Journal of Applied Statistics
Vol. 37, No. 7, July 2010, 1089–1111
Detection of changes in a random financial
sequence with a stable distribution
Dong Hana , Fugee Tsungb∗ , Yanting Lic and Jinguo Xiana
a Department
of Mathematics, Shanghai Jiao Tong University, Shanghai, 200030, People’s Republic
of China; b Department of Industrial Engineering and Logistics Management, Hong Kong University
of Science and Technology, Hong Kong; c Department of Industrial Engineering and Management, School
of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China 200030, People’s
Republic of China
(Received 19 February 2008; final version received 19 March 2009)
Quick detection of unanticipated changes in a financial sequence is a critical problem for practitioners in
the finance industry. Based on refined logarithmic moment estimators for the four parameters of a stable
distribution, this article presents a stable-distribution-based multi-CUSUM chart that consists of several
CUSUM charts and detects changes in the four parameters in an independent and identically distributed
random sequence with the stable distribution. Numerical results of the average run lengths show that the
multi-CUSUM chart is superior (robust and quick) on the whole to a single CUSUM chart in detecting the
shift change of the four parameters. A real example that monitors changes in IBM’s stock returns is used
to demonstrate the performance of the proposed method.
Keywords: logarithmic moment estimators; multi-CUSUM charts; detection of changes; random
sequence with stable distribution
1.
Introduction
Tremendous amounts of time series data from the financial markets that contain important information for investors and financial analysts can be collected. Investment bankers, derivatives traders,
stockbrokers, and securities and bonds investors make their decisions based on their experience
and historical data. They are always exposed to high risk of economic loss. Thus, one of the
most important tasks of financial institutions is to evaluate exposure to market risks. A common
methodology that they use for estimation of market risks is the value at risk (VaR), which is the
highest possible loss over a certain period of time at a given confidence level.
For instance, a daily VaR for a given portfolio of assets of 2 million dollars at a 95% confidence
level implies that without abrupt changes in the market conditions, there is a 5% chance that
one-day losses will exceed 2 million dollars. The VaR is obtained based on the the cumulative
∗ Corresponding
author. Email: [email protected]
ISSN 0266-4763 print/ISSN 1360-0532 online
© 2010 Taylor & Francis
DOI: 10.1080/02664760902914433
http://www.informaworld.com
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D. Han et al.
distribution function of portfolio returns in one period. In order to obtain reliable and precise VaR
estimates, financial firms must be keenly alert to changes in the underlying distribution of portfolio
returns. Ignoring changes in the distribution leads to incorrect VaR estimates, which eventually
could cause bad decisions and enormous economic losses. Therefore, the ability to detect changes
in financial time series is essential. In order to avoid or prepare for the unanticipated changes, it
is critical to have a proper tool that monitors such financial data efficiently.
A number of researchers have studied tests for structural breaks, in particular, those in the tail
index. Perron [34] and Andrews [2] deal with tests for structural breaks in model parameters with
known and unknown dates (see also [17] and the references therein for a review on this subject).
Quintos et al. [35] present methods for testing structural breaks in tail index of heavy-tailed
(possibly dependent) time series with possibly unknown break date. Ibragimov [18] studied the
efficiency, peakedness, and majorization properties of linear estimators under heavy-tailedness
assumptions.
On the other hand, to monitor and detect changes in a sequence, various statistical process
control (SPC) schemes have been developed in industrial quality control, such as the Shewhart
chart [43], the cumulative sum chart [33], the exponentially weighted moving average [39], the
Shiryayev–Roberts procedure [40,44], Bayes-type statistics [7], the regression control chart [28],
residual-based control chart [1]), the GLR test [45], and its applications [3,48], the autoregressive
moving average chart [20], the proportional integral derivative chart [21], etc. These SPC schemes
have been extensively applied in industrial quality control and automated fault detection in dynamical systems. However, their dependence on the assumption that the underlying distribution of the
charting characteristics is Gaussian unfortunately limits their application to the financial world.
Mandelbrot [29,30] noticed long ago that many types of economic time series, such as stock
return (indexes, funds), do not fit the Gaussian distribution because of their heavy tails and strong
asymmetries. This led Mandelbrot to suggest a stable distribution, Sα (β, γ , μ), as a possible model
for the distributions of income and speculative prices. In recent years, the modeling of financial
processes and their analysis has become a very active and fast-developing branch of applied
mathematics and quantitative finance [4,5,8,11,36–38]. One reason for these developments, based
on stable distributions, is that the financial environment has tended to be more impulsive than a
Gaussian distribution can describe.
The popular stable distribution, being a generalization of the Gaussian distribution, shares many
“nice” properties with the Gaussian distribution. For instance, the sum of independent stable
random variables has only a stable distribution, which is similar to the central limit theorem
for distributions with a finite second moment. Also, it is known that all one-dimensional stable
distributions can be uniquely characterized by their characteristic function [47], which is given by
⎧
πα ⎪
α
⎪
exp
iμt
−
γ
|t|
1
−
iβsign(t)
tan
, if α = 1
⎨
2
φ(t) =
⎪
2
⎪
⎩exp iμt − γ |t| 1 + iβ sign(t) log |t| , if α = 1
π
(1)
where sign(t) is the sign of t defined by sign(t) = 1 if t > 0, sign(0) = 0, and sign(t) = −1
otherwise.
There are four parameters in the distribution function. The parameter α (0 < α ≤ 2) is the
characteristic exponent that controls the heaviness of the tails. The number μ(−∞ < μ < ∞) is
the location parameter that corresponds to the mean for 1 < α ≤ 2 and the median for 0 < α ≤ 1
and β = 0. The number γ (γ > 0) is the scale parameter that sets the dispersion around its location
parameter and, therefore, is analogous to the standard deviation. The number β(−1 ≤ β ≤ 1) is
the symmetry parameter that characterizes the skewness.
Journal of Applied Statistics
1091
Note that in almost all cases, the stable distributions denoted by Sα (β, γ , μ) do not have a closed
probability density function. The only exceptions to this are the cases when α = 2 (Gaussian
distributions), α = 1, β = 0 (Cauchy distributions) and α = 0.5, β = ±1 (Lévy distributions).
Since the distribution of a stable random variable X ∼ Sα (β, γ , μ) with α ∈ (0, 2) obeys the
power decay P (|X| > x) ∼ Cx −α , it follows that the stable distributions have infinite second (or
higher) moments except for the case when α = 2. In fact, the stable distributions with α < 2 have
finite moments only when order p is lower than α; that is,
⎧
⎪
⎨= ∞ if p ≥ α, α < 2,
p
E(|Xα | ) < ∞ if 0 ≤ p < α < 2,
⎪
⎩
< ∞ if p ≥ 0, α = 2,
where the random variable is Xα ∼ Sα (β, γ , μ). For more details about the properties of stable
distributions, see Uchaikin and Zolotarev [47].
It should be stressed that most of the recent empirical papers report that financial return series
show tail indices around three, that is, variances of the variables are finite but their fourth moments
are infinite (see [15,24,26] and the book by [14]). We believe that this conclusion is true when the
proportion, (Tail length)/(Size of simple), is small in Hill’s estimation (see Table 6 in Section 5).
When the proportion is nearly 0.15 ≈ 310/2070, the 95% confidence interval for α is (1.79, 2.24).
This means that the tail indices may be less than two. Moreover, Ibragimov et al.’s [19] paper
shows that economic losses from natural disasters and the time series in catastrophe insurance
markets have infinite variance tails.
Because stable distributions have no closed probability density function with infinite variance,
conventional SPC charting schemes are not easily applicable to monitoring and detecting changes
in sequences that have stable distributions. Our main purpose in this article is to present a stabledistribution-based multi-CUSUM chart to detect simultaneously the changes in four parameters,
α, β, γ and μ, in an independently and identically distributed (i.i.d.) random sequence with a
common stable distribution, Sα (β, γ , μ). The multi-CUSUM chart can, to a considerable extent,
achieve the following three goals in monitoring and detecting unknown changes in a stabledistributed random sequence: it can (1) signal an alarm as quickly as possible when there is a
change in the sequence, (2) accurately indicate the possible type/amount/size/etc. of the change,
and (3) easily handle computational complexity [16].
