The correspondence between a survival function and its hazard or failure-rate is a central idea in survival analysis and in the theory of counting processes. This correspondence is shown to be a special case of a more general correspondence between multiplicative and additive matrix-valued measures on the real line. Additive integration of the survival function produces the hazard, while the multiplicative integral, or so-called product-integral, of the hazard yields the survival function. The easy generalization to the matrix case (noncommutative multiplication) allows an elegant and completely parallel treatment of intensity measures of Markov processes, with many possible applications in multistate survival models. However, the difficulties and multiplicity of theories of product-integration in multivariate time explain why so many different multivariate product-limit estimators exist. We give a complete and elementary treatment of the basic theory of the product-integral $\pi(1 + dX)$ together with a discussion of some of its applications. New results are given on the compact differentiability of the product-integral, to be used along with the functional $\delta$-method for getting large-sample results for product-limit estimators.

A bivariate Weibull model is proposed for censored survival data when there may be dependence between the survival and censoring random variables. The model is based on marginal transformations of the Pickands class of bivariate exponential distributions. Assuming the model to hold, the joint distribution of survival and censoring times is shown to be identifiable, and the maximum likelihood estimator of the parameters is shown to be consistent.

This paper presents lower bounds, derived from the information inequality, for the Bayes risk under scaled quadratic loss. Some numerical results are also presented which give some idea concerning the precision of these bounds. An appendix contains a proof of the information inequality without conditions on the estimator. This result is a direct extension of an earlier result of Fabian and Hannan.

For a group model in which the group $\mathbf{G}$ acts freely on the parameter space $\mathbf{\Omega}$, this paper considers a prior which is a product of right Haar measure on $\mathbf{G}$ and a limiting form of Jeffreys' prior for the maximal invariant. When the parameter of interest is the orbit of $\mathbf{G}$ in $\mathbf{\Omega}$, it is shown that such a prior is the reference prior defined by Bernardo. A method of calculating this reference prior is given which avoids the necessity of working in a parameterization of $\mathbf{\Omega}$ which expresses $\mathbf{\Omega}$ as a product of $\mathbf{G}$ and a cross section. Examples of the multivariate normal distribution, with the parameter of interest being the correlation matrix or the eigenvalues of the covariance matrix, are discussed.

Any multiresponse estimation experiment requires a decision about the number of observations to be taken. If the covariance is unknown, no fixed-sample-size procedure can guarantee that the joint confidence region will have an assigned shape and level. Double-sampling procedures use a preliminary sample of size $m$ to determine the minimum number of additional observations needed to achieve a prescribed accuracy and coverage probability for the parameter estimates. The triple-sampling procedures of this paper, less sensitive to the choice of $m$, revise the sample size estimate after collecting a fraction of the additional observations prescribed under double sampling. Second-order asymptotic results relying on conditional inference show that triple sampling is asymptotically consistent; in addition, the regret for triple sampling is a bounded function of the covariance structure and is independent of $m$.

Biological models for the independent action of two drugs imply that if the drugs are administered in various combinations of doses, then the corresponding probabilities of response must satisfy certain inequalities if the drugs are acting independently. The hypothesis that the probabilities do satisfy these inequalities can be tested using the likelihood ratio test or using the bootstrap test proposed by Wahrendorf and Brown in 1980. In the simplest dose design, only one dosage of each drug is used. The three combination doses required to test the hypothesis are each drug singly and the combination of the two drugs. The asymptotic distribution of the bootstrap test is derived. The asymptotic distribution of the likelihood ratio test is obtained by applying Feder's results. The calculation of the asymptotic critical values and powers is presented.

We consider the consistency and weak convergence of $S$-estimators in the linear regression model. Sufficient conditions for consistency with varying dimension are given which are sufficiently weak to cover, for example, polynomial trends and i.i.d. carriers. A weak convergence theorem for the Hampel-Rousseeuw least median of squares estimator is obtained, and it is shown under rather general conditions that the correct norming factor is $n^{1/3}$. Finally, the asymptotic normality of $S$-estimators with a smooth $\rho$-function is obtained again under weak conditions on the carriers.

A general approach to the first order asymptotic analysis of penalized likelihood and related estimators is described. The method gives expansions for the systematic and random error. Asymptotic convergence rates in a family of spectral norms are obtained. The theory applies to a broad range of function estimation problems including nonparametric density, hazard and generalized regression curve estimation. Some examples are provided.

