aa r X i v : . [ s t a t . O T ] O c t Statistical Science (cid:13)
Institute of Mathematical Statistics, 2014
A Conversation with Howell Tong
Kung-Sik Chan and Qiwei Yao
Abstract.
Howell Tong has been an Emeritus Professor of Statistics atthe London School of Economics since October 1, 2009. He was appointedto a lectureship at the University of Manchester Institute of Science andTechnology shortly after he started his Master program in 1968. He re-ceived his Ph.D. in 1972 under the supervision of Maurice Priestley, thusmaking him an academic grandson of Maurice Bartlett. He stayed atUMIST until 1982, when he took up the Founding Chair of Statistics atthe Chinese University of Hong Kong. In 1986, he returned to the UK, asthe first Chinese to hold a Chair of Statistics in the history of the UK, byaccepting the Chair at the University of Kent at Canterbury. He stayedthere until 1997, when he went to the University of Hong Kong, firstas a Distinguished Visiting Professor, and then as the Chair Professorof Statistics. At the University of Hong Kong, he served as a Pro-Vice-Chancellor and was the Founding Dean of the Graduate School. He wasappointed to his Chair at the London School of Economics in 1999. He isa pioneer in the field of nonlinear time series analysis and has been a sci-entific leader both in Hong Kong and in the UK. His work on thresholdmodels has had lasting influence both on theory and applications. He hasdrawn important connections between time series and deterministic dy-namical systems, linking statistics with chaos theory, and the models hehas developed have found significant applications in fields as diverse aseconomics, epidemiology and ecology. He has made novel contributionsto nonparametric and semi-parametric statistics, especially in model se-lection and dimension reduction for time series data. He has written fourbooks (one with Kung-Sik Chan and another with Bing Cheng) and over162 papers (sometimes with collaborators) in Statistics, Ecology, Actu-arial Science, Control Engineering, Reliability, Meteorology, Water En-gineering, Engineering Mathematics and Mathematical Education. His1990 monograph
Non-linear Time Series Analysis—A Dynamical Sys-tem Approach is a classic. He is a Foreign Member of the NorwegianAcademy of Science and Letters, an elected member of the ISI, a Fellowof IMS and an Honorary Fellow of the Institute of Actuaries (UK). Hewon a Chinese National Natural Science Prize (Class II) in 2000 and theRoyal Statistical Society awarded him the Guy Medal in Silver in 2007.The following conversation is partly based on an interview that tookplace in the Hong Kong University of Science and Technology in July2013.
Kung-Sik Chan is Professor of Statistics, Department ofStatistics & Actuarial Science, University of Iowa, IowaCity, Iowa 52245, USA e-mail:[email protected]. Qiwei Yao is Professor of Statistics, Department of Statistics, London School ofEconomics and Political Science, United Kingdome-mail: [email protected]. K.-S. CHAN AND Q. YAO
Fig. 1.
Howell with his childhood hero, Professor Loo KengHUA, and Mary Tong, at Tong’s home in Poynton, Cheshire,UK, 1979.
QY:
You were supervised by Maurice Priestley foryour doctorate. What was your thesis on?
HT:
My doctoral thesis was entitled “Some prob-lems in the spectral analysis of bivariate nonsta-tionary stochastic processes.” It was an extension ofMaurice Priestley’s evolutionary spectral analysis,which he proposed in 1965, from the univariate caseto the bivariate case, including both the open-loopand close-loop systems. The contents of the thesisformed the basis of a joint paper which Maurice andI read to the Royal Statistical Society in 1972. I canstill remember the occasion well, as it was my firsttaste of academic subtlety in Britain.I must tell you that a British statistician can do aclean demolition job at an RSS discussion meeting,without even showing his hammer. (I hope you willforgive me for being gender blind when I speak.) Ithas been said that one has to be courageous or fool-hardy to read a paper to the RSS. I have learnt a lotbesides the demolition skill since then, by attendingRSS discussion meetings in London. The franknessof views is very helpful, as much for the readers as forthe authors, because it enables everybody to have amore critical assessment of the strengths and weak-nesses of the presented work. Of course, there willalways be cases of premature euphoria as well ascases of misplaced cold shoulder. Despite its imper-
This is an electronic reprint of the original articlepublished by the Institute of Mathematical Statistics in
Statistical Science , 2014, Vol. 29, No. 3, 425–438. Thisreprint differs from the original in pagination andtypographic detail. fection, I do not think that I am alone in saying thatthe forum remains the best in the statistical world.In many ways, it has made the RSS unique.Returning to my doctoral thesis, I think much ofit is now out of date and mostly of little practicalsignificance. I am especially disappointed with thefact that the evolutionary coherency spectrum fornonstationary time series turns out to be time in-variant. However, there is perhaps a curious littleresult in the thesis which you might find interest-ing. It concerns the function exp { i ( kωt + ω t ) } , ω being a fixed constant. I showed that this frequency-modulated wave admits no generalized frequencyin Priestley’s sense. In fact, I am inclined to takethe view that for frequency-modulated waves thewavelet approach is more natural. In the 1990s, BingCheng and I developed a wavelet representation fora general stochastic process.For the modelling of nonstationary time series,I think that the piecewise stationary approach intro-duced by Tohru Ozaki and myself in 1975 is a verypractical one. Specifically, as each new “short” blockof data arrives, we check if the AR model fitted tothe latest block needs to be changed. If it does, thena new AR model is the latest state of the system,otherwise the previous state stays. This approachis ideally suited for real-time implementation. I un-derstand that Professor Genshiro Kitagawa and hismarine engineering colleagues have built many suc-cessful auto-pilots for boats based on this approach,under the guidance of the late Professor Akaike. QY:
Can you tell us something about the earlypart of your career in higher education?