To this end, a critical step is to solve the problem of sequentially estimating the parameters of a
stable distribution, since an unbiased and consistent parameter estimation approach would be the
key to obtaining effective statistical functions and establishing an efficient multi-CUSUM chart. In
the next section, we present the proposed estimators and discuss their unbiasness and consistency.
The corresponding multi-CUSUM chart is presented in Section 3. Section 4 presents the numerical
simulation results that compare the average run lengths (ARLs) of detecting changes in the four
parameters of a stable distributed sequence. Section 5 provides a real example that illustrates
the step-by-step procedure used to detect changes in IBM’s stock returns, and gives a simple
comparison of detection performance between the multi-CUSUM chart with other CUSUM tests
defined by Hill’s estimator which was used by Quintos et al. [35] to estimate α. Conclusions and
suggestions for future research are presented in Section 6, with the proofs of two propositions
given in Appendix.
2.
Refined logarithmic moment estimators
There have been many studies in the literature addressing the problem of estimating the parameters
of stable distributions. Most of these studies focus on the special case of symmetric stable distributions with β = 0 [16,13,27]. Various estimation techniques for estimating the parameters of
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general (possibly skewed) stable distributions and further references can be found in Kuruoğlu [23]
and Nolan [31,32]. Here, we are especially interested in the logarithmic moment (LM) estimators
suggested by Kuruoğlu [23], since the estimators not only have closed-form solutions and require
less computation, but they can also compete with the characteristic function techniques. However,
the LM estimators for the parameter β can only estimate |β|; that is, they cannot detect the sign
of β. This weakness will limit their applications to some extent.
In this section, we first refine the LM estimators to resolve the existing weakness in
Kuruoğlu [23]. We also assess the unbiasness and consistency properties of the estimators.
2.1 Refined LM estimators
Let Xk , k ≥ 1, be i.i.d. stable random variables with the parameters α, β, γ and μ, that is, Xk ∼
Sα (β, γ , μ) for k ≥ 1. Since the sum of independent stable random variables is also stable, the
distribution of a weighted sum of i.i.d. stable variables with weights, ck , can be obtained from the
characteristic function [41]:
n
n
n
n
<α>
k=1 ck
α
(2)
ck Xk ∼ Sα β n
,γ
|ck | , μ
ck ,
α
k=1 |ck |
k=1
k=1
k=1
where we denote the signed pth power of the number x as
x p = sign(x)|x|p .
From Equation (2), we can further generate three new sequences, {XkL }, {XkS } and {XkLS }, of i.i.d.
stable variables with a zero location parameter, a zero symmetry parameter, or zero values for
both the location parameter and the symmetry parameter (except when α = 1), respectively:
2 − 2α
L
α
Xk = X3k + X3k−1 − 2X3k−2 ∼ Sα β
, γ (2 + 2 ), 0
(3)
2 + 2α
XkS = X3k + X3k−1 − 21/α X3k−2 ∼ Sα (0, 4γ , μ(2 − 21/α ))
XkLS
= X2k − X2k−1 ∼ Sα (0, 2γ , 0)
(4)
(5)
for k ≥ 1. The random variables XkLS and XkS are usually referred to as symmetrized random
variables.
Next, we list three formulae given by Kuruoğlu [23], one for signed fractional moments and
the others for LMs of skewed stable distributions, that are used in the following discussion. Let
X ∼ Sα (β, γ , 0). Then
(1 − p/α) γ p/α sin(pθ/α)
E(X <p> ) =
(6)
(1 − p) cos θ
sin(pπ/2)
for p ∈ (−2, −1) ∪ (−1, α), and
γ 1
1
E(log |X|) = λ0 1 −
+ log α
α
cos θ
θ2
π2 1
1
E([log |X| − E(log |X|)]2 ) =
− 2,
+
2
6 α
2
α
where (.) is the gamma function, λ0 = −0.57721566 . . . and
πα θ = arctan β tan
.
2
(7)
(8)
(9)
Journal of Applied Statistics
1093
Since | cos θ | and θ 2 occur, respectively, in Equations (7) and (8), one can only estimate |θ |
and therefore, |β|. To estimate θ , we need to introduce a number that is relative to θ . In fact, the
required number is just E(sign(X)), which has been considered by Zolotarev [49]. It follows from
Equation (6) that
E(sign(X)) = E(X 0 ) = lim E(X p ) =
p→0
2θ
.
πα
(10)
Thus, by using Equation (7), (8) and (10), we can estimate θ as well as the other parameters.
Now, we introduce five sample statistics for estimating the parameters:
An =
1
sign(XkL ),
n k=1
Bn =
1
log |XkL |
n k=1
Cn =
1
log |XkLS |,
n k=1
Dn =
1 (log |XkLS | − Cn )2
n − 1 k=1
n
n
n
and
n
E(n) =
S
X(n+1/2)
if n is odd,
S
S
X(n+2/2)
+ X(n/2)
if n is even,
(11)
S
,1 ≤ k ≤
where {XkL }, {XkS } and {XkLS } are defined in Equations (3), (4) and (5), respectively, X(k)
S
S
S
n, is the increasing order of arranging Xk , 1 ≤ k ≤ n; that is, X(k) , 1 ≤ k ≤ n satisfies X(1)
<
S
S
< · · · < X(n)
.
X(2)
In this article, we mainly consider the case: α = 1, 0 < α < 2. The resulting estimators, α̂, β̂,
γ̂ and μ̂, respectively, for α, β, γ and μ are summarized as follows.
LM Estimator for α (α = 1, 0 < α < 2)
Note that {XkLS } is i.i.d. with XkLS ∼ Sα (0, 2γ , 0). It follows from Equation (8) that
π2
E([log |X| − E(log |X|)] ) =
6
2
that is,
α=
1
1
+
;
α2
2
6E([log |X| − E(log |X|)]2 ) 1
−
π2
2
(12)
−1/2
.
Thus, we may use the following statistic
α̂ =
1
6Dn
−
2
π
2
−1/2
(13)
as an estimator of the parameter α.
Note that if Xk ∼ Sα (0, 1, 0), by Equation (7) we can choose
α̂ =
as an estimator of the parameter α.
λ0 − (1/n)
λ0
n
k=1
log |Xk |
(14)
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D. Han et al.
LM Estimator for β (|β| ≤ 1)
Since
XkL
∼ Sα
2 − 2α
α
β
, γ (2 + 2 ), 0 ,
2 + 2α
(15)
it follows from Equation (10) that
θ=
παE(sign(X))
.
2
We therefore take
θ̂ =
π α̂An
2
(16)
as an estimator of the parameter θ . Thus, the estimator of parameter β can be written as
β̂ =
2 + 2α̂ tan(θ̂)
.
2 − 2α̂ tan(π α̂/2)
(17)
LM Estimator for γ (γ > 0)
By Equations (7), (13), (15) and (16), we may take
γ̂ =
1
| cos(θ̂ )| exp{(Bn − λ0 )α̂ + λ0 }
2 + 2α̂
(18)
as an estimator of the parameter γ .
Median Estimator for μ
It is known that the parameter μ is the mean of the stable distributions when 1 < α ≤ 2 and
γ = 1 [47]. Thus, the parameter μ can be estimated by the simple mean
1
Xk ,
n k=1
n
μ̂ =
where Xk , k ≥ 1, is an i.i.d. stable random variable with 1 < α ≤ 2. However, the dissipation
of the value of the sample mean will be large since the variances of the stable distributions with
α < 2 is infinite. Moreover, it is not applicable to use the simple mean for an estimator of the
parameter μ when 0 < α ≤ 1. Next, we will use the median technique presented by Zolotarev [49
p. 240] to estimate μ. It follows from Equation (5) that the median of XkS coincides with μ̃, where
μ̃ = μ(2 − 21/α ). Taking the sample median E(n) in Equation (11) as an estimator of the parameter
μ̃, we can obtain the median estimator for μ in the following:
μ̂ =
E(n)
.
2 − 21/α̂
(19)
Remark 1 The estimator θ̂ in Equation (16) requires substantially less computation than that
suggested by Kuruoğlu [23].
Remark 2 The estimators above require either 3n + 2 or 2n + 1 observations.
