Asymptotic approximations to the expected sample size are given for a class of tests of power one introduced in [10]. Comparisons are made with the method of mixtures of likelihood ratios, and an application is given to Breiman's gambling theory for favorable games.

Using the model of random censorship, a necessary and sufficient condition for the consistency of the standard (actuarial) life table estimate of a survival distribution is derived. We establish the asymptotic normality of this estimate, showing that Greenwood's variance formula is nearly correct. In the case of a continuous survival distribution we establish limiting normality for the product limit estimate and for the closely related cumulative hazard process. Some applications of these results are outlined.

Let $f(x)$ be a continuous, strictly positive probability density function over an interval $\lbrack a, b\rbrack$ and $F(x)$ its associated $\operatorname{cdf}$. Suppose $\{\phi_i(x)\}^\infty_{i=0}$ is a complete orthonormal basis for $L_2\lbrack a, b\rbrack$ and that $f(x)$ and $\log f(x)$ have orthogonal series expansions, in the $\phi_i$'s, over $\lbrack a, b\rbrack$. Estimators for $f(x)$ and $F(x)$ are chosen from the canonical exponential family of distributions generated by $\{\phi_i(x)\}^\infty_{i=0}$, and convergence theorems are presented for these estimators in the special case of Legendre polynomials over $\lbrack -1, 1\rbrack$.

For a random variable $X$ with possible distributions indexed by a parameter $\theta$, and for real-valued $T = T(X)$ and $V = V(X, \theta)$ with $\operatorname{Var} T < \infty$ and $0 < \operatorname{Var} V < \infty$, Schwarz's inequality gives $\operatorname{Var} T \geqq \{\operatorname{Cov} (T, V)\}^2/\operatorname{Var} V$. Necessary and sufficient conditions are given for this inequality to be of Cramer-Rao type: $\operatorname{Var} T \geqq \{a_m(\theta)\}^2/\operatorname{Var} V$ where $m(\theta)$ is a notation for $ET$ and $a_m(\theta)$ is a notation for $\operatorname{Cov} (T, V)$. Specialized to $V = \{\partial p\theta(X)/\partial\theta\}/p_\theta(X)$, where $p_\theta$ is a probability density function for $X$, these conditions are necessary and sufficient for validity of the Cramer-Rao inequality. The use of these inequalities in proving an estimator minimum variance unbiased is shown to be superfluous. The use of these inequalities in proving admissibility is discussed, with examples.

Let $f$ be a density which vanishes for negative values of its argument and varies regularly with exponent $\alpha - 1$ at zero, where $1 < \alpha < 2$. Further, let $f_\theta$ denote $f$ translated by $\theta$. We find and study the asymptotic distribution of the MLE $\hat{\theta}_n$ based on a sample size $n$ as $n \rightarrow \infty$.

We consider an autoregressive linear process $\{x_t\}$, a one-sided moving average, with summable coefficients, of independent identically distributed variables $\{e_t\}$ with zero mean and fourth moment, such that $\{e_t\}$ is expressible in terms of past values of $\{x_t\}$. The spectral density of $\{x_t\}$ is assumed bounded and bounded away from zero. Using data $x_1,\cdots, x_n$ from the process, we fit an autoregression of order $k$, where $k^3/n \rightarrow 0$ as $n \rightarrow \infty$. Assuming the order $k$ is asymptotically sufficient to overcome bias, the autoregression yields a consistent estimator of the spectral density of $\{x_t\}$. Furthermore, assuming $k$ goes to infinity so that the bias from using a finite autoregression vanishes at a sufficient rate, the autoregressive spectral estimates are asymptotically normal, uncorrelated at different fixed frequencies. The asymptotic variance is the same as for spectral estimates based on a truncated periodogram.

