Athens Conference on Applied Probability and Time Series Analysis: Volume I: Applied Probability In Honor of J.M. Gani

459 Pages·1996·31.53 MB·English

by  F. Thomas Bruss

Lecture Notes in Statistics 114
EditedbyP. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,
I. Olkin, N. Wermuth, S. Zeger
Springer Science+Business Media, LLC
C.C. Heyde
Yu V. Prohorov
R. Pyke
S.T. Rachev
(Editors)
Athens Conference on Applied Probability
and Time Series Analysis
Volume I: Applied Probability
In Honor ofJ.M. Gani
, Springer
C.C. Heyde
Stochastic Analysis Program. SMS
Australian National University
Canberra ACT 0200
Australia
Yu V. Prohorov
Academy of Sciences
Mathematical Institute
Vavilov Street 42
Moscow 333
Russia
RonaldPyke
University of Washington
Department of Mathematics
Seattle. WA 98195
S.T. Rachev
University of California. Santa Barbara
Department of Statistics and Applied Probability
SantaBarbara.CA 93106
CIP data available.
Printed on acid-free paper.
© 1996 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1996
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ISBN 978-0-387-94788-4 ISBN 978-1-4612-0749-8 (eBook)
DOI 10.1007/978-1-4612-0749-8
Editorial
The AthensConferenceon Applied Probabilityand TimeSeries Analysis was held at
the Titania Hotel in Athens over the period March 22-26, 1995. It was jointlysponsored
by the University ofCalifornia, Santa Barbara, the UniversityofPireus, Greece, and the
AustralianNational University, Canberra. It wasdesigned tobringtogetherresearchers in
applied probabilityand timeseriesanalysisfrom acrossthe world. However, there was the
specific intention to honour J. Gani and E.J. Hannan for their pioneering work in these
fields. Unfortunately, ProfessorHannanpassed away beforetheconferencewas held,soits
timeseries component became a memorial to him.
Thepublished proceedingsappear intwovolumes: Volume 1, Applied Probability. In
HonorofJ.M.Gani, and Volume 2, Time Series. In Memory ofE.J.Hannan.
BiographicalinformationconcerningJ.M.Ganicanbefoundin Austral. J.Statist. 30A
(1988), Preface and pp. 1-13 and Statistical Science 10 (1995), 214-230. For a detailed
Obituary ofE.J.Hannansee Historical Records ofAustralian Science 10 (1994), 173-185.
The Editors
March 1996
Preface
It isover 30 yearssince three enthusiastic Australians and the London Mathematical
Society (LMS) founded the Applied Probability Trust (APT) in 1964, and launched its
first publication, the Journal of Applied Probability (JAP). By the early 1960s, it had
become obvious that a journal dedicated to the applications ofprobability theory to the
biological, physical, social and technological sciences was very much needed. It remained
only to create it.
I recognized this need only very gradually, and began in 1962-3 by convincing two
Australiancolleagues at the Australian National University, where Iwas then working, to
helpmesatisfyit. Aftersomediscussion, theyagreed toassist meinraisingfinance for the
journal. Their names were Norma McArthur ofthe Department ofDemography and Ted
Hannan ofthe Department ofStatistics. Eventually, after a meetingbetween the Council
of the LMS and me in 1963, we raised enough financial support to create the APT and
start JAP. The founding trustees ofthe APT were Sir Edward Collingwood for the LMS,
Ted Hannan, Norma McArthur and myself; sadly, I find myself the sole survivor of this
group. But the APT lives on; its current trustees, apart from me, are Daryl Daley, Chris
Heyde and Sir John Kingman for the LMS.
Thefirst volumeofJAP published in1964consistedoftwo issues totallingjustunder
400pages;thelastcompletevolumeof4issuesin1995numbered1149pages. Itscompanion
volume,Advances in Applied Probabilitybegun in 1969, also had 4issues of1191 pages in
1995. Thus, after 31 years, the number of pages in applied probability published in the
APT'sjournals alone had increased by a factor of6.
But this hardly represents the true measureofgrowthofthesubject in thepastthree
decades. There arenow another30or morejournals, many ofthem published only within
the last decade, whichcontain some material in applied probability, including time series
v
analysis. Among the better known ofthese are
Annales de l'Institut Henri Poincare, Section B. Probabilites et Statistique
Annals ofAppliedProbability
AppliedStochastic Models and Data Analysis
Communications in Statistics, Stochastic Models
International Journal ofForecasting
Journal ofForecasting
Journal ofthe RoyalStatistical Society, Series B
Journal ofTime Series Analysis
Mathematics ofOperations Research
Operations Research
Probability in the Engineering and Informational Sciences
Probability Theory andRelated Fields
Queueing Systems, Theory and Applications
Scandinavian Actuarial Journal
Sequential Analysis
Stochastic Analysis and Applications
Stochastic Processes and their Applications
Stochastics and Stochastics Reports
Theory ofProbability and its Applications.
