time version. For the latter, the paper also gives more explicit calculations: the evaluation of $\tau$ in terms of $p^\ast$, for example. But no attempt is made to determine the critical level $p^\ast(N)$ for the optimal policy. In fact, as we shall see, a very simple relation holds: $p^\ast (N) = (N + 1)^{-1}.$ However, the aim here is to show how the optimal policy can be found for a more realistic specification of the minimization problem, involving given delay and inspection costs. We shall concentrate entirely on the continuous time model and suppose that each inspection incurs an instantaneous cost $K > 0$, not depending on its outcome, whereas any delay in detecting the arrival of the disorder leads to a cost $c > 0$ per unit time. Hence the total expected cost is $K(N + 1) + c\tau$, which depends both on the decision procedure and on the initial condition $p(0) = 0$. The minimization will be based on the calculation of the minimum expected future cost $f_\ast(\varphi)$, as a function of the current state $\varphi(t) = p(t)/(1 - p(t))$, by solving a certain differential equation with special boundary conditions. A heuristic argument, in which one simply assumes that $f_\ast(\varphi)$ is suitably differentiable, can be given without much difficulty. But, strictly speaking, it is not clear that risk functions such as $f_\ast(\operatorname{var})$, each of whose values is defined as the infimum of a class of expectations, are sufficiently well behaved. This difficulty is often encountered in statistical applications of dynamic programming to processes in continuous time. Typically, it is extremely difficult to establish the required differentiability properties directly and it is necessary to seek an indirect justification by means of existence and uniqueness theorems. In our case the formal solution can be produced explicitly and, because of this, its justification is much easier. Nevertheless, the approach is complicated by the need to establish several preliminary results, and the discussion of these special properties is limited to a brief indication of the main steps, in the hope that the essential structure of the argument will be more generally useful. Section 2 is concerned with the information process $\{\varphi(t)\}$. Its relation to the observed process $\{\eta(t)\}$ is described and certain properties of its increments are collected for later use. The main argument begins in Section 3 with a discussion of sub-optimal decision procedures defined by specifying an open continuation set $\mathscr{C}$ within the space $\lbrack0, \infty)$ of possible "initial" states $\varphi$. It is shown that any risk function $f(\varphi)$ can be determined for each sub-interval of the corresponding set $\mathscr{C}$ by solving the basic differential equation appropriately. Sections 4 and 5 consider the special solution $f_\ast(\varphi)$ which represents the optimal decision procedure, and give the required verification that $f_\ast(\varphi)$ is uniformly minimal. The final section contains a brief analysis of the operating characteristics of the optimal policy and indicates the importance of evaluating the particular minimum risk $f_\ast(0)$.

Let $X_N$ for $N = 1, 2, \cdots$ be an independent sequence of random variables with finite first absolute moments; let $a_N = \{a_{N,k} : k = 1, 2, \cdots\}$ for $N = 1, 2, \cdots$; let $A_N = 1/N \sum^N_{k = 1} (X_k - EX_k)$; and let $A_N = \sum_\infty^{k = 1} a_{N, k} (X_k - EX_k)$. Early work in probability dealt with the convergence (almost everywhere and in probability) to zero of the sequence. $A_N$. More recent work has dealt with the convergence to zero of sequences of the form $S_N$ under various assumptions on the coefficients $a_{N,k}$ and the distributions of the $X_N's$. In most cases the assumptions made about the $X_N's$ have been not much stronger or weaker than the assumption of a finite upper bound on their $\gamma$th absolute moments for some $\gamma \geqq 1$. The classical result giving exponential convergence rates in the law of large numbers was established by Cramer [6] (see also [4]) and states that if the $X_N's$ are identically distributed, and if their common moment generating function is finite in some interval about the origin, then for each $\epsilon > 0$ there exists $0 \leqq \rho < 1$ such that $P\{| A_N| \geqq \epsilon\} \leqq 2\rho^N$. Baum, Katz, and Read [3] investigated this exponential convergence further. However, their investigation was restricted to sequences of the form $A_N$. Koopmans [17] dealt with averages of the form $1/N \sum^N_{k = 1} \sum^\infty_{j=-\infty} a_jX_{k-j}$. In [11] the exponential rate was obtained for sequences $S_N$ provided $\sum_k \mid a_{N, k}\mid \leqq M < \infty$ for all $N$ and $\max_k |a_{N,k}| \leqq O(1/N)$. A corresponding result was obtained for continuous time stochastic processes in [12]. More recently Chow [5, Section 2] obtained similar results under stronger assumptions on the moment generating functions involved. The results obtained here generalize and unify the results of [11], [12], and [5, Section 2]. The results are stated in Section 2 and proved in Section 2 and proved in Section 3. Corollaries and details of the relationships between these results and previous results are contained in Section 4.