HT:
My first job in higher education was a lec-tureship at the then Northern Polytechnic, London,UK, in 1967. Remember I had only a B.Sc. degree!I took the job for two reasons: (1) To help my fatherfinancially because my mother had just joined us inLondon from Hong Kong, having waited for sevenlong years; (2) I lost my passion for Algebra.When I graduated from the University of Manch-ester Institute of Science and Technology (nowmerged with the University of Manchester) in 1966,I was very keen on Algebra. So I went to QueenMary College of the University of London on a post-graduate studentship funded by the UK Science andEngineering Research Council. The general expec-tation was to do a Ph.D. in Algebra.At that time, QMC was the hot house of Algebrain the UK, under the inspiring leadership of Profes-sor Kurt Hirsch. He came to the UK to escape from
CONVERSATION WITH HOWELL TONG Hitler’s Germany, like many of his contemporariesincluding Bernard Neumann, Hanna Neumann, PaulM. Cohn and others. He was my mentor. I re-member attending courses on Homological Algebra,Group Representation Theory and others. I even at-tended seminars given by Saunders MacLane andother leading algebraists. One of the first things thatProfessor Hirsch asked me was “Have you studiedLebesgue integration at UMIST?” When he heardthat I had not, he said, “In that case, Mr. Tong, youare only half-educated. I suggest that you attend acourse on it in our inter-collegiate postgraduate pro-gramme.”As there was no dedicated Lebesgue Integrationin that year’s programme, I chose a course on Prob-ability Theory (via Measure Theory). The lecturerwas none other than Professor Harry Reuter fromImperial College, London. Much later I learned thathe was famous for his collaborative work with DavidKendall on birth-and-death processes etc. Again, he,the son of the Socialist Mayor of Magdeburg, cameas a young man to the UK to escape Hilter’s Ger-many; he was looked after by the Cambridge math-ematical analyst, Professor Charles Burkill, and hischaritable wife, Greta Braun. Professor Reuter wassuch a wonderful lecturer that he got me hooked. Infact, his course made me reconsider my entire aca-demic direction.I decided that Probability would be far morefun and useful. The decision to quit Algebra wasnot painful. One must always follow one’s passion.So, I can honestly claim that I was facilitated bya famous algebraist into Statistics. (In doing so,I dropped from the 13th generation of academic de-scendants of Sir Issac Newton to the 14th, accordingto the Mathematics Genealogy Project!) You see,I have had experiences of discontinuous decisionsmore than once in my life. Thresholds have beentruly an integral part of my life in more senses thanone.As it turned out, I stayed at the Northern Poly-technic for just one year. My teaching duty was notheavy and I had free time to read around. I read sev-eral books on probability and stochastic processes.For example, I came across the delightful book onthe theory of time series by Akiva Yaglom, whichkindled my interest in the subject. Many years later,I was able to thank Akiva in person for his introduc-tion. I met him in 1986 at the First Bernoulli WorldCongress held at Tashkent in the former USSR;we were both walking up the Heavenly Mountain
Fig. 2.
With Akiva Yaglom and his wife at the foot of theTian Shan Mountain on the Tashkent side, 1986. (or Tianshan) from the Soviet side. We became in-stant friends. Do you know that the theoretical un-derpinnings of the ARIMA model made popular byGeorge Box and Gwilym Jenkins were already laidrather fully by him in 1955? I learned this fact fromPeter Whittle’s charming book
Prediction and Regu-lation , first published in 1963, when he was Professorof Statistics at Manchester. The book contains manygems and has remained one of my favourites sincemy days at the Northern Polytechnic. Another bookthat also captured my attention was the one by UlfGrenander and Murray Rosenblatt entitled
Analysisof Stationary Time Series (1957). You know, in myday there were not too many books on time series.One could probably count them on the fingers of oneor two hands.At the Northern Polytechnic, there was then asmall study group on forecasting led by Dr. War-ren Gilchrist, who later moved to head the Statis-tics Department at the Sheffield Polytechnic, nowcalled the Sheffield Hallam University. I went alongmainly to listen. Then one day I was asked if I wouldlike to speak to the group on a paper of my choice.I happened to be studying Jim Durbin’s
Biometrika paper on the fitting of a moving average modelvia a long autoregression. I remember showing thegroup all my calculations, which helped me under-stand the paper and survive my first seminar. Littlecould I foresee at the time that my path would crossDurbin’s several times later in my life. When Priest-ley’s name was mentioned at one of the meetings of
K.-S. CHAN AND Q. YAO
Fig. 3.
