We consider generalized sequential probability ratio tests (GSPRT's), which are not necessarily based on independent or identically distributed observations, to distinguish between probability measures $P$ and $Q$. It is shown that if $T$ is any test in a wide class of GSPRT's, including all SPRT's, and $T'$ is any rival test possessing error probabilities and sample sizes no greater than those of $T$, then $T'$ must be equivalent to $T$. This notion of optimality of $T$ is weaker than that of Kiefer and Weiss but the results are stronger than theirs. It is also shown that, if an SPRT $T'$ has at least one error probability strictly less than that of another SPRT $T$ with the other error probability no larger, $T'$ requires strictly more observations than $T$ some of the time, under both $P$ and $Q$, and never fewer observations. This assertion generalizes Wijsman's conclusions. The methods used in this paper are quite general, and are different from those of the earlier authors.

Let $\{Z_i\}$ be i.i.d., let $\{\varepsilon_i\}$ be i.i.d. Bernoulli, independent of $\{Z_i\}$, let $T_0 = z$ and $T_n = \varepsilon_n(T_{n-1} + Z_n)$ for $n \geqq 1$. Under a moment condition, optimal stopping rules are found for stopping $T_n - nc$ where $c > 0$ (the cost model), and for stopping $\beta^nT_n$ where $0 < \beta < 1$ (the discount model). Special cases are treated in detail. The cost model generalizes results of N. Starr, and the discount model generalizes results of Dubins and Teicher.

This paper is concerned with confidence sequences, i.e., sequences of confidence regions which contain the true parameter for every sample size simultaneously at a prescribed level of confidence. By making use of generalized likelihood ratio martingales, confidence sequences are constructed for the unknown parameters of the binomial, Poisson, uniform, gamma and other distributions. It is proved that for the exponential family of distributions, the method of using generalized likelihood ratio martingales leads to a sequence of intervals which have the desirable property of eventually shrinking to the population parameter. The problem of nuisance parameters is considered, and in this connection, boundary crossing probabilities are obtained for the sequence of Student's $t$-statistics, and a limit theorem relating to the boundary crossing probabilities for the Wiener process is proved.

A simple combination of one-sided sequential probability ratio tests, called a 2-SPRT, is shown to approximately minimize the expected sample size at a given point $\theta_0$ among all tests with error probabilities controlled at two other points, $\theta_1$ and $\theta_2$. In the symmetric normal and binomial testing problems, this result applies directly to the Kiefer-Weiss problem of minimizing the maximum over $\theta$ of the expected sample size. Extensive computer calculations for the normal case indicate that 2-SPRT's have efficiencies greater than 99% regardless of the size of the error probabilities. Accurate approximations to the error probabilities and expected sample sizes of these tests are given.

Approximation of the covariance matrix $\mathbf{\Sigma}$ of $T$ consecutive observations from a second-order stationary process with continuous positive spectral density $f(\lambda) = \lbrack\sigma^2/(2\pi)^2\rbrack|\sum^\infty_{j=0}\delta_je^{i\lambda j}|^2$ is considered. If $\mathbf{\Sigma}^\ast$ is the covariance matrix corresponding to a process with spectral density $1/\lbrack(2\pi)^2f(\lambda)\rbrack$, then $\mathbf{\Sigma}^\ast - \mathbf{\Sigma}^{-1} \geqq 0$. A matrix $\sigma^{-2}\mathbf{A'A}$ with the property that $\mathbf{\Sigma}^\ast - \sigma^{-2}\mathbf{A'A} - \mathbf{\Sigma}^{-1} \geqq 0$ is also considered. For autoregressive-moving average processes of order $(p, q), \mathbf{\Sigma}^\ast - \sigma^{-2}\mathbf{A'A}$ and $\sigma^{-2}\mathbf{A'A} - \mathbf{\Sigma}^{-1}$ are shown to have rank $\min \lbrack\max (p, q), T\rbrack$ and $\mathbf{\Sigma}^\ast - \mathbf{\Sigma}^{-1}$ to have rank $\min \lbrack 2\max (p, q), T\rbrack$. Some results concerning the covariance determinant are also discussed. If $D_T$ is $\sigma^{-2T}|\mathbf{\Sigma}|$ for sample size $T$ and $D_0 = 1$, then $D_T < D_{T+1}, T = 0, 1, \cdots$, unless the process is autoregressive of order $p$, in which case $1 < D_1 < \cdots < D_p = D_{p+1} = \cdots$.

