Richard D. Gill's home page

Mathematical Statistics

Mathematical Institute

Faculty of Mathematics and Natural Sciences

Leiden University

To contact me quickly, try email (surname at math dot leidenuniv dot nl), or mobile phone. Click here for my postal and visiting addresses, office and mobile phone numbers, email addresses, and further contact possibilities. If you are an Indian IT student looking for a summer interneeship then read this.

Loophole free Bell experiment

26 August, 2015. They did it at last! A first reaction. What did they do, exactly, and what was my (modest) scientific contribution.

Illustrations: Figure 7 from "Bertlmann's socks and the nature of reality", Bell (1981); and the set-up in Delft, 2015. (Sorry I initially gave the figure from Bell's paper the wrong name, I have now changed it to figure7.jpg).

See "Bell games" for a comparison of the standard Bell experiment and the Delft "event-ready detectors" version built on top of entanglement swapping.

The UK Lucia: Ben Geen

I was asked by defense lawyers working on behalf of Ben Geen, to analyse statistical data of occurrence of respiratory arrest in accident and emergency at numerous UK hospitals (draft R html notebook, draft report). After I had done my statistical work, I dug more deeply into other aspects of the case. The more I discovered, the more I was shocked that this case was a carbon copy of our own (Netherlands) case of Lucia de Berk. However there are three important differences. (1) The UK media did an even more perfect hatchet job on him than the Dutch media did on Lucia. (2) The Criminal Cases Review Commission is understaffed and underfunded, and takes a formalistic (legalist) view: give us a "new legal fact" or we will do nothing. (3) There is no Metta de Noo: no medical whistleblower. In the Lucia case, Metta de Noo fought for 7 years to get Lucia the fair (re-) trial which Lucia deserved. Metta is a senior medical expert, well connected in society, and she had inside information about the case. In the Lucia case, what was needed (and finally happened at the re-trial in Arnhem, 2009--2010) was an independent and thorough re-appraisal of existing (medical) evidence.

Ben has already sat out 8 years of his 30 year sentence. Because he claims to be innocent he is denied any "good prisoner" benefits and will also never get parole or early release. A small supporters' group has set up a website Justice for Ben Geen.

At last some UK journalists, at "The Times" no less, are showing some interest: Nurse 'was victim of Shipman hysteria'

Lucia de B at TEDxFlanders 2014 Naked; and remarks on Lucia de B: the movie

YouTube video. Slides of my talk Murder by Numbers. The naked truth about the case of Lucia de B.

A historical document: statement by Haga-Hospital, 2010, regarding the acquittal of Lucia de Berk: English translation, original.

Four days after the TEDx event, I saw the movie Lucia de B. on its premiere night in Amsterdam. Here is my personal film review: Splendid acting, very moving, beautifully told human story, centering around Lucia herself. Despite compression of the story line and focus on Lucia's personal experiences, it still contained such key features as: the close personal links between key people from hospital, justice and experts (image right); the mental illness and mental breakdown of the chief-paediatrician at JKZ ... There was a vain and ambitious hospital director. A bad statistician. Real life heroine Metta de Noo and hero Ton Derksen were concentrated in the film into the imaginary person of one imaginary whistleblower at the last place you might expect to find them: in the Public Prosecution service. But on the other hand: it wasn't black and white. There were good medics and bad medics, good nurses and bad nurses, good cops and bad cops ... Apparently, even some people in the Public Prosecution service found the witch hunt deeply disturbing.

For more (much, much more) on the Lucia case, see my collected past home-page writings on Lucia.


Some years ago I offered a prize for the person who remasters the logo of the VVS: the Dutch statistical society (top image) in the most beautiful postscript. An exercise in curve fitting with splines, perhaps? Better still would be a mathematical/statistical story of the curves themselves, providing an elegant parametric family which reproduces the whole logo. Finally I decided to do it myself, and I think I am getting close with this perspective image of some very simple 3-dimensional curves, with indeed a statistical story behind them (bottom). The R script which draws this logo can be found here. It should generate a rotatable 3-d image...

See the slides of my Amst-R-dam R users group meetup 2011 (updated 2012) talk R-fun: part 1, the VVS logo in R; part 2, R on an iDevice. For the latest news on "R on an iDevice" see the 2014 talk R on an iDevice, given at a Data Science NL meetup.

For more R fun: I am nowadays an enthusiastic user of RStudio and RPubs. You can find all kinds of R work by me at my RPubs site.

