Chan (2009) studied the tail probabilities of the maxima of moving sums with a wide choice of scanning sets. He first developed a theory parallel to the study of tail probabilities in Gaussian or Gaussian-like random fields in the classical framework of Pickands (1969) and Qualls and Watanabe (1973) [see also Piterbarg (1996) and Chan and Lai (2006)]. Motivated by recent developments in molecular biology [see Examples 3.1 and 3.2], he considered a more general marked Poisson random field.
Consider here {ˇti, i ≤ 1} to be a homogeneous Poisson point process on Rm with intensity λ > 0 and letX1, X2,ã ã ã be i.i.d. random varibles with cumulative distribution function F and independent of the Poisson point process. Let ρ and K(θ) be defined as in Section 2.2. To consider the scan statistic, define first some notations.
Letσm(ã) be the volume of set inRm. For anyA⊂Rm, vectorq∈Rm and real number η, let q+ηA ={q+ηα :α∈A}.
For any B ⊂ Rm, define the score ˇS(B) = P
ˇti∈BXi. Let D be a bounded subset of Rm. Define the scan statistic to be
MD,B = sup
v∈D
S(vˇ +B).
Assume that Θ = {θ : K(θ) < ∞} is an open neighborhood of 0. For c >
ρσm(B), choose ˇθc>0 and distribution Fθˇc to satisfy K0(ˇθc) =c/σm(B) and Fθˇc(dx)
F(dx) = eθˇcx K(ˇθc). Define the large deviation rate function to be
Iθˇc = ˇθcc−σm(B)[K(ˇθc)−1].
Chan (2009) provided the following tail probability approximation of MD,B. Theorem 3.3. Let B be convex and bounded. Define xλ = ˇθc(λc−dbλc/dc) if F is arithmetic with span d and xλ = 0 ifF is nonarithmetic. Then as λ→ ∞,
Pλ
MD,B ≥λc
∼[2πσm(B)K00(ˇθc)]−1/2e−λIˇθc+xλλm−1/2σm(D)ωB
for some positive and finite constant ωB.
The constant ωB above is specified in Chan (2009). When B is rectangular,ωB has an explicit expression in terms of the overshoot constant.
Example 3.3. Let B = Qm
j=1[0, βj] with βj > 0 for all j. Consider ˇY1,Yˇ2,ã ã ã to be i.i.d, random variables satisfying
P( ˇY1 ∈dx) = [ ¯F(dx) +K(ˇθc)Fθˇc(dx)]/[1 +K(ˇθc)],
where ¯F denotes the cumulative distribution function of−X1. Let ˇSn= ˇY1+ã ã ã+ ˇYn and ˇτb = inf{n≥1 : ˇSn≥b}. Define the overshoot constant
ˇ
νc= lim
b→∞E
e−θˇc( ˇSτbˇ −b) .
Chan (2009) showed that ωB has the following expression:
ωB = ˇ
νc[cσ−1m (B)−ρ] m χc
m
Y
j=1
βjm−1
,
where χc = ˇθc for F nonarithmetic and χc=d−1(1−e−dθˇc) for F arithmetic with span d.
Although the window B in Chan (2009) can take arbitrary shape, it still has fixed size. For similar reasons, a useful extension is to consider variable window sizes.
Chapter 4
Examples and Numerical Studies
Example 4.1. Consider again X1 in Example 2.1. Then K(θ) = eθ and φ(z) = zlogz−z + 1. Choose za satisfying φ(za) = c/a. Then θa = φ0(za) = logza. By Theorem 3.1, as λ→ ∞,
Pλ{M(a0, a1)≥λc} ∼(2π)−1/2λ3/2e−λcc2 Z a1
a0
θa−1e−θa/2νa2a−5/2(1−a)da, (4.1)
where νa is given by (2.4),
νa = aza
c expn
−
∞
X
n=1
n−1h
1−χ22n2zaθan za−1
+χ22n 2θan za−1
io .
To evaluate the above complicated formula, we first provide following straight- forward plots ofνa.
Figure 4.1: Plots ofνa against a when c= 0.01 andc= 0.1.
0.2 0.4 0.6 0.8
0.880.900.920.94
a
nu
0.2 0.4 0.6 0.8
0.750.800.85
a
nu
Figure 4.2: Plots of νa against a whenc= 1 andc= 10.
