Shailaja R Deshmukh

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Strong Completeness and its Application to
Random Thinning of Random Processes Shailaja
Deshmukh 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

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• 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)

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• 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

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• 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

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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
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• 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

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• 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 ?0-1 and ?1-1
  respectively.

• Tk S1 Sk V0 V1, where Vi G (?i,
  k ),
• ?i scale parameter and k is the shape
  parameter, i 0,1.

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• 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) PXi(tj)
xj j 1,, n,
x1,x2,, xn - real numbers, t1,t2,, tn -
positive real numbers,
t1 lt t2 lt lt tn.
Family of finite dimensional distribution
functions of Z(i) (k)
Gi PZi(K(j)) xj, j 1, 2, ,n , K(j) k1
k2 kj
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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 PZi(K(j)) xj, j 1,, n PXi(TK(j))
xj, j 1,, n PXi(Uj) xj,
j 1,, n E PXi(Uj) xj, j
1,, n U1,U2,,Un
EPXi(Uj) xj, j 1,, n
EFi(L1, L1L2, , L1L2Ln)

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G1 G2 implies EF1(L1, L1L2, ,
L1L2Ln) EF2(L1, L1L2, ,
L1L2Ln) EF1(L1, L1L2, , L1Ln)- F2(L1,
L1L2, , L1Ln) 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 (?0, kj ) and V1kj G (?1, kj ),
?0 and ?1 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
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X G ( a, k), a scale parameter, k shape
parameter f (x) ak e-ax xk-1/?(k), x gt
0
a known, k ? I
Not a one parameter exponential family, parameter
space is not an open set
Complete family
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Ek( h(x)) 0 for all k ? I
? ? h(x) ak e-ax xk-1/?(k)dx 0 , for all k ?
I ? g(k) 0 , for all k ? I
? S zk g(k) 0, 0 lt z lt 1
? ? h(x) e-ax (S (a zx)k-1/(k-1)!) dx 0
? ? h(x) e-?x 0, ? a (1 z), 0 lt ? lt a
? ? h(x) e-?x 0, for all ? gt 0, by analytic
continuation
? h(x) 0, a.s. Pk for all k ? I
G(a , k), k ? I is a complete family
Strongly complete
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• Definition A family of distributions F?, ? ? T
  is called strongly
• complete if there exists a measure µ on (T ,
  ?) such that for every
• subset T of T for which µ(T T) 0, ?
  h(x) F?(dx) 0 for all
• ? ? T implies that h (x) 0 a. s. P? for
  every ? ? T. (Zacks, 1971)

• Strong completeness implies completeness by
  taking T T

• Suppose T1 and T2 are independent random
  variables. If F?T1,? ? T
• is complete and F?T2, ? ? T is strongly
  complete then the family
• of joint distributions F?, ?T1,T2 ? ? T, ?
  ? T is complete.
• (Zacks, 1971)

• Gamma family is strongly complete

• Parameter space - I, ? - sigma field, µ is a
  measure induced by
• geometric distribution

• For A ? ?, µ(A) S d (1 d) k 1, sum being
  taken over k ? A

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Suppose T is a subset of I such that µ(I - T
) 0
? h(x) ak e-ax xk-1/?(k)dx 0 , for all k ? T
? g(k) 0, for all k ? T
µ(k) 0 , for all k ? (I - T)
g(k) µ(k) 0 , for all k ? I
S d (1 d) k 1( ? h(x) ak e-ax xk-1/?(k)dx)
0, sum being taken on I
Using Fubinis theorem, summation and integration
can be interchanged
? h(x) e-?x 0, for all ? gt 0, by analytic
continuation
h(x) 0, a.s. Pk for all k ? I
Gamma family is strongly complete
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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 ?8PX(t) ? B lim n?8 PZn ? 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

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• 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
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• 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

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• Suppose X has negative binomial distribution

• Pk X x (x k -1)C(k-1) pk qx-k , x
  0, 1, ,

• p known, k ? I,

• not a one parameter exponential family

• Complete

• Strongly complete

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• Risk models in insurance

• U (t) reserve/ value of the fund/ insurers
  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) lt 0

• Distribution of S (t) or its discrete version
  Sn S Xi, i running from 1 to n

• Observed data are the claim amounts in various
  time periods - weeks or months

• Uk S 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

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• 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

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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
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Work in progress

• X(t), t ? T Original process under study
• Zn X(Tn), Tn, n 1 Renewal process

G1 G2 implies EF1(L1, L1L2, ,
L1L2Ln) EF2(L1, L1L2, ,
L1L2Ln) EF1(L1, L1L2, , L1Ln)- F2(L1,
L1L2, , L1Ln) 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 ? I family of L
S d (1 d) k 1( ? h(x) f(x,k)dx) 0, sum
being taken on I ? ? h(x) S (1 d)k f(x,k)
0 ? ? h(x) A(x, d) 0, A(x, d) S
(1 d)k f(x,k) Can we conclude that h(x) 0
a.s.?
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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.
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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 Eberhardts 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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Thank You