This list is by no means exhaustive. One is led to conclude that applied probability
and timeseriesanalysisremainveryactivefieldsofresearch, whosegrowthseemsdestined
to continue wellinto the future.
Randomness is an integral part of many natural and social phenomena, so that it
is not surprising to find probabilistic methods applied to a wide range of problems. As
probability theory itself becomes deeper and more sophisticated, it is used increasingly
to model such problems. Two recent examples are the use ofthe Black-Scholes equation
in probabilistic financial models, and of martingale methods in estimating the infection
parameters ofepidemics.
The contributions to this Conferenceattest tothebreadth ofapplied probabilityand
timeseriesanalysis. Ishallleaveit to PeterRobinsonandMurray Rosenblatttocomment
indetailonthetimeseriesvolume,andwillrestrictmyselftoabriefsummaryoftheapplied
probability proceedings. These include sections on probability and probabilistic methods
in recursive algorithms and stochastic models, Markov and other stochastic models such
as Markov chains, branching processes and semi-Markov systems, biomathematical and
genetic models, epidemiological models including S-I-R (Susceptible-Infective-Removal),
householdandAIDSepidemics, financial modelsfor optionpricingandoptimizationprob
lems, random walks, queues and their waiting times, spatial models for earthquakes and
inferenceonspatialmodels. Therewouldhardlyseemtobeanyareaofreallifeorscientific
research that is resistant to probabilistic analysis.
IshouldliketothinkofLaplaceasoneoftheearlygodfathersofappliedprobability. In
his Theorie Analytique de Probabilites(1812), heappliedprobabilisticreasoningtovarious
demographic models. In Chapter 6 of his book, on the probability of causes of future
VI
events based on observed events (pp. 377-401), for example, Laplace considered three
topical problems. First he analyzed the different ratios of boys to girls born in London,
Naples and Paris; he went on to discuss mortality tables, probabilities ofsurvival and a
method for estimating the "population ofa large empire". He concluded his Chapter by
considering the ratios of christenings of boys and girls in Paris, which had been 25/24
between 1745and 1784,and predictedthat theprobabilitythatit wouldbegreater than 1
for 100 years thereafter was P = 0.782. Ido not know ifthis result has ever been verified
against subsequent data, but I canattest to the fascination ofLaplace's arguments.
Letmeendonanoteofgratitude. Ipersonallyenjoyedthe AthensConferencegreatly,
not only because of the quality of the participants' papers, and the liveliness of their
discussions, but also becauseofthewarmwelcome we received from ourGreekcolleagues.
Ted Hannan would have loved the occasion, had he only been with us.
Toalltheparticipantswhohelpedtocelebratethevigourandinventivenessofapplied
probability and time series analysis, to our generous Greek hosts, and to Bessy Athana
sopoulos for her help in the organizationofthe Conference, Ioffer mywarmest thanks.
Reference
Laplace, P.-S. (1812) Theorie Analytique des Probabilites. Vol. 7 of the Oeuvres
Completes de Laplace, Paris.
