This paper considers and unifies two sequential problems which have been extensively discussed. A class of sequential problems is proposed that includes both. An arbitrary partial ordering constraint is permitted to restrict possible strategies. An algorithm is proposed for finding the optimal strategy, and we prove that a strategy is optimal for the class of problems if and only if it can be found by the algorithm. The main tool is a set of functional equations in strategy space.

This is Part I of a two-part paper; the purpose of this two-part paper is (a) to develop new concepts and techniques in the theory of majorization and Schur functions, and (b) to obtain fruitful applications in probability and statistics. The main theorem of Part I states that if $f(x_1, \cdots, x_n)$ is Schur-concave, and if $\phi(\lambda, x)$ is totally positive of order 2 and satisfies the semigroup property for $\lambda_1 > 0, \lambda_2 > 0: \phi(\lambda_1 + \lambda_2, y) = \int \phi(\lambda_1, x)\phi(\lambda_2, y - x) d\mu(x)$, where $\mu$ is Lebesgue measure on $\lbrack 0, \infty)$ or counting measure on $\{0, 1, 2, \cdots\}$, then $h(\lambda_1, \cdots, \lambda_n) \equiv \int \cdots \int \Pi^n_1 \phi(\lambda_i, x_i)f(x_1, \cdots, x_n) d\mu(x_1) \cdots d\mu(x_n)$ is also Schur-concave. This theorem is then applied to obtain renewal theory results, moment inequalities, and shock model properties.

This is Part II of a two-part paper. The main purpose of this two-part paper is (a) to develop new concepts and techniques in the theory of majorization and Schur functions, and (b) to obtain fruitful applications in probability and statistics. In Part II we introduce a stochastic version of majorization, develop its properties, and obtain multivariate applications of both the preservation theorem of Part I and the new notion of stochastic majorization. This leads to a definition of Schur families of multivariate distributions. Generalizations are obtained of earlier results of Olkin and of Wong and Yue; in addition, new results are obtained for the multinomial, multivariate negative binomial, multivariate hypergeometric, Dirichlet, negative multivariate hypergeometric, and multivariate logarithmic series distributions.

Let $\mathbf{a}$ and $\mathbf{b}$ be $r$-dimensional real vectors. It is shown that if $\mathbf{a}$ majorizes $\mathbf{b}$, then $E(\Pi^r_{j = 1} X_j^a j) \geqq E(\Pi^r_{j = 1} X_j^b j)$ holds for nonnegative random variables $X_1, \cdots, X_r$ whose joint pdf is permutation symmetric. If in addition the components of $\mathbf{a, b}$ are nonnegative integers, then for every Borel-measurable set $A$, $\Pi^r_{j = 1} P\lbrack\cap^{a_j}_{i = 1} \{Z_i \in A\}\rbrack \geqq \Pi^r_{j = 1} P\lbrack\cap^{b_j}_{i = 1} \{Z_i \in A\}\rbrack$ holds for conditionally i.i.d. random variables $Z_i$. Applications are considered.

Let $\mathscr{P}$ be a family of distributions on a measurable space such that $(\dagger) \int u_i dP = c_i, i = 1, \cdots, k$, for all $P\in\mathscr{P}$, and which is sufficiently rich; for example, $\mathscr{P}$ consists of all distributions dominated by a $\sigma$-finite measure and satisfying $(\dagger)$. It is known that when conditions $(\dagger)$ are not present, no nontrivial symmetric unbiased estimator of zero (s.u.e.z.) based on a random sample of any size $n$ exists. Here it is shown that (I) if $g(x_1, \cdots, x_n)$ is a s.u.e.z. then there exist symmetric functions $h_i(x_1, \cdots, x_{n - 1}), i = 1, \cdots, k$, such that $g(x_1, \cdots, x_n) = \sum^k_{i = 1} \sum^n_{j = 1} \{u_i(x_j) - c_i\}h_i(x_1, \cdots, x_{j - 1}, x_{j + 1}, \cdots, x_n);$ and (II) if every nontrivial linear combination of $u_1, \cdots, u_k$ is unbounded then no bounded nontrivial s.u.e.z. exists. Applications to unbiased estimation and similar tests are discussed.