0.2 0.4 0.6 0.8
0.550.600.650.70
a
nu
0.2 0.4 0.6 0.8
0.350.400.450.50
a
nu
All the above figures are based on a0 = 0.1, a1 = 0.9. For fixed c, a0 and a1, νa appears to be a increasing function of a exhibiting negative concavity. On the other hand, the scale of νa significantly decreases and the shape of plots slightly flattens as c increases. More experiments also show that the choices of a0 and a1 mainly influence the scale rather than the shape of plots.
Figure 4.3: Plots of νa against θa when c= 0.01 andc= 0.1.
Figure 4.4: Plots of νa againstθa when c= 1 and c= 10.
It appears that there exists a linear relationship between νa and θa. We there- fore provide corresponding linear regression equation in each plot. R2 from the regression slightly decreases (from 1 to 0.9952) as c increases (from 0.01 to 10) which indicates the strong explanatory power of linear regression equations. Since the computation of νa is complicated, we may simply estimate νa using θa. How- ever, the regression coefficients also significantly change as c changes. Therefore
the linearity may be just spurious and violated in other cases.
We then provide the following plot to illustrate the tail distribution approxi- mation given by (4.1).
Figure 4.5: The tail probability approximation.
4 6 8 10 12
0.000.050.100.15
100 * c
p
The plot is base on λ= 100. The p-values decay dramatically at the beginning space(from 4 to 6) and the shape is not smooth, which are unfavorable to numer- ical studies. The p-values approach slowly and smoothly to 0 after 8. To check those approximations, we provide the following comparison between Monte Carlo simulations and analytical results in (4.1).
Table 4.1: Monte Carlo simulations and analytical p-values.
λc Direct Monte Carlo Analytical Estimate in (4.1) 8 (8.4±0.8)×10−3 8.00×10−3
9 (3.5±0.3)×10−3 3.36×10−3 10 (1.4±0.1)×10−3 1.44×10−3
11 0 5.86×10−4
The results in the second column of Table 4.1 are based on 104 paths of Monte Carlo simulations with λ = 100, a0 = 0.1, and a1 = 0.9. The related R code is provided in the Appendix. We have to run the R code during several days to get each simulation result, which is extremely time consuming compared to that we only need several minutes to derive analytical p-values. For even smaller probabil- ities, we need even more time to get desired results of accuracy (i.e. standard error
= 10% of simulated probabilities). By the comparison, the analytical p-values by (4.1) appear to agree well with the simulations.
We next examine the accuracy of upper bound given by Corollary 3.1.
Example 4.2. LetX1 be an exponential random variable with parameter 1. Then K(θ) = 1−θ1 and φ(z) = z −2√
z + 1. Choose za satisfying φ(za) = c/a. Then θa =φ0(za) = 1−√1z
a. By Corollary 3.1, as λ→ ∞,
Pλ{M(a0, a1)≥λc} ≤[1 +o(1)]2−1π−12λ32e−λcc2 (4.2)
× Z a1
a0
θ−1a (1−θa)32a−52(1−a)da.
We provide following comparison between Monte Carlo simulation results and upper bounds given by (4.2).
Table 4.2: Monte Carlo simulations and upper bounds by (4.2).
λc Direct Monte Carlo Upper bounds by (4.2) 7 (1.2±0.1)×10−2 2.48×10−2 8 (4.9±0.4)×10−3 1.10×10−2 9 (1.9±0.2)×10−3 4.75×10−3
10 0 2.00×10−3
The simulation results in the second column in Table 4.2 are based onλ= 100, a0 = 0.1 anda1 = 0.9. It appears that upper bounds by (4.2) are not bad. Taking the first row in Table 4.2 for example, the simulated p-value and the upper bound are in the same order. On the other hand, it also appears that upper bounds do not converge to the tail probabilities as λ → ∞. This indicates again the importance of the overshoot constant.
Chapter 5
Proof of Theorem 3.1
We first examine the local behavior of segmental scores in a window by using a change of measure approach and then combine all these windows together to provide the tail probability of M(a0, a1). Specifically we define the basic window
W =n
(x, y) :x=x0−v1λ−1, y =y0 +v2λ−1,0≤v1, v2 ≤mo ,
where m→ ∞ as λ→ ∞ and m=o(λ). Choose x0 ≤y0 to be multiples of mλ−1 in [0,1] satisfying a0 ≤a=y0−x0 ≤a1.
