Strong Completeness in Random Thinning of Processes and Its Applications

1. Strong Completeness and its Application to Random Thinning of Random Processes ShailajaDeshmukh University of Pune, Pune – India Visiting professor, University of Michigan, Ann Arbor • Outline • Random Thinning of stochastic process • Identifiable sampling schemes • Markov sampling of a continuous parameter stochastic process • Strong completeness of gamma family • Applications in spatial sampling • Negative binomial sampling of a discrete parameter stochastic • process and its applications in risk analysis

2. {X(t) , t  T} , T : Discrete or continuous • Various observational schemes • Complete observation for a fixed time interval Keiding (1974, 1975), Athreya (1975, 1978) • Observing a process till a fixed no. events occur (inverse sampling) Moran (1951), Keiding (1974, 1975) • Observing the process at specified deterministic epochs t1, t2, …tn Prakasa Rao (1988), Su & Cambanis (1993)

3. Kingman (1963, Ann. Math. Statist ) • Fixed epoch sampling suffers from non- identifiability Observed data may come from different processes • Kingman (1963) advocated selecting epochs t1, t2, …tn randomly • Criterion – Process derived from the original process randomly, • should determine the stochastic structure of the original process • uniquely • The process used for sampling should be identifiable • {X(t), t  T}: Original process under study • Zn = X(Tn), Tn : Random variables • {Zn, n  1} : Derived process or randomly thinned process

4. Identifiability of a sampling scheme d {X(1) (t)} {X(2) (t)} =    d {Z(1) (n)} {Z(2) (n)} = • Derived process determines the original process uniquely • Identifiability is essential for justfication of inference based • on randomly derived process • Basawa (1974) , Baba (1982) • Identifiable sampling schemes

5. continuous discrete • Tn : n th success in • independent Bernoulli trials. • Bernoulli sampling • Deshmukh (1991), Austr. J. of Statistics • Tn : n th event • in Poisson process. • Poisson sampling • Kingman (1963) • Ann. Math.Statist • Tn : n th visit to state 1 • in two state Markov chain • Markov sampling • Deshmukh (2000), Austr.&New • Zealand J. of Statistics • Tn : n th visit to state 1 • in two state Markov process. • Markov process sampling • Strong completeness of gamma • family. Deshmukh (2005) • Stochastic modelling & Applications Tn: n-th epoch of k-th success in Bernoulli trials Strong completeness of negative binomial family Extension of PASTA

6. Markov sampling of a continuous parameter stochastic process • {X(t), t ≥ 0}: continuous parameter stochastic process • {Y(t), t ≥ 0}: Markov process with state space {0,1} and Y(0) = 1 • {Y(t)} is independent of {X(t)} • Observe {X(t)} at the epochs of visits to state 1 of {Y(t)} • Tk = S1 + … + Sk : epoch of k-th visit to state 1 of {Y(t)} • {X(t)} is observed at Tk, k ≥ 1, • {Z(k) = X(Tk), k ≥ 1} is derived from the original process by MS

7. Aim: Whether {Z(k)} determines the stochastic structure of the original • process uniquely • Waiting time Tk for the k-th visit to the state 1 of the Markov process 1 W1 0 W0 1 • Waiting time for the first visit to the state1: S1 = W0 + W1 • W0 and W1 are independent random variables having exponential • distribution with mean λ0-1 and λ1-1 respectively. • Tk = S1 + … + Sk = V0 + V1, where Vi ~ ~ G (λi, k ), • λi : scale parameter and k is the shape parameter, i = 0,1.

8. Sampling scheme is identifiable d {X(1) (t)} {X(2) (t)} =  Markov sampling   d {Z(1) (k)} {Z(2) (k)} = Family of finite dimensional distribution functions of {X(i) (t)} Fi ≡ Fi(t1, t2,…, tn) = Fi(t1, t2,…, tn; x1,.., xn) = P[Xi(tj) ≤ xj; j = 1,…, n], x1,x2,…, xn - real numbers, t1,t2,…, tn - positive real numbers, t1 < t2 < … < tn. Family of finite dimensional distribution functions of {Z(i) (k)} Gi = P[Zi(K(j)) ≤ xj, j = 1, 2, …,n ], K(j) = k1 + k2 + … + kj 8

9. k1 k2 kn 0 K(1) K(2) K(n-1) K(n) TK(1) TK(2) TK(n-1) TK(n) U1 U2 Un-1 Un L1 L2 Ln Let TK(j) =Uj and Lj = Uj – Uj-1 , Lj = U1 + U2 + … + Uj Gi ≡ P[Zi(K(j)) ≤ xj, j = 1,…, n] = P[Xi(TK(j)) ≤ xj, j = 1,…, n] = P[Xi(Uj) ≤ xj, j = 1,…, n] = E{ P[Xi(Uj) ≤ xj, j = 1,…, n | U1,U2,…,Un] } = E{P[Xi(Uj) ≤ xj, j = 1,…, n ]} = E{Fi(L1, L1+L2, …, L1+L2+…+Ln)}