Edinburgh Workshop on Nonlinear Time Series Howell organised in 1989 (left to right ignoring row number: Wai-Ke-ung Li, Ruey Tsay, Colin Sparrow, Russell Gerrard, John Lane, Murray Rosenblatt, Gudmundur Gudmundsson, John Petruc-celli, Tze-Liang Lai, Tony Lawrance, Peter Robinson, Dominic Guegan, T. K. Brown, Pham Dinh Tuan, Timo Terasvirta,Rodney Wolff, Clive Granger, Peter Fisk, David Cox, Martin Casdagli, Jonathan Tawn, Tohru Ozaki, Granville TunnicliffeWilson, Howell Tong, Dag Tjostheim, Ed McKenzie, Peter Lewis, Richard Smith, Neville Davies, David Jones, Kung-SikChan, Zhao-Guo Chen). the study group, I looked up some of his papers,after which I knew that I would have to return toUMIST!You see, Maurice came to UMIST just when Iwas starting my final undergraduate year; he lec-tured to us on mathematical statistics and stochas-tic processes. We at UMIST had excellent exposure to Statistics through Peter Wallington and MauricePriestley. The former worked on queuing theory un-der Dennis Lindley. The only trouble was that theymade the subject LOOK so easy that two of themore academically inclined students, including my-self, opted for something more abstract like Alge-bra!
Fig. 4.
ISI meeting in Paris, 1989. Left to right: Maurice Priestley, Tata Subba Rao, Mary Tong, Anna Tong, Ritei Shibata,Haruku Shibata, Nancy Priestley, Howell Tong.
CONVERSATION WITH HOWELL TONG Fig. 5.
International Conference on Financial Statistics, Hong Kong, 1999.
To cut a long story short, Maurice welcomedme back. In fact, thanks to an oversight on thepart of the head of department (Maurice was notthe Ho.D.), I was appointed as a demonstrator tocompensate for the SERC postgraduate studentshipthat the Ho.D. forgot to apply on my behalf. The up-shot was that I started my university teaching careeras a postgraduate student and joined the universitypension scheme at quite a young age. This turnedout to be very beneficial many years later when myuniversity pension (based on defined benefits) wascalculated.
QY:
What made you shift from frequency-domainto time-domain in your research in time series anal-ysis?
HT:
As we all know, the history of time seriesanalysis switches to and fro between the time do-main and the frequency domain. I started my re-search from the frequency-domain end. I stayed withit for a few years. Then in 1973, Maurice, Subba Raoand I got a research grant, with which Professor Hi-rotugu Akaike of Japan was invited to visit us for sixmonths. Hiro’s visit marked the beginning of the endof my frequency-domain research. Let me elaborate.The first phase of Hiro’s time series research hadbeen almost exclusively frequency-domain. He wasin fact an international figure in the area. Then he started his collaborative research in designing a feed-back controller for a cement kiln. To his dismay,he discovered that in the presence of feedback, thefrequency-domain approach was inadequate due toa serious bias problem associated with the estima-tion of the frequency-response function. His findingswere recorded in the
Proceedings of Spectral Analy-sis of Time Series edited by Bernard Harris in 1967.This impressive piece of work led to the invitationfrom UMIST.His visit gave me ample opportunities to learnfrom his experiences. He was working on his fun-damental state–space work at the time, which cul-minated in identifying a state as a basis vector of thepredictor space of a second-order stationary multi-variate time series. His vast knowledge impressed medeeply, so I decided to visit his institute in Tokyo,Japan. He was very supportive of my wish. In theevent, I was awarded a Royal Society Japan Fellow-ship without any trouble. I guess that I could wellhave been the only applicant, as the fashion of theday in the UK was to go westwards. The six monthsI spent at Hiro’s institute completed my (inverse)Fourier transform and I returned to the UK as apredominantly time-domain person. I have alreadyrelated the transformation process in my obituary ofProfessor Akaike published by both the Royal Sta-tistical Society and the Institute of Mathematical
K.-S. CHAN AND Q. YAO
Fig. 6.
Hirotugu Akaike enjoying Howell’s after-dinnerspeech at a conference honoring Akaike, Yokohama, 2003.
Statistics. Therefore, I shall not repeat the accounthere, except to say that his personal mini-libraryplayed a vital role.
KSC:
Your earlier works in time series analysiswere all linear. What made you decide to switch tononlinearity?