The stopping time $N$ of a sequential test based on an i.i.d. sequence $X, X_1, X_2, \cdots$ with common distribution $P$ is defined as the first integer $n \geqq 1$ such that $l_1 < L_n < l_2$ is violated, where $L_n$ is a statistic depending only on $X_1, \cdots, X_n$. If for all $l_1, l_2$ there exist constants $c > 0, \rho < 1$ such that $P(N > n) < c\rho^n, n = 1, 2, \cdots,$ then $N$ is called exponentially bounded under $P$. In the contrary case $P$ is termed obstructive. A general theorem is proved whose conclusion is of the from that $N$ is exponentially bounded under $P$ unless $P\{f(X) = 0\} = 1$, with $f$ a function that depends on the particular testing problem. Among the applications presented there are two new results. The first is for the sequential linear hypotheses $F$-test. The function $f$ is found, and the distributions for which $P\{f(X) = 0\} = 1$ are shown to be supported on spheres. All these $P$'s, and only these, are obstructive. The second result concerns the sequential two-sample Wilcoxon test for the equality of the distributions of random variables $X$ and $Y$. Here the result is simple: $P$ is obstructive if and only if $P(X = Y) = 1$. Two auxiliary results of independent interest are presented. The first is a generalization of the exponential convergence of empirical to theoretical distribution function. The second one shows that if random variables $X$ and $Y$ have joint distribution $P$ and marginal distribution function $F, G$, respectively, then $P\{F(Y) = G(X)\} = 1$ if and only if $P(X = Y) = 1$.

For a broad class of jackknife statistics, it is shown that the Tukey estimator of the variance converges almost surely to its population counterpart. Moreover, the usual invariance principles (relating to the Wiener process approximations) usually filter through jackknifing under no extra regularity conditions. These results are then incorporated in providing a bounded-length (sequential) confidence interval and a preassigned-strength sequential test for a suitable parameter based on jackknife estimators.

Material in Chapter VI of Hajek and Sidak's book is extended to a sequential analysis setting: conditions are given under which a sequence of log-likelihood-ratio processes (log-likelihood-ratios for sequential sampling, represented as jump processes in continuous time) converges weakly to a Wiener process with drift, the drift parameter depending on which hypothesis, in a suitable neighborhood of a null hypothesis, prevails. Conditions for convergence of other "test statistic" processes, related to likelihood ratios, are also given. Asymptotic sequential tests can thereby be constructed. Some "two-sample problem" examples are treated.

The sequential estimation of $p$, the probability of success in a sequence of Bernoulli trials, is considered for the case where loss is taken to be symmetrized relative squared error of estimation, plus a fixed cost $c$ per observation. Using $s_n/n$ as a terminal estimator of $p$, where $s_n$ is the number of successes in $n$ trials, a heuristic rule is derived and shown to perform well for any fixed $0 < p < 1$ as $c \rightarrow 0$. However for any fixed $c > 0$, this rule performs badly for $p$ close to 0 or 1. To overcome this difficulty a uniform prior on $p$ is introduced, and the optimal Bayes procedure is shown to exist and to have bounded sample size. The optimal Bayes risk is shown to be $\sim 2\pi c^{\frac{1}{2}}$ as $c \rightarrow 0$, and is computed for various values of $c$, along with the expected loss for various values of $p$.

A Berry-Esseen bound is obtained for trimmed linear combinations of order statistics. These linear combinations are written as the sum of a linear and a quadratic combination of independent exponentially distributed random variables plus a remainder term. The remainder term is shown to be of negligible order and Fourier methods are then employed to handle the linear and quadratic terms. The main theorem is also given in a version that more easily lends itself to applications.