Let $X = (X_1, \cdots, X_n)$ be an $n$-dimensional random vector with density $f(x - \theta)$. Assume the loss incurred in estimating $\theta$ by $d$ is $L(d - \theta)$ and is convex. Let $F$ be a generalized prior density. The question of inadmissibility of generalized Bayes estimators of $\theta$ is considered. As in Brown (1974c), the crucial role played by the "moment structure" in determining inadmissibility results is indicated (moments defined as the quantities $m_{i,j(1), \cdots, j(l)} = \int \prod^l_{k=1} x_{j(k)} \cdot \lbrack(\partial/\partial x_i)L(x)\rbrack f(x) dx)$. Detailed inadmissibility results are given for a particular moment structure, one which arises most naturally in trying to estimate a single coordinate or a linear combination of coordinates of $\theta$. For example, suppose $\theta_1$ is to be estimated by the generalized Bayes estimator $\delta_F$. (Thus $\theta_2, \cdots, \theta_n$ are nuisance parameters.) Under certain conditions, the most important being that the moment structure be of a certain form, it is shown that if there exist constants $\lambda > 0$ and $T > 0$ such that $\delta_F(x) \geqq x_1 - (n - 3 - \lambda)\big/ \big\lbrack 2x_1 \int\big(\frac{\partial^2}{\partial x_1^2} L(x)\big) f(x) dx\big\rbrack \quad \text{for} x_1 > T,$ then $\delta_F$ is inadmissible. Thus, for example, under certain quite general assumptions, the best invariant estimator, $\delta_0(x) = X_1$, of the first coordinate of a location vector is inadmissible if $n \geqq 4$.

Let $X$ be an $n$-dimensional random vector with density $f(x - \theta)$. It is desired to estimate $\theta_1$, under a strictly convex loss $L(\delta - \theta_1)$. If $F$ is a generalized Bayes prior density, the admissibility of the corresponding generalized Bayes estimator, $\delta_F$, is considered. An asymptotic approximation to $\delta_F$ is found. Using this approximation, it is shown that if (i) $f$ has enough moments, (ii) $L$ and $F$ are smooth enough, and (iii) $F(\theta) \leqq K(|\theta_1| + \sum^n_{i=2} \theta_i^2)^{(3-n)/2}$, then $\delta_F$ is admissible for estimating $\theta_1$. For example, assume that $F(\theta) \equiv 1$ and that $L$ is squared error loss. Under appropriate conditions it can be shown that $\delta_F(x) = x_1$, and that $\delta_F$ is the best invariant estimator. If, in addition, $f$ has 7 absolute moments and $n \leqq 3$, it can be concluded that $\delta_F$ is admissible.

Percentage points are given for the asymptotic distributions of the goodness-of-fit statistics $W^2, U^2$ and $A^2$, for the cases where the distribution tested is (a) normal, with mean or variance, or both, unknown; (b) exponential, with scale parameter unknown. Some exact means and variances are also given. The distributions can be expressed as a sum of weighted chi-square variables; the weights are calculated, and the higher cumulants can then be found. The first four cumulants are used to approximate the distributions and give the percentage points.

Associated with any Borel gambling model $G$ or dynamic programming model $D$ is a corresponding class of stochastic processes $M(G)$ or $M(D)$. Say that $G(D)$ is regular if there is a $D(G)$ with $M(D) = M(G)$. Necessary and sufficient conditions for regularity are given, and it is shown how to modify any model slightly to achieve regularity.

The approximate Bahadur slope of the Chernoff-Lehmann $\chi^2$-test-of-fit to a scale-location family on $R^k$ is computed. The goal is to select cells (whose number is independent of sample size) to maximize this slope. The supremum is found and is shown to be a maximum only in trivial cases. If the $\sup$ is finite there is always a best selection for a fixed number of cells. Equally likely cells are shown to be admissible when the alternative is large.

The general mixed linear model can be written $y = X\alpha + Zb$, where $\alpha$ is a vector of fixed effects and $b$ is a vector of random variables. Assume that $E(b) = 0$ and that $\operatorname{Var} (b) = \sigma^2D$ with $D$ known. Consider the estimation of $\lambda_1'\alpha + \lambda_2'\beta$, where $\lambda_1'\alpha$ is estimable and $\beta$ is the realized, though unobservable, value of $b$. Among linear estimators $c + r'y$ having $E(c + r'y) \equiv E(\lambda_1'\alpha + \lambda_2'b)$, mean squared error $E(c + r'y - \lambda_1'\alpha - \lambda_2'b)^2$ is minimized by $\lambda_1'\hat{\alpha} + \lambda_2'\hat{\beta}$, where $\hat{\beta} = DZ'V^{\tt\#}(y - X\hat{\alpha}), \hat{\alpha} = (X'V^{\tt\#}X) - X'V^{\tt\#}y$, and $V^{\tt\#}$ is any generalized inverse of $V = ZDZ'$ belonging to the Zyskind-Martin class. It is shown that $\hat{\alpha}$ and $\hat{\beta}$ can be computed from the solution to any of a certain class of linear systems, and that doing so facilitates the exploitation, for computational purposes, of the kind of structure associated with ANOVA models. These results extend the Gauss-Markov theorem. The results can also be applied in a certain Bayesian setting.

A central limit theorem is proved for the sample serial covariances of an ergodic, stationary, purely nondeterministic process whose linear innovations have their first four moments as for a sequence of independent random variables. The necessary and sufficient condition for the theorem is then that the spectra be square integrable.

Conditions to ensure the asymptotic normality of rank statistics having a scores generating function with finitely many jumps are obtained. These conditions are derived by studying rank statistics with a scores generating function $J$ such that $J(u) = 1$ or $0$ as $u \geqq s$ or $u < s$ for a fixed $s, 0 < s < 1$. No differentiability conditions are imposed on the underlying distribution functions at the jump points of the scores generating function.