10. G1 = G2 implies E{F1(L1, L1+L2, …, L1+L2+…+Ln)} = E{F2(L1, L1+L2, …, L1+L2+…+Ln)} E{F1(L1, L1+L2, …, L1+…+Ln)- F2(L1, L1+L2, …, L1+…+Ln)} = 0 Expectation is with respect to the joint distribution of (L1, L2,…,Ln). L1, L2,…,Ln are independent and Lj ~ Vokj + V1kj ,where Vokj ~G (λ0, kj ) and V1kj ~ G (λ1, kj ), λ0 and λ1 are known, kj j= 1, …,n are the only unknown parameters . Expectation is with respect to the joint distribution of (Vokj ,V1kj j= 1, …,n). If the joint distribution of (Vokj ,V1kj j= 1, …,n) is complete G1 = G2 implies F1 = F2 Strong completeness of family of Vokj /V1kj for any j implies completeness of the joint distributions

11. X ~ G ( α, k), α : scale parameter, k : shape parameter f (x) = αk e-αx xk-1/Г(k), x > 0 α : known, k Є I+ Not a one parameter exponential family, parameter space is not an open set Complete family

12. Ek( h(x)) = 0 for all k Є I+ ⇔ ∫ h(x) αk e-αx xk-1/Г(k)dx = 0 , for all k Є I+ ⇔ g(k) = 0 , for all k Є I+ ⇔ Σzk g(k) = 0, 0 < z < 1 ⇔ ∫ h(x) e-αx (Σ (αzx)k-1/(k-1)!) dx = 0 ⇔ ∫ h(x) e-θx = 0, θ = α (1 – z), 0 < θ < α ⇔ ∫ h(x) e-θx = 0, for all θ > 0, by analytic continuation ⇔ h(x) = 0, a.s. Pk for all k Є I+ {G(α , k), k Є I+} is a complete family Strongly complete

13. Definition: A family of distributions {Fθ, θЄΘ} is called strongly • complete if there exists a measure μ on (Θ , fi) such that for every • subset Θ* of Θ for which μ(Θ – Θ*) = 0, ∫ h(x) Fθ(dx) = 0 for all • θЄΘ* implies that h (x) = 0 a. s. Pθfor every θЄΘ. (Zacks, 1971) • Strong completeness implies completeness by taking Θ* = Θ • Suppose T1 and T2 are independent random variables. If {FθT1,θЄΘ} • is complete and {FθT2, θЄΘ’} is strongly complete then the family • of joint distributions {Fθ, θ’T1,T2θЄΘ, θ’ ЄΘ’} is complete. • (Zacks, 1971) • Gamma family is strongly complete • Parameter space - I+,fi-sigma field, μ is a measure induced by • geometric distribution • For A Є fi, μ(A) = Σδ (1 – δ) k – 1, sum being taken over k Є A

14. Suppose Θ* is a subset of I+ such that μ(I+ - Θ* ) = 0 ∫ h(x) αk e-αx xk-1/Г(k)dx = 0 , for all k ЄΘ* ⇔ g(k) = 0, for all k ЄΘ* μ({k}) = 0 , for all k Є (I+ - Θ*) g(k) μ(k) = 0 , for all k Є I+ Σδ (1 – δ) k – 1( ∫ h(x) αk e-αx xk-1/Г(k)dx) = 0, sum being taken on I+ Using Fubini’s theorem, summation and integration can be interchanged ∫ h(x) e-θx = 0, for all θ > 0, by analytic continuation h(x) = 0, a.s. Pk for all k Є I+ Gamma family is strongly complete

15. Thus, the joint distribution of ((Vokj ,V1kj j= 1, …,n) ) is complete. Further using continuity of F we get G1 = G2 implies F1 = F2 Markov sampling is an identifiable sampling scheme • {Z(k), k ≥ 1} is a Markov process iff {X(t)}is a Markov process. • {Z(k), k ≥ 1} is a stationary process iff {X(t)}is a stationary • process. • limt →∞P[X(t) Є B] = lim n→∞ P[Zn Є B] • Fraction of time the process {X(t)} is in set B (a measurable • subset of a state space of {X(t)}) is the same as the fraction of • time the process {X(t)} is in B when observed at the epochs of • visits to the state 1 of {Y(t)} • Parallel to the Poisson Arrivals See Time Averages (PASTA) • property

16. Application : Identifiable sampling designs in spatial processes to • select the locations. • {Z(s) , s  D} : Spatial process • s: Locations, D : Study region • Aim : To select locations at which the characteristic under study is • to be measured, thickness or smoothness of powder coating, • nests of birds • Most common scheme: Regular sampling, Cressie,1993 Non-identifiability 16

17. Study region – Continuous • Aim : Selection of locations (s1,s2) • If both coordinates are selected by Poisson sampling, it generates • CSR pattern. If both coordinates are selected by Markov Process • sampling, it generates aggregated pattern. • Spatial process observed at these locations determines the original • process uniquely • Study region – Discrete • Adopt Bernoulli sampling or Markov sampling • Deshmukh (2003), JISA (Adke Special volume) • Prayag & Deshmukh (2000) : Environmetrics • Test for CSR against aggregated pattern 17

