Asymptotic expansions are derived for the behavior of the optimal sequential test of whether the unknown drift $\mu$ of a Wiener-Levy process is positive or negative for the case where the process has been observed for a short time. The test is optimal in the sense that it is the Bayes test for the problem where we have an a priori normal distribution of $\mu$, the regret for coming to the wrong conclusion is proportional to $|\mu|$ and the cost of observation is one per unit time. The Bayes procedure is compared with the best sequential likelihood ratio test and with the procedure which calls for stopping when no fixed additional sampling time is better than stopping. The derivations allow for generalizing to variations of this problem with different cost structure.

The problem of sequentially testing whether the mean of a normal distribution is positive has been approximated by the continuous analogue where one must decide whether the mean drift of a Wiener-Levy process is positive or negative [3]. The asymptotic behavior of the solution of the latter problem has been studied as $t \rightarrow \infty$ and as $t \rightarrow 0$ [1], [2], [4], [5]. The original (discrete) problem, can be regarded as a variation of the continuous problem where one is permitted to stop observation only at the discrete time points $t_0, t_0 + \delta, t_0 + 2\delta, \cdots$. Especially since the numerical computation of the solution of the continuous version can be carried out by solving the discrete version for small $\delta$, it is important to study the relationship between the solutions of the discrete and continuous problems. These solutions are represented by symmetric continuation regions whose upper boundaries are $\tilde x_\delta(t)$ and $\tilde x(t)$ respectively. The main result of this paper is that \begin{equation*}\tag{(1.1)}\tilde x_\delta(t) = \tilde x(t) + \hat z\sqrt\delta + o(\sqrt\delta).\end{equation*} This result involves relating the original problem to an associated problem and studying the limiting behavior of the solution of the associated problem. This solution corresponds to the solution of a Wiener-Hopf equation. Results of Spitzer [6], [7] can be used to characterize the solution of the Wiener-Hopf equation and yield $\hat z$ as an integral, which, as Gordon Latta pointed out to the author, is equal to $\zeta(\frac{1}{2})/(2\pi)^{\frac{1}{2}} = -.5824$. The associated problem referred to above is the following. A Wiener-Levy process $Z_t$ starting at a point $(z, t), t < 0$ is observed at a cost of one per unit time. If the observation is stopped before $t = 0$, there is no payoff. If $t = 0$ is reached, the payoff is $Z^2_0$ if $Z_0 < 0$ and 0 if $Z_0 \geqq 0$. Stopping is permitted at times $t = -1, -2, \cdots$.

N. R. Goodman [4] has discussed some aspects of the complex multivariate normal distribution, in particular, the analogue of the Wishart distribution and of multiple and partial correlations. We shall obtain maximum likelihood estimates for certain parameters and likelihood ratio tests of certain hypotheses arising in the study of such complex multivariate normal distributions. Although the general principles involved in the derivation of the distribution of the associated statistics are well known to mathematicians working on group representations (see, for example, Gelfand and Naimark [3], p. 24), it has seemed desirable to derive the needed results on this type in a way parallel to one method of obtaining the distributions of real multivariate analysis (Deemer and Olkin [2], Olkin [6] and Roy [7]). With the help of the results derived, we can handle the complex variates in the same manner as we do for real variates in the case of Gaussian distributions. Moreover, it can be noted that for every distributional result of classical multivariate Gaussian statistical analysis obtainable in closed (explicit) form, the counterpart analysis for complex Gaussian is also obtainable in closed (explicit) form with necessary changes. It may be pointed out that the non-central distributions in this connection were derived independently by A. T. James [5] with the help of zonal polynomials of hermitian matrices, but we feel that sometimes the derivation of distributions with the help of Jacobian transformations may be useful.

Consider absolutely continuous unimodel distributions on the real line. A class of estimates is proposed for estimating the mode of the probability distribution. The large sample behavior is studied, and it is found that in the case of main interest the estimates are not even consistent. To remedy this a modification is suggested and it is shown that the new estimates are consistent.

In this paper a new set of inequalities and bounds for the incomplete gamma function are obtained. These inequalities and bounds are based on continued fraction expansions of the incomplete gamma function (Sections 2 and 3). Comparisons between the two sets of inequalities and some other known inequalities are made (Section 4). Bounds are also obtained for the Mills' ratio for the normal integral (Section 5) and an analogue of Mills' ratio (Section 6) for the gamma distribution. Some other applications of these bounds to distribution theory problems arising in multiple decision theory are described (Section 6).