HT:
Again it had to do with an RSS discussionmeeting. On 18th May, 1977, I read a very short pa-per to the RSS, as one of three discussion papers. Atthe meeting, two features were highlighted, namely,time-irreversibility and limit cycles. I can rememberthe challenging problem posed by Dr. Granville Tun-nicliffe Wilson: “Would we not prefer a model whichin the absence of such (he meant random) distur-bances would exhibit stable periodic deterministicbehavior—a limit cycle?” I decided to take up thechallenge.Coincidentally, around the same time, the Swedishcontrol engineer, Professor K. Astr¨om, gave a semi-nar at UMIST. He described a bilinear control sys-tem, in which the output is not just a simple lin-ear function of past (control) input and past outputbut also their cross products. For time series ana-lysts, an obvious way to imitate this framework isby replacing the control input by a stochastic noise.(Of course, in doing so we are replacing a manip-ulated variable by an unobservable one!) I playedaround with this idea for a bit and even publishedsomething on it.However, very quickly I convinced myself that wasprobably not the best way to address Granville’schallenge: if I switch off the driving noise, the systemwould grind to a halt! One day, as I was mowing my
Fig. 7.
Howell receiving the 2007 Guy Medal in Silver fromPresident Tim Holt. lawn, strip by strip, it dawned on me that a piece-wise linear model would be a good candidate. Therest is history, which you know I have recounted inthe article “Birth of the threshold time series model”in
Statistica Sinica (2007).Actually, the earliest mention of the idea can betraced to my contribution to the discussion of TonyLawrance and N. T. Kottegoda on modelling ofriverflow time series in 1976. There was an interest-ing follow-up. At the time, it seems that my friendTony could not see any relevance of the thresholdidea to riverflow time series modelling. I am surethis was my fault. So, understandably he complainedthat I and one other contributor were “followinga tradition of the Society in taking the opportu-nity to publicize their forthcoming works—at theexpense of other authors’ reprint charges.” I hopethat subsequent applications of the threshold modelin riverflow time series modelling and linking of theLawrance–Lewis’s exponential autoregressive modelto the threshold model have convinced him that theadditional reprint charges were perhaps not unjus-tified.
KSC:
Can you tell us more about the developmentof the threshold models, including their impact onecology, economics and finance and other areas?
HT:
I have given a fairly detailed overview in myarticle “Threshold models in time series analysis—30 years on” in
Statistics and Its Interface (2011).I sincerely hope that the model will continue to en-joy its popularity with users from diverse disciplines.It makes me a very happy man when I see applica-tions of the model in econometrics, economics, fi-nance, ecology, epidemiology, psychology, hydrology
CONVERSATION WITH HOWELL TONG Fig. 8.
Nils Christian Stenseth and Howell, in Hong Kong,2008. and many others. Frankly, some of the applicationareas are beyond my wildest dream. For example,just the other day my attention was drawn to coversong detection and bipolar disorder via the thresh-old model.It would be wonderful if somebody could put allthe most successful applications in book form. Hint,hint. . .Now the basic idea of the threshold model is verysimple: divide the state space into regimes and ruleeach with a simple linear model. It has a non-parametric flavor within a parametric framework.Of course, if we divide the state space arbitrarilyfinely, as in a spline approach, we gain generalityat the expense of loss of parsimony or interpretabil-ity. Successful applications of the threshold modelhave shown that, in many real applications, two orthree regimes will often suffice. Especially encourag-ing is the fact that quite often the regimes are inter-pretable. In mathematics, the idea of piecewise lin-earization is, of course, very old. In oscillations the-ory, the former Soviet mathematicians, Andronovand Khaikin, had introduced and studied (nearly)
Fig. 9.
Mary, Peter Whittle and Howell, in Hong Kong,2009. exhaustively piecewise linear differential equationsin the 1930s. In statistics, we had two-phase linearregressions and Tukey’s regressogram a long timeago, but it seems that they had made no or littleimpact on time series modelling, till the launchingof the threshold autoregressive model and more gen-erally the threshold principle. I must say that fromthe standpoint of stochastic dynamical systems, theincorporation of time in a regression framework is aparadigm-shifting step because without time thereis no dynamics. This is why I hail Yule’s inventionof the autoregressive model as one of the greatestrevolutions in statistical modelling because it ush-ered in the era of dynamic (as against static) mod-elling. I find it unfortunate that some recent text-books have blurred the distinction between a dy-namic model and a static model.Bruce Hansen (2011) has given an extensive re-view of the impact of the threshold model in econo-metrics and economics. Without any doubt, it is ineconometrics/economics that the threshold modelhas made its greatest impact. More recently, the in-fluence seems to be spilling into the field of financeincluding actuarial science.Another significant area of application is ecology.Of course, you, Kung-Sik, have done some marvel-lous joint work with our dear friend, Nils ChristianStenseth of Norway. You have covered so much ofthe animal kingdom: mink, lynx, rodent, lemmingand so on. Your more recent work with your for-mer doctoral student, Noelle Samia, and Nils Chris-tian’s team on plague epidemics using data fromKazakhstan is truly wonderful. As your papers haveshown yet again, often it is through real applicationsthat real progress on the implementation of what I
