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Work supported by U. S. Army Research Office (Durham) Grant No. DAH C04-70-G0006
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EFFICIENT ESTIMATION OF mE MEAN OF AN EXPONENTIAL
DISTRIBUTION WHEN AN OUTLIER IS PRESENT
by
Prakash C. Joshi
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 655
January 1970
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Efficient Estimation of the Mean of an Exponential
Distribution when an Outlier is Present*
Prakash C. Joshi
Department of Biostatistics
University of North Carolina at Chapel Hill
This paper extends some of the results obtained in a recent paper by Kale
and Sinha [3] for the exponential distribution.
The problem of selecting an
efficient estimator of the expected value in the presence of an outlying observation with higher expected value is discussed.
An iterative procedure for the
estimation of the mean is provided and the method is illustrated by considering
an example.
1.
INTRODUCTION AND SillM\RY
In a recent paper [3], Kale and Sinha have considered the following problem:
Given n independent observations xl'x Z' •.• ,xn ' n-l of which are from
p(x,a) = (l/a) exp( -x/a), x~O, a>O
and one of which is from p(x,a/a.), 0<0.2::.1, we wish to estimate the parameter a.
For 0<0.<1, the single observation from p(x,a/a.) essentially represents an
outlying observation with a higher expected value.
If 0.=1 (termed as the
homogeneous case by Kale and Sinha), then there is no outlying observation and
in this case
n
T
n
=
I
. 1
1=
x. / (n+ 1)
1
(1.1)
* Supported by U. S. Army Research Office (Durham) Grant No. DAH C04-70-G0006.
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is the best linear estimator of a in the sense that it has the smallest Mean
Square Error CMSE) in the class of all linear estimators Csee e.g. [4]).
To
begin with, we do not know the a parameter and the outlying observation.
How-
ever, in many situations we can reasonably assume that each X.1 has an equal
chance of being the outlying observation.
Sinha [3] have suggested an estimator of a based on the smallest m «n) order
statistics of the form
m-l
Tm = C. L1
xC') 1
+ Cn-m+l)x
))/(m+l) ,
m
C
1=
where XCI)
~
X(2)
,.
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~
...
~X(n)
Cl.2)
represent the order statistics obtained by re-
arranging the random variables Xl'X 2 , •.. ,Xn . They arrived at this estimator by
using the fact that the set {XCm+l)'XCm+2)"",XCn)} contains the outlying observation with a high probability.
However, they have given no method of determining
m*, the opt:i.nnJm value of m in the sense that the MSE of Tm* is smallest. In this
paper, we first derive an exact expression for the MSE of Tm and then tabulate m*
for various values of n and a.
~,
Our computations show that even for a as small as
m* is equal to n for all n>6.
equal to n in equation Cl.2).
Therefore, we allow the possibility of m being
Note that for m=n, equation Cl.2) reduces to (1.1).
The use of this table requires the value of a.
However, a is not known.
So, we first give a simple estimator of a based on intuitive grounds and then
provide an iterative procedure for the estimation of a and a.
Finally, we illustrate
our method by considering an example.
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Under this assumption, Kale and
2.
BASIC DISTRIBUTION TIIEORY RESULTS
In this section we will obtain an exact expression for the MSE of Tm in
estimating a. Let y.1 = X.la'
Ci=1,2, ... ,n). Then, exactly n-l of these random
1
variables are from
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p(y,l) = e- y ,
y?"O,
while the remaining one is from p(y,l/a).
Now by our assumption each Y. has
1
an equal chance of being distributed as p(y,l/a).
Hence the joint p.d.f. of
Yl 'Y 2""'Yn is [3]
n
= (a/n)
2
i=l
n
exp[-
2
y. + (l-a)y.].
j=l J
1
If we now let Y(i) = x(i)/a, then the joint p.d.f. of Y(1)'Y(2)""'Y(n)
is the sum of n components corresponding to n mutually exclusive and exhaustive
cases of Y(i) being the outlier.
Hence
n
n
g(y(1),y(2),···,y(n)) = (n-l)! ai~leXP[-j~lY(j) + (l-a)y(i)]'
where
°
~ y (1) ~ Y(2) ~ ..•
.s. y (n)
<
00.
(2.1)
Further
m-l
y(.) 1
+ (n-m+l)y(
Tm = a[ . 21
m)]/(m+l).