In a previous paper [2] the author investigated some alternative estimates of shift in the $p$-variate one-sample problem. This paper examines the properties of tests for shift similar to Hotelling's $T^2$, based on (I) asymptotically normal estimates, in particular those of the type considered in [2], and (II) the originating univariate test statistics of the latter group. The notation, similar to that used in [2], and the tests are introduced in Section 2. In the third section the asymptotic distribution of such test statistics for sequences of alternatives tending to $\mathbf{0}$ as $n^{-\frac{1}{2}}$ is found to be noncentral $\chi^2$. The tests of type I and, often simpler, tests of type II have the same asymptotic distribution. In Section 4 we find that the Pitman efficiency of two such tests depends, in general, on the direction in which the origin is approached. The efficiency, in terms of "generalized variance," of the estimates of [2] lies between the maximum and minimum Pitman efficiencies of the corresponding tests of type I (maximum and minimum being taken over direction). This "generalized variance" efficiency is found to be equal to the efficiency of the tests as defined in terms of a criterion of goodness introduced by Isaacson [9] ($D$-optimality). In case the coordinates are identically distributed, if correlation effects alone are taken into account, it is shown that whatever two tests are considered, there always exists a direction in which one improves the other. Section 5 continues with a discussion of the tests based on the estimates introduced in [2] in relation to Hotelling's $T^2$. Their desirable properties and pathologies are found to be similar to those of the parent estimates. The remaining sections deal with the case $p = 2$. Under normality the minimum Pitman efficiency with respect to $T^2$ of the tests mentioned above behaves like the efficiency of the parent estimates with respect to the mean.

In linear models with several observations per cell, a class of estimates of all contrasts are defined in terms of rank test statistics such as the Wilcoxon or normal scores statistic, which extend the results of Hodges and Lehmann (1963) and Lehmann (1963). The asymptotic efficiency of these estimates relative to the standard least squares estimates, as the number of observations in each cell gets large, is shown to be the same as the Pitman efficiency of the rank tests on which they are based to the corresponding $t$-tests.

For the two-sample problem, Wilcoxon [21], Fisher and Yates [6], Terry [19], Hoeffding [10], Hodges and Lehmann [8], Savage [17], Chernoff and Savage [3], Lehmann [13], Capon [2] and others have considered rank-sum statistics equivalent to $S_N(H) = m^{-1} \sum E(Z(R(X_i)) \mid H) - n^{-1} \sum E(Z(R(Y_i)) \mid H)$, where $E(Z(j) \mid H)$ is the expectation of the $j$th order statistic of a sample of size $N = m + n$ from a population with cpf (cumulative probability function) $H$ and $R(X_i) \lbrack R(Y_j)\rbrack$ is the rank of $X_i\lbrack Y_j\rbrack$ in the combined sample of $X$'s and $Y$'s. In order to perform tests based on these statistics one needs special tables of expected values as well as tables of the hypothesis distribution. Further, in general, exact desired significance levels can only be achieved through randomization. The object of this note is to introduce rank-sum statistics in which one randomizes initially and circumvents the necessity of two special tables. These new randomized statistics, which are formed by deleting the expectation signs "$E$" in $S_N(H)$, generally satisfy the same asymptotic goodness criteria as their nonrandomized counterparts. Moreover, they have the added advantage that for appropriate choices of the parameters they have null hypothesis distributions which are continuous, known and tabulated (e.g., normal, $\chi^2$, etc.) In particular, one of the new two-sample statistics has an exact normal distribution and is asymptotically uniformly more efficient than the $t$-test for translation alternatives. This idea is extended to obtain randomized rank-sum statistics for the independence, randomness, $k$-sample and two-factor problems analogous to the statistics of Friedman [7], Puri [15], Stuart [18] and others. As in the references listed above, prime interest will be in those cases for which $H$ is normal, uniform or exponential. However, the methodology is equally applicable to other continuous cpf's $H$.

$n$ balls on a circle are colored white or black according to $n$ mutually independent binomial trials. It is shown here that, when their expectations converge with $n$, (a) counts of runs of various lengths are asymptotically independent Poisson; (b) counts of certain configurations other than runs yield asymptotic correlated Poisson distributions; (c) counts of configurations with structure independent of $n$ can be partitioned into equivalence classes, with asymptotic equivalence (equality with probability one) and asymptotic independence respectively within and among classes. It is also shown that (d) there cannot, essentially, exist configurations whose counts, asymptotically, are marginally, but not multivariate, Poisson.

This is primarily a review of combinatorial problems connected with ballot problems, runs, records, and amalgamation, but numerous new results and applications occur throughout the paper. The early history of the classical ballot problem is clarified, and many recent generalizations and applications are noted. For runs and records, the main emphasis is on the derivation of the null-hypothesis distributions, with only passing reference to asymptotics and non-null distributions. An appendix lists recent work on the Kolmogorov and Smirnov statistics. There are 183 references.