The problem of automatic bandwidth selection for a kernel regression estimate is studied. Since the bandwidth considerably affects the features of the estimated curve, it is important to understand the behavior of bandwidth selection procedures. The bandwidth estimate considered here is the minimizer of Mallows' criterion. Though it was established that the bandwidth estimate is asymptotically normal, it is well recognized that the rate of convergence is extremely slow. In simulation studies, it is often observed that the normal distribution does not provide a satisfactory approximation. In this paper, the bandwidth estimate is shown to be approximately equal to a constant plus a linear combination of independent exponential random variables. In practice, the distribution of a weighted sum of chi-squared random variables can be approximated by a multiple of a chi-squared distribution. Simulation results indicate that this provides a very good approximation even for a modest sample size. It is shown that the degrees of freedom of the chi-squared distribution goes to infinity rather slowly (at the rate $T^{1/5}$ for a nonnegative kernel). This explains why the distributions of the bandwidth estimates converge to the normal distribution so slowly.

A method for robust nonparametric regression is discussed. We consider kernel $M$-estimates of the regression function using Huber's $\psi$-function and extend results of Hardle and Gasser to the case of random designs. A practical adaptive procedure is proposed consisting of simultaneously minimising a cross-validatory criterion with respect to both the smoothing parameter and a robustness parameter occurring in the $\psi$-function. This method is shown to possess a theoretical asymptotic optimality property, while some simulated examples confirm that the approach is practicable.

We consider in this paper the estimation problems for both orders and coefficients of linear feedback control systems, described by ARMAX models. The estimation algorithms are inspired by the Hannan-Rissanen method used for the estimation of stationary ARMA models, while the convergence analyses are based on limit theorems for both double array martingales and nonnegative supermartingales, and on techniques of stochastic Lyapunov functions. Traditionally used assumptions, such as the strictly positive real condition and the requirement of known upper bounds for true orders, are not imposed here.

Let $\{X_n\}$ be the branching process with immigration and let $m$ and $\lambda$ be the means of the offspring and immigration distributions, respectively. Estimation results for $m$ and $\lambda$ were obtained in the literature for the subcritical $(m < 1)$ and supercritical $(m > 1)$ cases, but no unified estimation procedure was developed, which would allow inference without knowledge of the range of $m$. The goal of this paper is to investigate this problem.

A linear process is represented as a driving white noise convolved with a system response sequence. The concept of natural peakedness of a system response sequence is defined and its properties are investigated. Utilizing natural peakedness, the convergence theory of maximum standardized cumulant deconvolution is established and the uniqueness theorem of non-Gaussian linear process representations is proved. In addition, autoregressive models on a countable abelian group are defined and the relation between cumulant deconvolution and autoregressive models is given.

In the class of polynomial regression models up to degree $n$ we determine the design on $\lbrack -1, 1\rbrack$ that maximizes a product of $n + 1$ determinants of information matrices weighted with a prior $\beta$, where the $l$-th information matrix corresponds to a polynomial regression model of degree $l$, for $l = 0, 1, \cdots, n$. The designs are calculated using canonical moments. We identify a special class of priors $\beta(z)$ depending on one real parameter $z$ so that analogous results are obtained as in the classical $D$- and $D_1$-optimal design problems. The interior support of the optimal design with respect to the prior $\beta(z)$ is given by the zeros of a Jacobi polynomial and all the interior support points have the same masses. The masses at the boundary points $-1$ and $1$ are $(z + 1)/2$ times bigger than the masses of the interior points. The results found in one dimension are generalized to the problem of determining optimal product designs in the case of multivariate polynomial regression on the $q$-cube $\lbrack -1,1\rbrack^q$. Explicit solutions are obtained for the $D$- and $D_1$-optimal product designs in the polynomial model of degree $n$ for all $n \in \mathbb{N}$ and $q \in \mathbb{N}$.

For the class of repeated measurements designs based on $t$ treatments, $n$ experimental units and two periods, the following results are obtained. 1. The equivalence of the information matrices of such repeated measurements designs and of certain block designs is established. The implication of this equivalence on the optimality of both repeated measurements designs and block designs is explored. 2. A family of universally optimal designs or $A$-optimal designs is constructed depending whether or not $n$ divides $t$. 3. Families of optimal designs for residual effects and for comparing test treatments with a control are constructed.