1=
In order to evaluate the MSE of Tm, we now need EcY(i)) and E(Y(i)Y(j))'
However, equation (2.1) is not very convenient for this purpose. Note that the
marginal distributions of the extreme order statistics can be obtained by using
some simple probabilistic arguments.
Fn (y)
Thus, for example, for Y(n) we have
= Pr (Y (n )
=
-< y)
n
II Pr(Y. < y)
. 1
11=
-y n-l .
= (l-e -ay)(l-e)
From this we can obtain the density, m.g.f. and hence the moments of Yen)'
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Similar results are valid for Y(l) also.
We now express T in terms of another set of random variables which are
m
easier to handle.
To this end, make the transformation
zr = (n-r+l)(y (r) -y (r-l))' r=l, 2, ... ,n,
where y (0) = O.
Then
i
=
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(2.2)
zI!(n-k+l)
k=l
and
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m
T
I a = . I 1z.1 I (m+1) .
1=
(2.3)
m
MOreover, the joint p.d.f. of ZI,Z2, ... ,Zn is
n
= (a/n) I
r
I
z.
j=l J
i=l
b
i
n
exp[-
+ (I-a)
I
zk/(n-k+l)],
k=l
= (n-r+a)/(n-r+I), r=1,2, ... ,n
(2.4)
we see that
+
(2.5)
Before proceeding any further, we now evaluate the probability that Y(r)
is distributed as p(y ,l/a), while the remaining random variables, viz.,
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Y(l)' ... 'Y(r-l)'Y(r+l)' ... 'Y(n) are distributed as p(y,l).
This is precisely
the integral of the rth term in the sum given by the R.H.S. of (2.1) or
equivalently of (2.5) (see e.g. [3]).
Denoting this probability by u , we then
r
have
0000
r
00
ur = (a/n)f f ····f exp(- I b.z.
o0
0
i=l 1 1
n
I
.
1
1=r+
z.)dz
l ·· .dzn
1
(2.6)
An alternative method of finding u
r is also given in [3], where it is shown that
it can also be expressed as
00
f (l_e-y)r-l
o
n-l)
ur = a (r-l
I
e-(n-r+a)Ydy
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(2.7)
(2.8)
Using equation (2.4), the equivalence of (2.6) and (2.8) is easy to prove.
We now state a lenuna involving the weighted sums of the above mentioned
probabilities.
compact form.
Lemma 1.
This is needed to express the marginal distribution of Zi in a
The proof of this is given in the Appendix.
For m=l,2, ... ,n
n
I
u
r=m r
= (n-m+a)umla
(2.9)
and
n
(n-m+a) (n+IDa)um
I rur =
a (a+l)
r=m
(2.10)
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\,m-l
\,m-l
The sums Lr=lu
r and Lr=lrur can be obtained by noting that u l = a/(n-l+a)
and \' n u =1 Lr~lrur = (n+a)/(a+l).
Lr=l r '
As mentioned previously, we now obtain the marginal distribution of Z. in
1
terms of the probability u i introduced above. Integrating out Zl,ZZ"",Zi_l'
z.1+l""'Z n from (2.5) and using (2.6), we have the marginal p.d.f. of Z.1
=e
on us ing Lerruna 1.
-z.
1
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1
E(Zi) = 1 +
where 6 = (l-a)/a.
1
-b.z. -z.
1 l_ e 1)
Hence
I_
I
I
I
+ [(n-i+a)u./a] (b. e
(n-i+a)u.1 1
(0. - 1)
a
1
=1 +
(n-i+a)u.1 n-i+l
(n-i+a - 1)
a
=1
eu.,
+
(2.11)
1
Similarly
E(Z~)
1
= 2 + 26u.(2n-2i+l+a)/(n-i+a).
1
Next, for i<j the joint p.d.f. of Z. and Z. from (2.5) is
J
1
-z.-z.
h(zi'zj) = (ul +... +ui_l)e 1 J
-b.z.-z.
+ (u.+ ... +U. l)b. e l l J
1
J-
1
-b.z.-b.z.
+ (u. +... +u ) b. b. e l l J J
J
n
1
J
(2.12)
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Hence, for i <j
E(Z.Z.) = (u1+... +u. 1) + (u.+ ... +u. l)/b.
1 J
11
J1
+ (u.+ ... +u )/(b.b.)
n
J
1
J
1 1 1
= 1 + (-b
- l)(u.+
... +un )+(~
- -b)
(u.+
... +Un )
•
1
D.D.
