The Cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life-time distribution of an individual. In this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process. This permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates. Furthermore, this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model. Finally an example of a statistical analysis is included.

We consider the problem of estimating the state of a large but finite number N of identical quantum systems. As N becomes large the problem simplifies dramatically. The only relevant measure of the quality of estimation becomes the mean quadratic error matrix. Here we present a bound on this quantity: a quantum Cramér-Rao inequality. This bound succinctly expresses how in the quantum case one can trade information about one parameter for information about another. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states—a striking example of nonlocality without entanglement for mixed but not for pure states. Cramér-Rao bounds are generally only derived for unbiased estimators. Here we give a version of our bound for biased estimators, and a simple asymptotic version for large N. Finally we prove that when the unknown state belongs to a two-dimensional Hilbert space our quantum Cramér-Rao bound can always be attained, and we provide an explicit measurement strategy that attains it. Thus we have a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. The same is true for qubits in the same mixed state if one restricts oneself to separable measurements, but nonseparable measurements allow dramatic increase of efficiency. Exactly how much increase is possible is a major open problem.

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.

Weak convergence results are proved for the product-limit estimator on the whole line. Applications are given to confidence band construction, estimation of mean lifetime, and to the theory of $q$-functions. The results are obtained using stochastic calculus and in probability linear bounds for empirical processes.