Control Charts are the main tools of Statistical Process Control. They are used for deciding whether a process is statistically stable or not. Much theory and many applications have been developed for the Gaussian (Normal) distribution in this area. However, in real data sets we usually face up nonnormal processes. Consequently, this theory does not apply. In the present paper, we focus attention on the lognormal distribution that can be considered as a special nonnormal case. In particular, we present the Shewhart Control Charts developed up to now, under such distributional assumptions and a new Control Chart based on the CUSUM theory.Control chart, Nonnormality, Shewhart, CUSUM, Average run length, Lognormal

Maravelakis, Petros

Panaretos, John

Psarakis, Stelios

English

Research Papers in Economics

  1 
CONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION 
 
Petros Maravelakis, John Panaretos and Stelios Psarakis 
Department of Statistics 
Athens University of Economics and Business 
76 Patision St., 104 34, Athens, GREECE 
 
 
1. Introduction 
 
Control charts are the tools of statistical process control for detecting a 
change in a process. The most known are the Shewhart control charts that where 
developed for controlling the mean and variance of the distribution our 
observations follow. Shewhart charts perform satisfactorily for detecting large 
shifts in the process. However, for small shifts they had poor results. This fact led 
Page in 1954 to construct CUSUM control chart for detecting small and moderate 
shifts in the process. 
Control charts have found great applications in industry and especially the 
ones for the normal distribution. Nevertheless, a process may not follow the 
normal distribution-it may be positive or right skewed. Morrison (1958), Joffe and 
Sichel (1968) and Kotz and Lovelace (1998) provided examples of processes 
that follow the lognormal distribution. 
Morrison (1958), Kotz and Lovelace (1998) have developed Shewhart control 
charts for the lognormal distribution. Joffe and Sichel (1968) constructed a 
control chart for testing sequentially arithmetic means from a lognormal 
population. In this paper, CUSUM procedures are developed for the lognormal 
for testing the mean and variance. In addition, a chart based on the sequential 
probability ratio test (SPRT) is illustrated.  
 
 
 
   2 
2. Lognormal  distribution 
 
A positive random variable X is said to be lognormally distributed with two 
parameters µ and  2 σ  if Y=lnX is normally distributed with mean µ and variance 
2 σ . The probability density function is  
 
()
 



 



≤
>






− =
−
0 x , 0
0 x , exp ) x ( f 2
2
σ 2
µ x ln
π 2 x σ
1
 
 
There are four forms of the lognormal  distribution  as  Crow  and  Shimizu        
(1988) present in their book: 
(a) The two-parameter distribution Λ(µ,  2 σ ) that describe positive skew data with 
a lower threshold of zero. 
(b) The three-parameter distribution Λ(τ, µ,  2 σ ) that describe positive skew data 
with a lower threshold of τ. This form can be reduced to the two-parameter 
case by the transposition Y=X-τ. 
(c) The three-parameter distribution Λ(θ, µ,  2 σ ) that describe negative skew data 
with an upper threshold of θ. This form can be reduced to the two-parameter 
case by the transposition Y=θ-X. 
(d) The four-parameter distribution Λ(τ, θ, µ,  2 σ ) that describe skew data with 
upper and lower thresholds of θ and τ. 
 
In the following, we will deal with the two-parameter distribution. The three-
parameter may occur in practice but with the suitable transformations as it is 
shown in (b), (c) it reduces to the two-parameter one. The four-parameter is not 
likely to have applications in process control. 
 