•
J
1
1
J
1
= 1 + eu. + eu.(n-i+1)/(n-i+a).
1
(2.13)
J
Variances and co-variances of Zl,Z2""'Zn can now be obtained from
equations (2.11)-(2.13).
Likewise, the means, variances and co-variances of
Y(1)'Y(2)""'Y(n) can be obtained by using equation (2.2).
Thus for example
i
E(Y (i))
= l. E(Zk/(n-k+1)).
k=l
Now from equation (2.11)
and for k>2
(l-a)uk
1
= n-k+1 + a(n-k+l)
From equation (2.6), it is easy to show that for k>2
~-uk-l
Hence
and
=
(l-a)~(n-k+l).
(2.14)
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ui
=a +
i
I
k=2
1
n-k+1 '
where, by convention, the second term of the R.H.S. is zero for i=l.
Finally, from equation (2.3) we have the MSE of Tm in estimating cr given
by
(2.15)
m-1 m
m
1
m
= (m+l)2 [ I E(Z~)+2 I
I E(Z.Z.)-2(m+1) I E(Z.)+(m+1)2].
i=l 1
i=l j=i+1 1 J
i=l 1
Using equations (2.11)-(2.13) and Lemma 1 repeatedly, we get
(2.16)
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The details of the proof are given in the Appendix.
Note that for a=l
and for m=n
(2.17)
Equation (2.17) has been also obtained by Kale and Sinha [3] by using a different
approach.
They have also given same special cases of (2.16).
sidered by them are:
(i) any n and m=l,
(ii) n=3 and m=2,
The cases con(iii) n=4 and m=2,3 .
The result given in (2.16) is valid for all values of m and n.
This allows us
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9
to find a value of m for which the MSE of Tm is smallest. We will denote this
optimum value of m by m*. It is clear that among all the estimators Tm
(m=1,2, ... ,n), Tm* has the maximum efficiency. A numerical evaluation of m*
is considered in the next section.
3.
OPTIMUM VALUE OF m
The problem of finding m* from equation (2.16) theoretically is quite difficult.
However, for given a, the problem can always be solved numerically (at
least for small values of n).
We need only find the n Mean Square Errors
corresponding to n different possible values of m* and then pick the one with
the smallest MSE.
by this method.
In Table 1, we tabulate m* for n = 2(1)10(5)20(10)50 obtained
The efficiency of Tm* relative to Tn defined by
is also tabulated.
The a parameter was increased in steps of 0.05.
it was observed that m*=n and these values are not tabulated.
For a>0.55
Our computations
also revealed that the values for the MSE and efficiency as tabled in [3] for
the case n=4, m=3 are in error.
The correct values for this case are given in
Table 2.
In their paper, Kale and Sinha suggest to take m<n.
Table 1, we allow the possibility of m* being equal to n.
as
However, in view of
In fact for a as low
we reconunend the estimator Tn for estimating cr. A theoretical justification for this can be seen from equation (2.16) by comparing the Mean Square
~,
Errors for Tn- 1 and Tn for
a=~.
Now e=(l-a)/a = 1,
2 _
1
2
MSE ( Tn) / cr - n+l + (n+l)2
and
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where
For moderate values of n, the first term of the R.H.S. in both of these expressions will be dominant and hence the MSE (Tn ) will be less than the MSE (Tn- 1).
4.
For the estimation of
parameter a.
a values.
ESTIMATION OF
AND a
0
we use the estimator Tm*. But m* depends on the
Now from Table 1 it is clear that m* is same for a wide range of
0
The problem of finding an efficient estimator of a and some tests con-
cerning it will be considered in another paper.
Here, we only give a simple esti-
mator of a based on intuitive grounds.
It is easy to show that
n
E( LX.)
·11
1=
= (n-l
+
l/a)o.
However, the equation
n
I
x.1
.1
1=
where & is an estimator of
0,
= (n-l
+
l/a);,
(4.1)
leads to an under estimate of a.
Thus, for
example, in the homogeneous case (a=l) we will use the estimator Tn for estimating
But if we put Tn for & in (4.1) then we only get
suggest to use the equation
0.
n Tn
= (n-l
+
l/a)&
a=~.
We therefore
(4.2)
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for the estimation of a..
involves
a.
This is consistent with the homogeneous case but also
So we give the following iterative procedure for the estimation of
a and a..
