1. Introduction
In this contribution we investigate orderly point processes on the state-space R+ = (0,∞). Let T(i) [i ∈ N = {1,2,3, . . .}] be the occurrence time of the i-th point event after t = 0 and ti the realized value of this random variable and put T(0) = 0 with probability 1. We denote by{U(i), i= 1,2, . . .}the sequence of gap lengthsU(i) = T(i)−T(i−1) between consecutive point events. A point process is called orderly if the conditionP(N(t, t+δ)>
1) =o(δ) for allt≥0 and forδ >0 andδ→0 is satisfied, whereN(t, t+δ) stands for the random number of point events in (t, t+δ]. Such an orderly point process has with probability 1 no point events occurring simultaneously.
An important role in the theory of point processes plays the complete intensity function λ∗(t|Ht) = limδ>0,δ→0[δ−1P(N(t, t+δ)>0|Ht)],
where Ht :={N(t);T(1), . . . , T(N(t))}specifies the evolution of the point process up to and including the time t andN(t) stands for the random number of point events in (0, t]. It is assumed that E[N(t)]<∞for allt >0.
In this paper we consider the case where the complete intensity function is given by the formula
λ∗(t:ρ;λ, c) =λ+ρ1(c,∞)(t−ti−1) forti−1< t≤ti andi∈ N andλ, c >0 and 1A(x) = 1 ifx∈Aand 1A(x) = 0 otherwise.
We note that in the important special caseρ= 0
λ∗(t: 0;λ, c)≡λand the point process is the homogeneous Poisson point process with global rateλand that under this condition λ∗ does not depend on Ht. If ρ 6= 0λ∗ does depend on the last point event observed in (0, t) and provided that ui =ti−ti−1 > c the point process changes its rate from λto λ+ρ at time t=ti−1+c. Thus if ρ >0 the point process increases the value of its intensity as soon as the gap ui exceedsc and exhibits therefore a self-correcting feature. Augmenting the value of the intensity when a gap gets long generates a tendency of the process to avoid that the empirical event rate becomes very low.
From the point of view of model selection it is of interest to dispose of good tests for the hypotheses H0:ρ= 0
versus H1:ρ >0
for knownλ andc. We assume that the point process is observed during the time period (0, T(n0)], i.e. from the start of the experiment up to the occurrence time of then0-th point event after t= 0, wheren0 is a fixed natural number.
In§2 we present a locally most powerful test and in §3 we look at this hypothesis testing problem from the point of view of the asymptotic optimality of the procedure. In§4 the results are extended to deal with the detection of other self-regulation properties of Poisson type point processes.
2. A locally most powerful test forH0 versus H1
We seek thus a test with maximal first derivative of the power function at ρ= 0 for given level of significance. In order to find such a test, we determine the likelihood function
L(ρ;λ, c:t1, . . . , tn0) =Qn0
i=1{λ∗(ti−ti−1:ρ;λ, c)}exp[−Rtn0
0 λ∗(t:ρ;λ, c)dt] = Qn0
i=1{λ+ρ1(c,∞)(ui)}exp[−λPn0
i=1ui−ρPn0
i=1(ui−c)+],
which indicates the density of the observed random eventT(1) =t1, . . . , T(n0) =tn0 in dependence ofρfor known λandc. Here (v)+:=v forv >0 and (v)+:= 0 forv≤0. The log likelihood function is then given by
ln(L(ρ;λ, c:t1, . . . , tn0)) =Pn0
i=1{ln(λ+ρ1(c,∞)(ui))} −λtn0−ρPn0
i=1(ui−c)+. Derivation with respect toρyields
∂ln(L(ρ;λ, c:t1, . . . , tn0)/∂ρ=Pn0
i=11(c,∞)(ui)/(λ+ρ)−Pn0
i=1(ui−c)+.
The efficient score statistic is obtained by evaluating this expression atρ = 0 and substituting the corresponding random variables in replacement of the observed values which leads to
Tλ,c;n0(0) =Pn0
i=1{1(c,∞)(U(i))(1/λ−(U(i)−c))}. The following properties ofTλ,c;n0(0) are of interest:
E[Tλ,c;n0(0) :ρ= 0] =Pn0
i=1(P(U(i)> c:ρ= 0)/λ−E[(U(i)−c)+]) =n0e−λc/λ−n0e−λc/λ= 0 and
V ar[Tλ,c;n0(0) :ρ= 0] =n0e−λcV ar[U(i) :ρ= 0] =n0e−λc/λ2.
