20320130406030

International Journal of Advanced Research in Engineering RESEARCH IN ENGINEERING
INTERNATIONAL JOURNAL OF ADVANCED and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME

AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 4, Issue 7, November - December 2013, pp. 258-265
© IAEME: www.iaeme.com/ijaret.asp
Journal Impact Factor (2013): 5.8376 (Calculated by GISI)
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IJARET
©IAEME

MOMENT PROPERTIES OF TWO MAXIMUM LIKELIHOOD
ESTIMATORS OF THE MEAN OF TRUNCATED EXPONENTIAL
DISTRIBUTION
Faris M. Al-Athari
Professor and Head, Department of Mathematics, Faculty of Science and
Information Technology, Zarqa University, Jordan

ABSTRACT
Properties of two types of maximum likelihood estimators of the mean of truncated
exponential distribution are presented in this article. The estimators include those based on truncated
and non- truncated exponential data. The Monte Carlo method with the help of MATLAB version
6.5, based on 50,000 trials for each alternative, is used to evaluate the moment properties of the
estimators. The simulated variances and meansquared errors of the estimators are compared with
each other and with Cramer Rao lower bounds. The results for the truncation points ξ = 0.05, 0.25,
0.5, 1.0(1.5)10.0 and sample sizes n = 20, 30, 50, 100, 200 indicate that the maximum likelihood
estimator based on the random sample from the exponential population is always more efficient than
the maximum likelihood estimator that based on the random sample from the truncated exponential
population.
KEYWORDS: Truncated Exponential; Maximum Likelihood; Fisher Information; Asymptotic
Relative Efficiency; Simulation Technique.
1. INTRODUCTION
Frequently, in engineering and other scientific disciplines, an estimate is desired of the mean
among the elements of the population belonging to a certain group. For example, in life testing
problems, separate estimate for the life time mean might be required for bulbs of certain quality of
the survival times. In this case, the survival times are limited to be less than b and might follow a
truncated exponential distribution. Besides, the exponential distribution is very important and widely
used distribution in statistics, engineering and in the field of life- testing (see [1] and [2]). In such
estimation, the intuition suggests that the proper approach is to specify a model for that part of the
population, obtain a sample from that part of the population and proceed with standard statistical
258

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –

methods. There are different approaches for sampling selection from a subset of a larger population
(see [3] pp. 551- 560, and [4] pp. 199- 206).
This paper deals with two different maximum likelihood estimators for the mean of the truncated
exponential distribution. The maximum likelihood estimation using a sample from the (complete)
exponential distribution is compared to the case where a sample from the truncated exponential
distribution is available. The Monte Carlo simulation is used to evaluate the variances, the meansquared errors of the estimators. The relative and the asymptotic relative efficiencies of the
estimators are considered. The Fisher information for the two sampling methods is given. It turns out
that sampling from an exponential distribution is more efficient than sampling from a truncated
exponential distribution in estimating the truncated exponential population mean, and the sample
from the exponential distribution carry more information about the mean than the sample from the
truncated exponential distribution. The results of large scale simulation investigations evaluating the
moment properties of the estimators are presented for the case of truncation from right.
2. THE TRUNCATED EXPONENTIAL DISTRIBUTION
Let X be a random variable with parameterized probability density function (p.d.f.) f (.; θ)
and cumulative distribution function (c.d.f.) F (.; θ ) , where θ is a vector of parameters. Then the
p.d..f, g ( y; θ) , of X conditional on a<X<b is simply defined by

g ( y; θ) =

f ( y; θ) I (a , b) ( y)

... (1)

F(b; θ) − F(a; θ)

The case b=∞ corresponds to truncated only from left and the case a= -∞ corresponds to
truncation only from right (see [3] p.559). As special case, if X is a random variable with
exponential p.d.f. of mean θ , the p.d.f. of Y, the truncated version of X truncated on the right at b, is
given by

g(y;θ) =

e − y / θ I (0, b) (y)

... (2)
θ (1 − e - ξ )
b
where ξ = .
θ
The truncated exponential distribution can occur in a variety of ways. It may directly seem to
be a good fit as a distribution for a given available data set, or it may result from the type of
sampling used when the underlying distribution is assumed to follow the exponential distribution
(see [5]-[6]). Generally, the truncated distributions are very important and widely used in statistics.
[7]- [12], among others, treated the truncated distributions in different disciplines.
3. MEAN AND VARIANCE
To find the mean, µ(θ, b) , of the truncated exponential distribution given in equation (2), we
have,

µ(θ, b) = θ − b(e ξ − 1) −1

(3)

by using the integration by parts technique [13].
259


Similarly, the variance σ 2 (θ, b) of the truncated exponential distribution is given by

σ 2 (θ, b) = θ 2 −

b 2eξ

...(4)

(e ξ − 1) 2

By using the fact that σ 2 (θ, b) >0, it can be shown that µ(θ, b) is monotonic increasing in θ;
as θ tends to 0, the function tends to 0 and as θ tends to ∞ the function tends to b/2. Therefore, the
range values of µ(θ, b) is the open interval (0, b/2) and hence the maximum value of the likelihood
function if it exists, occurs at a stationary point and does not occur at any boundary point of the
interval (0, b/2) (see [13] p. 255).