K.-S. CHAN AND Q. YAO have called the Threshold Principle can be made.You have implemented the principle for count data.I don’t want to embarrass you, Kung-Sik, but I mustsay that the implementation is a truly remarkablecontribution.Of course, regimes can be delineated either sharplyor smoothly. Coming from Hong Kong, I am ratherhappy with a sharp border! Well, the self-excitingthreshold autoregressive (SETAR) model uses asharp delineation. However, some people are lessreceptive to sharp delineations. In this case, we canconsider a softer delineation, for example, a smooth(perhaps “soft” is a better word) threshold autore-gressive model. You, Kung-Sik, and I have actuallydeveloped quite a comprehensive methodology andwe have even given it the acronym of STAR model.The idea has apparently attracted considerableattention in the econometrics literature, under thesame acronym. I could perhaps make one or tworemarks here. For simplicity, let us consider a one-threshold model. If the estimated threshold is in thevicinity of small probability, for example, near thetail or an anti-mode of the marginal distribution,then it tells us that there is probably insufficient in-formation in the data on the functional form of themodel there. In that case, whether we use an indica-tor function as in the SETAR model or a more so-phisticated smooth function as in the STAR modelis of secondary importance. After all, all models arewrong. When choosing between a SETAR model anda STAR model, a more relevant question is whichone is more useful and interpretable.More recently, you, Kung-Sik, Shiqing Ling, DongLi and I have shown systematically how the thresh-old approach can provide powerful tools to modelconditional heteroscedasticity in finance, environ-ment, ecology and others. We have exploited themixture of distributions in the driving noise of thethreshold approach.So far I have focused my answer on a univariatetime series. Although there are generalizations of thethreshold model to multivariate time series, I thinkmuch work remains to be done. One key question isthe delineation of regimes for a p -dimensional statespace. The topography can be quite vast. Too vastperhaps? My gut feelings are that it is still possibleto construct an efficient search algorithm.Besides the question of sharp and smooth delin-eation, there is also the one to do with observableor hidden threshold variables. I must tell you that I wasted an excellent research problem of Markov-chain driven TAR model in 1983 by assigning it tothe wrong student; I should have passed it to you,Kung-Sik, and you would have cracked it in threemonths. The idea was there in the paper I read tothe RSS in 1980 (page 285, line 12 from below).Sometimes, we can even consider partially observ-able and partially hidden threshold variables. I havegiven a discussion in my 2011 recount in Statistics& Its Interface . KSC:
On looking back, the threshold idea is verynatural. Nowadays the idea is applied in many ar-eas, for example, ecology, economics and so on. Andthe TAR models are often featured very substan-tially in elementary text-books, for example, WalterEnder’s
Applied Econometric Time Series Analysis and Cryer and Chan’s
Time Series Analysis: WithApplications in R . Yet, the idea seems to have takenquite some time before it was universally accepted.Don’t you think that this is a little odd?
HT:
Well, it was probably my fault as much asyours for not being good salesmen! More seriously,as I have hinted at earlier, the history of statisticsis full of cases of belated recognition as well as pre-mature enthusiasm. Of course, there are also casesof instant recognition that have withstood the testof time. Like many other professions, value judg-ments by statisticians can sometimes be more sub-jective than scientific. I prefer to let TIME be thejudge. I can remember Hiro Akaike saying to memany years ago (perhaps it was in the 1970s), “Ireckon that AIC could probably survive 30 years.”You see, even he had made the wrong predictionabout his own baby!
QY:
You have also had keen interest in chaos. Howdoes chaos fit in with statistics in general and timeseries in particular?
HT:
The primary object of study in Statistics ischance or, equivalently, randomness. The traditionalview in statistics seems to place randomness at oneend and determinism at the other. And it wouldbe heresy to mix the two. In fact, every statisticiancarries with him ε ’s everywhere, as if he owes hisentire existence to them. If you ask him where his ε ’scome from, he would give you a long list of sources,which is usually all right as far as it goes, except forthe likely absence of one very significant ingredient.Let me digress first.Suppose I toss a coin in this room. I hope you willagree that it is a reasonably close system free fromexternal disturbances. Now, I can write down the CONVERSATION WITH HOWELL TONG Fig. 10.
Qiwei Yao and Howell in Hong Kong, 2009, with Wai-Keung Li and Mike So in the background. precise equations of motion of the coin by appealingto Newtonian mechanics. But I also know I cannotpredict its outcome with certainty, if I give it a goodthrow. Why? Where is the source of randomness?As long ago as the beginning of the 20th Century,H. Poincar´e already included sensitivity to initialconditions as a significant source of randomness. So,even the most basic generator of randomness used by a statistician is a deterministic system; its ran-domness is due to what is called chaos by the dy-namicists. Thus, what excuses can statisticians haveto ignore chaos? Rather than burying our heads inthe sand, I suggest that it is more constructive for usstatisticians to learn more about chaos and make ourcontributions. Another interesting example is to dowith point processes. Within the setup discussed in
Fig. 11.
