Does Extinction distribution determine the offspring distribution in Simple Branching?

Transcript of Does Extinction distribution determine the offspring distribution in Simple Branching?

From Extinction to Reproduction inBienayme-Galton-Watson processes

Daniel Tokarev

Monash University

11 July, 2012

Daniel Tokarev From Extinction to Reproduction in Bienayme-Galton-Watson processes
BGW Processes

Let be some a random variable supported onnon-negative integers with pmf {pi} (reproductiondistribution)

Let Z0 = 1 and Zn+1 =Z (n)

i=0 i, n, where i, n are iid like and also independent of the pastThe information about the process is encoded inprobability generating function

f (s) =

i=0

pisi .

Recall that E = f (1) := , E( 1) = f (1) and thefunctional iterates fn(s), n = 1,2, . . . are the probabilitygenerating functions of the process at time n, while f (s)k ,k -integer is a pgf of a process started with k individuals.

BGW Processes




f (s) =

i=0

pisi .


BGW Processes




f (s) =

i=0

pisi .


BGW Processes




f (s) =

i=0

pisi .


BGW Processes




f (s) =

i=0

pisi .


BGW Processes




f (s) =

i=0

pisi .


Classification and Extinction Time

BGW processes are broadly divided into three typesSubcritical ( < 1), critical ( = 1) - extinction certain andsupercritical ( > 1) - extinction uncertainSince the iterated function fn(s) is the PGF of Z (n) inparticular fn(0) is the Pr of extinction after n steps andtaking the limit as n, gives the Pr of eventualextinction



subcritical.pdf

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0



0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Extinction, iterates and PGFs

Suppose two individuals have extinction time distributionsno more than apart (wrt some sensible norm):

E.g. {fn(0)} {gn(0)} := supn1{|fn(0) gn(0)|} .

Can we then deduce a similar statement about thecorresponding {pi} and {qi}?Specifically if = 0, will it follow that {pi} and {qi} are thesame?Must be true, otherwise two distinct PGFsintersect ininfinitely many points! Or is it?Easy to construct two PGFs that share artibrarily manyiterates:



E.g. {fn(0)} {gn(0)} := supn1{|fn(0) gn(0)|} .




E.g. {fn(0)} {gn(0)} := supn1{|fn(0) gn(0)|} .




E.g. {fn(0)} {gn(0)} := supn1{|fn(0) gn(0)|} .




E.g. {fn(0)} {gn(0)} := supn1{|fn(0) gn(0)|} .




E.g. {fn(0)} {gn(0)} := supn1{|fn(0) gn(0)|} .


Finitely many points in common

Let f (s) = ex1 and denote its extinction pmf by{fn(0)} =: {tn}. For some integer j , and small > 0 leth(s) := s

ji=1(s ti).

Then for sufficiently small = (j), f (s) + h(s) will be aPGF with the same first j iterates as f (s). More generally

TheoremLet 0 a1 < a2 < < an = 1 be a finite ordered sequencewith f (ai) =: bi , i = 1, . . . ,n and bn = 1. Further letk := #{j 0 : pj > 0} , so that f (s) =

ki=1 pji s

ji , whereji s are the indices of strictly positive probabilities pj . Thereexists a distribution {qi} on Z+ with {qi} 6= {pi}, such that for itsPGF g(s) =

i0 qis

i , g(ai) = f (ai) = bi , for i = 1, . . . ,n if andonly if n k.



ji=1(s ti).



ki=1 pji s


i0 qis




ji=1(s ti).



ki=1 pji s


i0 qis




ji=1(s ti).



ki=1 pji s


i0 qis


Supercritical case

Recall that in supercritical case the iterates accumulate toa point inside the unit intervalHence by Identity principle, we cannot have zeroesaccumulating to a point inside the region of analyticity.More generally

Theorem

For any sequence of extinction probabilities {f ki (0)} of a mortalsupercritical BGW process with Z (0) = r , there is a uniquenon-lattice offspring distribution {pi}.

Indeed if f (s) a PGF of a lattice RV on N, letg(s) = f(s1/), then the BGW process corresponding to fand starting with Z0 = will have the same extinction distas that corresponding to g.

Supercritical case


Theorem



Supercritical case


Theorem



Supercritical case


Theorem



Supercritical case


Theorem



When moments exist

So if the PGF is analytic at accumulation point of theiterates at 0 (call it q), the question is settledFor the case q = 1, what if all moments exist? Then allfactorial moments exist, ie left-sided derivatives at 1 existBut existence of moment, factorial moments and left-sidedderivative does not imply that the PGF is analytic at 1, eglet pi = c2

k , c = 1/

2

k , easy to check that allmoments

cpk2

k exist but the PGF f (s) =

pisi

cannot be continued beyond 1 since

(1 + a)k2

k =for all a > 0.

When moments exist


k , c = 1/

2


cpk2


pisi


(1 + a)k2

k =for all a > 0.

When moments exist


k , c = 1/

2


cpk2


pisi


(1 + a)k2

k =for all a > 0.

When moments exist


k , c = 1/

2


cpk2


pisi


(1 + a)k2

k =for all a > 0.