4. MAXIMUM LIKELIHOO ESTIMATORS
Case1. Assume that X1 , X 2 , ..., X n be a random sample taken from an exponential distribution with
mean θ . Now by using the maximum likelihood estimator X of θ , the invariance property of the
maximum likelihood method (see [14]-[15]), and the equation (3), we find that the maximum
ˆ
likelihood estimator, µ1 of µ is given by

ˆ
µ1 = X − b ( e b / x − 1) −1

... (5)

Under the regularity conditions ([16] p.194, [17] pp. 143- 144 and [18] pp. 156- 158), this
estimator is consistent, asymptotic efficient and best asymptotically normal with mean µ and,
ˆ
asymptotic variance, avar( µ1 ), attains the Cramer Rao lower bound.

Case 2.
Assume that Y1 , Y2 , ..., Yn be a random sample of size n taken from the truncated
exponential distribution given by equation (2). The likelihood function, say L2, is

L 2 = θ − n (1 − e − ξ ) − n exp(−n θ −1 y)

(6)

where y is the sample mean. It follows

[

∂ log L1 / ∂ µ = −n 1 − ξ 2 e ξ (e ξ − 1) −2

]

−1

(µ − y) / θ 2

(7 )

It can be shown that 0 < 1 − ξ 2 e ξ (e ξ − 1) −2 < 1 and hence when 0 < y < b / 2, the maximum
value of L2 occurs at a stationary point y. Clearly y is the unique maximum likelihood estimator of
µ when 0 < y < b / 2. When y ≥ b / 2 , the likelihood function L2 does not have a maximum.
Therefore, the proper definition of the maximum likelihood estimator of µ is:

Y
if Y ≤ b / 2
ˆ
µ2 = 
does not exist if Y > b / 2

...(8)

This estimator is also consistent, asymptotic efficient and best asymptotically normal with
ˆ
mean µ and, asymptotic variance, avar ( µ 2 ), attains the Cramer Rao lower bound.
260


5. ASYMPTOTIC VARIANCES OF THE ESTIMATORS

ˆ
ˆ
The asymptotic variance, avar ( µ1 ), of µ1 is the reciprocal of the Fisher information
I1 = −E(∂ 2 Log L1 / ∂ θ 2 ) (∂µ / ∂θ) −2 , where L1 is the likelihood function of the exponential p.d.f.
Thus:
ˆ
avar ( µ1 ) = θ 2 [1 − ξ 2 e ξ (e ξ − 1) −2 ]2 / n

(9)

ˆ
ˆ
Similarly, the asymptotic variance, avar ( µ 2 ), of µ 2 is:
ˆ
avar ( µ 2 ) = θ 2 [1 − ξ 2 e ξ (e ξ − 1) −2 ] / n

(10)

ˆ
ˆ
For the comparison issue, the asymptotic relative efficiency of µ 2 relative to µ1 can easily be found
by using equations (9) and (10) such that

ˆ ˆ
ARE ( µ 2 , µ1 ) = 1 − ξ 2 e ξ (e ξ − 1) −2

(11)

Which is less than or equal to 1 and converges to 1 as ξ → ∞.

6. SOME PROPERTIES OF THE ESTIMATORS
Property: Let Z1 , Z 2 , ..., Z n be a random sample from either the exponential or the truncated
exponential distribution, then for any positive real number a and and b with i=1, 2:

ˆ
(a ) µ i (

Z1 Z 2
Z b
1
ˆ
,
, ..., n , ) = µ i ( Z1 , Z 2 , ..., Z n , b )
a a
a a
a

ˆ
(b) var (µ i (

Z1 Z 2
Z b
1
ˆ
,
, ..., n , )) = 2 var ( µ i ( Z1 , Z 2 , ..., Z n , b ))
a a
a a
a

Z Z
Z b  1

ˆ
ˆ
(c) B  µ i ( 1 , 2 , ..., n , )  = B (µ i ( Z1 , Z 2 , ..., Z n , b ) )
a a
a a  a