Consider the tail probability in W conditional on the starting point (x0, y0).
Pλn sup
(x,y)∈W
S(x, y)≥λco
= Pλn
S(x0, y0)≥λco
(5.1) +Pλn
S(x0, y0)< λc, sup
(x,y)∈W
S(x, y)≥λco .
To evaluate s(x, y), apply the Taylor expansion. By (2.2),
S(x, y) = λ(y−x)φ
N(x, y) λ(y−x)
(5.2)
= λ(y−x)h
φ(za) +θa N(x, y)
λ(y−x) −za
+O N(x, y)
λ(y−x) −za2i
= θaN(x, y)−λ(y−x)ψ(θa) +λ(y−x)O N(x, y)
λ(y−x)−za2 .
Define a probability measure Qλ under which X = {(ti, Xi}ni=1 is a non-uniform compound Poisson process with rate λM(θa) and mark distribution Fθa inside [x0, y0] and rateλ and mark distribution F outside [x0, y0]. Then
dQλ dPλ
(X) =
e−λaK(θa)(λaK(θa))N(x0,y0) eθaN(x0,y0)
K(θa)N(x0,y0)
e−λa(λa)N(x0,y0) (5.3)
= exp{θaN(x0, y0)−λaψ(θa)}.
By this change of measure approach, we examine the local behavior of S(x0, y0) under Qλ. Specifically θaN(x0, y0)−λaψ(θa) is asymptotically normal with mean θaλaK0(θa)−λaψ(θa) =λc, and variance θ2aλaK00(θa) under Qλ,
Qλ θaN(x0, y0)−λaψ(θa)∈du
(5.4)
= [1 +o(1)] 1 θap
2πλaK00(θa)expn
− (u−λc)2 2θ2aλaK00(θa)
o ,
where o(1) is uniform over bounded values of u. Combining (5.2), (5.3) and (5.4),
Pλn
S(x0, y0)≥λco
= Pλ
n
θaN(x0, y0)−λaψ(θa)≥λc o
= Z ∞
0
Pλn
θaN(x0, y0)−λaψ(θa)∈λc+duo
= Z ∞
0
exp{ưλcưu}Qλn
θaN(x0, y0)−λaψ(θa)∈λc+duo
= Z ∞
0
exp{ưλcưu}[1 +o(1)] 1 θap
2πλaK00(θa)expn
− u2 2θ2aλaK00(θa)
o du.
Since limλ→∞exp n
−2θ2 u2 aλaK00(θa)
o
= 1, Pλn
S(x0, y0)≥λco
= [1 +o(1)]e−λc θap
2πλaK00(θa). (5.5) Similarly,
Pλn
S(x0, y0)< λc, sup
(x,y)∈W
S(x, y)≥λco
(5.6)
= Z ∞
0
Pλn sup
(x,y)∈W
S(x, y)≥λc|S(x0, y0) = λcưuo Pλn
S(x0, y0)∈λc−duo
= Z ∞
0
[1 +o(1)]e−λc+u θap
2πλaK00(θa)Pλ
n sup
(x,y)∈W
S(x, y)≥λc|S(x0, y0) = λcưu o
du.
By (5.2), the linear approximation ofS(x, y)−S(x0, y0) isθa[N(x, x0) +N(y0, y)]−
λ(y−x−a)ψ(θa), which is independent of S(x0, y0). Hence,
Pλn sup
(x,y)∈W
S(x, y)≥λc|S(x0, y0) =λcưuo
= Pλn sup
(x,y)∈W
θa[N(x, x0) +N(y0, y)]−λ(y−x−a)ψ(θa)≥u−o(1)o
= Pλn sup
0≤v1,v2≤m
θa[N(x0−v1λ−1, x0) +N(y0, y0+v2λ−1)]
−(v1+v2)ψ(θa) ≥u−o(1) o
,
where o(1) is uniform over bounded values of u. Let
Cu = Pλ
n sup
0≤v1,v2≤m
θa[N(x0−v1λ−1, x0) +N(y0, y0+v2λ−1)]
−(v1+v2)ψ(θa) ≥u−o(1)o .