The classical stochastic approximation problem can be regarded as choosing design points so that the responses are close to some target level in the expected squared distance. Motivated by different loss criteria, a family of stochastic approximation algorithms is proposed. This family has the same simplicity as the classical Robbins-Monro procedure does and contains the latter as a special case. Using appropriate representations and martingale limit theorems, we establish asymptotic properties for this family. Using the semiparametric formulation, lower bounds are obtained for estimating the desired parameters under any adaptive design, showing that the proposed algorithms with appropriate scaling are asymptotically efficient.

Suppose $X_1,\cdots, X_n$ are distributed according to a probability measure under which $X_1,\cdots, X_n$ are independent, $X_1 \sim F_0$, for $i = 1,\cdots, \lbrack\theta_n n\rbrack$ and $X_i \sim F^{(n)}$ for $i = \lbrack\theta_nn\rbrack + 1, \cdots, n$ where $\lbrack x\rbrack$ denotes the integer part of $x$. In this paper we consider the asymptotic efficient estimation of $\theta_n$ when the distributions are not known. Our estimator is efficient in the sense that if $F^{(n)} = F_{\eta_n}, \eta_n \rightarrow 0$ and $\{F_\eta\}$ is a regular one-dimensional parametric family of distributions, then the estimator is asymptotically equivalent to the best regular estimator.

Three well-known partial orderings and one new one, designated by $\geq_{b_t}, t = 1, 2, 3, 4$, are defined on permutations of $\{1, 2, \cdots, n\}$ through a unified approach. Various formulations of these partial orderings are also considered. With the aid of these formulations we show that the four orderings on permutations are equivalent to positive dependence orderings defined over empirical distributions based on rank data. In particular, we show that the orderings $b_1, b_2, b_3$ and $b_4$ are equivalent, respectively, to more concordant, more row regression dependent, more column regression dependent and more associated.

An importance sampling method studied by Siegmund and Asmussen in the case of waiting time probabilities is extended to the mean and other functionals. The ideas involve exponential families of queues and control variates (regression), and it is found both from theory and practice that the method is dramatically better than standard tools like regenerative simulation or sample-mean estimation, not least under heavy traffic conditions.

Let $(X, \mathscr{A})$ be a measurable space and $\{P_{\vartheta,\tau}\mid\mathscr{A}: \vartheta \in \Theta, \tau \in T\}$ a family of probability measures. Given an appropriate estimator sequence for $\vartheta$, we define a sequence of asymptotic maximum likelihood estimators for $\tau$ and give bounds for its large deviation probabilities under conditions which are natural for the application to the estimation of mixing distributions. This paper generalizes earlier results of Pfanzagl to the following cases: (i) estimator sequences restricted to a sieve; (ii) estimator sequences using a given estimator sequence for a nuisance parameter; (iii) convergence under the "wrong model;" (iv) large deviation probabilities instead of consistency.

Additive interactions of $n$-dimensional random vectors $X$, as defined by Lancaster, do not necessarily vanish for $n \geq 4$ if $X$ consists of two mutually independent subvectors. This defect is corrected and an explicit formula is derived which coincides with Lancaster's definition for $n < 4$. The new definition leads also to a corrected Bahadur expansion and has certain connections to cumulants. The main technical tool is a characterization theorem for the Moebius function on arbitrary finite lattices.

We consider the problem of determining whether a threshold autoregressive model fits a stationary time series significantly better than an autoregressive model does. A test statistic $\lambda$ which is equivalent to the (conditional) likelihood ratio test statistic when the noise is normally distributed is proposed. Essentially, $\lambda$ is the normalized reduction in sum of squares due to the piecewise linearity of the autoregressive function. It is shown that, under certain regularity conditions, the asymptotic null distribution of $\lambda$ is given by a functional of a central Gaussian process, i.e., with zero mean function. Contiguous alternative hypotheses are then considered. The asymptotic distribution of $\lambda$ under the contiguous alternative is shown to be given by the same functional of a noncentral Gaussian process. These results are then illustrated with a special case of the test, in which case the asymptotic distribution of $\lambda$ is related to a Brownian bridge.

The average run length (ARL) is a key variable for assessing the properties of process control procedures. For continuous sampling procedures that are based on finite weighted sums (such as the moving sum technique) closed form expressions of the ARL are not available in the literature. For normally distributed random variables, Lai gives bounds for the ARL. In this paper we derive a lower bound of the ARL that (1) does not depend on normality and (2) in many situations is much sharper than the one obtained by Lai. Our results also imply that Lai's upper bound deviates from the true value less than the number of terms in the sum. Furthermore, we show that the applicability of Lai's bounds is not restricted to normally distributed control variables.