Journal of Applied Statistics
1095
2.2 Unbiasness and consistency of the LM estimators
In this section, we assess the unbiasness and consistency of the proposed estimators. However,
these properties cannot be studied directly. This is because the statistic α̂ cannot be computed if
6Dn /π 2 − (1/2) is less than or equal to zero, and β̂, γ̂ and μ̂ cannot be computed also when
θ̂ = π/2 and α̂ = 1 or α̂ = 2. To ensure that the required statistics cannot only be computed but
also are consistent and asymptotically unbiased estimators of the parameters, we have to modify
the five estimators: α̂, θ̂ , β̂, γ̂ and μ̂.
Let A = E(sign(X)). Since α = 1, it follows from Equations (9) and (10) that θ = π/2, i.e.
−π/2 < θ < π/2, and therefore, −1 < Aα < 1.
The Modified Estimator α̂ˆ
To avoid α̂ = 1 or α̂ = 2, we take two positive numbers a, b such that a <
and define
⎧
6Dn
1
⎪
⎪
−
⎪
2
⎪
π
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1 + n−a
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪1 − n−a
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 6D
1
n
−
Dn (a, b) =
2
π
2
⎪
⎪
⎪
⎪
⎪
⎪1
⎪
−a
⎪
⎪
⎪4 + n
⎪
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
− n−a
⎪
⎪
4
⎪
⎪
⎪
⎪
⎪
⎪
6Dn
1
⎪
⎪
− ,b
⎩max
π2
2
if
√
b/2 and b < 1/9,
1
6Dn
− > 1 + n−a ,
2
π
2
if 1 ≤
6Dn
1
− ≤ 1 + n−a ,
2
π
2
if 1 − n−a ≤
6Dn
1
− < 1,
π2
2
if
1
6Dn
1
+ n−a < 2 − < 1 − n−a ,
4
π
2
if
1
6Dn
1
1
< 2 − ≤ + n−a ,
4
π
2
4
if
1
6Dn
1
1
− n−a ≤ 2 − ≤ ,
4
π
2
4
if
1
6Dn
1
− < − n−a ,
2
π
2
4
where n ≥ n0 such that 1/4 − b > n−a . Thus, we replace the estimator α̂ with the following
statistic
α̂ˆ = (Dn (a, b))−1/2
(20)
√
as an estimator of the parameter α. Obviously, the modified estimator α̂ˆ is bounded, i.e. α̂ˆ ≤ 1/ b,
and not equal to 1 or 2. Since the estimator
α̂ =
λ0 −
1
n
λ0
n
k=1
log |Xk |
has no definition when 1/n nk=1 log |Xk | = λ0 for Xk ∼ Sα (0, 1, 0), so we may modify it by
the same way such that the modified estimator α̂ˆ is finite when 1/n nk=1 log |Xk | = λ0 .
1096
D. Han et al.
The Modified Estimators θ̂ˆ and β̂ˆ
To avoid θ̂ = ±π/2, we take the following statistic θ̂ˆ to replace θ̂ as an estimator of the parameter θ :
⎧
ˆ n
⎪
π α̂A
⎪
ˆ n > 1 + n−a
⎪
if α̂A
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
π(1 + n−a )
⎪
⎪
ˆ n ≤ 1 + n−a
⎪
if 1 < α̂A
⎪
⎪
2
⎪
⎪
⎪
−a
⎪
⎪
⎪ π(1 − n )
ˆ n≤1
⎪
if 1 − n−a ≤ α̂A
⎪
⎪
2
⎪
⎪
⎪
⎨ ˆ
ˆθ̂ = π α̂A
n
ˆ n | < 1 − n−a
if |α̂A
⎪
⎪
2
⎪
⎪
⎪
⎪
π(1 − n−a )
⎪
⎪
ˆ n ≤ −(1 − n−a )
⎪
−
if − 1 ≤ α̂A
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪ π(1 + n−a )
⎪
ˆ n < −1
⎪
−
if − (1 + n−a ) ≤ α̂A
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
ˆ n
⎪
π α̂A
⎪
ˆ n < −(1 + n−a ).
⎩
if α̂A
2
Thus, the modified estimator of parameter β can be written as
ˆ
2 + 2α̂ tan(θ̂)
.
β̂ˆ =
ˆ
2 − 2α̂ˆ tan(π α̂/2)
ˆ
(21)
ˆ is also bounded.
The resulting estimator, β̂,
The Modified Estimators γ̂ˆ and μ̂ˆ
To make the modified estimator γ̂ˆ bounded, we introduce a bounded statistic
a log n if Bn > a log n,
Bn (a) =
Bn
if Bn ≤ a log n.
Thus, we take the following bounded statistic
γ̂ˆ =
1
| cos(θ̂ˆ )| exp{(Bn (a) − λ0 )α̂ˆ + λ0 }
2 + 2α̂
(22)
as a modified estimator of the parameter γ . Similarly, the modified estimator μ̂ˆ can be written as
μ̂ˆ =
E(n)
2 − 21/α̂ˆ
.
(23)
On the unbiasness and consistency of the modified estimators, we give two propositions in the
following (see Appendix for the proofs):
ˆ β̂,
ˆ γ̂ˆ and μ̂,
ˆ θ̂,
ˆ are consistent and asymptotically unbiased
Proposition 1 All five statistics, α̂,
estimators of the parameters, α, θ, β, γ and μ.
Journal of Applied Statistics
1097
Proposition 1 shows that the modified estimators indeed have good statistical properties,
although they need more computation.
Proposition 2 For any small > 0,
P(|α̂ˆ − α̂| ≥ ) −→ 0,
P(|θ̂ˆ − θ̂ | ≥ ) −→ 0,
P(|β̂ˆ − β̂| ≥ ) −→ 0,
P(|γ̂ˆ − γ̂ | ≥ ) −→ 0,
P(|μ̂ˆ − μ̂| ≥ ) −→ 0
as n → ∞.
Proposition 2 shows that there is actually little difference in the sense of probability convergence
between the modified estimators and the previous refined estimators. That is, the five refined
estimators, α̂, θ̂ , β̂, γ̂ and μ̂, have the same statistical properties as well as the five modified
estimators do in the sense of probability convergence.
The main reason for us to introduce and study the the modified estimators is that the unbiasness
and consistency of the five refined estimators, α̂, θ̂ , β̂, γ̂ and μ̂, cannot be studied directly since
they cannot be computed in some cases. From the Propositions 1 and 2, we see that the five
refined estimators, α̂, θ̂ , β̂, γ̂ and μ̂, may have the same good statistical properties such as the
unbiasness asymptotically and consistency as well as the five modified estimators do in the sense
of probability convergence. Moreover, the five refined estimators require less computation than
the modified ones do. Thus, we will use the refined estimators to construct the multi-CUSUM
charts in the next section.
3.
Multi-cusum charts for random sequences
A multi-chart is a combination of several control charts or a set of control charts used together for
obtaining a better detection power [9,10,25]. Specifically, a multi-chart with multiple CUSUM
charts, i.e. the multi-CUSUM chart, is known to have particularly good properties in detecting and
diagnosing an abrupt change in a process [16]. In this section, we apply the multi-CUSUM chart
to monitoring changes in random sequences with stable distributions based on the LM estimators
discussed in the previous section.
Let Xk , k = 1, 2, . . . be the kth independent observation of a random sequence with a known
common stable distribution Sα0 (β0 , γ0 , μ0 ). For simplicity, we take α0 = 1.5, β0 = 0, γ0 = 1 and
μ0 = 0. Here, we take α0 = 1.5 since the tail index α of heavy-tailed time series in financial
markets is usually between 1 and 2, and the number μ in the stable distribution Sα (β, γ , μ)
is the mean when 1 < α ≤ 2. Suppose that at some time period, τ , which is usually called a
change point, the probability distribution of Xk changes from S1.5 (0, 1, 0) to Sα (β, γ , μ). In other
words, from time period τ onwards, Xi has the common distribution Sα (β, γ , μ). In practice, the
change point and the post-change distributions are usually unknown, that is, τ and the changed
parameters vector (α, β, γ , μ) are unknown. To identify the multi-CUSUM chart for detecting
unknown abrupt changes from the parameter vector, ξ0 = (1.5, 0, 1, 0) to ξ = (α, β, γ , μ), let
ᾱ1 , β̄1 , γ̄1 and μ̄1 be the four positive, pre-specified known reference values that describe the
change values from ξ0 = (α0 , β0 , γ0 , μ0 ) = (1.5, 0, 1, 0) to ξ = (α1 , β1 , γ1 , μ1 ), where ᾱ1 =
(1/α0 ) − (1/α1 ), β̄1 = β1 − β0 , γ̄1 = γ1 − γ0 and μ̄1 = μ1 − μ0 .
