797
Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 797-804
BOUNDS FOR THE MEAN SQUARE ERROR OF RELIABILITY ESTIMATION
FROM GAMMA DISTRIBUTION IN PRESENCE OF AN OUTLIER OBSERVATION
M.E. GHITANY and W.H. LAVERTY
Department of Mathematics
University of Saskatchewan
Saskatoon, Saskatchewan
Canada STN OW0
(Received August 6, 1987 and in revised form December 5, [988)
In this paper we discuss the behavlor of the statistic
ABSTRACT.
minimum variance unbiased (UMVU)
with unknown scale
R(t).
as
a
the uniformly
estimate for the reliability of gamma distribution
parameter a when an outlier observation is
outlier effect on c,
(t),
Given the
present.
we determine bounds for the mean and mean square error (MSE) of
A semi-Bayeslan approach is discussed when the outlier effect on
prior distribution of
random variable having a
beta
is treated
of
Results
type.
the
exponential distribution (Sinha [I]) are given as particular cases of our results.
KEY WORDS AND PHRASES.
UMVU estimation, gamma distribution, reliability function,
outlier observation, confluent hypergeometric series.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
62F33.
INTRODUCTION.
be such that n-I of them are
(XI,X2,...,X n)
Let the independent random variables
distributed as
f(x:o)
[(N
I)’
q(’]-I
e
.-’x/
(
x
N-I
x
>
0,
o"
>
(1.1)
0,
where N is a natural number, and one of these random variables Is distributed as
f(x; (;/(*)
while each
X.i has
[(N
I)’
(o/ooN] -I
a priori probability
e
x
N-I
x
>
0,
0
<
< I,
I/n of being distributed as (1.2).
context of outlier studies the model (I.I) is known as the "homogeneous
case".
(1.2)
In the
M. E. GHITANY AND W. H. LAVERTY
798
The
reliability
probability law f(x;
a)Is
f
R(t)
time"
"mission
at
of
t
a
system
llfe
whose
follows
the
given by
f(x); o) dx
t
e
-t/e
(t/)
k
(1.3)
k!
k=O
Basu [2] and Nath [3], considering different approaches, obtained the unique UMVU
R(t), namely,
estimate of the reliability function
N-I
(t)
J--0
where
Aj
I(I
(t/s) N-j
t/s) (n-l)N+j,
(nN-1)
(N-j-l)! [n-l]N+J)!
and
X
s
0,I,...,N-I,
having p.d.f
i
[(nN-l)’
fl(s;)=
The
(1.4)
s,
t
nN]-1
e
-s/o
s
nN-l,
s
>
(I.6)
0.
finding UMVU estimate for the reliability function from the gamma
problem of
distribution
r()
f(x;;,)
#’]-I
e
-x/
x
-I
x
>
O,
>
0, o
>
0
with unknown parameters ,has not yet been solved.
2.
VARIANCE OF
(t),
HOMOGENEOUS CASE.
Since the second moment around the origin of R(t) is
E[(t) 12
t
we find that
I
E[(t) 12
where for any v
L (t)
v
[(t) ]2 fl(S; c0
ds,
2
A.Ak
3
0j ekN-
( J+k)/2 (t)
0
[(nN-l)!
onN] -I
2(N-v-I)
t
f
e
-s/
Os-(nN-l)
(s
t)
2[(n-l)N+v] ds. (2.2)
t
The integral I (t) can be simplified as follows
v
Iv(t)
where
l(1)(t)
v
(2)
(t)
v
I
.
(1)(t)
v
+
nN-1
r=O
B
.
r:
1
(2)(t)
v
o)u
v(t) 0f e-(t/ (l+u)-(nN-r-1) du
2[ (n- l) N+v
rffinN
B
r:v
(t)
f e -(t/ )U(l+u)-(nN-r-1)du,
0
(2.3)
(2.4)
BOUNDS FOR THE MEAN SQUARE ERROR OR
RELIABILITY ESTIMATION
799
with
B
r’v
(t)
(2[
(2[(n-l)N + v]))
(n-|)N + v
r)!
(n--f)-!-(-l)r
e
-t/o (t/ ) nN
for every r --0, l,...,2[(n-l)N+v].
A direct simplificat[on of the expressions in (2.3) and (2.4) gives us
I
e
t/a
nN-r-2
nN-3
(If(t)
v
[
[.
r=O
k--I
(-t/
l)nN-r-2
B
r: v
.
[
r=nN
whe re
-Ei(-Y)
exponent [al
e
-Z
z
nN- r-k-2
BnN_l:v (t)
(t/
)-I
(t/o) -( r-nN- k+2)
k!
(2.6)
(2.7)
Idz,
(2.8)
Now,
function.
integral
Ei(-t/ o) +
(r-nN+
Br:v(t)
k=O
[(k-l)! (-t/)
BnN_2:v (t)
2[(n-I)N+v] r-nN+l
l(2)v (t)
the
(nN- r-2)
Ei(-t/ )}
a nd
is
(t)
var[(t)]
Eft(t)] 2
R2(t)
can
be
computed.