P. S. Wong, C. K. Ing, N. H. Chan, W. Wu, K. L. Tsui, Peter Hall, T. L. Lai, R. Liu and Howell, at the ChineseUniversity of Hong Kong, in 2009. K.-S. CHAN AND Q. YAO
David Cox and Walter Smith (1954), we can identifya connection between point processes and chaos viathe circle map: x n = x n − + Θ, x = 0 ( n = 1 , , . . . ),where we observe y n = x n mod 1. Note that for irra-tional Θ, y is uniformly distributed on [0 , Scandinavian Journalof Statistics .You asked about time series. It turns out thatmany nonlinear time series models in statistics dogenerate chaos when we switch off the driving noise.That is what makes them so endearing! In a sense,there is the inherent randomness due to chaos of theunderlying deterministic system (I have called it theskeleton elsewhere), as well as the other randomnessdue to the random driving force, perhaps reflectingthe fact that we are dealing with a complex sys-tem with multiple sources of randomness, some, butusually not all, of which can be explained with somedegree of precision.If we accept the above argument, then a natu-ral question is how to define initial-value sensitivityof a stochastic dynamical system. Of course, Qiwei,you know the answer very well, as we have writtenabout the topic. It turns out that the conventionalapproach adopted by the deterministic dynamicistsis inadequate, as it ignores the diffusion due to theexistence of multiple sources of randomness. Insteadof looking at the movement of state x from one timeinstant to the next as they do in deterministic dy-namics, we now look at the movement of one dis-tribution F ( x ) from one time instant to the next.Since the focus is now on the distribution, we haveto generalize the way we measure the sensitivity ofthe movement to initial values (i.e., initial distri-butions). We introduced a stochastic counterpart ofthe Lyapunov exponent. This experience shows thebenefit of having statisticians involved in the studyof deterministic chaos. KSC:
You interacted with people outside statis-tics. How did that come about?
HT:
Mostly by chance and more importantly bytaking advantage of it. It is important to enjoy lis-tening and have a sense of curiosity. For example,I collaborated with Dr. Gudmundsson of Iceland be-cause I remembered that he was working on geo-physical problems when he was a post-doctoral re-search fellow at UMIST. I met him there when Iwas a research student, and I listened to him andremembered what he had told me. So, many years later, I contacted him when I was interested in river-flow time series. Another example is Professor NilsChristian Stenseth. I met him via his doctoral stu-dent Ottar Bjørnstad, who contacted me and invitedme to visit his department. I went to Oslo, listenedto him and his colleagues and found the team thereideally placed for collaborative research. Nowadays,the internet is wonderfully convenient. Sometimes,I have not even ever met my co-authors in person.
QY:
Besides time series analysis, you have alsoworked in other areas of statistics, for example,Markov chain modelling, reliability, dimension re-duction. What motivated you?
HT:
They were mostly my part-time activitiesfor a bit of fun, except for dimension reduction,which was serious business. By about the mid-1990s,I knew I had to get into nonparametrics and semi-parametrics. But they were developing very rapidly.It was not easy for me to keep up, especially at atime when I was heavily involved with administra-tion. Luckily, Bing Cheng and you, Qiwei, arrived inCanterbury, UK. I have learned so much from you.Thank you very much! As for dimension reduction,there is an interesting story behind it. As you know,the area actually laid outside my normal expertisein the 1990s. I was starting my sabbatical leave atthe University of Hong Kong from the Universityof Kent, UK, initially for three years—I was lucky.I knew that Dr. Lixing Zhu of the department (nowchair professor at Baptist University, Hong Kong)was an expert in semi-parametrics. So, I discusseddimension reduction with him. I was not impressedwith the need in the literature to under-smooth theestimator of the nonparametric function. It mightalso be then or perhaps a little later when I ques-tioned the efficacy of techniques such as the slicedinverse regression estimation for time series becausetime-irreversibility is the rule in real time series. Lix-ing shared my concerns but was himself very busywith other research problems, so he mentioned theproblem to one of Professor Wai-Keung Li’s new re-search students, Yingcun Xia. Yingcun was an ex-ceptionally bright student. To cut a long story short,his doctoral thesis formed the basis of a joint discus-sion paper on MAVE which I, on behalf of the fourauthors, read to the RSS in 2002. The trick was toestimate both the nonparametric part and the para-metric part jointly. In this way under-smoothing isrendered unnecessary.
KSC:
We all know that you have held senior ad-ministrative positions in five universities across two
CONVERSATION WITH HOWELL TONG Fig. 12.
Howell with colleagues at the Nonlinear Time Series Workshop in National Singapore University, 2011; from leftto right and ignoring rows: Dong Li, Qiwei Yao, Kung-Sik Chan, Mike So, Peter Brockwell, Ken Siu, Rainer Dahlhaus, ZudiLu, Marc Hallin, Cheng Xiang, Richard Davis, Yingcun Xia, Ying Chen, Rong Chen, Howell Tong, Myung Seo, Shiqing Ling,Simone Giannerini, Cathy Chen, Azam Pirmoradian.
Fig. 13.
Howell and Murray Rosenblatt, after the formerreceived the Distinguished Achievement Award from the In-ternational Chinese Statistical Association at the Joint Sta-tistical Meeting at San Diego, USA, in July 2012. continents. Can you share your experience with usplease? Perhaps you could begin with the ChineseUniversity of Hong Kong.