Joe Gani
Stochastic Analysis Group, SMS
Australian National University
March 1996
vii
VOLUME 1. APPLIED PROBABILITY
CONTENTS
EDITORIAL
PREFACE
A. PROBABILITY AND PROBABILISTIC METHODS
F.T. Bruss and T.S. Ferguson. Half-prophets and Robbins' problemofminimizing
the expected rank 1
M. Cramer and L. Rueschendorf. Analysisofrecursive algorithms by the
contraction method 18
K.R. Parthasarathy. Comparisonofcompletely positive maps ona C*-algebra
and a Lebesgue decomposition theorem 34
P. Picard and C. Lefevre. Abelexpansions and generalized Abel polynomials
instochastic models 55
M. Scarsini and M. Shaked. Positivedependence orders: asurvey 70
B. MARKOV AND OTHERSTOCHASTIC PROCESSES
O. Chryssaphinou and S. Papastavridis. APoisson limiton the number of
appearances ofa pattern ina Markov chain 92
D. Ghoshand A.P. Godbole. Palindromes in random lettergeneration: Poisson
approximations, rates ofgrowthand Erdos-Renyi laws 99
A.G. Hart and P.K. Pollett. Direct analytical methodsfor determining
quasistationarydistributions for continuous-time Markov chains 116
G. Kersting and F.C. Klebaner. Explosions in Markov processesand submartingale
convergence 127
J.S. Kwon and R. Pyke. Probabilitybounds for product Poisson process 137
C. Lefevre and P. Picard. On the first-crossing ofa Poisson processona
lower boundary 159
D.J. Scott and R.L. Tweedie. Explicitrates ofconvergenceofstochastically
ordered Markovchains 176
M.N. Slavtchova-Bojkova. Multi-type age-dependent branching processes with
state-dependent immigration 192
P.-C.G. Vassiliou. The nonhomogenous semi-Markov systemin a stochastic
environment 206
G.P. Yanev and N.M. Yanev. Branching processes with two types ofemigrationand
state-dependent immigration 216
C. BIOMATHEMATICAL MODELS
W.J. Ewens. Remarkson the lawofsuccession 229
F. Papangelou. Large deviationsofthe Wright-Fisher process 245
D. EPIDEMIC MODELS
F. Ball. Threshold behaviourinstochasticepidemics amonghouseholds 253
IX
N.G. Becker and K. Dietz. Reproduction numbersand critical immunity levels
for epidemics in a communityofhouseholds 267
J. Gani. Modelling the spread ofHIV in prisons 277
M. Neuts and J.-M. Li. An algorithmicstudy ofS-I-Rstochasticepidemic models 295
E. FINANCIALMODELS
B. Gamrowski and S.T. Rachev. Testing the validity ofvalue-at-work measures 307
A. Rejman and A. Weron. Option pricing for hyperbolic CRRmodel 321
G. Samorodnitsky. Aclassofshot noise models for financial applications 332
P. Whittle. Why discount? The rationaleofdiscounting in optimization
problems 354
F. RANDOM WALKS AND QUEUES
J.W. Cohen. On periodic Pollaczekwaiting time processes 361
M. Kotulski and K. Weron. Random walk approach to relaxation indisordered
systems 379
G. SPATIALMODELS
LV. Basawa. Inference for a classofcausalspatialmodels 389
Y.Y. Kagan and D. Vere-Jones. Problems in the modellingandstatistical
analysis ofearthquakes 398
H. INFERENCE
A. Dannwerth and D. Plachky. Onthe existenceofUMVU estimatorsfor Bernoulli
experiments in the non-identicallydistributed case withapplications to the
randomized response methodand the unrelated question model 426
L. Takacs. On a three-sample test 433
x
HALF-PROPHETS AND ROBBINS' PROBLEM
OF MINIMIZING THE EXPECTED RANK
F. Thomas Bruss', Universite Libre de Bruxelles
Thomas S. Ferguson", University ofCalifornia, Los Angeles
Summary: Let Xl,X2,•..,Xn bei.i.d. random variables with a known
continuous distribution function. Robbins' problem is to find a sequential stop
ping rule without recall which minimizes the expected rank ofthe selected ob
servation. An upper bound (obtained by memoryless threshold rules) and a
procedure to obtain lower bounds of the value are known, but the difficulty is
that the optimal strategy depends for all n > 2 in an intractable way on the
whole history of preceding observations. The goal of this article is to under
stand better the structure of both optimal memoryless threshold rules and the
(overall) optimal rule. We provethat theoptimal rule is a "stepwise" monotone
increasing threshold-function rule and then study its property of, what we call,
full history-dependence. For each n, we describe a tractable statistic of pre
ceding observations which is sufficient for optimal decisions of decision makers
with half-prophetical abilities who can do generally better than we. It is shown
that their advice can always be used to improve strictly on memoryless rules,
and we determine such an improved rule for all n sufficiently large. We do not
know, however, whetheronecanconstruct,as n -+ 00, asymptotically relevant
improvements.
Keywords: Sequential selection - Full information - Memoryless threshold
rules - "Stepwise" monotonicity - Prophets -Order statistics.
AMS 1991 Subject Classification: Primary 60G40, Secondary 62L15.
§1. Introduction.
1.1 The Problem. Let XI,X2,..•,Xn be i.i.d. r.v.'s with c.d.f. F. We
assume F to be continuoussothat the Xk'sare uniquely rankable(a.s.). Sincethe
payoffs we consider depend only on the ranks of the Xi, we may and do assume,
w.l.o.g., that F is uniformon [0,1]. The relative rank ofobservation Xk is defined
as
Lk :s :s :s
rk = [(Xj Xk), 1 k n, (1.1)
j=l
* Address: Universite Libre de Bruxelles, Departement de Mathematique et In
stitut de Statistique, CP 210, B-1050 Brussels, Belgium, e-mail: [email protected]
•• Address: Department of Mathematics, University of California, Los Angeles,
CA 90095, USA, e-mail: [email protected]
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