When moments exist continued

So the previous result does not guarantee that the iteratesuniquely determine reproduction distribution {pi}Divided differences come to the rescue and give us more!

TheoremLet {Zn} be either a supercritical or a non-supercritical BGWprocess for which the moment generating function exists. Then{fi(0)} =: qi uniquely characterises {pi} which can bedetermined from the Taylor expansion of f around q given by

f (s) = q +

i=1 (qn, . . . ,qn+i)(s q)i , where (qi) := qi+1

and (qi , . . . ,qi+j) :=(qi+1,...,qi+j )(qi ,...,qi+j1)

qi+jqi




f (s) = q +



qi+jqi




f (s) = q +



qi+jqi

Approximation theory to the rescue!

We will need the following key result - evolution ofWeierstrass Approximation Theorem through to MntzsTheorem - Full Mntzs Theorem (Schwartz, Siegel):

TheoremLet {i}i=0 be a sequence of distinct positive real numbers

including 0, = Span{n

i=0

aixi |ai R}, and C[0,1] is the

space of continuous functions on [0,1]. Then

= C[0,1] iff i

2i + 1=.





i=0



= C[0,1] iff i

2i + 1=.





i=0



= C[0,1] iff i

2i + 1=.

Identity theorem for completely monotone functions

Recall that h(s) is completely monotone if h C[0,)]and for all n N, s R+, (1)nh(n)(s) 0.Given a family of functionsM with common domain D, wesay that a function is uniquely characterised by its valueson {i} D if for any two f, g M with f(i) = g(i) for alli = 1,2, . . ., implies f(s) = g(s) for all s D. We have

TheoremGiven a sequence of distinct non-negative real numbers{i} 3 0, a completely monotone function is uniquelycharacterised by its values on {i} iff i

2i + 1=. (1)




2i + 1=. (1)




2i + 1=. (1)




2i + 1=. (1)

And back to PGFs!

Observe that if f is a PGF and h is completely monotone,then f (g) is completely monotone, from this we easilyobtain

TheoremLet {qn}n=j , for some j N, l be a tail of a distribution ofextinction time of a BGW process {Zn}, with Z0 = r . Let Tdenote the RV time to extinction of {Zn}. Then {qn}n=j uniquelydetermines the reproduction distribution {pi} and r if

i=j

(1 qr ) = or equivalently ET =.

And back to PGFs!



i=j


And back to PGFs!



i=j


Towards a counter-example: Blaschke Products

Generalisation of Weierstrass products to functionsanalytic on the open unit disk

TheoremGiven a set of points {an} on the unit disk, there exists afunction analytic on the unit disk with zeros at {an} and uniqueup to a zero free analytic factor iff

i=1

(1 |ai |)

in which case it is given by

B(z) =

i=1

ai|an|

an z1 anz

.




i=1

(1 |ai |)


B(z) =

i=1

ai|an|

an z1 anz

.




i=1

(1 |ai |)


B(z) =

i=1

ai|an|

an z1 anz

.

The Number of Positive Zeros of Transforms

If f and g agree on the iterates, f (s) g(s) = E(s)B(s)where B(s) is a Blaschke product and E(s) is a zero freefunction analytic on the unit disk.For subcritical PGFs, we know that B(s) =

bisi with

|bi | 1/ig(s) would have Taylor coefficients = o(i2) and sincef (s) = g(s) + E(s)B(s), andWe need to find E(s) that would make the coefficient of theproduct E(s)B(s) decay faster than i2

The trouble is that we dont understand the pattern of signsin bi s - real Blaschke products are not well-understood



bisi with





bisi with





bisi with





bisi with





bisi with



Mixing AdvantageHamza, Jagers, Sudbury & Tokarev (2009), Extr.

How do mixed and unmixed populations compare?

Theorem (Hamza, Jagers, Sudbury and Tokarev (2009) Extr.)

Assume FSS that d = n and let Mi := Emax{(1)i , . . . ,

(n)i }

1n

ni=1

Mi Emax{1, . . . , n} 1n

ni=1

Mi +n 1

nmax

i=1,...,n{Mi}

In particular, if all the unmixed expected lifetimes are the sameand equal to M, then we have

M Emax{1, . . . , n} (2 1/n)M.

Mixing AdvantageHamza, Jagers, Sudbury & Tokarev (2009), Extr.

How do mixed and unmixed populations compare?

Theorem (Hamza, Jagers, Sudbury and Tokarev (2009) Extr.)

Assume FSS that d = n and let Mi := Emax{(1)i , . . . ,

(n)i }

1n

ni=1

Mi Emax{1, . . . , n} 1n

ni=1

Mi +n 1

nmax

i=1,...,n{Mi}

In particular, if all the unmixed expected lifetimes are the sameand equal to M, then we have

M Emax{1, . . . , n} (2 1/n)M.

What does tell us about ?

When does {fn(0)} uniquely characterizes f?When f is analytic in the neighbourhood of the fixed point q

From Identity Principle (an analytic function is uniquelycharacterized by its values at a set of points accumulatinginside the domain of analyticity).