HT:
After working at UMIST for 14 years,I thought it was high time for me to return to mybirth place, Hong Kong. There was a newly createdDepartment of Statistics at CUHK around 1981 anda new chair of statistics was advertised, to which Iapplied successfully. The new department in 1982consisted of 5 faculty members including myself, onesenior lecturer and three lecturers. (CUHK followedthe British system at that time.) There were also oneassistant computer officer (that was you Kung-Sik),one secretary and one messenger boy. Although Iwas the founding chair professor, actually I did notappoint them; all the faculty members were trans-ferred from the Department of Mathematics and allthe lecturers were formerly students of the seniorlecturer. Fortunately we got on very well indeed.Staff and graduate students had regular dim-sumlunches at a local restaurant. We shared the cost,the seniors paying more, of course—a workable so-cialist system! The biggest challenge was actuallycurriculum design. We decided that our first yearundergraduates should receive good groundings inthe guiding principles of our subject rather than rou- K.-S. CHAN AND Q. YAO
Fig. 14.
Howell with a group of post-graduate students at National University of Singapore, 2012. tine mathematical manipulations. I was voted to bethe guinea pig. It was fun and I learnt a lot myself!Professor George Tiao was our external examiner(another British practice) and he was most helpfuland supportive. He made plenty of constructive sug-gestions and gave us every encouragement. He hasbeen maintaining excellent relationship with CUHKand many other tertiary institutions in Hong Kongever since.
KSC:
What made you decide to leave CUHK in1986?
HT:
My decision to leave CUHK had nothing todo with local politics of the time. I was quite happyat CUHK and my vice-chancellor (equivalent to auniversity president in the US) was very happy toowith the development of my department and the de-partment has remained in very good shape to thisday. In fact, it all happened quite by chance whenI was visiting Professor David Cox’s department atImperial College, London. One day, David told methat a chair was to be advertised by the Univer-sity of Kent at Canterbury, UK. He suggested thatI could have a go if I was interested in returningto the UK. Well, I do not know to this day whyUKC decided to appoint me instead of any one ofthree other very strong candidates. As it turnedout, the biggest challenge was how to manage anot so united mathematics department, consisting ofpure mathematicians, applied mathematicians and statisticians. There were three sections, three bud-get holders and all in one department. A bit crazy!A year or two after my arrival, the vice-chancellorappointed me as the director of my department (di-rectorship was by appointment then). When I be-came aware of the wish of the university to buildup statistics and actuarial science by running down(pure and applied) mathematics, I reminded thevice-chancellor first the history of Thomas Becketand then my plan. As the director of my depart-ment, I could not possibly run down two sections tofatten up the third, especially when the latter wasassociated with me. However, I could build up statis-tics without harming mathematics by (i) taking ad-vantage of the donation secured by my predecessorfrom the Black Horse financial group to build a solidbase for actuarial science; (ii) taking over a majorportion of the management science group which wasbeing or about to be re-organised; (iii) consolidatingstatistical consulting activities and service provisionto Pfizer, whose UK base was nearby. By the timeI stepped down as director in 1993, the statisticsgroup (including actuarial science and the consult-ing arm) grew to more than 30 full-time staff work-ing under one roof, possibly the biggest in the UKthen. Our research rating also went up from 2 whenI joined to 4 when I stepped down.
QY:
Yes, I can remember those exciting days whenI joined you in 1990. Then you went to Hong Kongin 1997. Can you take us through that period please?
CONVERSATION WITH HOWELL TONG HT:
Again it was purely by chance that I wentto Hong Kong, this time to the University of HongKong. You see, HKU had a new and very enter-prising vice-chancellor, Professor Patrick Cheng. Hewas working very hard to turn HKU from a sleepyteaching-oriented university created in the colonialdays to a research-vibrant modern university. Hewas investing huge resources in attracting peoplefrom all round the world to HKU by creating po-sitions such as distinguished visiting professorships.A long-time fellow time series analyst, Dr. (nowChair Professor) Wai-Keung Li, seized the opportu-nity and was instrumental in getting me appointed.I arrived in HKU in 1997 on a 3-year sabbaticalleave (without pay, of course) from UKC. At thattime, UKC also had a new vice-chancellor, Profes-sor Robin Sibson. It was he who granted me theleave.
QY:
You were a visitor and yet you became thefounding dean of their graduate school. How did thatcome about?
