Let $\mathbf{B} = (N_1, \cdots, N_k)$ be a multivariate counting process and let $\mathscr{F}_t$ be the collection of all events observed on the time interval $\lbrack 0, t\rbrack.$ The intensity process is given by $\Lambda_i(t) = \lim_{h \downarrow 0} \frac{1}{h}E(N_i(t + h) - N_i(t) \mid \mathscr{F}_t)\quad i = 1, \cdots, k.$ We give an application of the recently developed martingale-based approach to the study of $\mathbf{N}$ via $\mathbf{\Lambda}.$ A statistical model is defined by letting $\Lambda_i(t) = \alpha_i(t)Y_i(t), i = 1, \cdots, k,$ where $\mathbf{\alpha} = (\alpha_1, \cdots, \alpha_k)$ is an unknown nonnegative function while $\mathbf{Y} = (Y_1, \cdots, Y_k),$ together with $\mathbf{N},$ is a process observable over a certain time interval. Special cases are time-continuous Markov chains on finite state spaces, birth and death processes and models for survival analysis with censored data. The model is termed nonparametric when $\mathbf{\alpha}$ is allowed to vary arbitrarily except for regularity conditions. The existence of complete and sufficient statistics for this model is studied. An empirical process estimating $\beta_i(t) = \int^t_0 \alpha_i(s) ds$ is given and studied by means of the theory of stochastic integrals. This empirical process is intended for plotting purposes and it generalizes the empirical cumulative hazard rate from survival analysis and is related to the product limit estimator. Consistency and weak convergence results are given. Tests for comparison of two counting processes, generalizing the two sample rank tests, are defined and studied. Finally, an application to a set of biological data is given.

This paper proves the apparently outstanding conjecture that the maximum likelihood estimate (m.l.e.) "behaves properly" when jackknifed. In particular, under the usual Cramer conditions (1) the jackknifed version of the consistent root of the m.l. equation has the same asymptotic distribution as the consistent root itself, and (2) the jackknife estimate of the variance of the asymptotic distribution of the consistent root is itself consistent. Further, if the hypotheses of Wald's consistency theorem for the m.l.e. are satisfied, then the above claims hold for the m.l.e. (as well as for the consistent root).

This paper provides an empirical Bayes approach to the problem of nonparametric estimation of a distribution (or survival) function when the observations are censored on the right. The results use the notion of a Dirichlet process prior introduced by Ferguson. The paper presents a generalization to the case of right censored observations of the rate result of an empirical Bayes nonparametric estimator of a distribution function of Korwar and Hollander in the uncensored case. The rate of asymptotic convergence of optimality is shown to be the best obtainable for the problem considered.

Let $X_1, \cdots, X_n$ be i.i.d. $F_0$ and let $Y_1, \cdots, Y_n$ be independent (and independent also of $X_1, \cdots, X_n$) random variables. Then assuming that $F$ is distributed according to a Dirichlet process with parameter $\alpha,$ the authors obtained the Bayes estimator $\hat{F}_\alpha$ of $F$ under the loss function $L(F, \hat{F}) = \int (F(u) - \hat{F}(u))^2 dw(u)$ when $X_1, \cdots, X_n$ are censored on the right by $Y_1, \cdots, Y_n,$ respectively, and when it is known whether there is censoring or not. Assuming $X_1, \cdots, X_n$ are i.i.d. $F_0$ and $Y_1, \cdots, Y_n$ are i.i.d. $G,$ this paper shows that $\hat{F}_\alpha$ is mean square consistent with rate $O(n^{-1}),$ almost sure consistent with rate $O(\log n/n^\frac{1}{2}),$ and that $\{\hat{F}_\alpha(u) \mid 0 < u < T\}, T < \infty,$ converges weakly to a Gaussian process whenever $F_0$ and $G$ are continuous and that $P\lbrack X_1 > u\rbrack P\lbrack Y_1 > u\rbrack > 0.$

Unbiased risk estimators are derived for estimators in certain classes of equivariant estimators of multinormal matrix means, $\xi,$ and regression coefficients $\beta.$ In all cases the covariance matrix is unknown. The underlying method, a multivariate version of that of James and Stein (1960), uses zonal polynomial expansions for the distributions of noncentral statistics. This gives, in one case, the required generalization of the Pitman-Robbins representation of noncentral chi-square statistics including the appropriate multivariate Poisson law. In the other case, a multivariate negative binomial law emerges. The result for regression coefficients suggests a new minimax estimator and, essentially, an extension of Baranchik's result.

Several problems involving multivariate generalized Bayes estimators are investigated. First, a characterization of admissible estimators as generalized Bayes estimators is developed for certain multivariate exponential families and quadratic loss. The problem of verifying whether or not an estimator is generalized Bayes is also considered. Next, an important class of estimators for a multivariate normal mean is considered. (The class includes many minimax, empirical Bayes, and ridge regression estimators of current interest.) Necessary conditions are developed for an estimator in this class to be "nearly" generalized Bayes, in the sense that if it were properly smoothed, it would be generalized Bayes. An application to adaptive ridge regression is given. The paper concludes with the development of an asymptotic approximation to generalized Bayes estimators for general losses and location vector densities. Using this approximation, weakened versions of the above results are obtained for general losses and densities.

Consider a random sample from an absolutely continuous multivariate distribution. Let $\mathscr{J}$ be a class of sets which are not too long and thin. A point $\mathbf{\theta}_n$ chosen from a minimum volume set $S_n \in \mathscr{J}$ containing at least $r = r(n)$ of the data may be used as an estimate of the mode of the distribution. In this paper, it is shown that $\mathbf{\theta}_n$ converges almost surely to the true mode under very minor conditions on $\{r(n)\}$ and the distribution. Convergence rates are also obtained. Extensions to estimation of local and/or multiple modes are noted. Finally, computational simplifications resulting from choosing $S_n$ from spheres or cubes centered at observations are discussed.

An asymptotic expression is given for the $\log$ error probability of a sequential test based on a random walk. This may be used to compute limiting relative efficiencies of such tests. The results are illustrated for the one-sided normal testing problem with an asymptotic Bayes test due to Schwarz. Some numerical comparisons are given for five sequential tests of a normal mean.

Let $(\mathscr{X}, \mathscr{B}_1, \mu)$ and $(\mathscr{Y}, \mathscr{B}_2, \nu)$ be $\sigma$-finite measure spaces and suppose $\Theta$ is a separable metric space. Let $f(x \mid y, \theta)$ be a family of conditional densities on $(\mathscr{X}, \mathscr{B}, \mu).$ Consider an action space $A$ which is a compact metric space with $\mathscr{B}_A$ the Borel $\sigma$-algebra and a loss function $W(\theta, a)$ such that $W(\theta, \bullet)$ is continuous. For any decision rule $\delta: \mathscr{B}_A \times \mathscr{X} \rightarrow \lbrack 0, 1\rbrack,$ assume the risk function $R(\delta, \bullet)$ is continuous on $\Theta.$ Suppose that a set of decision rules $\mathscr{M}_0$ is an essentially complete class for each $y \in \mathscr{Y}$ for the conditional decision problem. Let $\mathscr{M}^\ast$ be the set of decision rules $\eta: \mathscr{B}_A \times (\mathscr{X} \times \mathscr{Y}) \rightarrow \lbrack 0, 1\rbrack$ such that $\eta(\bullet \mid \bullet, y) \in \mathscr{M}_0 \mathrm{a.e.} \lbrack \nu\rbrack.$ Then $\mathscr{M}^\ast$ is an essentially complete class no matter what the family of marginal densities on the space $(\mathscr{Y}, \mathscr{B}_2, \nu).$

A reduced $U$-statistic (of order 2) is defined as the sum of terms $f(X_i, X_j),$ where $f$ is a symmetric function, $(X_1, \cdots, X_N)$ are independent and identically distributed (i.i.d.) random variables (rv's), and $(i,j)$ are drawn from a restricted, though balanced, set of pairs. (A $U$-statistic corresponds to summation over all $(i, j)$ pairs.) A limit normal distribution is found for the reduced $U$-statistic, and it follows that estimates based on reduced $U$-statistics can have asymptotic efficiencies comparable with those based on $U$-statistics, even though the number of steps in computing a reduced $U$-statistic becomes asymptotically negligible in comparison with the number required for the corresponding $U$-statistic. As an illustration, a short-cut version of the Hodges-Lehmann estimator is defined, and its asymptotic properties derived, from a corresponding reduced $U$-statistic. A multivariate limit theorem is proved for a vector of reduced $U$-statistics, plus another result obtaining asymptotic normality even when $(i, j)$ are drawn from an unbalanced set of pairs. The present results are related to those of Blom.

One-sample linear rank tests are considered for the case where the observations are not independent but come from an autoregressive process. It is proposed to apply the tests under these circumstances to certain transformations of the observations, rather than to the observations themselves. Then the tests have asymptotically the same properties as under independence, both under the hypothesis and under contiguous location alternatives. In particular, they are asymptotically distribution-free.

Asymptotically optimal empirical Bayes squared error loss estimation procedures are developed for three families of continuous distributions, uniform $(0, \theta), \theta > 0,$ uniform $\lbrack \theta, \theta + 1), \theta$ arbitrary, and a location parameter family of gamma distributions. The approach taken is to estimate the Bayes estimator directly. However, for the $\lbrack \theta, \theta + 1)$ case, it is shown that the indirect approach of applying the Bayes estimator, versus an almost sure weakly convergent estimator of the prior, also yields an asymptotically optimal procedure.

