International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 685

ISSN 2229-5518

The Distribution of the Average run Length (ARL)

of the Cusum Control Charts for Binomial Parameters when Observation are Poisson Distributed

Edokpa I.W., Osabuohien-Irabor Osa., Ogbeide E. EMichael

Abstract — In this study, the average run lengths (ARL) of the Cumulative Sum (CUSUM) charts for the binomial distribution when the underlying distribution is Poisson were obtained. It is observed that the parameters of the ARL changes considerably for a slight changes in the parameters o the underlying distribution.

Index Terms — Cumulative Sum chart, average run length, binomial distribution, Poisson distribution, distribution of ARL

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1.0 INTRODUCTION

Cumulative sum (CUSUM) procedures are often used to monitor the quality of manufacturing processes. The
major objective is to identify persistent causes of variation
If 𝑆𝑖 is plotted against the sample on a chart or recorded in tabulation, a change in the direction of the CUSUM path is
usually taken to indicate a change in process level. For a

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in the process average. This ability is attributed to the fact
that they have memory as they are based on successive sums of the observations minus a constant. Generally, we can say that CUSUM charts are able to detect small to moderate shifts whereas Shewhart charts are able to detect large shifts. A major problem in the applications of the Statistical Process Control (SPC) to the quality of materials produced by its process is that of making sure that the proportion of defective produced does not exceed the specified limit. A valuable tool of achieving this goal was proposed by Page [1]. The principal feature of the CUSUM control chart is that it cumulates the difference between
the observed value 𝑋𝑖 and the pre-determined target or reference value 𝑘, the CUSUM 𝑆𝑖 of the deviation of 𝑋𝑖 from 𝑘 is given in equation 1.1
one-sided CUSUM procedures in which the continuous or
discrete random variables 𝑋1 , 𝑋2 , 𝑋3 , … given in equation
(1.1) are taken successively and the cumulative sums
𝑆𝑖 = max(0, 𝑆𝑖−1 + 𝑋𝑖 ) , 𝑖 = 1,2,3, … 1.2
are formed, where 𝑆0 = 𝑤 (𝑤 ≥ 0). The process is
considered to being in - control until the first stage,
N, such that 𝑆𝑁 ≥ ℎ (0 ≤ 𝑤 ≤ ℎ)
The random variable N, referred to as the run length of
the procedure and it is defined as the stage at which
sampling terminates and necessary action is taken. We have to state here that in the case of standard normal data
with 𝑘 = 3 𝑎𝑛𝑑 ℎ = 3 we end up with the classical
Shewhart chart.
𝑆𝑖 = ∑𝑛
(𝑋𝑖 − 𝑘)

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1.10
The applications of CUSUM charts have received considerable attentions. The recognition of a CUSUM control scheme as a sequence of Wald Sequential

Edokpa I.W. is a senior Lecturer with a doctorate degree in statistics,

Specializing in Quality Control from Ambrose Alli University, Ekpoma, Nigeria. PH: +2348035787519. E-mail: wazirip1@gmail.com

Osabuohien-Irabor Osa. holds a Masters degree in Applied Statistics from Nnamdi Azikiwe University, Nigeria, and currently pursuing a doctorate degree in Statistics with specialty in Time Series Analysis.

He is a Lecturer at Ambrose Alli University, Ekpoma, Nigeria.

E-mail: osabuohien247@gmail.com, PH: +2348034861146

Ogbeide E.M. is a Lecturer at Ambrose Alli University, Ekpoma, Nigeria,

and currently pursuing a doctorate degree program in Industrial

Mathematics with an option in Statistics from the University of Benin,

Nigeria. Email: ogbeideoutreach@yahoo.com, PH: +2348035910189

Probability Ratio Test (SPRT) allows the optimality
properties of CUSUM procedures to be developed.
Johnson and Leone [2] gave a discussion on the CUSUM procedures using the relationship between SPRT’S and CUSUM. They also constructed the CUSUM charts for controlling the means of a Binomial and Poisson distribution. Lorden [3] gave the asymptotic optimality of CUSUM procedures for detecting a change in distribution.