Wald (1943) formulated criteria of optimality relative to experimental designs and proved that some known intuitively attractive designs are in fact optimal. Kiefer (1958), (1959), (1961), (1971) extended Wald's work on criteria of optimality and suggested some generalizations of experimental designs discussing their merits in respect to the formulated criteria. The purpose of this paper is to exhibit constructions of some families of Generalized Youden Designs (GYD) introduced by Kiefer. In his published work Kiefer gave some small examples of Generalized Youden Designs mainly for the purpose of illustrating the theory on criteria of optimality.

This paper extends some recent results on asymptotically optimal sequences of experimental designs for regression problems in stochastic processes. In the regression model $Y(t) = \beta f(t) + X(t), 0 \leqq t \leqq 1$, the constant $\beta$ is to be estimated based on observations of $Y(t)$ and its first $m - 1$ derivatives at each of a set $T_n$ of $n$ distinct points. The function $f$ is assumed known as is the covariance kernel of $X(t)$, a zero-mean $m$th order autoregressive process. Under certain conditions, we derive a sequence $\{T_n\}$ of experimental designs which are asymptotically optimal for estimating $\beta$.

Brunk, Franck, Hanson and Hogg (1966) ("Maximum likelihood estimation of the distributions of two stochastically ordered random variables," J. Amer. Statist. Assoc. 61 1067-1080) found and studied maximum likelihood estimates of a pair of stochastically ordered distribution functions. In this paper we discuss a generalization of this problem in which we do not require the domain of these "distribution functions" to be the real line. We think of the order restriction we impose on these "distribution functions" as an analogue of stochastic ordering on the line. Maximum likelihood estimates are found and strong uniform consistency properties are discussed.

In this note we apply the sequential theory developed by Chow and Robbins [1] and Gleser [3], [4] to the inverse linear regression problem. A two-stage sequential procedure has been proposed for the construction of a fixed-width confidence interval for $x$ (an unknown parameter). It is shown that the limiting probabilities of "correct decision" are equal to $P^\ast$ (pre-assigned).

In this paper, for the one and two sample location problems, a class of sequential tests based on robust rank order statistics is developed. The proposed tests terminate with probability one for square integrable score functions. Under more stringent regularity conditions and for local alternatives, the OC and ASN of the proposed tests are obtained, and the allied asymptotic relative efficiency results with respect to the sequential probability ratio and likelihood ratio tests are studied.

Let $X_1, X_2,\cdots, X_n$ be independent observations from an exponential distribution with an unknown location-scale parameter $(\mu, \sigma)$. Let $\bar{X} = n^{-1} \sum X_i$ and $M = \min X_i$. Under squared error loss the best location-scale equivariant estimator of $\sigma$ is $\bar{X} - M$, which agrees with the maximum likelihood estimator. Arnold (J. Amer. Statist. Assoc. 65 (1970) 1260-1264) and Zidek (Ann. Statist. 1 (1973) 264-278) have shown that $\bar{X} - M$ is inadmissible but the dominating estimator which they produce is probably inadmissible as well. In this paper a "smoother" dominating procedure is presented, and the risk functions of the various alternatives are plotted. Similar results are obtained for strictly bowl-shaped loss functions.

Two problems, classifying an individual into one of several populations and estimating the regression in that population, are simultaneously treated as one problem. This can be viewed as a problem of a missing observation on a categorical variable. When all variables are jointly distributed multivariate normal, the maximum likelihood solution is the intuitively appealing one: classify the individual using the usual likelihood ratio procedure, then estimate the regression using the observations from the selected population.

The purpose of this paper is to present robust estimates for the regression parameters in the general linear model. We start with a family of $M$-estimators, and using the observations, we estimate the asymptotic efficiency of each member in the family. Then we choose the estimate in the family with greatest estimated asymptotical efficiency. We prove that this procedure has the same asymptotical efficiency as the member of the family with the greatest asymptotical efficiency for the unknown distribution of the error.

Estimating equations for a real parameter $\theta$ which indexes a family of densities $p(x, \theta)$ were considered in the note by Godambe (Ann. Math. Statist. 31 (1960) 1208-1211). An optimality property of the equation $\partial \log p/\partial \theta = 0$ among unbiased estimating equations was established. In this paper an analogous result is proved for estimation of a real parameter $\theta_1$ in the presence of a nuisance parameter $\theta_2$.