Since Tλ,c;n0(0) is a sum ofn0 contributions, which are independent and identically distributed random variables, the standard central limit theorem implies under H0 and forn0→ ∞ that
Tλ,c;n0(0)∼N∗(0, n0e−λc/λ2),
where N∗(a, b) stands for the normal distribution with mean a and variance b. The critical region for large n0
of a locally most powerful test of H0 versus H1 may thus be based on the standardized test statistic Tλ,c;n∗ 0(0) = λ(n0)−1/2eλc/2Tλ,c;n0(0) and takes the form
λ(n0)−1/2eλc/2tλ,c;n0> z(1−α)
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where Φ(z(1−α)) = 1−αand Φ designates the distribution function of the standard normal random variable.
3. Asymptotic optimality of the test
The investigation of a testing problem for a given level of significance becomes particularly interesting if the power function tends forn0→ ∞to a non-degenerate limit, different from 0 and from 1. This may be achieved by allowing the parameter under the alternative to be a suitable function of n0. Thus instead of comparing the hypotheses H0 and H1 we are considering asymptotic hypotheses, formed by a sequence ˜H0 of null hypotheses H0,n0 and a corresponding sequence ˜H1 of alternative hypothesesH1,n0. For the problem discussed in§2 we choose
H0,n0:ρ=ζ = 0 andH1,n0:ρ=ζ/√n0withζ >0.
For such an asymptotic testing problem the performance of the test is judged in a purely asymptotic way by means of limits. Asymptotic optimal tests may thus exist even when finite tests with a corresponding optimality property are not available. The asymptotic level of significance of a test of ˜H0versus ˜H1is by definition equal toαprovided that the limit superior of the sequence of probabilities of rejectingH0,n0 whenH0,n0 is actually correct becomes for n0→ ∞smaller or equal toα. Furthermore a testϕis said to be asymptotically uniformly most powerful if the limit inferior of the power function ofϕminus the power function of any rival testϕ∗ of asymptotic level of significance αevaluated for any given positiveζ is allways non-negative.
The general theory of statistical inference for Poisson type point processes shows that a test statistic of an asymp- totically most powerful test may be obtained from the formula
∆T(n0)(ρ:T(1), . . . , T(n0)) =φT(n0)(ρ)RT(n0)
0 ∂(λ∗(t:ρ;λ, c))/∂ρ[λ∗(t:ρ;λ, c)]−1·[dN(t)−λ∗(t:ρ;λ, c)dt].
HereφT(n0)(ρ) =IT−(n1/20)(ρ), whereIT(n0)(ρ) is the Fisher information based on the observation of the point process.
Furthermore the family of probability laws of the observed part of the point process is LAN (locally asymptotic normal) with central statistic ∆T(n0) and ∆T(n0)∼N∗(0,1) for ζ = 0 [Kutoyants(1984), pp. 164–165]. In our case we find
IT(n0)(ρ) =n0e−λc/(λ+ρ)2.
The evaluation of ∆T(n0)(ρ) forζ = 0 leads to
∆T(n0)(0 :T(1), . . . , T(n0)) =λeλc/2Pn0
i=11(c,∞)(U(i))(1/λ−(U(i)−c))/√n0=Tλ,c;n∗ 0(0)
as choice of the standardized test statistic. It is easy to check that the regularity conditions 1–6 of this result are fulfilled in this application.
We come thus to the conclusion that a locally most powerful test of H0 versusH1 may be based on the same test statistic and the same critical region as the asymptotically most powerful test for the contiguous hypotheses ˜H0
versus ˜H1.
A sufficient condition in order that a test possesses the finite local and simultaneously the asymptotic optimality property is according to Witting and M¨uller-Funk (1995, p.348, theorem 6.152) that the family of probability laws isL2(0)-differentiable ( for this notion see [Witting (1985,§1.8.7)]. This is the case for our choice ofλ∗.
Finally the asymptotic power function of the test of ˜H0versus ˜H1 can be calculated by the third lemma of Le Cam [ see Witting and M¨uller-Funk(1995, p.347, 6.4.6) ] This leads to the result
P(Tλ,c;n∗ 0(0)> z(1−α) :ζ) = 1−Φ(z(1−α)−e−λc/2ζ/λ).