ˆ
ˆ
where µ i ( Z1 , Z 2 , ..., Z n , b ) and B ( µ i ( Z1 , Z 2 , ..., Z n , b ) ) are the MLEs of µ (θ, b) and their biases
based on the observations Z1 , Z 2 , ..., Z n .
b
ˆ
ˆ
, the relative efficiency of µ 2 with respect to µ1 is free of θ.
θ
ˆ
ˆ
Proof: It is known that the relative efficiency of µ 2 with respect to µ1 is defined as the ratio of the
meansquared errors and is given by

Theorem: For any fixed value ξ =

ˆ ˆ
RE(µ 2 , µ1 ) =

ˆ
MSE (µ1 (X1 , X 2 , ..., X n , b))
ˆ
MSE (µ 2 (Y1 , Y2 , ..., Yn , b))

(12)
261


where
ˆ
ˆ
MSE [µ1 (X1 , X 2 , ..., X n , b)] = E [µ1 (X1 , X 2 , ..., X n , b) − µ ] 2
ˆ
= E [θ µ1 (X1 / θ , X 2 / θ , ..., X n / θ , ξ) − µ ] 2
ˆ
= θ 2 E[ µ1 (X1 / θ , X 2 / θ , ..., X n / θ , ξ) − µ / θ] 2
Then

for

any

fixed

value

ˆ
E[ µ1 (X1 / θ , X 2 / θ , ..., X n / θ , ξ) − µ / θ] 2

of

ξ , and

letting Wi = X i / θ

it

is

clear

that

does not depend upon θ.

ˆ
ˆ
Similarly, MSE (µ 2 (Y1 , Y2 , ..., Yn , b)) = θ 2 E[µ 2 (Y1 / θ, Y2 / θ, ..., Yn / θ, ξ) − µ / θ] 2
and

ˆ
E[µ 2 (Y1 / θ, Y2 / θ, ..., Yn / θ, ξ) − µ / θ]2 does not depend upon θ for any fixed value of ξ . Hence
RE is free of θ .
7. SIMULATION

ˆ
ˆ
In order to investigate the properties and the values of the estimators µ1 and µ 2 a large
scale simulation investigation was made for the exponential p.d.f. truncated on the right. To get
ˆ
ˆ
the biases, variances and the meansquared errors of µ1 and µ 2 numerically, the simulation
technique with the help of MATLAB is used [19]. These are computed for 50,000 samples of sizes
(n= 20, 30, 50, 100, 200) generated from the exponential and the truncated exponential distributions
byusing the quantile functions of Psudo- uniform numbers that is X i = −θ ln(1 − U i ) and
y i = −θ ln[1 − u i (1 − e −ξ ) .
8. NUMERICAL RESULTS
The simulation results for the estimators are summarized in tables 1, 2, 3, and 4. Table 1
ˆ
ˆ
shows that the estimator µ1 has lower absolute bias than the estimator µ 2 when ξ ≤ 1 and has higher
absolute bias when ξ ≥ 2.5 , but its bias is small and in most cases is insignificant compared to the
variance in its combination to the mean-squared error. The results show that the absolute bias is
ˆ
generally too low. Tables 2 and 3 show that the estimator µ1 has lower variance and mean-srared
ˆ
error than the estimator µ 2 for all values of ξ and n and their variances are well approximated by the
asymptotic variances given by equations (9) and (10). Table 4 gives the percentage values of the
ˆ
ˆ
relative efficiency and the asymptotic relative efficiency of µ 2 relative to µ1 . It is obvious from this
table that for all sample size and all values of ξ , the relative efficiency is less than 1 and increasing
with ξ , and the asymptotic relative efficiency is a good approximation to the relative efficiency
even for a small sample size.

262


Table (1): Percentage values of the absolute biases of the estimators
n=20

n=30

n=50

n=100

n=200

ξ

ˆ
µ1

ˆ
µ2

ˆ
µ1

ˆ
µ2

ˆ
µ1

ˆ
µ2

ˆ
µ1

ˆ
µ2

ˆ
µ1

ˆ
µ2

0.05
0.25
0.50
1.0
2.5
4.0
5.5
7.0
8.5
10.

0.00
0.03
0.11
0.39
1.30
1.53
1.05
0.56
0.26
0.10

0.24
0.97
1.42
1.24
0.01
0.04
0.04
0.03
0.03
0.02

0.00
0.02
0.07
0.25
0.92
1.00
0.65
0.31
0.10
0.00

0.20
0.75
0.98
0.62
0.07
0.09
0.09
0.08
0.08
0.08

0.00
0.01
0.04
0.14
0.53
0.57
0.35
0.14
0.01
0.05

0.15
0.51
0.56
0.18
0.07
0.09
0.10
0.10
0.10
0.10

0.00
0.01
0.02
0.08
0.29
0.33
0.23
0.13
0.07
0.04

0.10
0.29
0.22
0.01
0.01
0.00
0.01
0.01
0.02
0.02

0.00
0.00
0.01
0.04
0.16
0.18
0.14
0.08
0.05
0.04

0.07
0.01
0.05
0.01
0.01
0.02
0.03
0.03
0.03
0.03

Table (2): Percentage values of the (n / θ 2 ) var , the ( n / θ 2 ) MSE and the (n / θ 2 ) a var of the
ˆ
estimator µ1 .
ξ
0.05
0.25
0.50
1.0
2.5
4.0
5.5
7.0
8.5
10