We study several combinatorial problems related to the choice of groups of subsets to be used in the subsampling procedure of Hartigan (J. Amer. Statist. Assoc. 1969). The complete separation property is shown to guarantee asymptotic efficiency relative to $t$. Problems associated with the relative variance criterion are discussed.

An algebraic technique which maps elements from distinct finite rings into subsets of another finite ring is defined, and a method for combining elements from distinct finite rings is demonstrated. The connection between this mapping and the constructing of confounding plans for the mixed factorial experiment with any number of factors at any number of levels is established, as well as the limitation of the procedure.

Using Pyke-Shorack (Ann. Math. Statist. (1968) 755-771) approach, based on weak-convergence properties of empirical processes, a Chernoff-Savage theorem concerning the asymptotic normality of two-sample linear rank statistics is proved for stationary $\phi$-mixing sequences $\{X_m\}$ and $\{Y_n\}$ of rv's. This main result (Theorem 4.1) is almost as strong as proved by Pyke and Shorack for sequences of independent rv's. The basic tool employed is the following new result concerning the behavior of empirical process $\{U_m(t): 0 \leqq t \leqq 1\}$ near 0 and 1 under $\phi$-mixing: For given $\varepsilon > 0$, the $P\lbrack(t(1 - t))^{-\frac{1}{2} + \delta}|U_m(t)| \leqq \varepsilon \forall 0 \leqq t \leqq \theta\rbrack, (0 < \delta < \frac{1}{2}, 0 < \theta < \frac{1}{2})$, can be made arbitrarily close to 1 by taking $m$ sufficiently large and $\theta$ sufficiently small.

If a sequence of distributions converges weakly to a distribution and if the associated densities are unimodal then the sequence of density functions converges to the density of the limiting distribution. This stronger form of convergence is useful in the investigation of limiting joint distributions.

A positive random variable $X$ whose mean exists, has a Pareto distribution if, and only if, $E(X \mid X > x) = h + gx$ for $g > 1$. This characterization was motivated by the fact that the Pareto distribution has been widely used to model income. Now, suppose that individuals under-report their income for income tax purposes. If one assumes that for a given income, the average under-reporting is a constant fraction of the amount by which the income exceeds the tax exempt level, then the average under-reporting error for a given reported income is a linear function of the reported income if, and only if, incomes follow a Pareto distribution.

A gambler seeks to maximize his probability of reaching a goal in a game where he is allowed at each stage to stake any amount of his current fortune. He wins each bet with a certain fixed probability $w$. Lester E. Dubins and Leonard J. Savage found optimal strategies for a gambler who knows $w$. Here strategies are found which are uniformly nearly optimal for all $w$ and, therefore, also for a gambler with an unknown $w$.

Let $Y_1, \cdots, Y_n$ be independent random variables with mean zero such that $|Y_i| \leqq i, i = 1, \cdots, n$, and let $\theta_1,\cdots, \theta_n$ be real numbers satisfying $\sum^n_1 \theta_i^2 = 1$. Set $S_n(\theta) = \sum^n_1 \theta_i Y_i$ and let $\varphi(x) = (2\pi)^{-\frac{1}{2}} \exp \lbrack - \frac{1}{2} x^2\rbrack$. THEOREM. For $\alpha > 0$, and for all $\theta_1,\cdots, \theta_n$, \begin{\align*}P\{|S_n(\theta)| \geqq \alpha\} &\leqq 2\inf_{0\leqq u \leqq \alpha} \int^\infty_u \frac{(x - u)^3}{(\alpha - u)^3} \varphi (x) dx \\ &\leqq 12 \frac{\varphi(\alpha)}{\alpha} \inf_{0\leqq \delta \leqq \alpha^2} \frac{\exp\lbrack\delta/2(2 - \delta/\alpha^2)\rbrack}{\delta^3(1 - \delta/\alpha^2)^4}.\\ \end{align*}