We consider a one-parameter exponential family generated by an arbitrary group of transformations of an abstract sample space. Topological assumptions about the group are not required. It is shown that such a family has densities either of the type of the normal distribution or of the type of the gamma distribution with respect to an invariant measure. This is a generalization of results of Dynkin (1951), Lindley (1958) and Ferguson (1962 and 1963).

Let $X_1, X_2, \cdots, X_n$ be a sequence of random variables (r.v.'s) and put $S_m = \sum^m_{\nu = 1} X_\nu, 1 \leqq m \leqq n$. It is well-known that \begin{equation*}\tag{(1)}E|S_n|^r \leqq n^{r - 1} \sum^n_{\nu = 1} E|X_\nu|^r\quad r > 1,\end{equation*} $E|S_n|^r \leqq \sum^n_{\nu = 1} E|X_|nu|^r,\quad r \leqq 1.$ However, if the r.v.'s satisfy the relations \begin{equation*}\tag{2}E(X_{m + 1} \mid S_m) = 0 \text{a.s.}\quad 1 \leqq m \leqq n - 1,\end{equation*} it is possible to improve the first inequality considerably. The case $r > 2$ with independent r.v.'s will be treated elsewhere by one of the authors, von Bahr. If $r = 2$, we have, under (2), \begin{equation*}\tag{3}ES^2_n = \sum^n_{\nu = 1} EX^2_\nu.\end{equation*} In the case $1 \leqq r \leqq 2$, we will show that under (2) \begin{equation*}\tag{(4)}E|S_n|^r \leqq C(r, n) \sum^n_{\nu = 1}e|X_\nu|^r,\end{equation*} where $C(r, n)$ is a bounded function of $r$ and $n$. In Theorem 2 we show that (4) is true with $C(r, n) = 2$. If the distribution of each $X_{m + 1}$ conditioned by $S_m$ is symmetric about zero, one can put $C(r, n) = 1$ (Theorem 1). Further, if the r.v.'s satisfy the following conditions \begin{equation*}\tag{(5)}E(X_i \mid R_{mi}) = 0\text{a.s.}\quad 1 \leqq i \leqq m + 1 \leqq n,\end{equation*} where $R_{mi} = \sum^{m + 1}_{\nu = 1, \nu \neq i} X_\nu$ it is possible to put $C(r, n) = 2 - n^{-1}$. The conditions (2) and (5) are fulfilled if the r.v.'s are independent and have zero means. In this case, however, it is possible to make $C(r, n)$ dependent on $r$, so that $C(r, n) \rightarrow 1$ as $r \rightarrow 2$. It is possible to show by an example, that (4) is not generally true with $C(r, n) = 1$ even in this case. If $1 \leqq r < s \leqq 2$ and $E|X_\nu|^s < \infty, 1 \leqq \nu \leqq n$, it is generally better not to use (4) directly, but to use it together with $E|S_n|^r \leqq (E|S_n|^s)^{r/s}$, so that $E|S_n|^r \leqq (C(s, n) \sum^n_{\nu = 1} E|X_\nu|^s)^{r/s}$. The case $r < 1$ is by (1) trivial.

Let $m$ and $n$ be the sizes, respectively, of two independent random samples both of which may be censored in an arbitrary manner so that $h(1 \leqq h \leqq m)$ and $k(1 \leqq k \leqq n)$ observations, respectively, remain. If arranged in ascending order, the remaining observations can appear in $\binom{h + k}{h}$ possible rank orders. In this paper we prove a theorem which is useful in obtaining the probability associated with any one of these rank orders, provided the two samples are drawn from populations with identical distribution functions.

Let $F_n(x)$ denote the empirical distribution function of a random sample of size $n$ drawn from a population having continuous distribution function $F(x)$. In Section 3 the limiting distribution of the supremum of the random variables $\{F_n(x) - F(x)\}/F_n(x), |F_n(x) - F(x)|/F_n(x), \{F_n(x) - F(x)\}/(1 - F(x)), |F_n(x) - F(x)|/(1 - F(x)), \{F_n(x) - F(x)\}/(1 - F_n(x)), |F_n(x) - F(x))|/(1 - F_n(x))$ is derived where sup is taken over suitable ranges of $x$ respectively. Relevant tests and some combinations of them are also discussed briefly in Section 3.

In an earlier paper [4], the author has given upper bounds for the number of disjoint blocks in (i) Semi-regular GD designs, (ii) certain PBIB designs with two associate classes having triangular association scheme, (iii) certain PBIB designs with two associate classes having $L_2$ association scheme and (iv) certain PBIB designs with three associate classes having rectangular association scheme. In this paper, we give bounds for the number of common treatments between any two blocks of the above-mentioned PBIB designs. The main tools used to establish the results of this paper are the theorems proved by (i) Bose and Connor [1], (ii) Raghavarao [3], and (iii) Vartak [6].