Applied Mathematics, 2013, 4, 1706-1708
Published Online December 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.412232
Open Access AM
Comments on “Average Life Prediction Based on
Incomplete Data”
Tachen Liang
Wayne State University, Detroit, USA
Received August 25, 2013; revised September 25, 2013; accepted October 6, 2013
Copyright © 2013 Tachen Liang. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We comment on the correctness of the article “Average life prediction based on incomplete data” by [1] (Applied
Mathematics, Vol. 2, pp. 93-105).
Keywords: Average Life Prediction; Censored Data
1. Introduction
Tang et al. [1] studied average life prediction based on
incomplete data assuming the prior distribution being
unknown. However, the paper contains serious errors and
the concluded results are incorrect. We shall point out the
errors in the following. To do so, we first describe the
considered statistical model as below.
Suppose that there are n different manufacture units
possessing the same technology and regulations. For a
known integer m > 0, a sample of m components are se-
lected from unit j, and put on life test at time t = 0, for
each . It is assumed that the lifetime of a
component arising from unit j follows a two-parameter
exponential distribution having probability density
1, ,jn
   
1
;, exp
j
jjj
j
fxxIx j



   .
Let 1,,
j
jm
X
X

1
min ,
denote the lifetimes of the m compo-
nents. The life test experiment will be terminated if one
of the m components fails. Thus,

,
j
j
XXjm is the lifetime of the ineffective
component from unit j. Let a > 0 be a pre-specified con-
stant. If j
X
X
a, then a second round sample is carried,
at which we sample one more component from unit j, and
denote its unknown lifetime by
j
Y.
Furthermore, it is assumed that

,,
jj

1, ,jn

,
are iid random parameters, and that
j
X
are possibly
censored from the right by a non-negative random cen-
soring variable
j
V, where V are iid, with a
common distribution W, and 1,,
n
V


,,
n
VV
1 are independ-
ent of

n1,,
X
X. Thus,
j
X
may not always be ob-
servable. Instead, one can only obser ve


min ,
j
jj
Z
XVand

j
jj
I
XV  
. Through the
preceding assumptions,


,,,,, ,
j
jj j
XYVZ jj j



,
1, ,jn

 are iid,


,1,,Vj n
j and




,,,, ,
jjj j1,
X
Yj  n are mutually independent.
Let

11
1n
j
j
XaY
j
SI
n

 and

a
21
1n
j
j
SIX
n


. 12
SSS

is the average life of
the second round sample. Tang et al. (2011) attempted to
predict 12
SSS

based on the data


,,1,,
jj
Z
jn 
. Let








11
11
1
ii
i
ii
i
IZ a
Snn WZ
a
m
Z
1
1
1
1
nn jj
iji j
n
i
i
Z
WZ
IZ
Za
nW










 






,



21
1
1
ni
ii
IZa
SnWZ
i




, and 12
SSS.
Tang et al. [1] proposed using S to predict . S
Tang et al. [1] claimed the following results:
, 1,2.
jj
ES ESj
 

  (1)
(see (2.8)-(2.9) of Tang et al. [1]).
Based on the identity property of (1), and some addi-
tional conditions, Tang et al. [1] claimed in their Theo-
rem 1 that 0SS

 in probability as n. They

T. C. LIANG 1707
further apply the identity property of (1) and Theorem 1
to claim their Theorem 2.
We now point out the errors of Tang et al. [1] as fol-
lows.
1) The sampling scheme is not well defined. Since
random censorship model is considered, we may be un-
able to observe the exact value of
j
X
. In case that
0
j

 and j
Z
a, it is possible that j
X
a or
j
X
a. In such a situation, shall we carry the second
round of sampling and sample one more component from
unit j? This is not discussed in the paper.
2) For the distribution W known case, unfortunately,
the claimed identity that


11
 is incorrect.
The error is pointed out as follows. In (2.8) of [1], it is
stated that (see Equation (2)):
ES ES



In (2) (or (2.8) of Tang et al. [1]), the first equality
is not true, where the notation

,

 is abused. The
correct computation is given below. Note that for each j,
given


,
j
j

,
j
X
follows a double exponential dis-
tribution having pdf


;,
j
j
fx

, and


,,,,,
j
jjjjj
XVZ



, 1, ,jn

 are iid. Using the iid
property, we can obtain E qua tion (3 )
Note that the expression of (3) is different from that of
(2). So, we see that


11
ES ES



This type of computational errors also occur at (5.3),
(5.4), (5.7), (5.8) and (5.9) of Tang et al. [1], where
2
1
ES



, 11
ESS



 and 2
1
ES



 were calculated.
Based on the preceding discussion, the correctness of
Theorem 1 in Tang et al. [1] is doubtful.
3) Tang et al. (2011) then applied the result of their
Theorem 1 for the case where the distribution W is un-






 
 
 


 

 
 
11
111
1
1,,
11
1
1,
1
d,d d,d
11
1d,
1
nn jj ii
iji i
j
xv xv
ZIZ a
ES EEE
nn WZ
WZ
ZaIZa
mEE WZ
Ix a
x
EWvlxxWvlx
Wx Wx
xaIxa
mE Wvlx
Wx

 

 





x

 

 


 
 
 








 

 













 

 


 


1 ,
d
1
exp exp
xv x
ma mma
EE
mm


 









   
 
 
  



 


S
(2)





 
 
 


 

 
11
111
11
1
()
1,,
11
1
1,
1
d,d d,d
11
1
nn jj ii
jj ii
iji i
j
jj ii
xv xv
ZIZ a
ES EEE
nn WZ
WZ
ZaIZa
mEE WZ
Ix a
x
EWvlxxWvlxx
Wx Wx
xaIx
mE

 

 







 


 
 

 






 
















 

 



 




 
,
, ,
,
d,d
1
1
exp exp
1
exp exp
ii
jj
xv
ji
ji
ji
ji
aWvlx x
Wx
ma mma
EE
mm
ma mma
EE E
mm

 













 




 
   

 
  
 
 
 

 
 
 
     

  
 


 


 
,, ,
1
exp exp
mam ma
EE E
mm
 











 

 
 
   


 
 





(3)
Open Access AM

T. LIANG
1708
known, and claimed their Theorem 2. However, since
Theorem 1 is dubious, the correctness of Theorem 2 is
also doubtful.
REFERENCES
[1] T. Tang, L. Z. Wang, F. E. Wu and L. C. Wang, “Average
Life Prediction Based on Incomplete Data,” Applied Ma-
thematics, Vol. 2, 2011, pp. 93-105.
http://dx.doi.org/10.4236/am.2011.21011
Open Access AM