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Kennett and Pollack [4] showed then superiority of a
CUSUM scheme for detecting a rare event over a non –
CUSUM scheme proposed by Chen [5]. Kenett and
Pollack [4] scheme can be improved upon by including the First Initial Response (FIR) feature, see Lucas and crosier[6].
In this paper, our aim is to develop a simple procedure of obtaining the ARL of a CUSUM control chart for Binomial
parameters when the observations are Poisson distributed using a method similar to [2] and compare the changes in the distribution of the ARL with the changes in the values of the CUSUM parameters as the sample sizes varies graphically.

2.0 The Average Run Length (ARL) of the CUSUM Control Chart

Cusum control schemes are usually evaluated by calculating their average run length (ARL), the ARL is defined as the average number of samples taken before an out- of- control signal is obtained. The ARL should be
large when the process is at the desired level and small
Brook and Evans [7] were the first to propose the new
method for computing the ARL based on a Markov chain.
This method applies to both discrete and continuous
variables.
The Markov Chain approach begins by approximating the
problem of obtaining the average run length (ARL) and
then obtains an exact solution to the approximate
problem. Champ [12] compared integral and Markov
chain approaches. They propose the integral equation approaches are more accurate than the Markov-chain approach, but it less versatile. The Markov-chain can calculate both the ARL and the distribution of the run length. It has also been used to calculate the properties of the modified CUSUM schemes such as the robust CUSUM schemes [6].
Assume that 𝑋1 , 𝑋2 , 𝑋3 , … are independent and identically
distributed random variables that are observed
sequentially. Let 𝑋1 , 𝑋2 , 𝑋3 , … , 𝑋𝑛 have (in-control) distributionfunction 𝐹0 and 𝑋𝑛+1 , 𝑋𝑛+2 , 𝑋𝑛+3 , … , 𝑋𝑘 have (out-of-control) distribution function 𝐹1 where 𝐹0 ≠ 𝐹1
The two distributions are known but the time of change is

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when the process shifts to an undesirable level. The in-
control ARL’s are often approximately closely by the
geometric distribution [7]. For a standard CUSUM, the
ARL distribution is nearly geometric except that there is a
lower probability of extremely short run lengths. When the FIR feature is used, the distribution is nearly geometric except that there is an increased probability of short run lengths due to the head start [7]. There are two majors methods for the computation of the ARL, there are the integral and Markov chain approach. Page[1] used integral equations for the computation of the ARL.
Let 𝐹(𝑥) be the distribution function of a single score
𝑥~𝑁(𝑚, 1),where N(m,1) denotes a normal distribution
with mean m and variance 1.0 and 𝐿(𝑧) be the ARL of the
one sided case, if 𝐿(0) is the ARL with an initial value of
zero. Then, for 0 ≤ 𝑧 < ℎ
assumed unknown.
Many schemes can detect such a change (e.g. Shewhart charts). These schemes are categorised by the expected time until the process signals while it remains in-control (false alarm rate). Among all procedures with the same false alarm rates, the optimal procedure is the one that detects changes quicker. Or we could say that among all procedures with the same in-control expected number of samples until signal, the optimal procedure has the smallest expected time until it signals a change when the process shifts to the out-of-control state.

2.1 Probability of the CUSUM Chart

Let 𝑋1 and 𝑋2 be two independent Poisson variables with parameters 𝜆1 and 𝜆2 respectively. Then the conditional distribution of 𝑋1 given 𝑋1 + 𝑋2 is given as

𝑃(𝑋1=𝑟∩𝑋2=𝑛−𝑟 )

𝑃(𝑋1 = 𝑟Ι(𝑋1 +𝑋2 = 𝑛)) =

=

𝑃 (𝑋 +𝑋 =𝑛)

𝜆

𝑟 𝜆1

1 2

𝑛−𝑟

𝐿(𝑧) = 1 + 𝐿(0)𝐹(−𝑧) + ∫0 𝑓(𝑥 − 𝑧)𝐿(𝑥)𝑑𝑥
2.10
𝑛

1 � �

� . 2.11

𝑟 � �𝜆 +𝜆

𝜆1+𝜆2

Van Dobben and de Bruyn [8] gave a discussion on the
derivation of this equation. Additionally, Wetherill [9]
gave an almost identical relationship but from a
Where 𝑟 = 0,1,2, … , 𝑛, then the mean of and variance of 𝑋1
are given by
𝐸(𝑋 ) = 𝑛𝑝 and variance 𝑉𝑎𝑟(𝑋 ) = 𝑛𝑝𝑞 where 𝑝 = 𝜆1
somewhat different way of thinking. Others that have
dealt with the same problem are Ewan and Kemp [10] and
recently Champ,Rigdon and Scharngi [11] gave a general