3.
MSE((t) ),
BOUNDS FOR
NONHOMOGENEOUS CASE.
For the nonhomogeneous case
[t can be shown that
the p.d.f of s[n this case is
given by
N-I
-n)N e- s/o s nN-I
(a
fct(S: )
r= 0
whe re
D
and
(N-l-r)! r! ([n-l]N-l)! [(n-l)N+r]
r
IF1(.;.;.)
(The notations
[
(;m:z)
k=O
(B)k
(-l)r
s>o,
(3.1)
(3.2)
r=0,1,...,N-l,
Kummer’s confluent hypergeometrlc series, i.e.
is the
IFI
and
Dr F1 (l;(n_l)N+r+l;(l_ct)_s,
(U)k
k
z
(m)
k!"
k
(3.3)
are shifted factorials defined by
()k
( + I)...( + k
I)
(B)O=I)
In particular s
implies
N-I
[
and we get
D
r=O
fl (s: q)
as given by (1.6).
f (s;o), the
final
r
F
(l:(n-1)N+r+l;O)
(nN-l)
.N..]-I
e
(nN-l)
s
nN-I
s> O,
Although MSE(R(t) a) can now be found explicitly by using
result
is
determine bounds for MSE(R(t) ).
not
of
practical
form.
Therefore,
our
For this purpose we consider the c.d.f
aim
is
to
800
M. E. GHITANY AND W. H. LAVERTY
F (n;o)
Pr(s>,11;c
where s is distributed as in (3.1).
Fa
o)
Fl(n;
It can be shown that
q
>
0.
(3.4)
s
>
O,
(3.5)
(; o),
It follows that
f (s;),
f (s;o)
and consequently
E JR(t)
(3.6)
R(t)
.
At the same time we have
m [(t)]
N--1
-n)N
(a
D
r=0
-n
(o
r
(t)
N-
)N
D
r-O
r
-s/ o S nN-I
IFI(I ;(n-|)N+r+l;(l-) )ds
(l;(n-l)N+r+l;(l- ).._ (t) e- s/ osnN- Ids
t
F
e
t
(3.7)
L(a,t) R(t),
where
N-I
L(,t)--N
D
(nN- l)!
r=O
IFIII;Cn-L)N+r+L;(I-).
r
(3.8)
Using (3.6)and (3.7), we obtain
E JR(t)]
L(,t) R(t)
"RCt).
(3.9)
By similar arguments as before, it can be shown that
Since
MSE(R(t)Ja)
E
^
LCa,t) E[R(t)]
El(t)]2
E[(t)]2
L(a, t)
[(t)]2
2 R(t) E
RZ(t)
MSE(R(t)Ia
2
E[(t)]2.
[(t)]
+
(3.10)
R2Ct)-
E[R(t)I
we finally obtain
2
[2 L(a,t)
II
R2(t)
(3.11)
2
^
where R(t), E[R(t)]
,L(a,t)are given by (1.3), (2.1) and (3.8), respectively. Note
and each of the bounds of (3.9) becomes the
implies that L(l,t)
that a-variance of
(t).
E
Since
[(t)]
m a [(t)]
where
J
r:
(t)
. .
(t)
t
it follows that
f (s; o) ds,
N-I
D
r=O
r
J
Ne-t/ (t/)nN
(3.12)
r: (t)
N-
J=O
([n-I]N + j)’ A
(1)k
[(l-o) t/
Jk=O ([n-l]N+j+l)k
V( [n-l]N+J+l
k
k!
[n-I N+J+k+2 t/
(3.13)
BOUNDS FOR THE MEAN SQUARE ERROR OF RELIABILITY ESTIMATION
801
and
/ e-PV v -I
o -<m(see Erdelyi [4]).
m r:a (t)
--(nN-l)!
r<.- )
)
o< [,o
+
r(p)
2FI(.,.;.;.)
m -2
),
m
>
(3. 4)
2,
Using (3.14) in (3.13), it can be shown that
N-l
N
e
(z
[(t/ o) N-j-I
j=0 (N-J-l)’.
[n-l]N+r+1" (I-,)) + (t/ o)
where
Idv
+v) m--
2
F I(I,[n-I]N+j+I;
(3.15)
[n-l]N+r+l;(l-a)t/ o)}
IFI(I
the Gauss’ hypereometrlc series, i.e.
[s
2 F[(I’ 2;
(l)k(P2)k
k=O"
re;z)
--)k
k
z
k’.
(3.16)
Further simplification leads to the approximation
E [R(t)
e
a
-t/o
(t/o)
r=0
for large n and small t, i.e.
r
r!
the presence of a sngle outlier has little effect on
the estimation of the reliability function R(t) of gamma distribution if there [s a
large number n of items testing over a short period of time t.
(Similar result is
proved by Sinha [l] for the exponential distribution).
4.
SEMI-BAYESI EN APPROACH.
Consider a as a random variable having prior distribut[on of beta type with non-
negative parameters p and q:
_r_(p+q)
g(a)
ap-i
r(p) r(q)
(l-a)
q-l,
0
<
a
< I.