 
   3 
3. CUSUM  Chart 
 
3.1.  The general exponential family. 
 
The probability density for any member of the exponential family with a single 
parameter θ can be written as: 
{} ) θ ( d ) y ( c ) θ ( b ) y ( α exp ) θ | y ( f + + =  
where θ is the parameter of the distribution and Y is the random variable. The 
joint density for a random sample of Y, where Y is a member of the exponential 
family, is given by: 
f(y|θ)= 







+ +∑ ∑
= =
) θ ( nd ) y ( c ) θ ( b ) y ( α exp
n
1 i
i
n
1 i
i . 
Suppose we want to test whether the process has gone from an in-control 
parameter value  0 θ  to an out-of-control value  1 θ . We will use Wald’s sequential 
probability ratio test (SPRT) because CUSUM is a sequence of Wald sequential 
tests [Page (1954)]. Then the variables  i Z  are: 
 
{} {} ) θ ( d ) θ ( d ) θ ( b ) θ ( b ) Y ( α ln Z 0 1 0 1 i ) Y ( f
) Y ( f
i
i 0
i 1 − + − = 

 

 =  
The CUSUM scheme is: 
() n 1 n n Z D , 0 max D + = −  
and it gives an out-of-control signal if 
A Dn > . 
Therefore the CUSUM scheme is: 
 
{} () ) θ ( d ) θ ( d ) θ ( b ) θ ( b ) Y ( α D , 0 max D 0 1 0 1 i 1 n n − + − + = −  
and it signals when  
A Dn > . 
Let ) Y ( α X n n =  and 
) θ ( b ) θ ( b
) θ ( d ) θ ( d
0 1
0 1 k
−
−
− = . If  ) θ ( b ) θ ( b 0 1 − >0 then we rescale 
CUSUM by dividing it with this quantity and    4 
( ) k X C , 0 max C n 1 n n − + = +
−
+  
 
where  () ) θ ( b ) θ ( b / D C 0 1 n n − = +  and the CUSUM signals if  + + > h Cn  where 
() ) θ ( b ) θ ( b / A h 0 1 − = + . If  ) θ ( b ) θ ( b 0 1 − <0 then we rescale CUSUM by dividing it 
with this quantity and  
 
( ) k X C , 0 min C n 1 n n − + = −
−
−  
 
where  () ) θ ( b ) θ ( b / D C 0 1 n n − = −  and the CUSUM signals if  − − − < h Cn  where 
() ) θ ( b ) θ ( b / A h 0 1 − = − . 
 
3.2. CUSUM schemes for the lognormal distribution 
 
3.2.1. σ known 
 
Let σ be fixed and known. The above density function is written in the form of 
the exponential family of distributions as follows: 
 
() ()






− + − − − =
2
2
2 2
2
σ 2
µ
σ
µ
σ 2
x ln x ln x ln π 2 σ ln exp ) µ | x ( f 
 
Then α(x)=lnx, b(µ)=
2 σ
µ , d(µ)= 
2 σ 2
µ − . Therefore 
) θ ( b ) θ ( b
) θ ( d ) θ ( d
0 1
0 1 k
−
−
− = =
2
µ µ
) µ µ ( 2
µ µ 0 1
0 1
2
0
2
1 +
−
−
=  
 
The CUSUM scheme for the mean when  0 1 µ µ >  will be: 
 
0 C0 = +  
) k X ln C , 0 max( C n n 1 n − + = + +
+  
+ k=
2
µ µ 0 1+
 
   5 
The scheme signals when  + + > h Cn  where  + h  is chosen to give a specified ARL. 
  The CUSUM scheme for the mean when  0 1 µ µ <  will be: 
0 C0 = −  
) k X ln C , 0 min( C n n 1 n − + = − −
+  
− k=
2
µ µ 0 1+
 
The scheme signals when  − − − < h Cn . 
 
3.2.2. µ known 
 
Let µ be fixed and known. The density function of the lognormal is written in 
the form of the exponential family of distributions as follows: 
() ()