For the experiment $\mathscr{E}$, given by $\mathscr{E} = \lbrack x$ (in space $X$) distributed with probability element $f(x \mid \theta) d\mu(x), \{\theta\} = \Theta$ the parameter space, $W(d, \theta)$ a non-negative loss function for decision $d \epsilon D\rbrack$, the definition of a (non-randomized) strict Bayes decision function (BDF) can, following Wald [7], be stated: $\delta_\pi$, given by the decision function $d_\pi(x)$, is a strict BDF with respect to the proper prior probability measure $\pi$ on $\Theta$ if $\delta_\pi$ minimizes $R(\pi, \delta) = \int d\pi(\theta)r(\theta, \delta) = \int d\pi(\theta) \int d\mu(x)W\lbrack d(x), \theta\rbrack f(x \mid \theta)$ [requiring, of course, that $R(\pi, \delta_\pi) < \infty\rbrack$. (As is well known, randomized decision functions can be excluded from standard Bayesian methods. We will not here attempt to justify their exclusion in the below deviations from the standard Bayesian formulations.) Relaxing the restriction that the prior measure be proper, we have: DEFINITION 1.1. $\delta_m$ is a normal generalized BDF (NGBDF) with respect to the generalized prior $m$ on $\Theta$ if $\delta_m$ minimizes $R(m, \delta)$ [requiring, of course, that $R(m, \delta_m) < \infty\rbrack$. DEFINITION 1.2. $\delta_m$ is an extensive generalized BDF (EGBDF) with respect to $m$ if $\delta_m = \{d_m(x)\}$ where $d_m(x)$ minimizes $\int dm(\theta)W(d, \theta)f(x) \mid \theta)$ a.e. $\mu$. (The use of the epithets "extensive" and "normal" is consistent with their use in Raiffa and Schlaifer [5]. Definition 1.2 differs from the definition of Sacks [6] in not requiring the finiteness of $\int dm(\theta)f(x \mid \theta)$.) Theorem 2.1 shows that if $\delta_m$ is an NGBDF then it is also an EGBDF. For some cases in which $\min_\delta R(m, \delta) = \infty$, the following generalization of EGBDF is useful: DEFINITION 1.3. $\delta_m^\ast$ is a comparative generalized BDF (CGBDF) with respect to $m$ if the quantity $\Delta_m(\delta, \delta_m^\ast)$ defined by $\Delta_m(\delta, \delta^\ast_m) = \int dm(\theta)\lbrack r(\theta, \delta) - r(\theta, \delta_m\ast)\rbrack$ is non-negative for all $\delta$. Admissibility considerations are unaffected by multiplying $W(d, \theta)$ by an arbitrary positive function of $\theta$. So, since the above definitions involve $m$ and $W$ in the composite element $W(d, \theta) dm(\theta)$, it is clear that any general sufficient condition for admissibility of $\delta_m$ or $\delta^\ast_m$ for $m$ proper may also be stated for $m$ general (that is, possibly improper). (The same point is made by Stein [4], p. 232.) Theorem 3.1 gives such a sufficient condition, suggested by the Lehmann-Blyth technique for proving admissibility [1], while Corollary 3.1 is an extension of the well known admissibility of strict BDF's under certain conditions. The above ideas are applied in Section 4 to the estimation with quadratic loss of the mean of the one dimensional exponential family. The very close links with Karlin's technique [3] are immediately apparent. The application clarifies Karlin's remark (p. 411 of [3]) that his results may be regarded as a refinement of the Lehmann-Blyth technique and also, finally, lends some support to his conjecture (p. 415 of [3]) that a certain condition for admissibility of the contracted estimator $\gamma x, 0 < \gamma \leqq 1$, is necessary as well as sufficient. An associated reference is Cheng Ping [2].