The empirical distribution function $P_n$ converges with probability 1 to a true distribution $P$ in $R^k$, uniformly over measurable convex sets, if and only if $P$ is a countable mixture of distributions, each of which is carried by a flat and gives zero probability to the relative boundaries of convex sets included in the flat.

For use in comparisons of sequential and nonsequential tests Berk (Ann. Statist. 3 991-998) has defined the limiting relative efficiency of sequential tests as the limiting ratio of the expected sample size under the null hypothesis and the supremum over the parameter set of the expected sample size. He has proved that for the symmetric binomial case the limiting relative efficiency of a class of SPR type tests coincides with a related quantity for SPR tests of drift in a Wiener process. He has also conjectured that this result applies to a more general class. In this note we prove that it holds for exponential families satisfying some mild regularity conditions.

It is desired to estimate the difference between the means of two independent normal distributions as accurately as possible and in a sequential manner when the total number of observations is fixed. The problem is posed in a Bayesian framework with conjugate prior distributions and squared error loss function. It is shown that the optimal sequential design depends on the ratio of the posterior variances of the two means. There exist constants (dependent on the prior parameters, the number of observations taken from each distribution, and the number of observations remaining) such that when the above-mentioned ratio exceeds this constant it is optimal to select the next observation from one distribution; otherwise it is optimal to select it from the other distribution.

Sequences of tests with error $\exp(-nA)$ of the first type are investigated. It is shown that the error of the second type of such a sequence of tests is bounded by $\exp(- nB)$ where $B$ is determined by the Kullback-Leibler information distance of the hypotheses tested. The information distance between the empirical measure and the null-hypothesis on a finite partition of the sample space is proposed to use as a test statistic. A sufficient condition is given which ensures that this test has error of the second type about $\exp(- nB)$ with the best possible $B$. The exact Bahadur slope of the proposed statistic is investigated.

Based on a random sample from a univariate distribution with density $f$, this note exhibits a class of kernel estimators of the $p$th order derivative $f^{(p)}$ of $f, p \geqq 0$ fixed. These estimators improve some known estimators of $f^{(p)}$ by weakening the conditions, sharpening the rates of convergence, or both for the properties of strong consistency, asymptotic unbiasedness and mean square consistency, each uniform on the real line.

It is shown that the limiting distribution of any regular estimator of a continuous cdf on [0, 1] can be represented as a convolution of the Brownian bridge process with another distribution on $C\lbrack 0, 1\rbrack$. The result is related to Hajek's representation for limiting distributions of regular parametric estimators.

In establishing weak convergence of von Mises' differentiable statistical functions to a normal distribution usually square integrability conditions with respect to the underlying kernel function are assumed. It is shown that these conditions can be weakened by assuming integrability of the von Mises' functional itself. In addition it is pointed out that in nontrivial cases the conditions of square integrability of the kernel do not hold whereas weak convergence of the von Mises' functional can still be proved.

Consider a sample from a multiple time series that is stationary and Gaussian. A test is presented for independence among the multivariate observations that comprise this sample. The test is a generalization of the Kolmogorov-Smirnov test for serial correlation in a single time series. In the test, pairs of spectral-matrix estimates are compared using the largest-root statistic. The comparisons, which are tested simultaneously, are between estimates obtained from upper and lower parts of the frequency band. Under the null hypothesis, the joint distribution of the largest roots is obtained in a form suitable for computation of significance levels.

Let $P_1$ be a finite population sampling plan and $V$ a collection of subsets of units. The inclusion probabilities for members of $V$ may be calculated. For example, if $V$ comprises all single units and pairs of units we obtain all first and second order inclusion probabilities $\pi_i, \pi_{ij}$. Another plan $P_2$ is called equivalent to $P_1$ with respect to $V$ if the corresponding inclusion probabilities for $P_1$ are equal to those for $P_2$. However, $P_2$ may have fewer samples with positive probability of selection, that is to say smaller "support." An upper bound is put on the minimum support size of all such $P_2$. For $P_1$ simple random sampling, some examples are given for $P_2$ with small support.