This applies when f is a PGF of a supercritical process withp0 6= 0

When E = - from a slight generalisation of Fellers proofof Mntzs TheoremNot when E
Athreya, K. B. and Ney, P.E. (1972) Branching Processes.Springer-Verlag.

Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987).Regular Variation. Cambridge University Press.

Carnicer, J. M., Pea, J. M. 1998. Characterizations of theoptimal descartes rules of signs. Math. Nachr. 189, 3348.

Feller, W. (1971) An Introduction to Probability Theory andIts Applications, Volume II, 2nd Ed., John Wiley & Sons,Inc.

Hamza, K., Jagers, P., Sudbury, A., Tokarev D. (2009)Mixing advantage is less than 2. Extremes, 12, 1931.

Klebaner, F., Tokarev, D. V., (2008). Generalised FractionalLinear Generating Function. (in preparation).

Lawton, J.H., May, R.M., Extinction rates, (1990).textitOxford University Press, Oxford, UK

Pakes, A. G. (1989) On the Asymptotic Behaviour of theExtinction Time of the Simple Branching Process. Adv.Appl. Prob. 21:470-471.

Seneta, E. (1974). Regularly varying functions in the theoryof simple branching processes. Adv. Appl. Probab. 6,408420.

Slack, R. S. (1968). A branching process with mean oneand possibly infinite variance, Zeitschrift frWahrscheinlichtkeitstheorie und Verwandte Gebiete9:139-145.

Slack, R. S. (1972). Further Notes on Branching Processwith Mean 1. Zeitschrift fr Wahrscheinlichtkeitstheorie undVerwandte Gebiete 25:31-38.

Chris D. Thomas, Alison Cameron, Rhys E. Green, MichelBakkenes, Linda J. Beaumont, Yvonne C. Collingham,Barend F. N. Erasmus, Marinez Ferreira de Siqueira, AlanGrainger, Lee Hannah, Lesley Hughes, Brian Huntley,

Albert S. van Jaarsveld, Guy F. Midgley, Lera Miles, MiguelA. Ortega-Huerta, A. Townsend Peterson, Oliver L. Phillips& Stephen E. Williams. Extinction risk from climate change,(2004), Nature 427, 145-148.

Tokarev, D. (2008). Growth of Integral Transforms andExtinction in Critical Galton-Watson Processes, Journal ofApplied Probability, 45, 1-9.

Tokarev, D.V., and Borovkov, K.A., (2009). On theexpectations of maxima of sets of independent randomvariables. Statistics and Probability Letters, 79, 2381 2388

Tokarev D.V., From Extinction to Reproduction inBienayme-Galton-WatsonProcesses. (in preparation).Other

Does Extinction distribution determine the offspring distribution in Simple Branching?

Documents

Transcript of Does Extinction distribution determine the offspring distribution in Simple Branching?

Continuous Probability Distributions Exponential, Erlang ...€¦ · •The Erlang distribution is a generalization of the exponential distribution. •The exponential distribution

MTH 2301 Méthodes statistiques en ingénierie...1 terminologie statistique rappels distribution de la moyenne: théorème central- limite distribution Khi -deux (χ2) distribution

Convergence in Distribution Central Limit Theorem · Convergence in Distribution Theorem. ... This statement of convergence in distribution is needed to help prove the ... of n Bernoulli’s)

Bioreport Distribution of Growth2

A New Approach for Distribution Testing

Normal Distribution - Mathlzhang/teaching/3070spring2009/Daily Updates... · Normal Distribution Proposition If X has a normal distribution with mean and stadard deviation ˙, then

Hypothesis Testing – Distribution

The infrared extinction law in various interstellar environments 1 Shu Wang 11, 30, 2012 Beijing Normal University Email: shuwang @ mail.bnu.edu.cn.

Inference for the Weibull Distribution

Service et distribution pour la Métrologie Tridimensionnelle.

EC4004 Lecture3: World Income Distribution and Convergence

1 Wind Power CUTTY SARK. 2 Weibull Distribution Weibull Distribution Plotter Programme.

Gamma Distribution Applications (See Wikipedia) The gamma distribution has been used to model the aggregate size of insurance claims. The gamma distribution.

Inference about means But We don’t know swhitlock/bio300/overheads/...its distribution to a standard normal distribution: € Y This would give a probability distribution of the

CURRICULUM VITAE - University of Crete · Web viewMaternal drug abuse versus maternal depression: Vulnerability and resilience among school-age and adolescent offspring. Development

DISTRIBUTION AND FEEDING ECOLOGY OF THE AFRICAN ...

Introductory Econometrics - Session 3 - Distribution … · DistributionsBasics of mathematical statsCon dence intervals Introductory Econometrics Session 3 - Distribution and con

Sampling Distribution & Confidence Intervalgchang.people.ysu.edu/class/mph/note/06_5_CI_Ch8_9_CLT_CI.pdfSampling Distribution & Confidence Interval CI -1 1 ... Sample Proportion, ...

Distribution Locational Marginal Pricing (DLMP) for ...

Exponential Distribution