Let $X_{ij}, i = 1, \cdots, n, j = 1, \cdots, k$, be independent with $X_{ij}$ having the continuous distribution function $P(X_{ij} \leqq x) = F_j(x - b_i)$ where $b_i$ is the nuisance parameter corresponding to block $i$. (These assumptions shall be called the $H_A$ assumptions.) This paper is concerned with procedures for testing the null hypothesis \begin{equation*}\tag{0.1}H_0 : F_j = F \text{(unknown)}, \quad j = 1, \cdots, k,\end{equation*} which are sensitive to the ordered alternatives \begin{equation*}\tag{0.2}H_a : F_1 \geqq F_2 \geqq \cdots \geqq F_k,\end{equation*} where at least one of the inequalities is strict. In particular, we introduce a test statistic $(Y)$ based on a sum of Wilcoxon signed-rank statistics. In Section 2 we develop the asymptotic distribution of $Y$ and find that, under $H_0, Y$ is neither distribution-free for finite $n$, nor asymptotically distribution-free. However, a consistent estimate of the null variance of $Y$ is used to define a procedure which is asymptotically distribution-free. In Section 3 we derive, under the $H_A$ assumptions, necessary and sufficient conditions for the consistency of $Y$ and two of its nonparametric competitors, viz., (1) Jonckheere's $\tau$ test [11] based on Kendall's rank correlation coefficient between observed order and postulated order in each block; (2) Page's $\rho$ test [17] based on Spearman's rank correlation coefficient between observed order and postulated order in each block. We find that (i) $Y$ is consistent if and only if $\sum_{u < v} \int H_u dH_v/k(k - 1) > \frac{1}{4}$ where $H_u = F^\ast_u F_u, u = 1, \cdots, k$, (ii) Jonckheere's test is consistent if and only if $\sum_{u < v} \int F_u dF_v/k(k - 1) > \frac{1}{4}$, and (iii) Page's test is consistent if and only if $\sum_{u < v} (v - u) \int F_u dF_v > k(k - 1) \cdot (k + 1)/12$. Section 4 is devoted to efficiency comparisons of the rank tests with respect to a normal theory $t$-test defined in Section 1. For a class of shift alternatives we show that the Pitman efficiency of $Y$ with respect to $t(E(Y, t))$ is greater than .864 for every $F$ and every $k$. When $F$ is normal, $E(Y, t) = .963$ for $k = 3$ and $\rightarrow .989$ as $k \rightarrow \infty$. These values compare favorably with the corresponding ones of Page's test (.716, .955) and Jonckheere's procedure (.694, .955). For these shift alternatives we also show that $.576 \leqq E(\rho, t) \leqq \infty$ and $.576 \leqq E(\tau, t) \leqq \infty$.

For block designs with one observation per cell, the model often used is the linear model in which the observations $X_{i\alpha} (i = 1, \cdots, r; \alpha = 1, \cdots, n)$ can be written \begin{equation*}\tag{1.1} X_{i\alpha} = v + \xi_i + \mu_\alpha + Y_{i\alpha} (\sum \xi_i = \sum \mu_\alpha = 0)\end{equation*} where the $\xi's$ are the parameters of interest (treatment effect) the $\mu's$ are nuisance parameters (block effect), and the $Y's$ are independent with common continuous distribution $F$. The purpose of this note is to discuss some new robust test statistics (e.g. 2.14 and 2.16) of the null-hypothesis $H_0 : \xi_1 = \xi_2 = \cdots = \xi_r$, and to discuss a new robust estimate (3.3) of the contrast $\theta = \sum c_i\xi_i$.

For testing hypotheses about $\alpha$ and $\beta$ in the linear regression model $Y_j = \alpha + \beta x_j + Z_j$, Brown and Mood [18] have proposed distribution-free tests, based on their median estimates. Daniels [6] has also given a distribution-free test for the hypothesis that the regression parameters have specified values. This latter test is an improvement on the Brown and Mood median procedure, although both are based on the signs of the observations. Recently Hajek [10] constructed rank tests, which are asymptotically most powerful, for testing the hypothesis that $\beta = 0$, while $\alpha$ is regarded as a nuisance parameter. In this paper, a class of rank score tests for the hypothesis $H : \alpha = \beta = 0$, is proposed in Section 2. This class includes as special cases, the Wilcoxon and the normal scores type of tests. In Sections 3 and 4 the limiting distribution of the test statistics is shown to be central $\chi^2$, under $H$, and non-central $\chi^2$, under a sequence of alternatives tending to the hypothesis at a suitable rate. In Section 5, the Pitman efficiency of the proposed tests relative to the classical $F$-test, is proved to be the same as the efficiency of the corresponding rank score tests relative to the $t$-test in the two sample problem.