(4.1)
The marginal p.d.f of s is given by
h
P’q
f f = (s; o)
(s; o)
.
g() dr
0
N-I
M(p,q)
fl(s;o)
=0
Dr 2F2
(l,q’[n-l]N+r+l,N+p+q" s/ ), (4.2)
whe re
M(p q)
)___r(N+p)r(nN)
2F2(I,
z;ml
is
the
generalized
corresponding to p
(4.3)
F(p) F(N+p+q)
and
m2;z)
hypergeometrlc
and q--
I,
(l)k (D2)k
k
z
Z
k=O (-m)k
(m)k
2 k
series.
For
we have
the
(4.4)
homogeneous
case,
which
is
802
M. E. GHITANY AND W. H. LAVERTY
2F2
I,
(l,l;(n-l)N+r+l, ; s)
and
r=O,l,...,N-l,
N-I
M((R),I)
which implies that h
,I
Denote by E
P,q
f (s; o).
(s; o)
[R(t)] the epectatlon of R(t) when
is distributed as in (4.1).
Uslng (3.5), we get
fl (s; a)
f (s;) gCa)da
a
0
that is
h
p,l
(s;a)
g(a)da,
0
(4 5)
f (s;o),
Consequently
E
P,q
[(t)]
Also, we have
E
P,q
[(tll
f
R(t)
t
)
(4.6)
(R(t).
M(p,q)
.
hp ’q (s: o)ds
D
rffiO
F (I q;[n-l]N+r+l N+p+q;tlo)
r 2 2
t
(t) fl(s;o)ds
(4.7)
L (p,q,t)R(t)
where
,
L (p,q,t)
N-I
M(p,q)
D
r=0
r
2F2
(l,q;[n-1]N+r+l,N+p+q;t/ a).
(4.8)
R(t).
(4.9)
Using (4.6) and (4.7), we obtain
L (p,q,t) R(t)
E
P,q
[(t)]
Similar ly
L (p,q,t)
E[(t)l 2
Ep, q [(t)l
2
Eta(t)12
(4.10)
Finally, we have
L (p,q,t)
Et(t)]Z-RZ(t)
MSE
[(t)
p,q]
ErR(t)]
z
{2L*(p,q,t) -l}R2(t).
It is easy to verify that for the homogeneous case, i.e.
bounds in (4.11) becomes the variance of R(t).
p=(R)
(4.11)
and q=l, each of the
BOUNDS FOR THE MEAN SQUARE ERROR OF RELIABILITY ESTIMATION
5.
803
EXPONENTIAL DISTRIBUTION AS A PARTICULAR CASE.
N=I,
When
i.e.
we
have an exponential distribution with scale parameter o, we
find that
R(t)
e
(t)
(1
(5.1)
s- )n-I
t
f (s O)
a
a
r(n)
on
t
e
(
-s/O s n-I
F (I ;n; (l-,)s/o)
IFl(l;n;(l-a))[R(t)]2-e -2t/
-{2
-]
p+q
(5.2)
s,
MSE((t)
4
4
1Fl(1;n;(1-e)t/o)-1}
(I q;n,p+q+l;t/o) E[R(t)] 2 -e 2t/o
2F2
)
E[(t)] 2
{2
(5 3)
E[(t)]
e
-2t/O
(MSE((t)
2F2(l,q;n,p+l+l;t/s)
(5.4)
p,q)
i} e -2t/
(5.5)
,whe re
E[(t)] 2
.
with
Il)(t)+ $2)(t)
n-3 n-r-2
(t)
r=O
k=1
Br:0(t)
(n-r-2)
{(k-l)’. (-t/o) n-r-k-2
-et/(-t/o)n-r-2
Ei(-t/o) +
(t/o)
Bn_2:o(t)
2(n-l) r-n+l
(t)
r =n
and
B
r:O
(t)
" Br:0
k =0
(t)
(5.6)
(r-n+l)Wk!
Bn_l:0(t)
-I
(t/O)-(r-n-k+2)
(2(n-l))
(-1) r e -t/o (t/o) n,
r! (2(n-1)-r)! (n-l)!
Ei -t / o)
(5.7)
(5.8)
(5.9)
The results in this section are those of Sinha’s [I].
ACKNOWLEDGEMENT
The authors are grateful to the editor Dr.Lokenath Debnath and the referee for
their helpful suggestions for improving the presentation of this paper.
804
M. E. GHITANY AND W.H. LAVERTY
REFERENCES
[.
SINHA,
2.
BASU,
S.K.,
Outlier Observat[on,
A.P.,
Estimation
Reliability
Estimates
per.
of
in
Life
Testing
In
the
Presence
of
an
Res. 20, (1972), 888-894.
Reliability
for
some
Distributions
Useful
in
Life
testing, Technoetrlcs 6, (1964), 215-219.
3.
4.
NATH,
G.B., Unbiased Estimates of Reliabillty Distribution, Scand. Actuarial
J.,(1975), 181-186.
ERDELYI, A., Higher Transendental Funct[ons, Vol. I, McGraw-Hill, New York, 1953.