18. Suppose X has negative binomial distribution • Pk[ X = x] = (x + k -1)C(k-1) pk qx-k , x = 0, 1, …, • p – known, k Є I+, • not a one parameter exponential family • Complete • Strongly complete

19. Risk models in insurance • U (t) : reserve/ value of the fund/ insurer’s surplus at time t • U (t) = initial capital + input via premiums by time t – output due to claims by t • S (t) = Output due to claim payments by t = 0∫t X(u) du,random part • Probability of ruin = P [ U (t) < 0] • Distribution of {S (t)} or its discrete version : Sn = Σ Xi, i running from 1 to n • Observed data are the claim amounts in various time periods - weeks or months • Uk = Σ Xi, i runs from 1 to Nk, Nk is the frequency of a claim in a fixed time period, • and Xi denotes the claim amount, Nk and Xi are random. • If Nk = 0, Uk = 0 • Nk – Poisson, negative binomial

20. {Tk, k ≥ 1}, Tk – Tk-1 are distributed as Nk with support I+ • Uk = S(Tk) – S(Tk-1) • Observed data are realization of the process {S (Tn), n ≥ 1}, a process observed • at random epochs • On the basis of these data we wish to study the process {Sn, n ≥ 1} • Identifiability of the random sampling scheme. • If {Sn} is modelled as a renewal process then identifiability of the random • sampling scheme is valid for any discrete distribution of Nn with support I+ . • (Teke & Deshmukh,2008, SPL) • If {Sn} is a discrete parameter process then identifiability of the random • sampling scheme is valid for negative binomial distribution of Nn • Strong completeness of the family of negative binomial distributions helps • to prove identifiability

21. {Sn , n  1}: Renewal process, f(s) : L.T. Renewal processes Cox process {Tn , n  1} : Renewal process Support – N, P(s) : p.g.f. Zn = S(Tn), Renewal process  g(s) : L.T. g(s) = P(f(s)) f(s) = P-1(g(s)) : Inversion formula {Zn} determines {Sn} gn(s) : Empirical L.T. fn(s) = P-1(gn(s)) Cox process Renewal processes P(s) – Geometric, Shifted geometric {Sn , n  1}: Random walk {S(t) , t  0}: Levy process P(s) – Geometric,Poisson & negative Binomial, both truncated at zero Bernstein, Stieltjes

22. Work in progress • {X(t), t  T}: Original process under study • {Zn = X(Tn)}, {Tn, n ≥ 1}: Renewal process G1 = G2 implies E{F1(L1, L1+L2, …, L1+L2+…+Ln)} = E{F2(L1, L1+L2, …, L1+L2+…+Ln)} E{F1(L1, L1+L2, …, L1+…+Ln)- F2(L1, L1+L2, …, L1+…+Ln)} = 0 Lj : sum of kj iid random variables, if the joint distribution of (L1, L2,…,Ln) is complete then, G1 = G2 implies F1 = F2. f(x, k), k Є I+}: family of L Σδ (1 – δ) k – 1( ∫ h(x) f(x,k)dx) = 0, sum being taken on I+ ⇔ ∫ h(x) Σ (1 – δ)k f(x,k) = 0 ⇔ ∫ h(x) A(x, δ)= 0, A(x, δ) = Σ (1 – δ)k f(x,k) Can we conclude that h(x) = 0 a.s.?

23. References 1.Baba, Y. (1982). Maximum likelihood estimation of parameters in birth and death process by Poisson sampling, J. Oper. Res. 15, 99-111. 2. Basawa, I.V. (1974). Maximum likelihood estimation of parameters in renewal and Markov renewal processes. Austral. J. Statist. 16, 33-43. 3.Cressie, N. A. C. (1993). Statistics for Spatial Data, Wiley, New York. 4.Deshmukh, S.R. (1991). Bernoulli sampling, Austral. J. Statist. 33, 167-176. 5.Deshmukh, S.R. (2000). Markov sampling, Aust. N. Z.J. Statist. 42(3), 337-345. 6.Deshmukh, S.R. (2003). Identifiable sampling design for spatial process. J. Ind. Statist. Assoc. 41(2) 261-274. 7.Deshmukh, S.R. (2005). Markov Arrivals See Time Averages, Stochastic Modelling and Applications. Vol. 8, 2, p. 1-20.

24. 8.Kingman, J.F.C. (1963). Poisson counts for random sequences of events. Ann. Math. Statist. 34, 1217-1232. 9.Prakasa Rao, B.L.S. (1988). Statistical inference from sampled data for stochastic process. Contemp. Math. 80, 249-284. 10. Prayag, V.R. & Deshmukh, S.R. (2000). Testing randomness of spatial pattern using Eberhardt’s index, Environmetrics, Vol. 11, p. 571-582. 11.Su, Y. and Cambanis, S. (1993). Sampling designs for estimation of a random process. Stochastic Process Appl. 46, 47-89. 12.Teke S.P. & Deshmukh, S.R.(2008) . Inverse Thinning of Cox and Renewal Processes, Statistics and Probability Letters, 78, p. 2705-2708.

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