Richard D. Gill

Coarsening at Random: Characterizations, Conjectures, Counter-Examples

The notion of coarsening at random (CAR) was introduced by Heitjan and Rubin (1991) to describe the most general form of randomly grouped, censored, or missing data, for which the coarsening mechanism can be ignored when making likelihood-based inference about the parameters of the distribution of the variable of interest. The CAR assumption is popular, and applications abound. However the full implications of the assumption have not been realized. Moreover a satisfactory theory of CAR for continuously distributed data—which is needed in many applications, particularly in survival analysis—hardly exists as yet. This paper gives a detailed study of CAR. We show that grouped data from a finite sample space always fit a CAR model: a nonparametric model for the variable of interest together with the assumption of an arbitrary CAR mechanism puts no restriction at all on the distribution of the observed data. In a slogan, CAR is everything. We describe what would seem to be the most general way CAR data could occur in practice, a sequential procedure called randomized monotone coarsening. We show that CAR mechanisms exist which are not of this type. Such a coarsening mechanism uses information about the underlying data which is not revealed to the observer, without this affecting the observer’s conclusions. In a second slogan, CAR is more than it seems. This implies that if the analyst can argue from subject-matter considerations that coarsened data is CAR, he or she has knowledge about the structure of the coarsening mechanism which can be put to good use in non-likelihood-based inference procedures. We argue that this is a valuable option in multivariate survival analysis. We give a new definition of CAR in general sample spaces, criticising earlier proposals, and we establish parallel results to the discrete case. The new definition focusses on the distribution rather than the density of the data. It allows us to generalise the theory of CAR to the important situation where coarsening variables (e.g., censoring times) are partially observed as well as the variables of interest.

Understanding Cox's Regression Model: A Martingale Approach

Abstract An informal discussion is given of how martingale techniques can be used to extend Coxs regression model and to derive its large sample properties.

Linear Nonparametric Tests for Comparison of Counting Processes, with Applications to Censored Survival Data, Correspondent Paper

This paper surveys linear nonparametric oneand k-sample tests for counting processes. The necessary probabilistic background is outlined and a master theorem proved, which may be specialized to most known asymptotic results for linear rank tests for censored data as well as to asymptotic results for oneand k-sample tests in more general situations, an important feature being that very general censoring patterns are allowed. A survey is given of existing tests and their relation to the general theory, and we mention examples of applications to Markov processes. We also discuss the relation of the present approach to classical nonparametric hypothesis testing theory based on permutation distributions.

An Elementary Approach to Weak Convergence for Quantile Processes, with Applications to Censored Survival Data

Abstract Let ξ be a continuously differentiable function with positive derivative, and let ξ n be a sequence of right-continuous increasing processes. We show that if n 1/2(ξ n − ξ) W, where W is continuous, then n 1/2(ξ−1 n − ξ−1) − W(ξ−1)/ξ′(ξ−1). This result is applied to classical processes such as the empirical distribution function, the Kaplan-Meier estimator, and some other situations. We also prove an analogous result for the bootstrapped version of n 1/2(ξ n − ξ) and show how this allows one to obtain confidence bands for the quantile function ξ−1, based on the bootstrap. Several examples are given.

Optimal full estimation of qubit mixed states

We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.

Maximal Violation of the Collins-Gisin-Linden-Massar-Popescu Inequality for Infinite Dimensional States

We present a much simplified version of the Collins-Gisin-Linden-Massar-Popescu inequality for the 2x2xd Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of d the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the 2x2xinfinity scenario.

CONDITIONS FOR FACTOR (IN)DETERMINACY IN FACTOR ANALYSIS

The subject of factor indeterminacy has a vast history in factor analysis (Guttman, 1955; Lederman, 1938; Wilson, 1928). It has lead to strong differences in opinion (Steiger, 1979). The current paper gives necessary and sufficient conditions for observability of factors in terms of the parameter matrices and a finite number of variables. Five conditions are given which rigorously define indeterminacy. It is shown that (un)observable factors are (in)determinate. Specifically, the indeterminacy proof by Guttman is extended to Heywood cases. The results are illustrated by two examples and implications for indeterminacy are discussed.

No time loophole in Bell's theorem: The Hess–Philipp model is nonlocal

Hess and Philipp recently claimed [(2001) Proc. Natl. Acad. Sci. USA 98, 14224–14227 and 14228–14233] that proofs of Bells theorem have overlooked the possibility of time dependence in local hidden variables, hence the theorem has not been proven true. Moreover they present what is claimed to be a local realistic model of the EPR correlations. If this is true then Bells theorem is not just unproven, but false. We refute both claims. First, we explain why time is not an issue in Bells theorem, and second, we show that their hidden variables model violates Einstein separability. Hess and Philipp have overlooked the freedom of the experimenter to choose settings of a measurement apparatus at will: any setting could be in force during the same time period.

On the estimation of multidimensional demographic models with population registration data

In this paper the estimation of multidimensional demographic models is investigated in situations where population registration data are available. With this kind of aggregate data, estimation by traditional methods is not possible. We look at two versions of the multidimensional model: the constant intensities model and the linear integration model. Some logical inconsistencies in the derivation of the latter are discussed. In particular, we argue that the linear integration model is not compatible with a Markov process. A new algorithm for the estimation of the constant intensities model with population registration data is proposed. Some preliminary results on the mathematical and statistical properties of this method are given. The algorithm is applied to Dutch nuptiality data.

Sequential models for coarsening and missingness

In a companion paper we described what intuitively would seem to be the most general possible way to generate Coarsening at Random mechanisms, a sequential procedure called randomized monotone coarsening. Counter-examples showed that CAR mechanisms exist which cannot be represented in this way. Here, we further develop these results in two directions. Firstly, we consider what happens when data is coarsened at random in two or more phases. We show that the resulting coarsening mechanism is not CAR anymore, but under suitable assumptions is identified and can provide interesting alternative analysis of data under a non-CAR model. Secondly, we look at sequential mechanisms for generating MAR data, missing components of a multivariate random vector. Randomised monotone missingness schemes, in which one variable at a time is observed and depending on its value, another variable is chosen or the procedure is terminated, supply in our opinion the broadest class of physically interpretable MAR mechanisms. We show that every randomised monotone missingness scheme can be represented by a Markov monotone missingness scheme, in which the choice of which variable to observe next only depends on the set of previously observed variables and their values, not on the sequence in which they were measured. We also show that MAR mechanisms exist which cannot be represented sequentially.