VVS stands for "Vereniging voor Statistiek". SMS stands for "Section Mathematical Statistics". The VVS also has an OR section, hence the common alternative name VVS-OR.

Quantum crackpots

I must admit to getting a lot of fun (and scientific stimulation) from interacting with Bell-deniers. More generally I consider this an important part of science-outreach: how can we explain Bell's theorem to the general public? Well a good start is to try to understand the mentality of very smart and well-educated people who believe that Bell made some fundamental but simple mistake, that they have exposed that mistake, and that there is an establishment conspiracy to suppress their findings. I wrote a tutorial paper intended to clarify the situation around the Joy Christian model. To my surprise it was *rejected* by on the grounds (a) that it was tutorial in nature, not scientific and (b) that it was personal, not scientific. I therefore joined the crackpots by posting it on where it drew (of course) fierce (but IMHO completely unscientific) criticism. Does Geometric Algebra provide a loophole to Bell’s Theorem? These activities also led this year (2015) to short papers in the Springer journal IJTP (International Journal of Theoretical Physics) and the Elsevier journal RinP (Results in Physics), refuting results in papers by J. Christian and by H. Geurdes which had appeared in those journals in the same year. I hope someone finds them useful. My "victims" were not amused, I'm sorry for that. I analyse and criticise their work, not their persons.


Spring 2016

Forensic Statistics and Graphical Models

FSG: Tuesdays 13:45--15:30 room 405

Master's level (or advanced Bachelor's)

Statistical Science for the Life and Behavioural Sciences

The master specialization Statistical Science for the Life and Behavioural Sciences is a collaboration of our group with others in biomedical statistics, biostatistics, and psychometrics.

Past courses

Here you can find links to various courses I have given in the past, in particular quantum statistics, statistics for astronomers, HOVO courses (adult education courses, in Dutch) on use and abuse of statistics, forensic science (Hovo-criminalistiek-statistiek-1, Hovo-criminalistiek-statistiek-2, Hovo-criminalistiek-statistiek-3).


For the most up to date impresion of my research interests, take a look at my recent talks on Slideshare.

My present PhD student is Giulia Cereda, working on forensic statistics, in particular the rare haplotype problem (aka the "fundamental problem of forensic mathematics"). Giulia is in a "cotutelle" arrangement: this is a joint project between Lausanne (Franco Taroni) and Leiden.

Interests, most active marked *

  • causality, graphical models, forensic statistics, forensic DNA, statistics and law, scientific integrity and scientific fraud, science and society ***
  • quantum statistics, probability, information, foundations *
  • statistical and computational learning *
  • missing data, censoring *
  • survival analysis, semiparametric models, martingale methods, counting processes, non parametric maximum likelihood
  • product integration, compact differentiation, the delta method, the bootstrap, empirical processes in statistics
  • spatial statistics and image analysis
  • random number generation
  • mathematical typography
  • foundations of statistics, probability, mathematics, quantum theory (see middle of this page) *
  • Recent talks and papers

    Scientific integrity
  • Worst Practices in Statistical Data Analysis, talk at Willem Heiser farewell symposium (includes material on Smeesters affair, and on Geraerts "Memory paper" affair)

  • Forensic Statistics
  • What is the chance that the match is a coincidence?"Talk (Sept 2013) on two problems from forensic statistics: the rare haplotype problem, and mobile phone colocation analysis
  • Talk on forensic statistics at Nordic Meeting of Statisticians, Umea, 2012
  • Talk on forensic statistics at Statistics Day conference, 2010

  • Quantum foundations
  • Schroedinger's cat meets Occam's razor: lecture slides, draft paper: quant-ph/0905.2723
  • A proof of Bell's inequality in quantum mechanics using causal interactions, with James Robins and Tyler VanderWeele
  • Refutation of Joy Christian's refutation of Bell's theorem
  • Statistics, causality and Bell's theorem, including a mathematical challenge in the design of Quantum Randi Challenges
  • Does Geometric Algebra provide a loophole to Bell's Theorem?, postmortem of Joy Christian's challange to Bell's theorem

    Coarsening at Random
  • Algorithmic and Geometric characterization of CAR (slides)
  • Algorithmic and Geometric characterization of CAR, math.ST/0510276; appeared in Annals of Statistics, with Peter Grunwald