− − − =
−
2
2
σ 2
µ x ln σ ln π 2 x ln exp ) σ | x ( f 
Then α(x)=() 2 µ x ln − , b(σ)=
2 σ 2
1 − , d(σ)=-lnσ. Therefore 
) θ ( b ) θ ( b
) θ ( d ) θ ( d
0 1
0 1 k
−
−
− = =
2
0 σ 2
1
2
1 σ 2
1
0 1 σ ln σ ln
−
−
−  
The CUSUM scheme for the variance when  0 1 σ σ >  will be: 
0 C0 = +  
() ) k µ X ln C , 0 max( C 2
n n 1 n − − + = + +
+  
2
0 σ 2
1
2
1 σ 2
1
0 1 σ ln σ ln
k
−
−
− =  
The CUSUM scheme for the variance when  0 1 σ σ <  will be: 
0 C0 = −  
() ) k µ X ln C , 0 min( C 2
n n 1 n − − + = − −
+  
2
0 σ 2
1
2
1 σ 2
1
0 1 σ ln σ ln
k
−
−
− =    6 
If one looks at the CUSUM schemes for the variance, he will notice that it is 
affected by changes in µ as well as by changes in σ. However, we monitor µ 
separately, therefore we are able to distinguish changes in the CUSUM chart for 
the variance caused by µ. 
 
3.3. A general result 
 
In the lognormal CUSUM for monitoring σ, I have shown that the lognormal is 
a member of the exponential family with parameters α(x)=() 2 µ x ln − , b(σ)=
2 σ 2
1 − , 
d(σ)=-lnσ. Let Z=() 2 µ x ln − . Then the distribution of Z is: 
 
[] () [ ] [] [] = ≤ − ≤ − = ≤ − = ≤ − = ≤ = z µ X ln z P z µ X ln P z µ X ln P z Z P ) z ( F 2
Z  
= [ ] ( ) ( ) z µ F z µ F z µ X ln z µ P X ln X ln − − + = + ≤ ≤ − .  
 
Therefore 
( ) ( )
() ()
z π 2 σ
exp
σ 2
µ z µ
π 2 σ
1
σ 2
µ z µ
π 2 σ
1
z 2
1
X ln dz
z µ d
X ln dz
z µ d
Z
2 σ 2
z
2
2
2
2
exp exp
) z µ ( f ) z µ ( f ) z ( f
 


 


−
− − − +
− +
=
=
 



 











− −








− =
= − − + =
         
           
 
But this is a  ( ) 2
2
1 σ 2 , Γ  distribution with density function  
 
()
) α ( Γ
β / x exp x β 1 α α
) β , α   ; x ( f
− − −
=  
 
Consequently, ~ ) x ( α ( ) 2
2
1 σ 2 , Γ . 
 
   7 
3.3.1. Monitoring  2 σ  in the lognormal CUSUM 
 
It is known that if Y~Γ(α,β) and Z=aY then Z~Γ(α,aβ). Therefore 
2 σ
1 α(x)~ ( ) 2 , Γ
2
1 . If we want to monitor  2 σ  we just have to monitor for a scale 
change of the gamma distribution from β to  + β . Hawkins and Olwell (1998) have 
constructed CUSUM schemes for the gamma distribution. 
 
3.3.2. Monitoring σ in the lognormal CUSUM 
 
From the definition of the lognormal distribution, it is known that  
lnx~N(µ,  2 σ ) and as a result  ) 1 , 0 ( Ν ~
σ
µ x ln − . Hence monitoring σ is just the case 
of monitoring for σ in the standard normal distribution. Hawkins and Olwell (1998) 
have also constructed CUSUM schemes for the normal distribution. 
 
3.4. Wald’s sequential probability ratio test (SPRT) 
 
CUSUM control charts, as Johnson (1961) pointed out, are roughly equivalent 
to the sequential probability ratio test (SPRT). SPRT, as it will be shown in the 
sequel, leads to an acceptance plan. This acceptance plan has been used for 
determining the in and out-of-control limits in CUSUM procedures. 
Suppose that we take a sample of m values  m x   ,..., 2 x   , 1 x , successively, from 
a population f(x,θ). Consider two hypotheses about θ,  0 θ θ : 0 Η =  and 
1 θ θ : 1 Η = . The ratio of the probabilities of the sample is: 
 
