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

Email: aa4156@wayne.edu

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, ,jn

1

;, exp

jjj

j

fxxIx j

.

Let 1,,

jm

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

XXjm

is the lifetime of the ineffective

component from unit j. Let a > 0 be a pre-specified con-

stant. If j

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

Y.

Furthermore, it is assumed that

,,

jj

1, ,jn

,

are iid random parameters, and that

are possibly

censored from the right by a non-negative random cen-

soring variable

V, where V are iid, with a

common distribution W, and 1,,

n

V

,,

n

VV

1 are independ-

ent of

n1,,

X. Thus,

may not always be ob-

servable. Instead, one can only obser ve

min ,

jj

XVand

jj

XV

. Through the

preceding assumptions,

,,,,, ,

jj j

XYVZ jj j

,

1, ,jn

are iid,

,1,,Vj n

j and

,,,, ,

jjj j1,

Yj

n are mutually independent.

Let

11

1n

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

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