A basic upward-side CUSUM chart, TC , can be defined as
n
TC = inf n : max
(Xi − δ/2) ≥ c ,
(24)
1≤k≤n
i=n−k+1
where δ/2 > 0 is the reference value related to the magnitude of the mean shift, δ, and c > 0 is a
control limit.
1098
D. Han et al.
We first present upward-side CUSUM charts, T (α1 ), T (β1 ), T (γ1 ) and T (μ1 ), in the following
for detecting the change in parameters, α, β, γ and μ, respectively.
Let
λ0 − log |Xk |
,
λ0
Yk =
Zk =
λ0
λ0 − (1/min{k, m}) ki=k−m+1 log |Xi |
and
Uk =
k
πZk
sign(Xi ).
2 min{k, m} i=k−min{k,m}+1
It follows from Equation (7) that {Yk , k ≥ 1} is i.i.d. with the mean E(Yk ) = 1/α0 = 1/1.5. Hence,
to detect the change from α0 = 1.5 to α, we use the following CUSUM chart:
⎧
⎨
⎡
⎛
n
⎝
T (α1 ) = min n : max ⎣
⎩ 1≤j ≤n
k=j
⎫
⎞⎤
⎬
Yi
ᾱ1 ⎠⎦
1
−
−
≥ lα1 ,
⎭
min{k, m} α0
2
i=k−min{k,m}+1
k
(25)
to detect the mean shift of {Yk } from 1/α0 to 1/α, where the positive number, lα1 , is the upward
control limit. Similarly, by Equations (9) and (16), the CUSUM chart T (β1 ) for detecting β is
given by
⎧
⎨
⎡
⎛
n
⎝
T (β1 ) = min n : max ⎣
⎩ 1≤j ≤n
k=j
⎫
⎞⎤
⎬
g(X1 , . . . , Xi )
β̄1 ⎠⎦
− β0 −
≥ lβ1 ,
⎭
min{k, m}
2
i=k−min{k,m}+1
k
(26)
where
tan(Uk )
.
tan(π Zk /2)
g(X1 , . . . , Xk ) =
By Equations (7) and (11), the CUSUM charts T (γ1 ) and T (μ1 ) for detecting γ and μ,
respectively, are written as
⎧
⎨
⎡
⎛
n
⎝
T (γ1 ) = min n : max ⎣
⎩ 1≤j ≤n
k=j
⎞⎤
k
⎫
⎬
h(X1 , . . . , Xi )
γ̄1
− γ0 − ⎠⎦ ≥ lγ1
⎭
min{k,
m}
2
i=k−min{k,m}+1
(27)
and
⎧
⎨
⎡
T (μ1 ) = min n : max ⎣
⎩ 1≤j ≤n
⎫
⎤
⎬
μ̄1 ⎦
E(k) − μ0 −
≥ lμ1 ,
⎭
2
n k=j
where
h(X1 , . . . , Xk ) = | cos(Uk )| exp
k
1
log |Xi | − λ0 Zk + λ0 .
k i=1
(28)
Journal of Applied Statistics
1099
Similarly, the downward-side CUSUM charts are written as
⎫
⎧
⎡
⎛
⎞⎤
n
k
⎬
⎨
Y
ᾱ
1
i
2
⎝
+ ⎠⎦ ≤ −lα2 ,
−
T (α2 ) = min n : min ⎣
⎭
⎩ 1≤j ≤n
min{k, m} α0
2
k=j
i=k−min{k,m}+1
⎫
⎧
⎡
⎛
⎞⎤
n
k
⎬
⎨
g(X
)
β̄
i
2
⎝
T (β2 ) = min n : min ⎣
− β0 + ⎠⎦ ≤ −lβ2 ,
⎭
⎩ 1≤j ≤n
min{k, m}
2
k=j
i=k−min{k,m}+1
⎫
⎧
⎡
⎛
⎞⎤
n
k
⎬
⎨
h(X
)
γ̄
i
2
⎝
T (γ2 ) = min n : min ⎣
− γ0 + ⎠⎦ ≤ −lγ2
⎭
⎩ 1≤j ≤n
min{k, m}
2
k=j
i=k−min{k,m}+1
and
⎧
⎨
⎫
⎡
⎤
n ⎬
μ̄
2 ⎦
E(k) − μ0 +
T (μ2 ) = min n : min ⎣
≤ −lμ2 ,
⎩ 1≤j ≤n
⎭
2
k=j
where ᾱ2 = (1/α2 ) − (1/α0 ), β̄2 = β0 − β2 , γ̄2 = γ0 − γ2 and μ̄2 = μ0 − μ2 are four positive
numbers and lα2 , lβ2 , lγ2 and lμ2 are the downward control limits. Since (α0 , β0 , γ0 , μ0 ) =
(1.5, 0, 1, 0), it follows that β̄1 = β1 , μ̄1 = μ1 , β̄2 = −β2 and μ̄2 = −μ2 .
Now, the multi-CUSUM chart, TM , for detecting the change of ξ = (α, β, γ , μ) can be
defined by
TM = min{T (α1 , α2 ), T (β1 , β2 ), T (γ1 , γ2 ), T (μ1 , μ2 )},
(29)
where T (α1 , α2 ) = min{T (α1 ), T (α2 )}, T (β1 , β2 ) = min{T (β1 ), T (β2 )}, T (γ1 , γ2 ) = min{T
(γ1 ), T (γ2 )} and T (μ1 , μ2 ) = min{T (μ1 ), T (μ2 )} are four two-sided CUSUM charts.
Remark 3 If Xk , k ≥ 1, has the common known distribution, Sα0 (μ0 , β0 , γ0 ), where β0 = 0 or
γ0 = 1 and μ0 = 0, then the above statistics, Yk , Zk , Uk , g(X1 , . . . , Xk ), h(X1 , . . . , Xk ), E(k) and
E(Yk ) should be replaced with
6
1
6Dk
1 −1/2
−
Yk = [log |XkLS | − E(log |XkLS |)]2 − , Zk =
π
2
π2
2
Uk =
π
Zk sign(XkL ),
2
g(X1 , . . . , Xk ) =
1
h(X1 , . . . , Xk ) =
| cos(Uk )| exp
2 + 2 Zk
2 + 2Zk tan(Uk )
2 − 2Zk tan(π Zk /2)
1
log |XiL | − λ0 Zk + λ0 ,
k i=1
k
E(k)
2 − 21/Zk
and E(Yk ) = 1/α02 , respectively.
4.
Performance evaluation by simulation comparison
For the convenience of discussion, we use the standard quality control terminology. Let P0 (·) and
E0 (·) denote the probability and expectation when there is no change in the mean. Denote P (·)
and E(·) as the probability and expectation when the change point is at τ = 1, and the mean is
changed from x0 to x. For a stopping time T as the alarm time with a detecting procedure, two most
frequently used operating characteristics are the in-control average run length (ARL0 ) and the
out-of-control ARL, defined by ARL0 (T ) = E0 (T ) and ARL(T ) = E(T ). Usually, comparisons
1100
D. Han et al.
of the control charts’ performance are made by designing the common ARL0 and comparing the
ARLs of the control charts for a given mean shift. The chart with the smaller ARL is considered
to have better performance.
In this section, we evaluate the performance of the proposed monitoring scheme by comparing the numerical simulation results of the ARL of the four separate two-sided CUSUM
charts, T (α1 , α2 ), T (β1 , β2 ), T (γ1 , γ2 ), T (μ1 , μ2 ), and the multi-CUSUM chart, TM . Here,
the statistics, Yk , Zk , Uk , g(X1 , . . . , Xk ), h(X1 , . . . , Xk ), and E(k) in the CUSUM charts
T (α1 , α2 ), T (β1 , β2 ), T (γ1 , γ2 ), T (μ1 , μ2 ) in Section 3 have been replaced, respectively, by
6
1
6Dk
1 −1/2
LS
LS 2
Yk = [log |Xk | − E(log |Xk |)] − , Zk =
−
π
2
π2
2
Uk =
π
Zk sign(XkL ),
2
g(X1 , . . . , Xk ) =
1
h(X1 , . . . , Xk ) =
| cos(Uk )| exp
2 + 2 Zk
2 + 2Zk tan(Uk )
2 − 2Zk tan(π Zk /2)
1
log |XiL | − λ0 Zk + λ0 ,
k i=1
k
E(k)
2 − 21/Zk
since we consider more general circumstances that it is not necessary to assume β0 = 0, γ0 = 1
and μ0 = 0. This has been pointed out in Remark 3.