1


and 𝑞 = 𝜆2

𝜆1+𝜆

1 𝜆1+𝜆2

2

method for obtaining integral equations used in the evaluation of many control charts.

2.2 Controlling the parameter 𝝀𝟏 where 𝝀𝟐 is unknown.

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Let 𝑋1 , 𝑋2 , … , 𝑋𝑘 be identical, independent distributed
Binomial random variable using the likelihood ratio test to
test the hypothesis 𝐻0 : 𝜆1 = 𝜆0 against the alternative hypothesis𝐻𝑎 : 𝜆1 = 𝜆𝑎 (> 𝝀𝟎 ), where 𝝀𝟐 is assumed known.
The change in the direction of the mean from 𝜆0 𝑡𝑜 𝜆 is
observed, if a point is plotted below line BC. The distance
between points AB is denoted by

ln� 1

𝐿(𝜆𝑎)


= 𝑓(𝑥1 ,…,𝑥𝑘Ι𝜆𝑎,𝜆2) = �𝜆𝑎

𝑆𝑘

𝜆0+𝜆2

, 2.21
𝜎 = 𝛼

𝑛 ln�𝜆𝑎+𝜆2

𝜆0+𝜆2

2.26

𝐿(𝜆0)

where 𝑘 = ∑𝑘

𝑓(𝑥1 ,…,𝑥𝑘Ι𝜆0,𝜆2)

𝑋𝑖

𝜆0

𝜆𝑎+𝜆2

and 𝜑 which is equal or less than angle 𝐴𝐵𝐶 is given by

𝑖=1

𝑛 ln�𝜆𝑎+𝜆2


𝜆0+𝜆2

The region of the SPRT discriminates between 𝐻0 : 𝜆1 = 𝜆0
aganst 𝐻𝑎 : 𝜆1 = 𝜆𝑎 (> 𝜆0 ) has the continuation region of
𝜑 = tan−1

ln�𝜆𝑎

𝜆0

� 2.27
ln � 𝛽 � < 𝑆

ln �𝜆𝑎� + 𝑘𝑛 ln �𝜆0+𝜆2 � < ln �1−𝛽� 2.22

2 1

2 𝜆𝑎+𝜆2

1−𝛼



𝑘 𝜆0

𝜆𝑎+𝜆2 𝛼


𝜎 = ln � � where 𝑔

𝑔 𝛼


= 2𝑛 ln �

𝜆

0+𝜆2

Where 𝛼 =probability of accepting 𝐻𝑎 when 𝐻0 is true)
and 𝛽 =probability (accepting 𝐻0 when 𝐻𝑎 is true). From
2.22 inequalities, we have
Varying the parameter 𝜆𝑎 when 𝜆2 is a constant. From
Johnson[2], the approximate formula for ARL to detect a
shift of 𝜆 from 𝜆0 to 𝜆𝑎 is given to be,

𝜆𝑎

𝝀𝟎 +𝜆2

1−𝛽




𝑆𝑘 ln �𝜆 � + 𝑘𝑛 ln �𝜆 +𝜆 � < ln � 𝛼 � 2.23

0 𝑎

1

2 𝐴𝑅𝐿 = ln 𝛼 , where 𝐴 = 𝐸 �ln 𝑓(𝑥Ι𝜆𝑎) Ι𝐴�

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If 𝛽 tends to zero, then equation 2.23 can be written as

𝐴 𝑓(𝑥Ι𝜆0)


ln�1 �+𝑘𝑛 ln 𝜆𝑎+𝜆2

The numerical values when 𝜆2 is constant (at 0.6), see table
1 for, n=5,10,15,…,60 when 𝛼 varies between 0.01 and
𝑆𝐾 <

𝜆𝑎

𝜆0+𝜆2

2.24

ln�𝜆1

The inequality 2.24 can be re-written as
0.1.The numerical values for varying 𝜆2 were obtained by
the authors and can be given on request.