  • Generalized Bell Inequalities
  • On the maximal violation of the CGLMP inequality for infinite dimensional states, with Stefan Zohren
  • Perfect Passion at a Distance: How to win Polish Poker (Use Quantum Dice!), pdf, html
  • Better Bell Inequalities (slides), at NATO Advanced Research Workshop on Quantum Communication and Security, Gdansk (among other places)
  • Better Bell Inequalities: Maximal Passion at a Distance, math.ST/0610115, to appear in Festschrift for Piet Groeneboom, IMS monographs series

    Optimal Quantum State Estimation
  • Local Asymptotic Normality in Quantum Statistics, Limerick research seminar, 2009; Stolen from Madalin Guta's 2009 Lunteren lectures: Part I and Part II and Part III (there is no part III).
  • Madalin Guta's Magic quantum statistics course
  • Jonas Kahn's PhD thesis Quantum Local Asymptotic Normality (and other questions of quantum statistics)
  • Conciliation of Bayes and Pointwise Quantum State Estimation,math.ST/0512443: pp. 239-261 in Quantum Stochastics and Information: Statistics, Filtering and Control, V.P. Belavkin and M. Guta (eds.), World Scientific (2008)
  • Asymptotic information bounds in quantum statistics, math.ST/0512443, to be revised and extended for Annals of Statistics (much delayed by my activities in the case of Lucia de B. - so a preliminary version appeared as the previous item in this list)
  • Conciliation of Bayes and Pointwise Quantum State Estimation (slides), at QUantum PRocess ESTimation 06, Budmerice, Slovakia
  • Optimal adaptive measurement of mixed qubits, Phys. Rev. Lett. 97 130501 (2006), quant-ph/0512177, with Manuel Ballester and Catalan friends Emili Bagan, Ramon Munoz-Tapia, Oriol Romero-Isart
  • Optimal collective measurement of mixed qubits Phys. Rev. (A) 73 032301 (2006), quant-ph/0510158, with Manuel Ballester and Catalan friends Emili Bagan, Alex Monras, Ramon Munoz-Tapia

    Lucia de Berk
  • Elementary statistics on trial (the case of Lucia de Berk) (joint with P. Groeneboom and Peter de Jong, rejected by Statistica Neerlandica ... new version for different journal in preparation).
  • On the (ab)use of statistics in the legal case against the nurse Lucia de B, preprint at, final version published (with discussion by David Lucy) in Law, Probability and Risk, 2007, joint with Marieke Collins, Michiel van Lambalgen, Ronald Meester.
  • Lucia talk at Vierhouten hackers conferencee Lies damned lies and legal truths
  • Astin day presentation: a story in a story Statistics and Ethics (Dutch outside, English inside)
  • Lies, damned lies, and legal truths (2010), pp. 39-50 in: L. Mommers, H. Franken, J. van den Herik, F. van der Klaauw and G.J. Zwenne, Het Binnenste Buiten (Liber Amicorum ter Gelegenheid van het Emeritaat van Aernout Schmidt), eLaw@Leiden, Law Faculty, University Leiden.
  • Remarks on the Lucia data - why the numbers keep changing (2009)

    The probiotica affair
  • Slides for talk at CCMO workshop, 11 December 2009
  • Statistics, ethics, and probiotica, (2009), Statistica Neerlandica
  • careless statistics costs lives, slides of talk
  • meten is weten, slides of a talk at the Science Cafe, Nijmegen, in Dutch
  • Publications

  • Papers in quantum statistics (arXiv:quant-ph)
  • Recent papers in mathematical statistics (arXiv:stat)
  • Publication list, including prepublications and unpublished work
  • Links to many of my older papers can be found via the MC and CWI repository
  • Collaboration

    Madalin Guta (Nottingham), Ole Barndorff-Nielsen (Aarhus), Jonas Kahn (Orsay), Peter Jupp (St. Andrews), Peter Grunwald (Amsterdam), Erik van Zwet (Leiden), Jan-Ake Larsson (Linkoping), Ramon Munoz-Tapia (Barcelona), Emili Bagan (Barcelona), Luis Artiles (Cuba/EURANDOM), Stefan Zohren (Leiden), Ronald Meester (Amsterdam), Marek Zukowski (Gdansk, Vienna), ...

    Foundational issues in quantum theory

    WARNING: Richard P. Feynmann said that attempting to understand quantum mechanics causes you to fall into a black hole, never to be heard from again

    The past is particles, the future is a wave

    Bell's fifth position

    During the academic year 2010-2011 I was Distinguished Lorentz Fellow (DLF) at the Netherlands Institute of Advanced Study in the Humanities and Social Science, NIAS. Here's my research proposal. The award ceremony was at NIAS, Wassenaar, late-afternoon of March 22, 2010. On the morning of the same day we held a complementary Breakfast symposium "Science, Media, Justice" at LUMC.

    The Smeesters affair (revised: July 4, 2013)

    Slides of talk on Smeesters case, and on the Geraerts-Merckelbach Memory paper affair. Talk originally given December 2012; slides updated March 2013; title "Integrity or fraud - or just questionable research practices?"
    Stimulated by media interest in the Geraerts-Merckelbach controversy on their "Memory" paper, I studied the published summary statistics in this paper using the same techniques as Simonsohn used for Smeesters, and found quite clear statistical evidence for "too good to be true". Without experimental protocols written up prior to the experiment, original data-sets, and laboratory log books detailing all the data selection and manipulation steps which resulted in the final data-set on which the summary statistics in the paper are based, one can only guess how these patterns arose. It certainly need not be fraud (fraud requires active intention to deceive).

    R-code for experiment with Simonsohn's fraud test (new version)
    Histogram of p-values of an honest researcher
    Histogram of p-values of a dishonest researcher
    You can continue reading here


    Biography and more ...

    First Leiden inaugural lecture

    Curriculum Vitae

    Past phd students

    Just for fun: things you wish your computer had (including the classic clippy's suicide note)

    The three doors problem

    A few years ago I discovered the enormous disussion on the Monty Hall (three doors) problem on wikipedia. My published writings on the subject are, in order of writing (and in order of insightfulness) an invited contribution to Springer's International Encyclopaedia of Statistical Science, 2010, a paper in Statistica Neerlandica, 2011, and contributions to the peer reviewed internet encyclopedias and In this manuscript you will find an expanded version the most recent published work, the article.
    In these works I distinguish between the original, somewhat ambiguous, real world question about a famous quiz show, and the many mathematizations of the question which have been proposed in the literature. Personally I prefer the lesser known game theoretic version. For me, the question is not "what is this probability?" or "what is that probability?", but: "what would you do?" And to me, the wikipedia controversy around the Monty Hall problems (concerning whether we should compute a conditional or unconditional probability of getting the car if we switch doors) is a warning against solution-driven science. I want to thank so many wikipedia editors for the inspiration they gave me.

    The holy grail of Monty Hall studies

    Suppose the car is hidden behind one of the three doors by a fair randomization. The contestant chooses Door 1. Monty Hall, for reasons best known to himself, opens Door 3 revealing a goat. We know that whatever probability mechanism is used by Monty for this purpose, the conditional probability that switching will give the car is at least 1/2. We know that the unconditional probability (ie not conditioning on the door chosen by the contestant, nor the door opened by Monty) is 2/3.
    Always switching gives the car with unconditional probability 2/3, always staying gives it with probability 1/3. Nobody in their right mind could imagine that there could exist some mixed strategy (sometimes staying, sometimes switching, perhaps with the help of some randomization device, and all depending on which doors were chosen and opened) which would give you a better overall (ie unconditional) chance than 2/3 of getting the car.
    This is true, of course. In fact, from the law of total probability, proving the optimality of (unconditional) 2/3 by always switching is equivalent to proving that all the six conditional probabilities of winning by switching, given door chosen and door opened, are at least 1/2. We can prove the latter using Bayes' theorem, or, better I think, using Bayes' rule in a smart way. However both these proofs require some sophistication.
    Is there an elementary proof? A short proof using words and ideas, no computations.
    Yes there is, and I learnt it from Sasha Gnedin.
    However you play there's always a door such that if the car is there, you'll miss it. Consider first deterministic strategies. We only need consider two cases: for "always switching" it's the door you initially chose, and for "sometimes switching" it's a door you won't switch to if you get the option. (If you never switch there are two such doors: just choose one). Ordinary readers won't be interested in randomized strategies but anyone who wants to include these will understand how to do it (now the door where you'ld certainly miss a car has to be a random door, determined by the same coin tosses used to implement the random choices in your own strategy).
    Note that the door which has been indicated in this way does not depend on where the car is actually hidden or how the host plays: it just depends on how the player plays. Therefore if the car is initially equally likely to be behind any of the three doors, we run a 1/3 chance that the car will be missed because it's behind this door. Therefore the 2/3 success-chance of always switching can't be beaten.
    I would call this a proof by coupling.