HT:
Well, it was all due to my big mouth as usual.My perpetual problem! After my arrival at HKU,one morning Wai-Keung (who was HoD) said to me,“Howell, as you are a chair professor, I’d suggestthat you attend our senate meeting this afternoonif you can spare the time. You see, I cannot go be-cause I have some departmental matters to attendto. Anyway, it might be fun for you to see how weoperate at HKU.” It turned out that the controver-sial item on the agenda was the establishment of agraduate school at HKU. The debate was gettingreally heated. It did not take me long to realize thatmany of those who opposed setting up a graduateschool were professors who came from Britain tenor twenty years previously during the colonial days.You can tell from their accents! I could see that thevice-chancellor and his team were getting nowhere.At this point, I thought I had to say something.So, I said, “As somebody who has just arrived fromBritain, I would like to inform senate members, es-pecially those who left that country many years ago,that the concept of a graduate school, no doubt anAmerican concept, is being adopted by a rapidly in-creasing number of universities in Britain. I feel thatthis is an irreversible trend world-wide.” After that,the debate subsided and the motion was carried. Thefollowing morning, the vice-chancellor rang me up.After thanking me for my intervention, he invitedme to be the founding dean. The rest is history. Mywife joked with me afterwards, saying “I thought you wanted to escape to Hong Kong in order to havepeace and quiet. See what you have done. Serves youright with your big mouth!” Well setting up a gradu-ate school at HKU was challenging, because my firstjob was to persuade nine faculties to relinquish theirpower to the graduate school, abide by some com-mon rules and regulations and to accept supervisionby the Graduate School. I had two associate deans(Professors John Malpas and Anthony Yeh) and onesenior administrator (Mrs. Yvonne Koo) from theregistry to assist me—we called ourselves the gang offour. We literally set down all the rules and regula-tions, down to the way we handled reference letters.We always sent a thank-you letter to each refereeenclosing a copy of his/her reference letter. This isa good way to uncover monkey business. In just afew years, we succeeded in improving our thesis com-pletion rate (after constant monitoring of progress)and employability of our graduate students (we rana small number of compulsory language-enhancingand skill-empowering courses).
KSC:
And you also became a pro-vice-chancellor(equivalent to a vice-president in the US system)!
HT:
Yes, I did serve as PVC to three VCs at HKU.My portfolio changed from one VC to the next and itincluded, at different times, research, administrationand development. The names did not mean much be-cause the dividing line was not sharp. My researchportfolio did mean that I was in charge of the uni-versity’s all important submission of research out-put to the Hong Kong University and PolytechnicGrants Committee, who decides our budget. Thework was tedious but it had to be done method-ically and colleagues had to be handled delicatelyand with compassion. I remember visiting a numberof departments and chatting to all the 60 or so headsof departments.
KSC:
You have collaborated with many people,mostly younger than you, in research. Can you shareyour experience with us?
HT:
I have always enjoyed young companies. Theyare without baggage, full of vitality and can thinkthe unthinkable. My experience suggests to me thatit is far easier sharing crazy ideas with the youngthan with the old. The old tends to react almostimmediately by saying, “They are wrong” or “Theyare trivial.” But the young would say, “Oh, that isinteresting. Let’s see!” I also think that it is the dutyof every statistician to work, from time to time, withsomebody younger than himself, for otherwise thereis no hope for the profession. K.-S. CHAN AND Q. YAO
QY:
Now that you have retired from the LondonSchool of Economics, how do you occupy your time?
HT:
Now that I have retired from the chair fromwhich Professor Jim Durbin also retired, it seemsthat I am as busy as ever. The freedom from admin-istration has given me more time to think (hopefullydeeper), travel and try other things. (I did enjoy ad-ministration when I had to do it. You see, I saw nopoint in complaining and making myself miserable.)Now, with my wife suddenly becoming a qualifiedkeep-fit instructor in her retirement, I have beenpersuaded to exercise more regularly than I usedto. I also try to keep up with the statistical liter-ature and continue doing some research. I am notdispleased with some of the recent results I sharedwith young colleagues. As a matter of fact, Yingcunand I published a discussion paper in
Statistical Sci-ence in 2011. We argue that, for dependent data, theMLE and its equivalents are not necessarily the mostefficacious when we know that the model is wrong.For example, for a wrong time series model, conven-tional methods still typically rely on functionals ofthe one-step-ahead predictors. We have challengedthem. More recently, Kung-Sik, Shiqing Ling, DongLi and myself have just had our paper on condition-ally heteroscedastic AR models with thresholds ac-cepted by
Statistica Sinica , to do with the thresholdapproach.I have joined the University for the 3rd Age,through which I have participated in activities thatI have never imagined I could do. For example, I en-joyed the course on book-binding. In fact, I have turned my copy of Peter Whittle’s charming littlebook
Prediction and Regulation from a poorly pro-duced paperback version into an acceptable hard-back. Do you know that Peter is also a bookbinder?I discovered this fact when I showed him the finishedproduct. Moreover, I am now able to indulge myselfmore in History, Literature and Philosophy. One re-gret is that I am not trilingual or better. I would loveto be able to enjoy, for example,
War and Peace inRussian. So much is often lost in translation. Justcompare Witter Bynner’s translation (possibly thebest available):“. . . Though I have for my body no wingslike those of the bright-coloured phoenix,Yet I feel the harmonious heart-beat ofthe Sacred Unicorn. . . ”with the famous original verse of Li Shangyin (ca.813–858).I have digressed!To me, retirement is one LONG (I hope) sabbati-cal leave that has opened doors into many fascinat-ing avenues. I recommend it!REFERENCES
Cox, D. R. and
Smith, W. L. (1954). On the superpositionof renewal processes.
Biometrika Hansen, B. E. (2011). Threshold autoregression in eco-nomics.
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