1 𝜆𝑎+𝜆2

𝑘𝑛

𝜆𝑎 −1



𝑆𝐾 < 𝑙𝑛 ��𝑎� �𝜆 +𝜆

� �ln � ��

𝜆

2.25

0 2 1

On plotting the sum 𝑆𝑘 against the sample number 𝑘, let A
be the last plotted point on the CUSUM chart, and B be
the vertex of the mask if C is obtained be drawing a

perpendicular line to line AB as shown in Fig 1,

A B

𝜑

𝑆𝑘

C

Fig 1: CUSUM Chart

Sample no.

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Table 1: The distribution of the ARL for CU UM chart of Binomial distribution when the underlying distribution is poisson

0.01 0.025 0.05 0.075 0.1

Values of Alpha








0.01 0.025 0.05 0.075 0.1

Values of Alpha

Fig 2.1: Dist. of ARL for varying values of θ for n = 05 Fig 2.2: Dist. of ARL for varying values of θ for n = 15

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0.01 0.025 0.05 0.075 0.1

Values of Alpha











0.01 0.025 0.05 0.075 0.1

Values of Alpha

Fig 2.3: Dist. of ARL for varying values of θ for n = 25 Fig 2.4: Dist. of ARL for varying values of θ for n =60

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CONCLUSION

It is observed that the parameters of the V-mask and ARL changes considerably for a slight shift in the parameters of the distribution and as n increases, the value of the ARL decreases. But for a fixed n, the value
of the ARL decreases as 𝜆𝑎 increases. This result
compares favourably with the result of Ashit and
Anwer [13] and the result of Johnson and Leone [2].

REFERENCES

[1] Page, E.S. (1954). Contionous Inspection Schemes, Biometrika, 41, 100-114
[2] JOHNSON N.L and LEONE F.C.(1962), “Cumulative
Sum Control Charts: Mathematical
Principles Applied to Their Construction and use, part III”, Industrial Quality Control, No 19(1),
pp 22-28
[3] LORDEN G. (1971).”Procedures for Reacting to a Change in Distribution”. Annals of Mathematical Statistics 42 pp 1887-1908.
[4] KENNETT, R. and POLLACK,M. (1983) “On a
Sequential Detection of a Shift in the
probability of a Rare Event”. Journal of the
American Statistical Association 78,pp389-395
[5] CHEN R(1978).” A Surveillance System for Congential Malformations”. Journal of the American Statistical Association 73,pp 323-327.
[6] LUCAS, J. M AND CROSIER, R. B (1982), “Fast Initial Response for CUSUM Quality Control Schemes”, Technometrics Vol. 24, pp 199 – 205.
[7] BROOK, D. AND EVANS, D.A. (1972). “An Approach to the Probability Distribution of Cusum Run Length”, Biometrika, 59, 539-549.
[8] VAN DOBBEN DE BRUYN, C.S. (1968). “Cumulative Sum Tests: Theory and Practice”. Griffin, London, United Kingdom.
[9] WETHERILL, B.G. (1977). “Sampling Inspection and
Quality Control”. Chapman and
Hall, New York.
[10] EWAN, W.D. and KEMP, K.W. (1960). “Sampling
Inspection of Continuous Processes
With No Autocorrelation Between Successive
Results”, Biometrika, 47, 363-380.

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[11] CHAMP, C.W., RIGDON, S.E. and SCHARNAGL,
K.A. (2001). “A Method for Deriving
Integral Equations Useful in Control Chart
Performance Analysis, Nonlinear Analysis”, 47,
2089- 2101.
[12] CHAMP C.W and RIGDON S.E.(1991). “The performance of exponentially weighted moving Average with estimated parameters. Technometrics
43,156-167.
[13] ASHIT B,CHAKRABORTY and ANWER KHURSHID: (2011) Cumulative Sum Control Charts for Binomial Parameters when the underly distribution is Poisson(2011) Revista Investigacion Operational Vol 32 no 1 pp 12-19

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