4. Investigations involving related self-regulation properties
So far we considered the possibility of point processes to react to long ongoing gap lengths with an increase of the intensity function. Similar considerations apply if the self-regulation property is linked not with the actual but rather with preceding, already realized gap lengths. For instance the complete intensity function may take the form λ∗1(t:ρ;λ, c) =λ+ρ1(c,∞)(ti−1−ti−2)
forti−1< t≤ti, i∈ N, P(T(0) =T(−1) = 0) = 1 andλ, c >0, n0>1.
The likelihood function for this choice is given by L1(ρ;λ, c:t1, . . . , tn0) =λQn0
i=2{λ+ρ1(c,∞)(ui−1)} ·exp[−λPn0
i=1ui−ρPn0
i=21(c,∞)(ui−1)ui], which leads to the score statistic
1Tλ,c;n0(0) =Pn0
i=2[(1(c,∞)(U(i−1))(1/λ−U(i)))]
with the propertiesE[1Tλ,c;n0(0) :ρ= 0] = 0 andV ar[1Tλ,c;n0(0);ρ= 0] =e−λc(n0−1)/λ2. Since1Tλ,c;n0(0) =Pn0
l=2 1Tλ,c;n(l) 0(0) with1Tλ,c;n(l) 0(0) = (1(c,∞)(U(l−1))(1/λ−U(l))
the efficient score statistic is the sum ofn0−1 uncorrelated, 1-dependent contributions generated from the sequence ofU-variables, which is stationary underH0. Application of a result of Billingsley(1968, p.177) yields underH0and forn0→ ∞the statement
1Tλ,c;n0(0)∼N∗(0, e−λc(n0−1)/λ2).
The critical region for testingH0 versusH1 for largen0 is thus given by IMS – JSM 2008
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1t∗λ,c;n0(0) = λeλc/21tλ,c;n0(0)/√
n0−1 > z(1−α), where 1t∗λ,c;n0(0) is the realization of the standardized test statistic1Tλ,c;n∗ 0(0).
The theorem of Kutoyants mentioned in§3 is again applicable and the asymptotically uniformly most powerful test of ˜H0versus ˜H1uses the same test statistic and the same critical region as the locally most powerful test.
If we think that the point process should only react in augmenting the value of the intensity function if several preceding consecutive gap lengths all have been relatively long, we may choose as complete intensity function λ∗r(t:ρ;c, λ) =λ+ρQr
l=11(c,∞)(ti−l−ti−l−1)
forti−1< t≤ti;i, r∈ N;P(T(0) =T(−1) =. . .=T(−r) = 0) = 1, λ, c >0, n0> r.
Applying the previous reasoning leads to the test with the efficient score statistic
rTλ,c;n0 =Pn0
i=r+1(Qr
l=1[1(c,∞)(Ui−l)((1/λ)−U(i))]
and forn0→ ∞to a critical region of the form (λeλcr/2/√n0−r)rtλ,c:n0 > z(1−α).
Instead of relying on a threshold-value on the gap length we may base the self-regulation property on a comparison of the length of the preceding gap with its mean value underH0. Then the choice of the complete intensity function λ∗(t:ρ, λ) =λexp[ρ(tn(t−0)−tn(t−0)−1−1/λ)(1−1(0,t1](t))]
with P(T(0) = T(−1) = 0) = 1, n0 ∈ N, λ > 0 and N(t−0) := limδ>0,δ→0N(t−δ) may be appropriate. The efficient score statistic is then given by
Tλ;n0(0) =Pn0
i=2[(U(i−1)−1/λ)(1−λU(i))]
and forn0→ ∞the critical region of the test takes the form λtλ;n0/√n0−1> z(1−α).
REFERENCES
Billingsley,P. (1968)Convergence of probability measures, J.Wiley, New York.
Kutoyants,Y. A. (1984)Parameter estimation for stochastic processes, Heldermann, Berlin.
Witting,H. (1985)Mathematische Statistik I, Teubner, Stuttgart.
Witting,H. & M¨uller–Funk,U. (1995)Mathematische Statistik IITeubner, Stuttgart. IMS – JSM 2008
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