Stochastic approximation procedures were shown by Sakrison to become asymptotically efficient estimators when used to minimize the Kullback-Leibler information, if certain conditions hold. Further results in this direction were obtained by Nevel'son and Has'minskij. This paper gives, first, alternative conditions for convergence and, secondly, shows that, under weaker conditions, asymptotic optimality is obtained by a modified stochastic approximation procedure. The modified procedure uses a consistent estimate which leads the approximating sequence to a proper local minimum of the Kullback-Leibler information. The conditions under which the procedure is asymptotically optimal are close to or weaker than those for asymptotic optimality of one-step-correction maximum likelihood methods.

A sequence of medians of posterior distributions is approximately median unbiased of order $o(n^{-1}) \operatorname{iff}$ the prior density is equal to the square root of Fisher's information function. It is shown that in this case the sequence of medians of posterior distributions is even an optimum sequence of estimates within the class of all estimator sequences being approximately median unbiased of order $o(n^{-1}).$ The result is proved by showing equivalence with an expansion of an optimum sequence given by Pfanzagl. In the case of a location parameter family the Bayesian representation is admissible.

Let $q_n(y), 0 < y < 1,$ be a quantile process based on a sequence of i.i.d. rv with distribution function $F$ and density function $f.$ Given some regularity conditions on $F$ the distance of $q_n(y)$ and the uniform quantile process $u_n(y),$ respectively defined in terms of the order statistics $X_{k:n}$ and $U_{k:n} = F(X_{k:n}),$ is computed with rates. As a consequence we have an extension of Kiefer's result on the distance between the empirical and quantile processes, a law of iterated logarithm for $q_n(y)$ and, using similar results for the uniform quantile process $u_n(y),$ it is also shown that $q_n(y)$ can be approximated by a sequence of Brownian bridges as well as by a Kiefer process.

Random variables $X$ and $Y$ are mutually completely dependent if there exists a one-to-one function $g$ for which $P\lbrack Y = g(X)\rbrack = 1.$ An example is presented of a pair of random variables which are mutually completely dependent, but "almost" independent. This example motivates considering a new concept of dependence, called monotone dependence, in which $g$ above is now required to be monotone. Finally, this monotone dependence concept leads to defining and studying the properties of a new numerical measure of statistical association between random variables $X$ and $Y$ defined by $\sup \{\operatorname{corr} \lbrack f(X), g(Y)\rbrack\},$ where the $\sup$ is taken over all pairs of suitable monotone functions $f$ and $g.$

For comparing two disease screening procedures with economic costs assigned to administration, false positives, and false negatives, the problem of testing a linear cell frequency constraint $\sum^K_{i = 1} a_i p_i \leqq 0$ arises with the multinomial $(n, (p_1, p_2,\cdots, p_K))$ model. An ad hoc statistic based upon the estimate of the $p_i$ values, $\sum^K_{i = 1} a_i X_i/n,$ is compared with the likelihood ratio statistic $-2\ln \lambda,$ the latter having an interesting form. For local (contiguous) alternatives the two statistics have similar large sample properties. However, the likelihood ratio statistic has greater large deviation efficiency for fixed alternatives and is recommended.

It is shown that for a test of a composite hypothesis on the parameter $\theta$ of an exponential family of distributions, mixture stopping rules are almost optimal with respect to certain criteria of optimality and a unique stopping rule is to be found among them which is optimal with respect to another type of optimality.

This note extends a result of Efron and Morris on domination of the maximum likelihood estimator for the mean of a multivariate normal distribution. We show that this result and our extension follow from a certain differential inequality. In a certain class of estimators having a unique unbiased estimator for the quadratic risk we find necessary and sufficient conditions for risk estimate dominance of a particular set of estimators. We show that, in the sense of risk estimates, these conditions imply that there are no estimators in this class which dominate the James-Stein or truncated James-Stein estimators.

It is shown that the problem of asymptotic efficiency (in the Wolfowitz-sense) admits a satisfactory solution for the class of strongly asymptotically median unbiased estimators. The concept of strongly asymptotic median unbiasedness furthermore links up the classical approach (concerning estimator-sequences whose distribution functions converge) with Pfanzagl's concept of median unbiasedness.

In this note it is shown how to some extent the Edgeworth expansion for the distribution function of Kendall's $\tau$ can be established by using a well-known general result on such expansions.

For symmetric univariate distributions the usual definition of unimodality due to Khintchine has several equivalent formulations. When these concepts are generalized to higher dimensions in an attempt to define multivariate unimodality questions concerning their equivalence naturally arise. Of particular interest in this area is the relationship between two concepts first studied by Sherman and more recently by Dharmadhikari and Jogdeo. They asked if requiring that a distribution belong to the closed convex hull of all uniform distributions on symmetric convex bodies was the same as requiring that the probability it assigns to a symmetric convex set decrease as the set is translated away from the origin in a fixed direction. Sherman conjectured that the two concepts were the same while Dharmadhikari and Jogdeo felt that this was not so and they suggested a possible counterexample to Sherman's conjecture. In this paper it is shown that their example is indeed a counterexample.

If $f_n(x)$ is any estimator of the density $f(x),$ it is proved that the mean integrated square error is no better than $O(n^{-1}).$