(i)
First estimate a by Tn- l' viz.
n-2
a = Tn- 1 = (i~l xCi)
+ 2 x(n_1))/n.
This serves as a good first approximation, since from equation (2.14) x(n) is
the outlying observation with highest probability.
Estimate & from equation (4.2).
(ii)
(iii)
Find m* from Table 1 and estimate
Repeat the steps (ii) and (iii) until a stable value of m* and hence
(iv)
of
a is
a by Tm*.
obtained.
We now illustrate our procedure by considering the data given by Sukhatme
[5] and analyzed for outliers by Carlson [2] and Basu [1].
Example 1.
The following ordered observations represent the length of interval,
in half minutes, between the successive telephone calls:
Here n=S and the first approximation of
gives &=0.66.
1,3,3,15,25,33,39,70.
a is
T7=19.75. Equation (4.2) then
Hence m*=S and the revised estimate of cr is TS=21.00.
Next suppose that xeS) is 90 instead of 70; i.e., the data is 1,3,3,15,25,
33,39,90.
and m*=7.
The value of T7 remains unchanged but
So the revised estimate of cr is again
TS=23.22.
T7=19.75.
This gives a=0.42
Note that a larger value of xeS) will lead to a smaller value of & and in
that case we may use T6=20.S6 or even T =20.33 for estimating cr.
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ACKNOWLEDGEMENTS
The author wishes to thank Professor Harry 9nith, Jr., the Editor of
Technometrics for providing an earlier draft of [3] and to Professor B. K.
Kale for his permission to use some of the results contained in [3] prior to
their publication.
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APPENDIX
We now sketch the proofs of the results mentioned in Section 2.
(i)
Proof of (2.9).
We have
n
L ur = un+un- l+"'+um
r=m
_
on using equation (2.8).
~
~
{af(a) + af(a+l)
fel)
f(2)
Adding the terms successively, we get
n
Lu = -::::+-'--o?-:-:..-;';""""'-Er=m r
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It should be noted that equation (2.9) is valid for any a>O and with the help
of (2.7) it can be rewritten as
(A. I)
(ii)
Proof of (2.10).
To prove (2.10), it is convenient to use equation (2.7).
Now consider the sum
Putting n-l=N and a+l=S we see that
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by (A.l)
= (n-m+a) (n-m)um/(a+l).
The result now follows on using equation (2.9).
It is clear that the same method can be applied to evaluate similar weighted
sums of ur . Thus, for example, we can first evaluate Lr=m
\ n (n-r) (n-r-l)ur and then
\ n r 2u.
However, we do not need them for our present work.
Lr=m
r
(iii)
Proof of (2.16).
Let
Then from equation (2.15) we can write
m
m-l m
m
L E(Z.Z.) - 2m L E(Z.)+(m+l)2.
S = L E(Z~-2Z.)+2 L
m 1=
. 1 1 1
· 1J=l+
·· 1 1 J
.1 1
1=
1=
From equations (2.11)-(2.13) we therefore have
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i I
m1
S = 2e
n-~+1 u. + 2
(l+eu. + n-~+1 eu.)
m
i=1 n-1+a 1
i=1 j=i+l
1 n-1+a J
m
-2m
L (l+eu.)
1
.1
1=
+ (m+l)2
m
. 1
m-l
= 2e L\ n-1+
'+ u. + m(m-l) + 2e L\ (m-i)u.
. 1
1
. 1 n-1 a 1
1=
1=
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IS
+ 26
m~1 n-i+l
~
2
~
2
u.-2m -2m6 L u. + (m+l) .
. . 1 J
. 1 1
. 1 n-1+a J=l+
1=
1=
L
•
L
After a rearrangement of terms, this can be written as
Sm = (m+l) +
26 (n-m+l)u
m-l
m
n-m+a
- 26 i~l i u i - 2m6um
m-l
,n-1'+1 m
,
.
L uo.
o 1 n-1+a . . J
1=
J=l
+ 26 L
(A. 2)
Now applying Lerrma 1 to Ij:iuj , it is easy to show that
m
I
- (n-m)u /a •
u. = (n-i+a)uo/a
1
m
j=i J
Substituting this in equation (A.2) we have
Sm = (m+1) +
26 (n-m+l)u
m-l
m
n-m+a
- 26 . I 1
i u.