In the linear regression model $Y_j = \alpha + \beta x_j + Z_j$, it is usual to estimate $\alpha$ and $\beta$ by the method of least squares. This method has, among other things, the nice property of providing "best" linear unbiased estimates, under very general conditions. Various other methods of estimation of the parameters are well known, see for example [12] and [14]. Most of these methods however make use of the actual values of the observations, and the estimates they yield are generally vulnerable to gross errors. For some alternative approaches to the problem, see [7], [11] and [13]. In a recent paper [9], Hodges and Lehmann proposed a general method of obtaining robust point estimates for the location parameter, from statistics used to test the hypothesis that this parameter has a specified value. In Section 1 of this paper, this method is used to define point estimates $\hat\alpha$ and $\hat\beta$ of $\alpha$ and $\beta$, in terms of certain test statistics. It is shown that the least squares estimates are obtainable as special cases from the general method of estimation discussed. In Section 2, the existence of `rank score' estimates is proved, and in Section 3, computing techniques are given and illustrated with an example. Both the small sample and asymptotic properties of the estimates are discussed. It is shown, for example, that the joint distribution of the estimates $\hat\alpha$ and $\hat\beta$ is symmetric with respect to the parameter point $(\alpha, \beta)$--and hence that $\hat\alpha$ and $\hat\beta$ are unbiased--if the underlying distribution of the observations is symmetric. In Section 5, the joint asymptotic normality of $\hat\alpha$ and $\hat\beta$ is proved, and in Section 6, it is shown that the asymptotic efficiency of $(\hat\alpha, \hat\beta)$ is the same as the Pitman efficiency of the rank tests [1], on which they are based, relative to the classical tests. Finally in Section 7, the $(\hat\alpha, \hat\beta)$-estimates are compared with the Brown and Mood median estimates with respect to their efficiencies.

Let there be two independent samples of sizes $m$ and $n$ respectively from two populations with continuous cdf's $F(x)$ and $G(y)$. To test the equality of the two populations Sobel [19], has proposed the statistic $V^{m,n}_r$ (to be defined later) based on the first $r$ ordered observations only. In this paper the large sample properties of $V^{m,n}_r$ have been studied. The test is compared with other "$r$ out of $N$" tests by computing the appropriate asymptotic relative efficiencies. The test statistic is found to be quite satisfactory in all the cases considered and is particularly suitable for location alternatives.

A certain class of functions of Cauchy variates whose members are also Cauchy variates is described. Some independence properties of Cauchy variates are also discussed.

Heller has given necessary and sufficient conditions that a stochastic process be induced from a Markov chain. We consider a process induced by a Markov chain to be a probabilistic finite automaton with one input. With each state of a probabilistic finite automaton, we may associate a function $p(u \mid v)$, which tells us the probability that, if we apply the input sequence $v$ to the machine started in the state, we should observe output sequence $u$. We give a necessary and sufficient condition that a function $p(u \mid v)$ be realizable as much as input-output function. Finally, we show Heller's result is extended by our condition.