For comparison purposes, the in-control ARL (ARL0 ) of all candidate charts are forced to be
equal; in other words, the control limits of each control chart are determined to correspond to the
same level of type-I error. The chart that has the lowest out-of-control ARL at the desired shift
size presents the highest detection power to the pre-specified shift. Here, for illustration purposes,
the common ARL0 for the five control charts is chosen to be around 500, which is a typical value
in practice. We denote control limits of four two-sided CUSUM charts, T (α1 , α2 ), T (β1 , β2 ),
T (γ1 , γ2 ) and T (μ1 , μ2 ) as (lα 1 , lα 2 ), (lβ 1 , lβ 2 ), (lγ 1 , lγ 2 ) and (lμ 1 , lμ 2 ), respectively. For the eight
control limits of the multi-CUSUM chart, TM , we denote them as (lα1 , lα2 , lβ1 , lβ2 , lγ1 , lγ2 , lμ1 , lμ2 ).
Here, we take α1 = 1.3, α2 = 1.7, β1 = 0.5, β2 = −0.5, γ1 = 2, γ2 = 0, μ1 = 1 and μ2 =
−1. The main reason for us to take such reference values is that they are symmetric about ξ0 =
(1.5, 0, 1, 0) since α0 − α1 = α2 − α0 , β0 − β1 = β2 − β0 , γ0 − γ1 = γ2 − γ0 and μ0 − μ1 =
μ2 − μ0 . In order to make T (α1 , α2 ), T (β1 , β2 ), T (γ1 , γ2 ), T (μ1 , μ2 ) and TM have a common
ARL0 = 500, the values of the control limits are taken as follows:
lα 1 = 12,
lα 2 = 8,
lα1 = 28,
lα2 = 15.1,
lβ 1
lβ 2
lβ1 = 155.1,
lβ2 = 155.1,
= 75.1,
= 75.1,
lγ 1 = 66,
lγ 2 = 3,
lγ1 = 145.1,
lγ2 = 5.2,
lμ 1
lμ 2
lμ1 = 173.2,
lμ2 = 173.2,
= 102,
= 102,
where the common ARL of the CUSUM charts, T (α1 , α2 ), T (β1 , β2 ), T (γ1 , γ2 ), T (μ1 , μ2 ), in
TM need to satisfy
ARL0 (T (α1 , α2 )) = ARL0 (T (β1 , β2 )) = ARL0 (T (γ1 , γ2 )) = ARL0 (T (μ1 , μ2 )) = 2200.
That is, we can get the ARL0 (TM ) ≈ 500 for the multi-chart TM by numerical computation when
the ARL0 s of its four component CUSUM charts all approximate to 2200. Of course, we can
also obtain the ARL0 (TM ) ≈ 500 when the four component CUSUM charts of TM have different
ARL0 . In this case, the ARL0 of some charts must be larger than 2200, some must be smaller than
2200 among the four component charts. We make the four component CUSUM charts have the
same ARL0 here, since the four parameters α, β, γ and μ are considered in the article to have
Journal of Applied Statistics
1101
the same importance. The control limits can be determined by repeated computation for taking
different control limits.
Tables 1 and 2 illustrate the numerical simulation results of the ARLs of the four two-sided
CUSUM chars, T (α1 , α2 ), T (β1 , β2 ), T (γ1 , γ2 ), T (μ1 , μ2 ) and the multi-CUSUM chart, TM .
The numerical results of the ARLs were obtained based on 1,000,000-repetition experiments. The
in-control value of the four parameters are α = 1.5, β = 0.0, γ = 1, μ = 0.0. The true values
of α, β, γ and μ of a stable distribution are listed in the first four columns in each table. Concerning the out-of-control scenario, we consider two extreme cases. Table 1 lists the simulation
results when only one parameter out of α, β, γ and μ changes and the other three parameters
do not change. Table 2 contains the simulation results of the ARLs when all four parameters
change. The notation over the ARL values in the two tables indicates that the value is greater
than the in-control ARL of 500. The ARL values in boldface mean that it is the minimal value
among the four ARL values of T (α1 , α2 ), T (β1 , β2 ), T (γ1 , γ2 ) and T (μ1 , μ2 ) in each row of
Tables 1 and 2.
Table 1. Comparisons of the ARLs of the CUSUM charts and the multi-CUSUM chart with ARL0 ≈ 500.
Shifts
T (α
{α}
{β}
{γ }
{μ}
1.5
0
1
0
499.7
1.7
1.9
1.3
1.1
0
0
0
0
1
1
1
1
0
0
0
0
486.5
429.1
426.6
279.8
1.5
1.5
1.5
1.5
0.5
0.9
−0.5
−0.9
1
1
1
1
0
0
0
0
1.5
1.5
1.5
1.5
0
0
0
0
1.5
2.0
0.5
0.2
0
0
0
0
∗ 509.2
∗ 502.5
∗ 508.1
1.5
1.5
1.5
1.5
0
0
0
0
1
1
1
1
1
2
−1
−2
T (β1 , β2 )
1 , α2 )
T (γ1 , γ2 )
T (μ1 , μ2 )
TM
500.9
499.8
498.7
501.2
∗ 536.9
∗ 603.5
∗ 597.4
∗ 594.8
∗ 940.3
∗ 1672.0
∗ 562.0
∗ 507.1
482.4
∗ 519.2
389.7
299.2
257.6
142.6
300.2
162.3
215.6
216.4
215.4
216.6
349.0
213.9
368.8
231.4
272.2
282.5
270.0
280.1
98.2
82.5
97.7
84.3
109.5
97.8
111.7
102.4
498.5
497.5
494.9
493.9
500.8
153.4
82.7
78.9
25.3
143.3
65.8
∗ 3774.0
∗ 5821.0
138.0
69.6
107.2
39.1
498.5
497.1
496.9
491.1
496.3
225.4
100.9
223.2
100.4
334.7
185.8
332.8
184.3
∗ 506.4
500.0
∗ 504.5
∗ 504.4
497.3
494.3
493.0
Table 2. Comparisons of the ARLs of the CUSUM charts and the multi-CUSUM chart with ARL0 ≈ 500.
Shifts
{ξ }
{α}
{β}
{γ }
{μ}
ξ0
ξ1
ξ2
ξ3
ξ4
ξ5
ξ6
ξ7
ξ8
ξ9
1.5
1.7
1.7
1.7
1.9
1.9
1.9
1.1
1.1
1.1
0.0
0.5
0.9
−0.5
0.5
0.9
−0.5
0.5
0.9
−0.5
1.0
1.5
2.0
0.5
2.0
0.5
1.5
0.5
1.5
2.0
0.0
1.0
2.0
−1.0
−1.0
1.0
2.0
2.0
−1.0
1.0
T (α
1 , α2 )
499.7
215.9
214.5
215.4
214.7
216.1
215.7
215.4
215.1
211.3
T (β1 , β2 )
T (γ1 , γ2 )
T (μ1 , μ2 )
TM
500.9
358.3
213.8
370.8
213.9
214.7
372.1
350.8
214.4
368.5
499.8
116.2
73.7
177.5
72.2
184.0
116.7
138.3
118.1
74.4
498.7
35.6
16.5
127.9
22.2
101.8
43.5
258.5
16.4
15.9
501.2
49.2
25.3
129.4
30.3
104.0
54.8
139.4
25.8
22.7
1102
D. Han et al.
From Table 1, we see that
1. When only α changes, control charts T (β1 , β2 ) and T (γ1 , γ2 ) fail when there is a respective
increase in α. Although T (μ1 , μ2 ) responds to the decrease in α in the quickest manner, it
fails when α increases. Among the five control charts, although T (α1 , α2 ) has the best overall
performance for detecting decreases or an increase in α, the multi-CUSUM chart, TM , is also
good at detecting changes in α, except when there is an increase in α.
2. When only β changes, both T (μ1 , μ2 ) and TM are sensitive to changes in β. The other three
CUSUM charts lose detection capability no matter β increases or decreases.