Through a scaling transformation, we can look at the limiting probability of Pλ as one involving fixed Poisson rate λ0 = 1 and increasingly large scanning sets,
Cu = Pλ0n sup
0≤v1,v2≤m
θaN(λx0−v1, λx0)−v1ψ(θa) +θaN(λy0, λy0+v2)−v2)ψ(θa) ≥u−o(1)o
.
Let N1(v1) = N(λx0 −v1, λx0), N2(v2) = N(λy0, λy0+v2). Then N1 and N2 are independent and identically distributed compound Poisson processes with rate 1 and mark distribution F. Further let Y1(v1) = θaN1(v1)−v1ψ(θa) and Y2(v2) = θaN2(v2)−v2ψ(θa). Note that both Y1 andY2 are independent ofS(x0, y0). Hence,
Cu =Pλ0
n sup
0≤v1,v2≤m
[Y1(v1) +Y2(v2)]≥u−o(1) o
. (5.7)
Combining (5.6) and (5.7),
Pλn
S(x0, y0)< λc, sup
(x,y)∈W
S(x, y)≥λco
(5.8)
= Z ∞
0
[1 +o(1)]e−λc+u θap
2πλaK00(θa)Pλ0n sup
0≤v1,v2≤m
[Y1(v1) +Y2(v2)]≥u−o(1)o du.
By (5.1), (5.5) and (5.8), as λ→ ∞,
Pλn sup
(x,y)∈W
S(x, y)≥λco
(5.9)
= e−λc
θap
2πλaK00(θa) h
1 + Z ∞
0
euPλ0n sup
0≤v1,v2≤m
[Y1(v1) +Y2(v2)]≥uo dui
= e−λc
θap
2πλaK00(θa) Z ∞
−∞
euPλ0n sup
0≤v1,v2≤m
[Y1(v1) +Y2(v2)]≥uo du
= e−λc
θap
2πλaK00(θa)EPλ
0
h exp
sup
0≤v1,v2≤m
[Y1(v1) +Y2(v2)] i
= e−λc
θap
2πλaK00(θa)
EPλ0
exp sup
0≤v1≤m
[Y1(v1)]
2
= e−λc
θa
p2πλaK00(θa) Z ∞
−∞
euPλ0n sup
0≤v1≤m
[Y1(v1)]≥uo du2
.
LetTu = inf{v1 :Y1(v1)≥u}. Define a probability measureQλ0 under whichN1 is a compound Poisson process with rate K(θa) and mark distribution F(θa). Then
dQλ0
dPλ0(Y1(Tu)) =
e−K(θa)Tu(K(θa)Tu)N1(Tu) eθaN1(Tu)
K(θa)N1(Tu)
e−Tu(Tu)N1(Tu)
= exp{θaN1(Tu)−Tuψ(θa)}
= eY1(Tu).
Hence for (5.9), Z ∞
−∞
euPλ0
n sup
0≤v1≤m
[Y1(v1)]≥u o
du (5.10)
= Z ∞
−∞
eu Z ∞
−∞
Pλ0n
Y1(Tu)∈db, sup
0≤v1≤m
[Y1(v1)]≥uo du
= Z ∞
−∞
eu Z ∞
−∞
e−bQλ0n
Y1(Tu)∈db, sup
0≤v1≤m
[Y1(v1)]≥uo du
= Z ∞
−∞
euEQλ
0
h
e−Y1(Tu)I{ sup
0≤v1≤m
[Y1(v1)]≥u}i du
= Z ∞
−∞
EQλ
0
h
e−(Y1(Tu)−u)I{ sup
0≤v1≤m
[Y1(v1)]≥u}i du.
By (2.3),νa = limu→∞EQλ
0[e−(Y1(Tu)−u)] is the overshoot constant. Hence it follows from (5.10) that
Z ∞
−∞
euPλ0
n sup
0≤v1≤m
[Y1(v1)]≥u o
du (5.11)
∼ Z ∞
−∞
νaQλ0n sup
0≤v1≤m
[Y1(v1)]≥uo du
= νaEQλ
0
h sup
0≤v1≤m
[Y1(v1)i
= νaEQλ
0Y1(m) +o(1)
= νa[θaK(θa)m−mψ(θa)] +o(1)
= νamc/a+o(1).
By (5.9) and (5.11), as λ→ ∞, Pλn
sup
(x,y)∈W
S(x, y)≥λco
(5.12)
∼ e−λc θap
2πλaK00(θa)
νamc/a2
.
By Chan and Zhang (2007), the probability of joint externality in two disjoint windows is asymptotically negligible. Hence Theorem 3.1 follows by combining all the windows. Letx0 < y0 be multiples ofmλ−1 in [0,1] and a=y0−x0. Hence by (5.12), as λ→ ∞,
Pλn
M(a0, a1)≥λco
∼ X
a0≤a≤a1
Pλn sup
(x,y)∈W
S(x, y)≥λco
∼ Z a1
a0
e−λc θap
2πλaK00(θa)νa2λ2c2a−2(1−a)da
= (2π)−1/2λ3/2e−λcc2 Z a1
a0
θ−1a K00(θa)−1/2νa2a−5/2(1−a)da.
Chapter 6
Conclusions
The definition of score S(x, y) and therefore Theorem 3.1 are based on the rate function φ which further depends on the mark distribution F. In practice, however, it is more reasonable to replace φ with a general function g which is independent of F. This can be done in two steps. First consider g(x) = x, then S(x, y) =N(x, y), which is also used in Chan and Zhang (2007). Under the same definition of M(a0, a1), the methodology applied to φ should be adapted in such case. For example, we have to choose new θa without φ. A simple guess is to choose θa satisfying K0(θa) = c/a. The second step is to extend approximations derived when g(x) = x to general cases. We may need to place some regularity assumptions on g to derive corresponding results. For example, g is continuously differentiable in the neighborhood of origin.
Another future direction is to consider multi-dimensional time indexes. Chan (2009) provides a good example of studying scan statistics in a Poisson random field.
For simplicity, we can first examine the scan statistics in a cube instead of arbitrary indexing sets. The following question is then to describe the existence of overshoot constants, which is still at the heart of approximations in multi-dimensional cases.
The technique applied to marked Poisson processes should be adapted to marked Poisson random fields.
The complicated computation of overshoot constants may be an issue of concern in applications. Therefore it is necessary to study the linearity exhibited in Figure 4.3. If the linearity between νa and θa holds, we can simply use θa to derive approximations of νa. We can also apply some transformations (e.g. νa0 = eνa) and reexamine the relation between νa0 and a. If it shows valid linearity after the transformation, we then derive another simple approximation of νa.
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Appendix
Related R code
##main Monte Carlo simulation
sim1<-function(N1,N2,lambda,a0,a1,c){
##N1, number of paths used in the simulation
##N2, rounds of simulation in each path to derive p-value
##lambda, rate of Poisson process
##a0, a1, c are the same defined constants p<-numeric(N1);
var<-numeric(N1);
for(l in 1:N1){
m<-0
for(k in 1:N2){
n<-rpois(1,lambda);
##generate total number of variables if(n==0)break;
t<-runif(n);
t<-sort(t);
##generate n arrival times S<-matrix(0,n,n);
for(i in 1:n){
if(1/(lambda*a0)>=1) S[i,i]<-lambda*a0-log(lambda*a0)-1 if (i==n) break;
##generate the segmental scores consisting of only one variable for(j in (i+1):n){
if(t[j]-t[i]>=a0 & t[j]-t[i]<=a1){
if((j-i+1)/(lambda*(t[j]-t[i]))>=1){
S[i,j]<-(j-i+1)*(log((j-i+1)/(lambda*(t[j]-t[i])))-1) +lambda*(t[j]-t[i]);
##generated other segmental scores consisting of two or more variables }
} } }
if (max(S)>=lambda*c) m<-m+1;
##generate the scan statistic }
p[l]=m/N2;
var[l]=sqrt(m/N2*(1-m/N2)/N2);
##generate p-value and standard deviation in each path }
list(p=mean(p),var=mean(var));
##generate simulated p-value and standard deviation using all paths }
##analytical p-value calculation sim2<-function(lambda,a0,a1,c){
f<-function(a){
phi<-function(x)x*log(x)-x+1-c/a;
##the rate function
z<-uniroot(phi,c(1,max(c/a,9)));
z<-z$root;
theta<-log(z);
nu<-function(a){
##calculate the analytical overshoot constant sum=0;
for(i in 1:1e+04){
si<-1/i*(1-pchisq(2*z*theta*i/(z-1),2*i)