    The two envelopes problem

    From Three Doors to Two Envelopes (what will be next? One Coffin, perhaps?). Here is my fourth draft of the definitive article on the infamous two envelopes problem. The problem which Martin Gardner could not solve, and which many other famous people got wrong. Studied by probabilists, logicians, economists, philosophers. Now studied by me ...

    The mathematical heart of all exchange paradoxes is encapsulated in a little theorem which I call my "unified solution". It seems to be new.

    12 April, 2010: founding of the

    Bureau of Lost Causes

    This organisation (note added April 2015: presently dormant) has been set up inspired by the self-less efforts by so many people over the last six years, which only just now led to the extraordinary and total rehabilitation of Lucia de Berk. Now that the judicial authorities have apoligized personally and publicly, it is time to start finding out where avoidable mistakes were made. It is hard to believe that these can only be attributed to police investigation and legal procedures. However that is the implication of the recent public statement by the board of Lucia's hospital, (unauthorized rough English translation).

    Lucia de Berk
    The tunnel-vision which characterized Lucia's case was cemented in the two weeks around "the" nine-eleven inside a hospital in the Hague. Once by the end of those two weeks a major medical institution had (by implication) told the world that it had caught a serial killer, it must have been very hard for those who brought charges - a few individuals at the very top of the very same institution - not to have had some large influence, deliberately or innocently, on the results of police investigation, and on the "medical" interpretation of the medical dossiers which went to the courts. The events of the past year which came up during those two weeks of internal investigation and suddenly associated with Lucia had become unexpected and inexplicable, though previously every single one of them had been unremarkable.

    The hospital investigators into the crime were the same people earlier treating those patients, and making, as is completely natural, errors of diagnosis or treatment from time to time. The collegiality of the medical community means that mistakes by medical specialists within the Netherlands can hardly be admitted by others inside the same relatively tight, and extremely powerful, community. Highly placed medical authorities had to stand firm by their own previous and now provenly mistaken diagnoses. Others would be loath to criticise a highly regarded colleague's decisions in such a critical case.

    In the Netherlands, medical practitioners almost never admit to having made mistakes... consequently, they do not have to insure themselves agaist being sued for malpractice (which is good for their income), and in theory medical treatment should be less expensive than in other countries where lawyers and insurance companies profit from medical missers. However the Dutch arrangement has led to increasing distress among all those "victims" of medical errors, many of whom would probably be satisfied just to have an "accident" admitted! This June 16, a new code of practice has been introduced, by which medical practitioners will in future be able to apologize for errors without thereby admitting legal responsibility. A giant step for the medical profession, though only a very small step for their patients. Better than nothing, or merely a crumb to keep us "consumers" (the ones who pay for health care) quiet?

    Jose Booij
    One of the cases we have just taken on board is to unravel the unbelievable story of the illegal kidnapping of Jose Booij's six week old daughter Julia by a local child protection agency (Assen), and the ensueing cover-up by silencing of the mother through fair means and foul, now in it's sixth year. The kidnapping was judged illegal and a court order was given to return the child immediately. The judges of the courts for child-protection and family simply laughed, and did nothing. The child protection agency had acted on the basis of lies and insinuations of a jealous neighbour to local police and doctors. Her claims about Jose were believed. No attempt whatever was made to check these accusations, nor to hear Jose.

    In desparation, two and a half years ago, Jose wrote to the Cabinet of the Queen just before she was made homeless and all her remaining possessions were taken from her because she could no longer pay her bills (many of them fat lawyer's bills who did nothing except making a phone call and deciding to keep out of this mess), after losing her job, house, and health. Here is Jose's
    letter to the queen in Dutch (original) and in English (first rough translation), written just before she went underground.

    The cabinet of the queen forwarded her plea to the Ministry of Family and Welfare. Nothing was done for two and a half years. Her case was also brought at the same time to the European Court of Human Rights in Strasbourg. Nothing was done for two and a half years.

    On request I can show you an official report by well-known psychiatrist Bram Bakker, written five years ago, when Jose was up and fighting, though already suffering post-traumatic stress syndrome. It states that the allegations against Jose seemed quite unfounded and that considering the situation she was now in, she was standing up pretty well, psychologically. It still then seemed that it might not be difficult to get her baby back to her, provided she kept on fighting against the injustice which had been done her, and someone, somewhere, would stick out their neck for her. Still then, it could easily have been possible to save Jose's health and livelihood and future.

    Unfortunately, that would have required admitting that some mistakes had been made by some irresponsible local officials.

    Read more at Brian Sack's webpage Dutch Family Justice - the tragic case of Jose Booij.