n=20
var
MSE
0.5E-5
0.003
0.052
0.743
15.8
46.1
71.7
86.7
94.0
97.3

0.6E-5
0.003
0.054
0.773
16.2
46.5
71.9
86.7
94.0
97.3

n=30
var
MSE
0.5E-5
0.003
0.048
0.699
15.6
46.8
73.3
88.2
95.2
98.1

0.5E-5
0.003
0.049
0.718
15.9
47.1
73.4
88.2
95.2
98.1

n=50
var
MSE
0.5E-5
0.003
0.045
0.669
15.5
47.3
74.4
89.3
95.8
98.4

0.5E-5
0.003
0.046
0.679
15.6
47.5
74.5
89.3
95.8
98.4

n=100
var
MSE
0.5E-5
0.003
0.044
0.648
15.4
47.8
75.3
89.9
96.1
98.4

0.5E-5
0.003
0.044
0.654
15.5
47.9
75.3
89.9
96.1
98.4

n=200
var
MSE
0.4E-5
0.003
0.043
0.641
15.4
48.2
76.1
90.8
96.8
98.9

0.4E-5
0.003
0.043
0.644
15.4
48.3
76.2
90.8
96.8
99.0

avar
0.4E-5
0.003
0.042
0.629
15.3
48.4
76.6
91.2
97.1
99.1

Table (3): Percentage values of the (n / θ 2 ) var , the (n / θ 2 ) MSE and the (n / θ 2 ) a var of the
ˆ
estimator µ 2 .
ξ

0.05
0.25
0.50
1.0
2.5
4.0
5.5
7.0
8.5
10

n=20
var
MSE
0.008
0.02
0.23
0.42
1.07
1.47
5.59
5.89
38.98
38.98
69.71
69.71
87.56
87.56
95.45
95.45
98.41
98.41
99.42
99.42

n=30
var
MSE
0.008
0.02
0.24
0.40
1.16
1.45
6.25
6.36
39.33
39.34
69.96
69.96
87.84
87.84
95.70
95.70
98.63
98.63
99.61
99.61

n=50
var
MSE
0.008
0.02
0,25
0.38
1.30
1.46
7.04
7.06
39.09
39.09
69.58
69.59
87.53
87.54
95.49
95.50
98.48
98.49
99.50
99.50

263

var
0.008
0.28
1.54
7.75
38.98
69.36
87.22
95.14
98.10
99.11

n=100
MSE
0.02
0.37
1.59
7.75
38.98
69.36
87.22
95.14
98.10
98.4

n=200
var
MSE
0.009
0.02
0.33
0.37
1.84
1.84
7.92
7.92
39.10
39.10
69.59
69.59
87.54
87.54
95.51
95.51
98.50
98.51
99.52
99.53

avar
0.02
0.52
2.06
7.93
39.11
69.59
87.54
95.52
98.53
99.55


ˆ
ˆ
Table (4): percentage values of the relative and asymptotic relative efficiency of µ 2 relative to µ1
n

20

30

50

100

200

ARE

0.05
0.25
0.5
1.0
2.5
4.0
5.5
7.0
8.5
10.0

0.028
0.833
3.671
13.127
41.582
66.775
82.171
90.891
95.564
97.918

0.026
0.784
3.408
11.282
40.349
67.302
83.540
92.200
96.486
98.478

0.025
0.775
3.176
9.624
39.954
68.260
85.066
93.470
97.290
98.914

0.025
0.773
2.779
8.443
39.648
69.030
86.350
94.534
97.949
99.261

0.025
0.750
2.539
8.142
39.481
69.416
87.009
95.069
98.269
99.424

0.021
0.578
2.041
7.929
39.111
69.591
87.535
95.524
98.529
99.546

ξ

CONCLUSIONS
The maximum likelihood estimator using a sample from the (complete) exponential
distribution is compared to the case where a sample from the truncated exponential distribution is
considered. The simulation results turned out that the maximum likelihood estimator of the truncated
exponential distribution of a sample from the (complete) exponential distribution is more efficient
than the maximum likelihood estimator which obtained by using a sample from the truncated
exponential distribution. Moreover, this estimator is also better than the modified maximum
likelihood estimator which is mentioned by [20].

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