Let $\mathbf{S}_i : p \times p (i = 1,2)$ be independently distributed as Wishart $(n_i, p, \mathbf{\sigma}_i)$. Let the characteristic (ch) roots of $\mathbf{S}_1\mathbf{S}^{-1}_2$ and $\mathbf{\sigma}_1\mathbf{\sigma}^{-1}_2$ be denoted by $f_i (i = 1, 2,\cdots, p)$ and $\lambda_i(i = 1, 2, \cdots, p)$ respectively such that $0 < f_1 < f_2 < \cdots < f_p < \infty$ and $0 < \lambda_1 \leqq \lambda_2 \leqq \cdots \leqq \lambda_p < \infty$. The distribution of $f_1, f_2, \cdots, f_P$ as stated by James [5] is not convenient for further development and is slowly convergent for higher values of $f_i's$. The distribution of $(f_1, \cdots, f_p)$ mentioned by James [5] can be written as \begin{equation*}\tag{1} c|\mathbf{\lambda}|^{-\frac{1}{2}n_1}|\mathbf{F}|^{\frac{1}{2}(n_1 - p - 1} \alpha_p(\mathbf{F} \int_{O(p)}|\mathbf{I}_p + \mathbf{\lambda}^{-1}\mathbf{HFH}'|^{-\frac{1}{2}(n_1 + n_2)} d\mathbf{H}\end{equation*} where \begin{equation*}\tag{2}c = \pi^{\frac{1}{2}p^2} \Gamma_p(\frac{1}{2} n_1 + \frac{1}{2}n_2)\{ \Gamma_p (\frac{1}{2}P) \Gamma_p (\frac{1}{2}n_1)\Gamma_p(\frac{1}{2}n_2)\}^{-1}, \Gamma_p(t) = \pi^{\frac{1}{4}p(p - 1)} \Pi^p_{j = 1} \Gamma (t - \frac{1}{2}j + \frac{1}{2},\end{equation*} \begin{equation*} {3}\alpha_p(\mathbf{F}) = \Pi ^{p - 1}_{i = 1} \Pi^p_{j = i + 1} (f_j - f_i), \quad \mathbf{F} = \operatorname{diag} (f_1, f_2, \cdots, f_p), \mathbf{\lambda} = \operatorname{diag} (\lambda_1, \lambda_2, \cdots, \lambda_p) \end{equation*} and the intergral is over an orthogonal group $O(p)$ with $\int_{O(p)} d\mathbf{H} = 1$. For testing the null hypothesis $H_0(\lambda\mathbf{\lambda} = \mathbf{I}_p), \lambda > 0$ being given, we have two statistics given by \begin{equation*}\tag{4}(i) l = |\lambda\mathbf{F}|^{n_1}/|\mathbf{I}_p + \lambda F|^{n_1 + n_2} \text{and} (ii) \lambda f_P \text{or} \lambda f_p/(1 + \lambda f_p)\end{equation*}. $l$ is considered by Anderson [1] and $\lambda f_p$ is obtained by Roy [7]. (1) is rewritten in such a way that the joint density function of $(\lambda f_1, \lambda f_2, \cdots, \lambda f_p)$ has noncentral parameters $\mathbf{I}_p - (\lambda\mathbf{\lambda})^{-1}$ and it is given by \begin{equation*}\tag{5}c|\lambda\mathbf{\lambda} |^{\frac{1}{2}n_1}|\lambda\mathbf{F}|^{\frac{1}{2}(n_1-p-1} \alpha_p(\lambda\mathbf{F})| \mathbf{I}_p + \lambda\mathbf{F})| ^{-\frac{1}{2}(n_1 + n_2)} _1F_0^{(p)} (\frac{1}{2}n_1 + \frac{1}{2}n_2; \mathbf{I}_p - (\lambda\mathbf{\lambda}^{-1}, \lambda\mathbf{F}(\mathbf{I}_p + \lambda\mathbf{F})^{-1})\end{equation*}. Hence, similar to testing of means, we propose the statistics $T = tr (\lambda\mathbf{F})$ and $V = tr (\lambda\mathbf{F})(\mathbf{I}_p + \lambda\mathbf{F})^{-1}$ for testing the hypothesis $H_0$ and obtain their distribution of $T$ only while the moment generating function of $V$ is given. Moreover, if in the null hypothesis $H'_0(\lambda\mathbf{\lambda} = \mathbf{I}_p, \lambda > 0$ is unknown, then the test procedure will depend on the ratios of roots, and hence, we consider the joint distribution of $(x_1, x_2, \cdots, x_{p - 1})$, where $x_i = f_i/f_p$ for $i = 1, 2, \cdots, p - 1$, and $f_p$. Under null hypothesis $H'_0$, we obtain the density functions of $x_1$ (or $x_{p - 1}$) and of $(y_2, y_3, \cdots, y_{p - 1})$ for $y_i = (f_i - f_1)/f_p - f_1), i = 2, 3, \cdots, p - 1$.