-2m8u
1
m
1=
26 m-1
26
m-1
I-a
+I (n-i+1)u. - - (n-m)u
I (1 +
• ).
a 1=
. 1
1
a
mOl
n-1+a
1=
·
Lemma 1 t 0 ,m-1.
. 1) u andS1Illp
' l°f
APP1Y1ng
Li=1 1 u i and ,m-1(
Li=l n-1+
1 y1ng, we ge t
i
0
S
m
=
(m+1) + 28 2
-
28 2 (n-m)u [1:. + ~
1]
ma
.L n-i+a'
1=1
Recalling the definition of Sm' the result follows.
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Table 1.
0.05
E!n*
1 88.59
1 74.71
2 67.10
3 60.42
4 54.84
5 50.19
6 46.28
7 42.96
8 40.10
12 30.73
17 25.06
27 18.45
36 14.77
46 12.39
I~
mfe
I~
m*
2
3
4
5
6
7
8
9
10
15
20
30
40
50
1
2
3
4
5
6
7
8
9
14
19
29
39
49
2
3
4
5
6
7
8
9
10
15
20
30
40
50
m* and the efficiency em* -of Tm*
0.10
E!n*
1 21.96
2 17.58
2 15.97
3 14.51
4 13.27
5 12.22
6 11.33
7 10.57
8 9.91
13 7.65
17 6.36
27 4.87
37 4.03
47
3.49
mfe
0.35
0.30
em*
2.39
2.11
1.93
1.80
1. 70
1.63
1.56
1.51
1.47
1.33
1.25
1.17
1.12
1.10
m*
1
2
3
4
5
6
7
8
9
14
19
29
39
49
em*
1. 79
1.62
1.52
1.45
1.39
1.35
1.31
1.28
1.26
1.17
1.13
1.08
1.06
1.05
mfe
0.15
em*
1
2
2
3
4
5
6
7
8
13
18
27
37
47
9.66
7.88
6.83
6.29
5.81
5.40
5.05
4.75
4.49
3.59
3.06
2.46
2.14
1.93
0.40
em*
1 1.41
2 1.32
3 1.27
4 1.23
5 1.20
6 1.17
7 1.16
8 1.14
9 1.13
14 1.08
19 1.06
29 1.04
39 1.03
49 1.02
m*
mfe
0.20
em*
5.38
4.49
3.95
3.57
3.29
3.10
2.93
2.79
2.66
13 2.22
18 1.96
28 1.68
38 1.52
48 1.42
1
2
3
4
4
5
6
7
8
0.45
em*
1 1.16
2 1.13
3 1.11
4 1.09
5 1.08
6 1.07
7 1.06
8 1.05
9 1.05
14 1.03
19 1.02
29 1.01
39 1.01
49 1.00
mfe
mfe
0.25
em*
1
2
3
4
5
6
7
8
9
3.43
2.93
2.63
2.41
2.25
2.12
2.01
1.93
1.85
13 1.62
18 1.49
28 1.34
38 1.25
48 1.20
0.50
em*
1 1.00
2 1.00
3 1. 00
4 1.00
6 1.00
7 1.00
8 1.00
9 1.00
10 1.00
15 1.00
20 1.00
30 1.00
40 1.00
50 1.00
mfe
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
I·
I
17
Table 2.
MSE and efficiency of T for n=4
3
ex
MSE
Efficiency
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.4427
0.3749
0.3296
0.2993
0.2792
0.2661
0.2579
0.2531
0.2507
0.2500
15.09
3.95
1.93
1. 27
1.00
0.89
0.83
0.81
0.80
0.80
I
II
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
18
REFERENCES
[1]
Basu, A. P., 1965. "On some tests of hypotheses relating to the
exponential distribution when some outliers are present",
J. Amer. Statist. Assoc., 60, 548-559.
[2]
Carlson, P. G., 1958. '~ests of hypothesis on the exponential lower
limit", Skand. Aktuartidskr., 41, 47-54.
[3]
Kale, B. K. and Sinha, S. K., 1969. "Estimation of expected life in
the presence of an outlier observation", to appear in Technanetrics.
[4 ]
Kendall, M. G. and Stuart, A., 1967. The Advanced Theory of Statistics
Volume 2, Second Edition, Hafner Publishing Company, New York.
[5]
Sukhatme, P. V., 1936. "On the analysis of k samples from exponential
populations with especial reference to The problem of random
intervals", Statist. Res. Mem., !., 94-112.