For $N = 1, 2, \cdots$ let $X_{1N}, X_{2N}, \cdots, X_{NN}$ be independent rv's having continuous df's $F_{1N}, F_{2N}, \cdots, F_{NN}$. For the set $X_{1N}, X_{2N}, \cdots, X_{NN}$, let us denote by $X_{1:N} \leqq X_{2:N} \leqq \cdots \leqq X_{N:N}$ the order statistics, by $\mathbb{F}_N$ the empirical df and by $\bar{F}_N$ the averaged df, i.e. $\bar{F}_N(x) = N^{-1}\sum^N_{n=1} F_{nN}(x)$ for $x\epsilon(-\infty, \infty)$. It is shown that for each $\varepsilon > 0$ there exists a $0 < \beta(= \beta_\varepsilon) < 1$, independent of $N$, such that for $N = 1, 2, \cdots$, $(a) P(\mathbb{F}_N(x) \leqq \beta^{-1}\bar{F}_N(x), \text{for} x \epsilon (-\infty, \infty)) \geqq 1 - \varepsilon,$ $(b) P(\mathbb{F}_N(x) \geqq \beta\bar{F}_N(x), \text{for} x \epsilon \lbrack X_{1:N}, \infty)) \geqq 1 - \varepsilon.$ Moreover, these assertions hold uniformly in all continuous df's $F_{1N}, F_{2N}, \cdots, F_{NN}$. The theorem can be used to prove asymptotic normality of rank statistics and of linear combinations of functions of order statistics in the case where the sample elements are allowed to have different df's.

The Pearson-Fisher $\chi^2$ statistic is asymptotically chi-square under the null hypothesis with $M - m - 1$ degrees of freedom where $M =$ number of cells and $m =$ dimension of parameter. The Chernoff-Lehmann statistic is a weighted sum of chi-squares and the Kambhampati statistic is $\chi^2$ with $M - 1$ degrees of freedom. The approximate Bahadur slopes of the tests based on these statistics are computed. It is shown that the Kambhampati test always dominates the Chernoff-Lehmann and that no such dominance exists between the Pearson-Fisher test and Kambhampati test, or the Pearson-Fisher and Chernoff-Lehmann.

Uniformly most accurate level $1 - \alpha$ confidence procedures for a linear function $\mu + \lambda\sigma^2$ with known $\lambda$ for the parameters of a normal distribution defined by Land were previously shown for both the one-sided and two-sided procedures to be always intervals for $\nu \geqq 2, \nu$ being the number of degrees of freedom for estimating $\sigma^2$. These results are shown in this paper to hold also in the case $\nu = 1$. During the course of the argument a new inequality is obtained relating to the modified Bessel functions which is of independent interest.

In a two-factor design, a design which is optimal for one factor is shown to be optimal jointly for both the factors with respect to each of $A$-, $D$-, and $E$-optimalities. As an interesting consequence we have that a linked block design is optimal for the estimation of treatment differences. Similar results are also obtained for a class of three factor designs.

This paper generalizes the results of Paik and Federer (1970), Ehrenfeld and Zacks (1961) and Zacks (1963, 1964) regarding invariance and randomization in fractional replication. It is shown that (i) the characteristic roots of the information matrix of a design in the general factorial relative to an admissible vector of effects remain invariant under a permutation of levels; (ii) the unbiased estimation of a linear function of an admissible vector of effects can be obtained under equal probability randomization. In addition some applications of the results are indicated.

Empirical Bayes estimators, asymptotically optimal with rates, are proposed. In the component problem there is a pair $(X, \omega)$ of real valued random variables. The Lebesgue density of $X$, conditional on $\omega$, is of the form $u(x)C(\omega)e^{\omega x}$. Based on a realization of $X$, the problem is squared error loss estimation of $\omega$. Let $G$ be a prior distribution on $\omega$, and $R(G)$ be the Bayes optimal risk wrt $G$. Using $(X_1, \cdots, X_n)$, the observations in the past $n$ such problems, mean square consistent estimators of the derivative of $\log (\int C(\omega)e^{\omega x} dG(\omega))$ are proposed. Then these statistics and the present observation $X$ are used to exhibit estimators $\psi_n$ for the present problem whose risks $R_n$ converge to the Bayes optimal risk $R(G)$ as $n \rightarrow \infty$. In particular, with no assumption on the smoothness or on the form of $u$, a $\psi_n$ for each $\gamma$ in [0, 2) is exhibited. Sufficient conditions are given under which $c_1n^{-4/(4+3\gamma)} \leqq R_n - R(G) \leqq c_2n^{-2\gamma/(4+3\gamma)}$, where $c_1$ and $c_2$ are positive constants. The rhs inequality holds uniformly in $G$ with support in a bounded interval of the real line, while the other holds for a $G$ degenerate at a point and for all $n$ sufficiently large. (Thus with $\gamma$ close to $2, \psi_n$ achieves almost the exact rate.) Examples of families, including one whose $u$ function has infinitely many discontinuities, are given where conditions for the above inequalities are satisfied for $\gamma$ arbitrarily close to 2.