3. Both T (α1 , α2 ) and T (β1 , β2 ) are not sensitive to either decreases or increases in γ . T (μ1 , μ2 )
deteriorates when γ decreases. Although T (γ1 , γ2 ) is the quickest one to detect decreases in γ ,
the multi-CUSUM chart, TM , has the most satisfactory performance in detecting both increases
and decreases in γ .
4. The three CUSUM charts T (α1 , α2 ), T (β1 , β2 ) and T (γ1 , γ2 ) are not sensitive to either
decreases or increases in μ. Although T (μ1 , μ2 ) is the best among all charts in detecting the
mean shifts, the multi-CUSUM chart, TM , shows a performance similar to that of T (μ1 , μ2 ).
5. The main reason for some ARL1 larger than 500 is that the variances of the estimators became
smaller than that in the in-control case. For example, by 1,000,000-repetition experiments, the
variance of the estimator E(k) /(2 − 21/Zk ) changes from 4.51 to 3.44, 2.75, 8.17 and 19.8 when
the true value (α, β, γ , μ) changes from (1.5, 0, 1, 0) to (1.7, 0, 1, 0), (1.9, 0, 1, 0), (1.3, 0, 1,
0) and (1.1, 0, 1, 0), respectively. Thus, the corresponding ARL1 of T (μ1 , μ2 ) changes from
the in-control value 498.7 to 940.3, 1672, 257.6, and 142.6. When the true value (α, β, γ , μ)
changes from (1.5, 0, 1, 0) to (1.5, 0, 1.5, 0), (1.5, 0, 2.0, 0), (1.5, 0, 0.5, 0) and (1.5, 0, 0.2,
0), the variance of E(k) /(2 − 21/Zk ) changes from 4.51 to 6.96, 8.92, 2.30, 0.91, respectively.
So, the corresponding ARL1 of T (μ1 , μ2 ) changes from 498.7 to 143.3, 65.8, 3774 and 5821,
respectively.
In summary, when only one of the four parameters changes, the multi-CUSUM chart, TM , is
found to be superior to other charts on the whole for its robustness. It is not only robust (never
has the worst performance) but is also competent in its detection capability.
When all the four parameters change, we consider three shift sizes in each parameter. Here
α ∈ {1.1, 1.7, 1.9}, β ∈ {−0.5, 0.5, 0.9}, γ ∈ {0.5, 1.5, 2} and μ ∈ {−1, 1, 2}. Thus there will be
81 possible shifts from ξ0 = (1.5, 0, 1, 0) to ξ = (α, β, γ , μ). To reduce the number of simulation
experiments, we may use Taguchi’s orthogonal arrays [46], L9 (34 ), to choose the following nine
possible change vectors in the parameters:
ξ1 = (1.7, 0.5, 1.5, 1),
ξ2 = (1.7, 0.9, 2, 2),
ξ3 = (1.7, −0.5, 0.5, −1),
ξ4 = (1.9, 0.5, 2, −1),
ξ5 = (1.9, 0.9, 0.5, 1), ξ6 = (1.9, −0.5, 1.5, 2),
and
ξ7 = (1.1, 0.5, 0.5, 2),
ξ8 = (1.1, 0.9, 1.5, −1),
ξ9 = (1.1, −0.5, 2, 1).
Table 2 shows that T (α1 , α2 ), T (β1 , β2 ) and T (γ1 , γ2 ) all fail in some cases. T (μ1 , μ2 ) and
TM are the best two control charts among all the candidates. From ξ1 to ξ6 and ξ9 , the multiCUSUM chart, TM , can compete with T (μ1 , μ2 ) for quickly detecting the change. Moreover, TM
outperforms T (μ1 , μ2 ) in case ξ7 . Therefore, the multi-CUSUM chart is considered to show the
best and most robust performance when all the parameters change.
In summary, no matter which of the four parameters of a stable distribution changes, the
multi-CUSUM chart always shows the most satisfactory and robust performance.
Journal of Applied Statistics
1103
5. An illustration with real stock returns
We use IBM’s stock market returns from 1993 to 2007 to illustrate the implementation of the multiCUSUM chart for monitoring financial data. Figure 1 illustrates the daily IBM returns during
1993–2007. Statistical distributions of financial returns have been a primary area of investigation
in both statistics and financial economics. Many researchers [12,30,42] have supported the theory
that stock prices and/or stock returns follow stable distributions. The multi-CUSUM monitoring
scheme should therefore be useful. At the end of this section, the detection performance of other
kinds of CUSUM charts defined by Hill’s estimator are illustrated in Table 4.
The successive daily closing prices are P1 , . . . Pn , where Pk is the closing price on trading
day k. Let P0 be the closing price of the stock immediately before these n days. The stock returns
for IBM are denoted as
Pk
k ≥ 1.
Xk = 100 log
Pk−1
We consider IBM’s stock returns as an i.i.d random sequence with a stable distribution. To use
the multi-CUSUM chart to detect changes in IBM’s stock returns, we have to obtain an estimate of
the true values of the four parameters in the stock return distribution. We use the data before 2001
as Phase I data to estimate the four parameters for the stable distribution. The in-control values
of the four parameters are ξ0 = (α0 , β0 , γ0 , ξ0 ) = (1.5, 0.28, 0.25, 0). Then, we aim to evaluate
the detection capability of the multi-CUSUM chart to detect drops in IBM stock returns starting
from 2001.
First, we describe the step-by-step procedures for implementing the proposed multiCUSUM scheme:
1. Obtain the in-control value of the four parameters, α, β, γ , μ, of a stable distribution. If the
in-control values are unknown, estimate them from the Phase I historical data (e.g. the IBM
stock returns before 2001).
2. Based on the shift magnitudes in α, β, γ , μ, determine the reference values of ᾱ1 , ᾱ2 , β̄1 , β̄2 , γ̄1 ,
γ̄2 , μ̄1 , μ̄2 .
3. Determine the in-control ARL and find the control limits of the multi-CUSUM charts (as
described in Section 3).
4. Establish the multi-CUSUM chart and monitor the new data over time.
Figure 1. IBM stock price from 1993 to 2006.
1104
D. Han et al.
Table 3. Numerical results for monitoring IBM’s stock returns with ξ0 = (1.5, 0.28, 0.25, 0) and
ARL0 ≈ 500.
Control charts
Control limit
Reference values
Out-of-control
TM
T (α)
T (β)
T (γ )
T (μ)
lα = 9.8
α1 = 0.12
α2 = 0.12
lβ = 70.6
β1 = 0.12
β2 = 0.12
lγ = 2.4
γ1 = 0.15
γ2 = 0.15
lμ = 2.78
μ1 = 0.02
μ2 = 0.02
l = (59, 193.5, 6.7, 6.75)
234
294
54
255
60
In this illustrative case, by taking the reference values of ᾱ1 = ᾱ2 = 0.12, β̄1 = β̄2 = 0.12,
γ̄1 = γ̄2 = 0.15 and μ̄1 = μ̄2 = 0.02, we compare the detection ability of four two-sided CUSUM
charts, T (α1 , α2 ), T (β1 , β2 ), T (γ1 , γ2 ), T (μ1 , μ2 ) and the multi-CUSUM chart, TM , as defined
by Equations (24)–(28). The control limits of each control chart are determined by achieving an
in-control ARL0 of 500.
Table 3 presents the numerical results for monitoring changes in IBM’s stock returns starting
from 1 January, 2001. Using the CUSUM chart for γ is the quickest way to detect the changes in
the underlying distribution of IBM’s stock returns. It triggers an out-of-control signal at the end
of the second month in 2001, which could have given an early warning to the economic recession
in US in the first three quarters of 2001. The multi-CUSUM chart also outperforms the other three
single CUSUM charts by a large margin and has comparable detection capability with the the
CUSUM chart for γ .
Remark 4 In order to obtain the stable estimated values of the four parameters we ignore the
possible changes of the parameters, during the period 1993–2001. In fact, it is feasible to estimate
the distribution parameters by using data of one month.
We now discuss how to define the CUSUM charts by Hill’s estimator.
The Hill’s estimator
−1
mJ
1 J
J
log X(J −i+1) − log X(J −mJ +1)
α̂J =
mJ i=1
J
has been used by Quintos et al. [35] to estimate α, where X(i)
denote the ith ordered statistic of
the sample of size J , and the number mJ is the m largest observations of a sample of size J .
