Showing papers by "Richard D. Gill published in 2010"

History of applications of martingales in survival analysis

Abstract: The paper traces the development of the use of martingale methods in survival analysis from the mid 1970's to the early 1990's. This development was initiated by Aalen's Berkeley PhD-thesis in 1975, progressed through the work on estimation of Markov transition probabilities, non-parametric tests and Cox's regression model in the late 1970's and early 1980's, and it was consolidated in the early 1990's with the publication of the monographs by Fleming and Harrington (1991) and Andersen, Borgan, Gill and Keiding (1993). The development was made possible by an unusually fast technology transfer of pure mathematical concepts, primarily from French probability, into practical biostatistical methodology, and we attempt to outline some of the personal relationships that helped this happen. We also point out that survival analysis was ready for this development since the martingale ideas inherent in the deep understanding of temporal development so intrinsic to the French theory of processes were already quite close to the surface in survival analysis.

A tight Tsirelson inequality for infinitely many outcomes

Abstract: We present a novel tight bound on the quantum violations of the CGLMP inequality in the case of infinitely many outcomes. Like in the case of Tsirelson's inequality the proof of our new inequality does not require any assumptions on the dimension of the Hilbert space or kinds of operators involved. However, it is seen that the maximal violation is obtained by the conjectured best measurements and a pure, but not maximally entangled, state. We give an approximate state which, in the limit where the number of outcomes tends to infinity, goes to the optimal state for this setting. This state might be potentially relevant for experimental verifications of Bell inequalities through multi-dimenisonal entangled photon pairs.

Product-limit estimators of the gap time distribution of a renewal process under different sampling patterns

Abstract: Nonparametric estimation of the gap time distribution in a simple renewal process may be considered a problem in survival analysis under particular sampling frames corresponding to how the renewal process is observed. This note describes several such situations where simple product limit estimators, though inefficient, may still be useful.

Product-limit estimators of the gap time distribution of a renewal process under different sampling patterns.

Abstract: Nonparametric estimation of the gap time distribution in a simple renewal process may be considered a problem in survival analysis under particular sampling frames corresponding to how the renewal process is observed. This note describes several such situations where simple product limit estimators, though inefficient, may still be useful.

Elementary Statistics on Trial (the case of Lucia de Berk)

Abstract: In the conviction of Lucia de Berk an important role was played by a simple hypergeometric model, used by the expert consulted by the court, which produced very small probabilities of occurrences of certain numbers of incidents. We want to draw attention to the fact that, if we take into account the variation among nurses in incidents they experience during their shifts, these probabilities can become considerably larger. This points to the danger of using an oversimplified discrete probability model in these circumstances.

Lies, damned lies, and legal truths

Abstract: My acquaintance with Aernout Schmidt began when we were asked to be one anotherʼs opponent in a Leiden science-café debate on the celebrated case of the Dutch nurse Lucia de Berk, who at the time was serving a life sentence for seven murders and three attempted murders of her patients: children at a special childrenʼs hospital, and terminally ill old people in an ordinary hospital ward where she had earlier worked. The case was sparked when, on the early hours of 4 September 2001, for the so-manyʼth time (as it appeared) a young child died during one of her shifts. The statistical question of whether Luciaʼs repeated presence at a series of deaths and near-deaths could merely have been a coincidence was answered first for hospital authorities, then for police investigators, and finally in court (in 2003; answer: no, it could not have been chance).

The Monty Hall Problem is not a Probability Puzzle

Abstract: Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \Do you want to pick door No. 2?" Is it to your advantage to switch your choice? The answer is \yes" but the literature oers many reasons why this is the correct answer. The present paper argues that the most common reasoning found in introductory statistics texts, depending on making a number of \obvious" or atural" assumptions and then computing a conditional probability, is a classical example of solution driven science. The best reason to switch is to be found in von Neumann’s minimax theorem from game theory, rather than in Bayes’ theorem.

The Monty Hall Problem is not a Probability Puzzle (it's a challenge in mathematical modelling)

Abstract: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ``Do you want to pick door No. 2?'' Is it to your advantage to switch your choice? The answer is ``yes'' but the literature offers many reasons why this is the correct answer. The present paper argues that the most common reasoning found in introductory statistics texts, depending on making a number of ``obvious'' or ``natural'' assumptions and then computing a conditional probability, is a classical example of solution driven science. The best reason to switch is to be found in von Neumann's minimax theorem from game theory, rather than in Bayes' theorem.

The Three Doors Problem...-s

Abstract: I argue that we must distinguish between: (0) the Three-Doors-Problem Problem [sic], which is to make sense of some real world question of a real person. (1) a large number of solutions to this meta-problem, i.e., many specific Three-Doors-Problem problems, which are competing mathematizations of the meta-problem (0). Each of the solutions at level (1) can well have a number of different solutions: nice ones and ugly ones; correct ones and incorrect ones. I discuss three level (1) solutions, i.e., three different Monty Hall problems; and try to give three short correct and attractive solutions. These are: an unconditional probability question; a conditional probability question; and a game-theory question. The meta-message of the article is that applied statisticians should beware of solution-driven science.

The One and Only True Monty Hall Paradox

Abstract: Short rigorous solutions to three mathematizations of the famous Monty Hall problem are given: asking for an unconditional probability, a conditional probabiliity, or for a game theoretic strategy. It is concluded which mathematicization ought to be considere da s the Only True Solution of the True Monty Hall Problem: the little known Game Theoretical version.