∏
∏
=
=
=
m
1 i
0 i
m
1 i
1 i
) θ   , x ( f
) θ   , x ( f
m L 
   8 
We select two numbers A and B, which are related to the desired α and β errors 
in a way we will explain later. The sequential test is set up as follows: 
 
         as long as B< m L <A we continue sampling 
at the first i that  m L ≥A we accept  1 Η  
at the first i that  m L ≤B we accept  0 Η  
 
An equivalent way for computation is to use the logarithm of  m L . Then, the 
inequality becomes: 
 
logB <∑
=
m
1 i
) 1 θ   , i x ( f log -∑
=
m
1 i
) 0 θ   , i x ( f log < logA 
 
This family of tests is referred to as sequential probability-ratio tests. 
If  i z= l o g {} ) 0 θ   , i x ( f / ) 1 θ   , i x ( f  then sampling terminates when  
 
Α log i z ≥ ∑  
or 
Β log i z ≤ ∑  
 
The  i z ’s are independent random variables with variance, say,  2 σ >0. Obviously 
∑
=
m
1 i
i z  has a variance m 2 σ . As m increases the dispersion becomes greater and 
the probability that a value of ∑ i z  will remain within the limits logB and logA 
tends to zero. The mean z tends to a normal distribution with mean  2 σ /m and 
therefore the probability that it falls between (logB)/m and (logA)/m tends to zero. 
Consider a sample for which  m L  lies between A and B for the first n-1 trials 
and then  m L ≥A at the nth trial, so we accept  1 Η  (and reject  0 Η ). By definition, 
the probability of getting such a sample is at least A times as large under  1 Η  as   9 
under  0 Η . The probability of accepting  1 Η  when  0 Η  is true is α and that of 
accepting  1 Η  when  1 Η  is true is 1-β. Therefore: 
 
1-β ≥Aα 
or 
A
α
β 1−
≤ .                                                      (3.4.1) 
 
Similarly, we see that when we accept  0 Η  
 
) α 1 ( B β − ≤  
or 
α 1
β
B
−
≥ .                                                      (3.4.2) 
 
Wald (1947) showed that for all practical purposes the above inequalities hold as 
equalities. Thus: 
 
A
α
β 1−
=  and 
α 1
β
B
−
= . 
 
Suppose that a
α
β 1−
=  and b
α 1
β
−
=  and that the true errors of first and second 
kind for the limits a and b are  ' α  and  ' β . Then, from (3.4.1): 
 
β 1
α
a
1
' β 1
' α
−
= ≤
−
 
and from (3.4.2) 
α 1
β
b
' α 1
' β
−
= ≤
−
. 
Therefore 
β 1
α
β 1
) ' β 1 ( α
' α
−
≤
−
−
≤                                               (3.4.3)   10 
 
α 1
β
α 1
) ' α 1 ( β
' β
−
≤
−
−
≤                                               (3.4.4) 
Furthermore 
) ' α 1 ( β ) ' β 1 ( α ) α 1 ( ' β ) β 1 ( ' α − + − ≤ − + −  
or 
β α ' β ' α + ≤ + .                                                  (3.4.5) 
 
In practice α and β are small. From (3.4.3) and (3.4.4) we see that the amount 
that α’ can exceed α or β’ exceed β is negligible. In addition, from relation (3.4.5) 
we see that either α’≤α or β’≤β. Therefore the use of α and β in place of Α and Β 
can only increase one of the errors and only by a very small amount. 
 
3.4.1. Application of the SPRT to the CUSUM for the lognormal distribution 
 
Based on the theory of Wald we will derive a CUSUM scheme for the 
parameter µ of the lognormal distribution. Define  0 L  and  1 L  to be the likelihood 
functions of the random sample under  0 Η  and  1 Η , respectively, and let the 
likelihood ratio  0 1 L L be denoted by Λ. That is: 
 
() ()








− −








= ∑ ∏
=
−
=
n
1 i
2
1 i
σ 2
1
1 n
1 i
n
i 1 µ X ln exp π 2 σ X L
2  
() ()








− −








= ∑ ∏
=
−
=
n
1 i
2
0 i
σ 2
1
1 n
1 i
n
i 0 µ X ln exp π 2 σ X L
2  
and Λ is 
() ()
 



 











− − − − = ∑ ∑
= =
n
1 ι
2
0 i
n
1 ι
2
1 i
σ 2
1 µ X ln µ X ln exp Λ
2 . 
 