We consider four rolling estimators α̂j k , 1 ≤ j ≤ 4 which are modifications of Hill’s estimator.
−1
mj
1 Jj k
Jj k
α̂j k =
log X(Jj k −i+1) − log X(Jj k −mj +1)
mj i=1
where m1 = 12, m2 = 14, m3 = 26, m4 = 65 and
J1k = {k − 11, k − 10, . . . , k − 1, k}
J2k = {k − 13, k − 12, . . . , k − 1, k}
J3k = {k − 25, k − 24, . . . , k − 1, k}
J4k = {k − 64, k − 65, . . . , k − 1, k}.
By using the four rolling estimators, we can define four CUSUM charts in the following:
n ᾱ1
α̂j k − α0 −
≥ lj
Tj (α) = min n : max
1≤i≤n
2
k=i
Journal of Applied Statistics
1105
Table 4. Numerical results for monitoring IBM’s stock returns with α0 ≈ 1.5.
Control charts
Control limit
Reference values
T1 (α) (502)
l1 = 9.5
α1 = 0.2
α2 = 0.2
T2 (α)(501)
l2 = 8.8
α1 = 0.2
α2 = 0.2
T3 (α)(502)
l3 = 6.3
α1 = 0.2
α2 = 0.2
T4 (α)(498)
l4 = 1.6
α1 = 0.2
α2 = 0.2
280
255
290
1150
Out-of-control
Table 5. Estimation of the tail index using 2020 data points from January 1993 to January
2001 and the result of detection starting from January 2001.
Window size
Tail length
Tail index(α0 )
Control limit
Change point
ARL0
12
14
26
65
1.5094
1.5019
1.516
1.4871
9.5
8.8
6.3
1.6
280
255
290
1150
502
501
502
498
40
50
100
250
Table 6. Estimation of the tail index and its confidence interval using 2070 data points from
January 1993 to March 2001.
Tail length
Tail index
Confidence interval
310
2.0139
(1.79, 2.24)
200
2.6433
(2.28, 3.01)
100
3.4224
(2.75, 4.09)
50
3.6449
(2.63, 4.65)
for 1 ≤ j ≤ 4. Taking m1 = 12, m2 = 14, m3 = 26, m4 = 65 can make average values of the
four estimators α̂j k , 1 ≤ j ≤ 4 be equal to α01 = 1.5094, α02 = 1.5019, α03 = 1.516 and α04 =
1.4871, respectively, where the data {Xk } is the stock returns of IBM during the period 1993–
2001. The following Table 4 is the simulation results for the CUSUM chart Tj (α), 1 ≤ j ≤ 4 to
monitor changes in stock returns of IBM starting from 1 January, 2001.
The numbers in the parentheses in Table 4 denote the ARL0 of Tj (α), 1 ≤ j ≤ 4, respectively.
Table 4 indicates that the CUSUM chart T2 (α) shows the best performance among the four
charts, Tj (α), 1 ≤ j ≤ 4. As can be seen from Tables 3 and 4 that the CUSUM charts defined by
rolling Hill’s estimators can compete with the CUSUM charts considered in the paper in detecting
the change of the tail index α. In fact, the CUSUM charts defined by rolling Hill’s estimators may
have better performance than that in Table 4, if we take different reference values for each chart,
and the windows mj , 1 ≤ j ≤ 4, can vary in a suitable way with the simples.
Remark 5 Table 5 in the following illustrates the Hill’s estimation of the tail index using 2020
data points from January 1993 to January 2001 of IBM’s stock returns, and the results of detecting
the data starting from January 2001. Here, the tail lengths are taken to be 12, 14, 26 and 65
respectively in the four rolling estimators. As can be seen in Table 6 that the Hill’s estimations
of the tail indices may be less than two for (Tail length)/(Size of simple)≈ 0.1498 = 310/2070
(Hill’s estimation of the tail index is 2.0139 in the case).
6.
Discussion and conclusion
Effective monitoring schemes of financial time series provide powerful tools to decision makers in
the financial world. However, empirical observations show that the financial time series exhibit fat
1106
D. Han et al.
tails, excessive kurtosis, some time, infinite variation, and such series cannot be easily monitored
by conventional Gaussian-based SPC control schemes. On the other hand, stable distributions are
a rich class of probability distributions that allow skewness and heavy tails and have extensive
applications in finance and economics. We propose a stable-distribution-based control scheme
using a multi-CUSUM chart. In fact, except for economics and finance, a wide range of modeling
areas such as engineering, physics, astronomy, computer science, networks, and so on have been
modeled through stable distributions (see [32], in which about 1300 references relative to the stable
distributions are listed). We believe that the proposed stable-distribution-based multi-CUSUM
scheme can be extended to a variety of different practical problems.
To identify a multi-CUSUM chart for detecting changes in the four parameters, α, β, γ and μ in
an i.i.d. random sequence with the stable distribution Sα (β, γ , μ), we first refined LM estimators
for the four parameters, and then proved that the five estimators α̂, θ̂ , β̂, γ̂ and μ̂ not only have
unbiasness and consistency in the sense of probability convergence but also need less computation
than the modified ones.
Based on the refined LM estimators, we construct a multi-CUSUM chart. The numerical results
of the ARLs in Tables 1 and 2 illustrate that the multi-chart is superior (robust and quick) on the
whole to the single CUSUM charts in detecting the shifts of the four parameters in i.i.d. random
sequence with a stable distribution. Moreover, the example of monitoring changes in IBM’s stock
returns also shows that the multi-CUSUM chart indeed has good detection performance.
The numerical results for the multi-CUSUM charts are based on the condition that the constituent charts of the multi-CUSUM charts have a common ARL0 . It would be of interest to study
if the same results still hold for the multi-CUSUM chart when its constituent charts have different
ARL0 . In fact, if the change of one parameter, μ, is considered to be more important than the
other one, for instance, α, then the ARL0 of the control chart for detecting μ may be chosen to
be smaller than that of the control chart for detecting α, so that the change of the parameter μ
can be detected more quickly. It should also be possible to find a control chart that is sensitive
to changes in β. Other interesting problems that also warrant further research include how the
procedures considered in the article may be modified or adapted to account for dependence in the
time series, how to determine the sample size in order for the resulting control limits to have their
required properties when the known parameter values in the multi-CUSUM chart are replaced by
the estimated values, and how to make the multi-CUSUM chart such that it can be used efficiently
over a period of time that may include several out-of-control signals.
Acknowledgements
The authors are grateful to the editor and the anonymous referees for their valuable comments, which have helped improve
this article greatly. This research was supported by RGC Competitive Earmarked Research Grants 620707 and 620508.
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APPENDIX 1.
Proofs of Proposition 1 and Proposition 2
Proof of Proposition 1. By Equations (5) and (12), we can check that the statistic 6Dn /π 2 − 1/2
is a unbiased estimator of the parameter 1/α 2 . It is known that [22, P.105]
Var(Dn ) =
1
2
(E[log X1L − E(log X1L )]4 − [Var(log X1L )]2 ) +
[Var(log X1L )]2 .
n
n(n − 1)
As n → ∞, we have
E
6Dn
1
1
− − 2
π2
2 α
2
=
36
Var(Dn ) = O
π4
1
−→ 0.
n
(A1)
This means that the statistic 6Dn /π 2 − 1/2 is a consistent unbiased estimator of the parameter
1/α 2 . Note that 1/4 < 1/α 2 < 1 or 1/α 2 > 1 and 0 < b < 1/9. Let
amax = max{|1 − n−a − 1/α 2 |, |1 + n−a − 1/α 2 |},
amin = min{|1 − n−a − 1/α 2 |, |1 + n−a − 1/α 2 |}.
Obviously, amax → |1 − 1/α 2 |, amin → |1 − 1/α 2 | as n → ∞. It follows from the definition of
α̂ˆ that (Dn (a, b))−1 ≤ 1/b and
6Dn
1 2
a2
1
1 2
E Dn (a, b) − 2
E
≤ 1 + max
−
−
2
α
π2
2 α2
amin
(A2)
for large n, since, for large n,
1
Dn (a, b) − 2
α
2
a2
≤ max
2
amin
6Dn
1
1
− − 2
π2
2 α
2
for 1 − n−a ≤ 6Dn /π 2 − 1/2 ≤ 1 + n−a and
1
Dn (a, b) − 2
α
2
≤
6Dn
1
1
− − 2
π2
2 α
2
for 1/4 − n−a ≤ 6Dn /π 2 − 1/2 ≤ 1/4 + n−a . Thus, by Equations (A1) and (A2), the statistic α̂ˆ
is a consistent, asymptotically unbiased estimator of the parameter α since
!
1
(Dn (a, b) − 1/α 2 )2
1 2
2
2
ˆ
E(α̂ − α) = α E
−→ 0
=
O
=
O
E
D
(a,
b)−
√
n
2
2
α
n
Dn (a, b)( Dn (a, b) + 1/α)
as n → ∞.
Journal of Applied Statistics
1109
By the definition of θ̂ˆ , we have
2
" ˆ n
π α̂A
π
παA
ˆ
2
ˆ n − 1| ≤ n−a
E(θ̂ − θ ) ≤ E
+ (1 + n−a + |αA|)2 P |α̂A
−
2
2
2
#
ˆ n + 1| ≤ n−a
+P |α̂A
≤
# π
π2 "
E(An )2 E(α̂ˆ − α)2 + α 2 E(An − A)2 + (2 + |αA|)2
4
2
" #
−a
ˆ
ˆ
× P |α̂An − 1| ≤ n
+ P |α̂An + 1| ≤ n−a .
Note that −1 < αA < 1. It follows from Chebyshev’s inequality that
ˆ n − 1| ≤ n−a = P 1 − αA − n−a ≤ An (α̂ˆ − α) + α(An − A) ≤ 1 − αA + n−a
P |α̂A
≤ P |An (α̂ˆ − α)| ≥ (1 − αA − n−a )/2
(1 − αA − n−a )
+ P |α(An − A)| ≥
2
2
{[E(An )2 E(α̂ˆ − α)2 ]1/2 + α[E(An − A)2 ]1/2 }
|1 − αA − n−a |
for n ≥ n0 , where the number n0 satisfies | ± 1 − αA − n−a
0 | > 0. Similarly, we have
2
ˆ n + 1| ≤ n−a ≤
P |α̂A
{[E(An )2 E(α̂ˆ − α)2 ]1/2 + α[E(An − A)2 ]1/2 }
| − 1 − αA − n−a |
for n ≥ n0 . Since E(An )2 ≤ 2E(An − A)2 + 2A2 and, as in Equation (A1), the statistic An is a
consistent unbiased estimator of the parameter A, it follows that the statistic θ̂ˆ is a consistent
≤
asymptotically unbiased estimator of the parameter θ .
ˆ ≤ c na and |(2 −
Note that there exist two positive numbers c1 and c2 such that | tan(θ̂)|
1
−1
a
ˆ
2 ) tan((π α̂)/2)|
≤ c2 n . By using the following facts
1
1
x
y
x
y
|Z − Z | ≤ max{Z , Z }|x − y|, | tan(x) − tan(y)| ≤ max
,
|x − y|,
cos(x) cos(y)
(A3)
where Z > 1, we can check that there is a positive constant c that depends on c1 and c2 such that
⎛
⎞2
ˆ)
α̂ˆ
α
tan(
θ̂
tan(θ
)
2
+
2
2
+
2
⎠
(β̂ˆ − β)2 = ⎝
−
ˆ
2 − 2α tan(π α/2)
2 − 2α̂ˆ tan(π α̂/2)
α̂ˆ
≤ cn6a (α̂ˆ − α)2 .
√
Since a < b/2, that is, 6a < 1, it follows that
E(β̂ˆ − β)2 ≤ cn6a E(α̂ˆ − α)2 = O
n6a
n
−→ 0
as n → ∞. This means β̂ˆ is a consistent, asymptotically unbiased estimator of the parameter β.
Note that
√
a
exp{(Bn (a) − λ0 )α̂ˆ + λ0 } ≤ exp √ log n + λ0 ≤ eλ0 na/ b
b
√
and 2a/ b < 1. We can similarly show that γ̂ˆ is a consistent, asymptotically unbiased estimator
of the parameter γ .
1110
D. Han et al.
From Theorem 9.5.1. in Uchaikin and Zolotarev [47], it follows that the statistic E(n) is a consistent, unbiased estimator of the parameter μ̃ = μ(2 − 21/α ). Thus, the statistic μ̂ˆ is a consistent,
asymptotically unbiased estimator of the parameter μ since
E(μ̂ˆ − μ)2 ≤ O(n2a E(α̂ˆ − α)2 ) + O(n2a E(E(n) − μ̃)2 ) = O
n2a
n
−→ 0.
as n → ∞. This completes the proof.
Proof of Proposition 2
By the definition of α̂ˆ we have
6Dn
1
1
1
P |α̂ˆ − α̂| ≥ = P √
−$
≥ , 2 − − 1 ≤ n−a
1 ± n−a
π
2
6Dn /π 2 − 1/2 6Dn
1
1
1
1
+ P $
−$
≥ , 2 − − ≤ n−a
1/4 ± n−a
π
2 4
6Dn /π 2 − 1/2 1
6Dn
1
1
+ P √ − $
− <b .
≥ ,
2
2
b
π
2
6Dn /π − 1/2
Obviously,
1
1
lim P √
−$
≥ ,
n→∞
1 ± n−a
6Dn /π 2 − 1/2 6Dn
1
−a
=0
π 2 − 2 − 1 ≤ n
and
1
1
lim P $
−$
≥ ,
n→∞
1/4 ± n−a
6Dn /π 2 − 1/2 6Dn
1 1 −a
= 0.
π2 − 2 − 4 ≤ n
For the third term, we have
1
1
P √ − $
≥ ,
b
6Dn /π 2 − 1/2 6Dn
1
1
1
1 6Dn
>
−
−
−
b
.
<
b
≤
P
−
π2
π2
2
2 α2 α2
Note that 1/α 2 − b > 0. By Chebyshev’s inequality and (A1), it follows that
2 6Dn
1
1
1
1
1
6D
1
n
P 2 − − 2 > 2 − b ≤ O E
=
O
−
−→ 0,
−
2
2
π
2 α
α
π
2 α
n
as n → ∞. Thus, P(|α̂ˆ − α̂| ≥ ) → 0 as n → ∞.
Journal of Applied Statistics
1111
For θ̂ˆ , θ̂, we have
πAn
ˆ
−a
−a
ˆ
ˆ
ˆ
P |θ̂ − θ̂ | ≥ = P (α̂ − α̂) ≥ , |α̂An | < 1 − n , or |α̂An | > 1 + n
2
π
ˆ n − 1| ≤ n−a
+ P (1 ± n−a − α̂An ) ≥ , |α̂A
2
π
ˆ n + 1| ≤ n−a
+ P (1 ± n−a + α̂An ) ≥ , |α̂A
2
π
≤P
|An ||α̂ˆ − α̂)| ≥ 2
π
ˆ n | ≥ , |α̂A
ˆ n − 1| < n−a
+P
|1 ± n−a − α̂A
2
2
π
ˆ
+P
|An ||α̂ − α̂)| ≥
2
2
π
−a
ˆ
ˆ n + 1| < n−a
+P
|1 ± n + α̂An | ≥ , |α̂A
2
2
π
ˆ
+P
|An ||α̂ − α̂)| ≥
.
2
2
Since
π
ˆ n | ≥ , |α̂A
ˆ n − 1| < n−a = 0,
lim P
|1 ± n−a − α̂A
n→∞
2
2
π
ˆ
−a
−a
ˆ
lim P
=0
|1 ± n + α̂An | ≥ , |α̂A
n + 1| < n
n→∞
2
2
and
π
π
P
|An ||α̂ˆ − α̂)| ≥
≤ [E(A2n )E(α̂ˆ − α̂)2 ]1/2 ,
2
2
it follows that P(|θ̂ˆ − θ̂ | ≥ ) → 0 as n → ∞.
By using Equation (A3) and the fact that both Bn and E(n) are consistent, unbiased estimators
of the parameters B and μ̃, respectively, we can similarly prove that
P |β̂ˆ − β̂| ≥ −→ 0, P |γ̂ˆ − γ̂ | ≥ −→ 0, P |μ̂ˆ − μ̂| ≥ −→ 0
as n → ∞.
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