    Kevin Sweeney
    Another case we are studying, with all the same features, is the extraordinary story of Kevin Sweeney. More information on that case can be found below. The incredible similarities between the cases provide a worthy study in individual versus group mentality, and how a scape-goat is chosen when a society is feeling under threat. This will be researched by a multidisciplinary team of cultural anthropologists, ethologists, sociologists, historians, lawyers, psychologists and mathematicians during my DLF fellowship at NIAS and of course by the Bureau of Lost Causes.

    More Various

    Statistical ethics of the probiotica trial. This randomized triple-blind clinical trial of probiotics treatment for patients with predicted severe acute pancreatitis ended in controversy, when it transpired at the conclusion of the trial in December 2007, that rather more patients had died on the treatment arm of the trial than on the control arm.

    It seemed strange that the trial had not been terminated at the interim analysis. The researchers were using a a stopping rule of S.M. Snapinn, by which the trial would to be terminated early either if it were almost certain that the final result would be a significant positive effect of probiotica, or if it were almost certain that the final result would be insignificant. Here is a paper by myself, to appear in Statistica Neerlandica, and, in Dutch, a short article by probabilist Ronald Meester and microbiologist Pieter ter Steeg which appeared in the newspaper Trouw and an open letter to Meester and ter Steeg by biostatisticians Hans van Houwelingen and Theo Stijnen. Also in Dutch there are a series of interviews (early 2008) on the current affairs chat show 'Pauw and Witteman': chairman of the hospital board Geert Blijham, 23 January; patient Jochim Vromans, 24 Jaunary; probiotics expert Eric Claassen, 25 January; leader of the research team Hein Gooszen, 14 February.

    Later we obtained the data at the time of the interim analysis. It was given to journalists at a press conference on Feb. 13 2008, but never released to interested scientists. It turned out that the probiotica trial was not terminated for futility (following the Snapinn stopping rule) at the half way interim analysis, through a mis-reading of output of the SPSS package, which, without consulting the user, always reports the smaller p-value of the two one-sided Fisher's exact tests for equality of two binomial probabilities. Proper application of their own stopping rule would have led to early termination of the trial, since according to the criteria set in advance, there was no chance any more that it would result in a positive result for the probiotica treatment. The trial was de facto continued because there was a good chance that it would finally result in a negative result for probiotica. Here are slides of my talk careless statistics costs lives on the subject.

    Kevin Sweeney ... recently left a Dutch jail at the end of his sentence for murder of his wife by arson. He has always claimed innocence. Here is a link to his own site, Justice for Kevin Sweeney but seems to be defunct now: try the way back machine. Here is a short synopsis of the case, and here is my blog entry Justice in the Netherlands: Guilty until Proven Innocent. In May, 2008, he put in an application to revise the case (English translation) to the Supreme Court. The application is based on an analysis of the fire evidence by Fred Vos, entitled Het vergeten tijdspad (the forgotten timeline). This is the first time a careful reconstruction of the course of the fire has taken place, taking account of all evidence available to the courts. The evidence seems totally consistent with a fire accidentally started by smoking in bed; and is totally inconsistent with the prosecution's claim of arson using large quantities of white spirits (Dutch: terpentine). Vos is careful to distinguish observed facts from interpretations thereof. Many writers on the case, including myself, have been misled by such misinterpretations.

    Mathematical Centre (Amsterdam) publications are now available on internet. Here are two early works which had quite some impact, including the reprint of my 1979 PhD thesis:
    R.D. Gill (1980), Censoring and Stochastic Integrals, MC Tract 124.
    R.D. Gill (1983), The sieve method as an alternative to dollar-unit sampling: the mathematical background, Report SN 12
    Another useful link is to my Saint Flour lectures on survival analysis.

    Product-integrals are to products, as integrals are to sums. Though they have been around for more than a hundred years, they never became part of the standard toolbox, possibly because no-one invented the right mathematical symbol for them. I made a try quite some years ago, though they still have not caught on yet. With the crucial help of JC Loredo, my efforts resulted in, files for getting beautiful \prodi and \Prodi and \PRODI symbols in your LaTeX, and Loredo.ttf, a TrueType font for ordinary word processing. It is not that difficult these days to get new fonts into your latex, see for instance TUG's font installation instructions.

    My sanskrit name

    Sarasvati Leela dasa (dasa: a devotee; Leela: games; Sarasvati: goddess of science, music, self-knowledge)

    My Korean signature

    (Last updated: 27 January 2014)