We assumed that σ is known which is reasonable if control charts have been 
used to monitor the process for a period. Define A and B as follows: 
   11 
A
α
β 1−
=  and 
α 1
β
B
−
= . 
 
We accept the lot if Λ≤Β. We reject the lot if Λ≥A. If B<Λ<A we continue 
sampling. After some calculations, we conclude that 
 
(a) Accept the lot if Y≥ 1 Y =nS+ 1 h  
(b) Reject the lot if Y≤ 2 Y =nS+ 2 h  
(c) Continue to sample if  2 Y <Y< 1 Y  
 
where Y=∑
=
n
1 i
i X ln , S=
2
µ µ 1 0 +
, 
0 1
α 1
β 2
µ µ
ln σ
1 h
−


 


−
=  and 
0 1
α
β 1 2
µ µ
ln σ
2 h
−


 

 −
= . 
 
In the above three cases  1 h  and  2 h  are parallel lines referred as acceptance and 
rejection lines respectively. The results were developed for the case of  1 µ> 0 µ 
meaning that  1 µ  is an upper specification limit. In the same way, we can derive 
results for the case  1 µ< 0 µ.  
 
4. ARL performance 
  
Control charts are usually evaluated using the average run length (ARL). ARL 
is the average number of samples until signal. A computer program was written 
for the computation of the ARL based on the theory of Brook and Evans (1972). 
In this program the values of k and h are provided and we receive the value of 
the ARL. In the following several results of the program are presented and also 
the effect of the decision interval h is illustrated. The symbols µ and σ are the 
parameters of the lognormal distribution. 
 
 
 
   12 
µ=1  σ=0,6 
K  h  ARL 
11 5 632.6 
11 4 449.8 
11 3 316.1 
10.5 5  532.3 
10.5 4  376.6 
10.5 3  263.1 
µ=0  σ=1 
K  h  ARL 
11 5 358.5 
11 4 294.8 
11 3 240.2 
 
5. Conclusion 
 
In this paper, a new CUSUM control chart for the lognormal distribution is 
presented. The close relationship of the CUSUM and the SPRT is also shown in 
detail. Various versions of the new chart for monitoring σ are illustrated. Finally,  
a computer program for comparing different control charts using average run 
length (ARL) is provided. 
 
References 
 
Brook, D. and Evans, D.A. (1972). An Approach to the Probability Distribution of 
CUSUM Run Length, Biometrika, 59, 539-549. 
 
Crow, E.L. and Shimizu, K. (1988). Lognormal distribution, Theory and 
Applications, Marcel Dekker, Inc., New York and Bassel. 
 
Hawkins, D.M. and Olwell, D.H. (1998). Cumulative sum charts and charting for 
quality improvement (Statistics for engineering and physical science), Springer-
Verlag. 
   13 
Joffe, A.D. and Sichel, H.S. (1968). A chart for sequentially testing observed 
arithmetic means from lognormal populations against a given standard, 
Technometrics, 10, 605-612. 
 
Johnson, N.L. (1961). A Simple Theoretical Approach to Cumulative Sum 
Control Charts, Journal of the American Statistical Association, 835-840. 
 
Kotz, S. and Lovelace, C.R. (1998). Process capability indices in theory and 
practice, Edward Arnold. 
 
Morrison, J. (1958). The lognormal distribution in quality control, Applied 
Statistics, 7, 160-172. 
 
Page, E.S. (1954). Continuous Inspection Schemes, Biometrika, 41, 100-115. 
 
Wald, A. (1947). Sequential Analysis, John Wiley, New York. 

Control Charts for the Lognormal Distribution

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics