chapter: 5 shifted exponential distribution: different …€¦ · shifted exponential...

72

CHAPTER: 5

SHIFTED EXPONENTIAL DISTRIBUTION:

DIFFERENT METHODS OF ESTIMATION

5.1 INTRODUCTION

The distribution of the shifted exponential distribution (SE) is simply the distribution of

(X-T) where X is exponentially distributed therefore its pdf and cdf are given

respectively by;

( )[ ] 0>,T>;exp)( λλλ xTxxf −−= (5.1.1)

( )[ ] 0>,T>;exp1)( λλ xTxxF −−−= (5.1.2)

Therefore, SE distribution has a reliability function,

( ) ( ) ( )[ ] 0>,T>;exp1;; λλλ xTttFTtR −−=−= (5.1.3)

and a hazard function is given by,

Therefore, ( ) ( )( ) 0>,;; λλλ ==tRtfth (5.1.4)

The exponential distribution is most widely used distribution for lifetime data analysis,

because of its simplicity and mathematical feasibility. However, in real world, we rarely

come across the engineering systems which have constant hazard rate throughout their

life duration. Therefore, it seems practical to assume hazard rate as a function of time,

which led to the development of alternative model for lifetime data analysis.

A number of life time models (like Weibull, gamma, generalized exponential etc.) have

been proposed to model life time data that have monotonically increasing or decreasing

hazard rate function, though; non-monotonicity of the hazard rate has also been observed

73

in many situations. For example, in the course of the study of mortality associated with

some of the diseases, the hazard rate initially increases with time and reaches after a peak

after some finite period of times and then decline slowly (Singh et al., 2012). Several

researchers have considered Bayesian prediction problems for the shifted exponential

distribution. Dunsmore (1974) and Evans and Nigam (1980) discussed Bayesian

prediction of future observations based on type II censored sample. Madi and Tsui (1990)

derived a class of smooth estimators. Madi and Tom (1996) addressed the problem of

estimating the scale parameter and the parametric function and proposed Bayesian

estimators which are compared using Monte Carlo simulation technique.

On the basis of a doubly censored random sample Madi and Raqab (2003) have discussed

failure times drawn from a shifted exponential distribution, also address the problem of

Bayesian prediction of one-sample as well as two-sample problem. Raqab (2004) have

discussed simple approximation to one of prediction likelihood equations and derive

approximate predictors of missing failure times. He computes their mean square

prediction errors by simulation and compares them with the best linear predictors.

Further, presented two real examples to illustrate this method of prediction.

Kao (2010) demonstrates that a location parameter of an exponential distribution

significantly influences normalization of the exponential. Wang, et. al. (2014) proposed

an exact interval estimation procedure through the concept of generalized confidence

interval. Javed, et. al. (2014) represented shifted exponential as a likelihood function and

conjugate inverted gamma prior for making Bayesian inference comparatively robust

against a prior density.

The main aim of this chapter is to study how the different estimators of the unknown

parameter of an SE distribution can behave for different sample sizes and for different

parameter values. Here, we mainly compare the maximum likelihood estimators (MLE's)

with the other estimators such as the estimators based on percentiles (PCE's), least

squares estimators (LSE's), weighted least squares estimators (WLSE's) and the

estimators based on the linear combinations of order statistics (LME's), mainly with

74

respect to their root mean squared errors (RMSE's) using extensive simulation

techniques.

The remaining sections go as follows. In Section 5.2, we briefly discuss the MLE's and

their implementations. In Sections 5.3 to 5.8 we discuss other methods. Simulation

results, discussions, comparisons of different methods using graph and conclusions are

provided in Section 5. 9, 5. 10 and 5.11.

5.2 MAXIMUM LIKELIHOOD ESTIMATORS

In this section the maximum likelihood estimators of SED (λ) are considered, when λ is

unknown. If x1, x2... xn is a random sample from SED (λ) then the likelihood function

L (λ, T) is,

( ) ( ) ( ) ( )[ ]∏∏==

−−===n

i

n

iinn TxxfxxxfxxxL

112121 exp||,.....,,.....,; λλθθθ (5.2.1)

( ) ( )

−−== =

n

ii nTxnLL

1

lnlog λλλ (5.2.2)

On differentiating (5.2.2) with respect to λ and equating to zero,

The normal equation becomes:

0log

1

=+−=∂

∂ =

n

ii nTxnL

λλ (5.2.3)

After solving (5.2.3), we obtained the estimate λ as,

nTx

nn

ii +

=

=1

λ̂ (5.2.4)

Again differenting (5.2.3) with respect to λ, we have

22

2 log

λλnL −=

∂∂

Therefore, ( )nLE

V2

2

2 log

1ˆ λ

λ

λ =

∂

∂−= (5.2.5)

75

And T̂ is the smallest order statistics (i.e. nxT :1ˆ = ). Here, ( )T̂V is negative so we

define smallest order statistics for ( )T̂V , we define pdf of T = x(1), therefore we get,

( )( ) ( )[ ] ( ) ( ) TxenxfxFnxf Txnn

x >=−= −−− ,1 1

1

λλ

( ) ( ) ( ) ( ) ( )

−−

−

==== ∞ −∞−∞

−−∞

T

xn

T

xnTn

T

Txn

Tx dx

ne

nxeendxexndxxxfxE

λλλλμ

λλλλ 11

'1 (5.2.7)

Solving (5.2.7), we have, ( )λ

μn

TxE 1'1 +== (5.2.7 A)

( ) ( ) ( ) ( ) ( )

−−

−

==== ∞ −∞−∞

−−∞

T

xn

T

xnTn

T

Txn

Tx dx

nex

nexendxenxdxxfxxE

λλλλμ

λλλλ 2

22

122'

2 (5.2.8)

Solving (5.2.8) we have, ( )22

22'2

22

λλμ

nnTTxE ++== (5.2.8 A)

Substituting the values of (5.2.7 A) and (5.2.8A), in ( )2'1

'2 μμ − we have

222

2

222

2

1122 σλλλλ

μ ==

+−

++=

nnT

nnTT , therefore,

22

1)ˆ(

nTV

λ= (5.2.9)

In this method one can solve (5.2.4), (5.2.5) and (5.2.9) so as to get Maximum Likelihood

Estimates and root mean square error of parameters λ and T by using extensive

simulation technique.

5.3. ESTIMATION OF RELIABILITY

In statistics, reliability is a very important concept that determines the precision of

measurements. Statistical reliability determines whether or not the experiment is

reproducible

( )[ ] ( )[ ]TtTttFtR −−=−−+−=−= λλ expexp11)(1)( (5.3.1)

Differentiate (5.3.1) with respect to λ and squaring both sides,

( )( ) ( ){ } ( )[ ]{ }22

2

exp TtTttR −−−−=

∂∂ λ

λ (5.3.2)

76

Differentiate (5.3.1) with respect to T and squaring both sides,

( )( ) ( )[ ]{ }22

2

exp TtT

tR −−=

∂∂ λλ

(5.3.3)

From section (5.2), we have ;1

)ˆ(;)ˆ(22

2

nTV

nV

λλλ ==

Variance of estimate of reliability, ( )tR ˆ can be obtained from,

( )( ) )ˆ,ˆcov()()(

2)ˆ()(

)ˆ( )(

ˆ22

TTtRtRTV

TtRVtRtRV λ

λλ

λ

∂∂

∂∂+

∂∂+

∂∂= (5.3.4)

Now, Substituting the values of ( )λ,ˆ' tR ( )TtR ,ˆ' , ( )λ̂V and ( )TV ˆ from (5.3.2), (5.3.3)

and (5.2) in (5.3.4) ( )[ ]tRV ˆ is obtained as,

( )( ) ( )( )[ ] ( )

+−−−=

2

222 1

expˆn

ntTTxtRV λλ as 0)ˆ,ˆ( =TCOV λ (5.3.5)

In this method one can solve (5.3.1) and (5.3.5) so as to get estimate of reliability and

root mean square error of reliability by using extensive simulation technique.

5.4 ESTIMATION OF HAZARD RATE

The hazard rate function h(t), also known as the force of mortality or the failure rate, is

defined as the ratio of the density function and the survival function. That is,

( )1)( =

∂∂

=λ

λ thth (5.4.1)

Variance of estimate of hazard rate, ( )th ˆ can be obtained from,

( )( ) ( ) ( )λλ

ˆˆ2

VththV

∂∂=

(5.4.2)

Now, Substituting the values of ( )th ˆ' and ( )λ̂V , from (5.4.1) and (5.2.5) in (5.4.3)

( )( )thV ˆ is obtained as, ( )( )n

thV2

ˆ λ=

(5.4.3)

In this method one can solve (5.4.1) and (5.4.3) so as to get estimates hazard rate and root

mean square error of hazard rate by using extensive simulation technique.

77

5.5 METHOD OF MOMENT ESTIMATORS

In this section we provide the method of moment estimators (MME's) of the parameters

of SE distribution. If X follows SE (λ, T), then

( ) 1>;1 λλ

μ +== TxE and ( ) 2>;1

22 λ

λσ == xV (5.5.1)

Similarly, the mean and variance of the random sample x1, x2... xn from SE distribution

are, =

=n

i

i

nxX

1

and ( )

= −−

=n

i

i

nxx

S1

22

1 (5.5.2)

Therefore, equating the mean and variance of the sample with the mean and

variance of the population, from (5.5.1) and (5.5.2), we obtain, 1>;1ˆ λλ

−= XTMME (5.5.3)

From (5.5.1), we obtain, 1>;1ˆ σσ

λ =MME (5.5.4)

Then, the MME’s of λ and T, say, MMEλ̂ and MMET̂ respectively, can be obtained by

solving the two equations (5.5.3) and (5.5.4). Moreover one can solve equation (5.5.3)

and (5.5.4) so as to get moment estimates and root mean square error moment of

parameters λ and T by using extensive simulation technique.

5.6 ESTIMATORS BASED ON PERCENTILES

If the data comes from a distribution function which has a closed form, then it is quite

natural to estimate the unknown parameters by fitting a straight line to the theoretical

points obtained from the distribution function and the sample percentile points. This

method was originally explored by Kao (1958, 1959) and it has been used quite

successfully for Weibull distribution and for the generalized exponential distribution [see,

Murthy et al. (2004) and Gupta and Kundu (2001). In case of a SE distribution, it is

possible to use the same concept to obtain the estimator of λ based on the percentiles,

because of the structure of its distribution function.

( )[ ] 0>,T>;exp1)( λλ xTxxF −−−=

78

( ) ( )( )[ ]{ }TxTxF ii −−−= λλ exp1ln),;(ln (5.6.1)

Let X (i) denotes the i-th order statistic, i.e., X (1) < X (2) <.......... < X (n). If pi denotes some

estimate of F(x (i); λ, T), then the estimate of λ and T can be obtained by minimizing,

( ) ( )( )( )[ ]{ }=

−−−−=n

iii Txpf

1

2exp1lnln λ

(5.6.2)

with respect to λ and T . We notice that (5.6.2) is a non-linear function of x (i). It is

possible to use some non-linear regression techniques to estimate λ and T. We call these

estimators as percentile estimators (PCE's). Several estimators of pi can be used here

[see, Murthy et al. (2004)]. In this chapter, we mainly consider 1+

=n

ipi , which the

expected value of ( )( )TxF i ,,λ . Then the estimator of λ and T can be obtained by

minimizing (5.6.2) with respect to λ and T. The percentile estimator of λ and T, say,

PCEλ̂ and PCET̂ can be obtained as the solution of the non-linear equation;

Differentiating (5.6.2) with respect to λ, and equating to zero, we have non- linear

equation for x (i) and is given by,

( ) ( )( )[ ][ ][ ] ( )( ) ( )( )[ ] ( )( )( )[ ] 0exp1expexp1lnln2 1

1

=−−−−−−−−−−−=∂∂ −

= TxTxTxTxpf

iii

n

iii λλλ

λ

( ) ( )( ) ( )( )[ ]( )( )[ ]

( )( )( )[ ] ( )( ) ( )( )[ ]( )( )[ ] 0

exp1

expexp1ln2

exp1

expln2,

11

=−−−

−−−−−−+

−−−−−−

−=∂∂

==

n

i i

iiin

i i

iii

TxTxTxTx

TxTxTxpfOr

λλλ

λλ

λ

(5.6.3)

Again differentiating (5.6.3) with respect to λ, we get second non-linear equation for x (i)

is given by, where,

( )λλλλλλ ∂

∂+∂∂−=+−

∂∂=

∂∂

∂∂=

∂∂ BABAff

22222

2

(5.6.3A)

Let,

and

( ) ( )( ) ( )( )[ ]( )( )[ ]

= −−−−−−

−=n

i i

iii

TxTxTxp

A1 exp1

expln2

λλ

79

( )( )[ ][ ] ( )( ) ( )( )[ ]( )( )[ ]

= −−−−−−−−−

=n

i i

iii

TxTxTxTx

B1 exp1

expexp1ln2

λλλ

Again, let,

( )( )[ ] ( ) ( )( ) ( )( )[ ]TxTxuTxu iii −−−−=∂∂

−−= λλ

λ expexp

Since,

( ) ( )( )( )

( ) ( )( ) ( )

( )== −

∂∂−−

∂∂−−

−=∂∂

−

−−=

n

i

iin

i

ii

u

uuuuTxpA

uuTxp

A1

21 1

01ln

21

ln2

λλλ

(5.6.4)

Solving (5.6.4) by substituting values of u andλ∂

∂u, we get

( ) ( )( ) ( )( )[ ]( )( )[ ]{ }

= −−−−−−

=∂∂ n

i i

iii

TxTxTxpA

12

2

exp1

expln2

λλ

λ (5.6.5)

Since,

( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )

[ ]=

−

= −

∂∂−−−

∂∂−−+

∂∂−−−

=∂∂

−

−−=

n

i

in

i

i

u

uuuuuuuuuTxB

uuTxu

B1

2

1

1 1

01ln011ln1

21

1ln2

λλλλ

(5.6.6)


∂u, we get

( )( ) ( )( )[ ] ( )( )( )[ ] ( )( )[ ]{ }( )( )[ ]{ }

= −−−

−−−−−−−−−−=

∂∂ n

i i

iiii

Tx

TxTxTxTxB1

2

2

exp1

expexp1lnexp2

λλλλ

λ (5.6.7)

On putting values of λ∂

∂A and

λ∂∂B

, we have,

80

( ) ( )( ) ( )( )[ ]( )( )[ ]{ }

= −−−−−−

=∂∂ n

i i

iii

TxTxTxpf

12

2

2

2

exp1

expln2

λλ

λ

( )( ) ( )( )[ ] ( )( )[ ]{ } ( )( )[ ]{ }( )( )[ ]{ }

= −−−

−−−−−−−−−−

n

i i

iiii

TxTxTxTxTx

12

2

exp1

expexp1lnexp2

λλλλ

(5.6.8)

Differentiating (5.6.2) with respect to T, and equating to zero, we have non- linear


( ) ( )( )[ ]{ }{ } ( )( )[ ] ( )( )[ ]{ } 0exp1expexp1lnln2 1

1

=−−−−−−−−−=∂∂ −

= TxTxTxp

Tf

ii

n

iii λλλλ

Or,

( ) ( )( )[ ]( )( )[ ]{ }

( )( )[ ]{ } ( )( )[ ]( )( )[ ]{ } 0

exp1

expexp1ln2

exp1

expln2

11

=−−−

−−−−−−

−−−−−

=∂∂

==

n

i i

iin

i i

ii

TxTxTx

TxTxp

Tf

λλλ

λλ

(5.6.9)

Again differentiating (5.6.9) with respect to T, we get second order non-linear equation

for x (i) and is given by, where

( )TB

TABA

TTf

TTf

∂∂−

∂∂=−

∂∂=

∂∂

∂∂=

∂∂

22222

2

(5.6.9A)

( ) ( )( )[ ]( )( )[ ]{ }

= −−−−−

=n

i i

ii

TxTxp

A1 exp1

expln2

λλ

λ and ( )( )[ ]{ } ( )( )[ ]

( )( )[ ]{ }= −−−

−−−−−=

n

i i

ii

TxTxTx

B1 exp1

expexp1ln2

λλλ

λ

Let, ( )( )[ ] ( )( )[ ]TxTuTxu ii −−=

∂∂

−−= λλλ expexp ; Since,

( ) ( ) ( )

( )== −

∂∂−−

∂∂−

=∂∂

−

=n

i

in

i

i

uTuu

Tuup

TA

uup

A1

21 1

01ln

21

ln2 λλ

(5.6.10)

Solving (5.6.10) by substituting values of u andTu

∂∂

, we get

( ) ( )( )[ ]( )( )[ ]{ }

= −−−

−−=

∂∂ n

i i

ii

Tx

TxpTA

12

2

exp1

expln2

λλ

λ (5.6.11)

81

Since,

( )( ) ( ) ( ) ( )

[ ]=

−

= −

∂∂−−−

∂∂−−+

∂∂−−

=∂∂

−−=

n

i

n

i uTuuu

Tuuu

Tuuu

TB

uuuB

12

1

1 1

01ln011ln1

21

1ln2 λλ

(5.6.12)


∂∂

, we get

( )( )[ ] ( )( )( )[ ] ( )( )[ ]{ }( )( )[ ]{ }

= −−−

−−−−−−−−=

∂∂ n

i i

iii

Tx

TxTxTxTB

12

2

exp1

expexp1lnexp2

λλλλ

λ (5.6.13)

On putting values of TA

∂∂

andTB

∂∂

, we have,

( ) ( )( )[ ]( )( )[ ]{ }

( )( )[ ] ( )( )[ ][ ] ( )( )[ ]{ }( )( )[ ]{ }

=

=

−−−

−−−−−−−−

−−−−

−−=

∂∂

n

i i

iii

n

i i

ii

TxTxTxTx

TxTxp

Tf

12

2

12

22

2

exp1

expexp1lnexp2

exp1

expln2

λλλλ

λ

λλ

λ

(5.6.14)

Again differentiating (5.6.9) with respect to λ, we get second order non-linear equation

and is given by, where,

( )λλλλλ ∂

∂−∂∂=−

∂∂=

∂∂

∂∂=

∂∂∂ BABA

Tf

Tf

22222

(5.6.14A)

( ) ( )( )[ ]( )( )[ ]{ }

= −−−−−

=n

i i

ii

TxTxp

A1 exp1

expln2

λλλ

and ( )( )[ ][ ] ( )( )[ ]

( )( )[ ]{ }= −−−

−−−−−=

n

i i

ii

TxTxTx

B1 exp1

expexp1ln2

λλλλ

Let, ( )( )[ ] ( ) ( )( ) ( )( )[ ]TxTxuTxu iii −−−−=∂∂

−−= λλ

λ expexp

( ) ( )

( )== −

∂∂−−

+

∂∂

=∂∂

−

=n

i

in

i

i

u

uuuupA

uup

A1

21 1

0ln

21

ln2

λλ

λλ

λλ

(5.6.15)


∂u, we get

( ) ( )( )[ ] ( )( ) ( )( )[ ]{ }( )( )[ ]{ }

= −−−

−−−+−−−−=

∂∂ n

i i

iiii

Tx

TxTxTxpA1

2exp1

exp1expln2

λλλλ

λ (5.6.16)

82

Since,

( )[ ]

( ) ( ) ( ) ( )

( )

[ ]=

−

= −

∂∂−−−

−+

∂∂−−+

∂∂−−

=∂∂

−−=

n

i

n

i u

uuu

uuuuuuuu

Bu

uuB1

2

1

1 1

01ln

1ln011ln1

21

1ln2

λλ

λλ

λλ

λλ

(5.6.17)


∂u, so as to get

( )( )[ ] ( )( ) ( )( )[ ]( )( )[ ][ ] ( )( )[ ] ( )( ){ }

( )( )[ ]{ }= −−−

−−−−−−−−

+−−−−−−

=∂∂ n

i i

iii

iii

Tx

TxTxTxTxTx

TxB

12exp1

exp1exp1ln

expexp

2λ

λλλλλ

λ

λ

(5.6.18)

On putting values of TA

∂∂

andTB

∂∂

, we have,

( ) ( )( )[ ] ( )( ) ( )( )[ ]{ }( )( )[ ]{ } −

−−−−−−+−−−−

=∂∂

∂ =

n

i i

iiii

TxTxTxTxp

Tf

12

2

exp1

exp1expln2

λλλλ

λ

( )( )[ ] ( )( ) ( )( )[ ] ( )( )[ ][ ]( )( )[ ] ( )( ){ }

( )( )[ ]{ }= −−−

−−−−−

−−−+−−−−−−

n

i i

ii

iiii

Tx

TxTxTxTxTx

Tx

12exp1

exp1

exp1lnexpexp

2λ

λλλλλ

λ

(5.6.19)

Then, the PCE’s of λ and T, say, PCEλ̂ and PCET̂ respectively, can be obtained by

solving the non linear equations (5.6.8), (5.6.14) and (5.6.19). One can use Newton-

Raphson method to solve the non linear equations (5.6.8), (5.6.14) and (5.6.19).

Moreover one can find percentile estimates and root mean square error estimates of

parameters λ and T by using extensive simulation technique.

83

5.7 LEAST SQUARES AND WEIGHTED LEAST SQUARES ESTIMATORS

In this section, we provide the regression based method estimators of the unknown

parameter, which was originally suggested by Swain, et al. (1988) to estimate the

parameters of Beta distributions. The method can be described as follows: Suppose Y1 ,Y2

,...,Yn is a random sample of size n from a distribution function G( . ) and Y(1) < Y(2) <.....

< Y(n) denotes the order statistics of the observed sample. It is well known that,

( )( )[ ]1+

=n

iyGE i and ( )( )[ ] ( )( ) ( )

.,.......,3,2,1;21

12

ninn

iniyGV i =++

+−=

[see, Johnson, et al. (1995)]. Using the expectations and the variances, two variants of the

least squares methods can be used.

METHOD-5.7.1 LEAST SQUARES ESTIMATORS

Obtain the estimators by minimizing,

( )( )[ ]2

1 1=

+−=

n

ii n

iyGP (5.7.1)

with respect to the unknown parameter. Therefore, in case of SE distribution the least

squares estimator of λ and T, say, LSEλ̂ and LSET̂ can be obtained by minimizing,

( )( )[ ]{ }2

1 1exp1

=

+−−−−=

n

ii n

iTxP λ (5.7.2)

with respect to λ and T .



( )( )[ ]{ } ( )( ) ( )( )[ ] 0exp1

exp121

=−−−

+−−−−=

∂∂

=

TxTxn

iTxPii

n

ii λλ

λ (5.7.3)

Let,

( )( )[ ] ( ) ( )( ) ( )( )[ ]TxTxuTxu iii −−−−=

∂∂

−−= λλ

λ expexp

Then (5.7.3), becomes,

84

( )( )uTxn

iuPi

n

i−

+−−=

∂∂

=1 112

λ (5.7.4)

Now differentiating (5.7.4) with respect to λ, we have,

( )( )

λλ ∂∂−

+−−=

∂∂

=

uTxn

iuPi

n

i 12

2

1212

(5.7.5)


∂u, we have,

( )( )[ ] ( )( )[ ] ( )( )[ ]=

++−−+−−−−=

∂∂ n

iiii n

iTxTxTxP1

2

2

2

1exp21(exp2 λλ

λ (5.7.6)

This is non-linear equation in x (i).



( )( )[ ]{ } ( ) ( )( )[ ] 0exp1

exp121

=−−−

+−−−−=

∂∂

=

Txn

iTxTP

i

n

ii λλλ

( )( )[ ] ( )( )[ ]{ } 01

exp1exp21

=

++−−+−−−=

∂∂

=

n

iii n

iTxTxTP λλ

(5.7.7)

Let,

( )( )[ ] ( )( )[ ]TxTuTxu ii −−=

∂∂

−−= λλλ expexp

=

+++−=

∂∂ n

i niuu

TP

1 112 λ (5.7.8)

Differentiating (5.7.8) with respect to T, we have

=

+++−

∂∂=

∂∂ n

i niu

Tu

TP

1

22

2

1212 λ

(5.7.9)


∂∂

, we have,

( )( )[ ] ( )( )[ ]{ }=

++−−+−−−=

∂∂ n

iii n

iTxTxT

P1

22

2

1exp21exp2 λλλ

(5.7.10)

This is a non-linear equation in x (i)

85

Let,

( )( )[ ] ( ) ( )( ) ( )( )[ ]TxTxuTxu iii −−−−=∂∂

−−= λλ

λ expexp

=

+++−=

∂∂ n

i niuu

TP

1 112 λ

(5.7.11)

Differentiating (5.7.11) with respect to λ,

=

+

+

∂∂+

+

∂∂+

+

∂∂−=

∂∂∂ n

i niuuuuuuu

TP

1

22

122

λλ

λλ

λλ

λ (5.7.12)


∂u, we get,

( )( )[ ]( )( ) ( )( ) ( )( )[ ]

( )( )[ ] ( )( )=

++

+−−−−+

−−−−−−−−=

∂∂∂ n

i ii

iii

i

ni

niTxTx

TxTxTxTx

TP

1

2

11exp

exp21

exp2λλ

λλλλ

λ (5.7.13)


Then, the LSE’s of λ and T, say, LSEλ̂ and LSET̂ respectively, can be obtained by

solving the non linear equations (5.7.6) (5.7.10) and (5.7.13). One can use Newton-Raph

son method to solve the non linear equations (5.7.6) (5.7.10) and (5.7.13). Moreover one

can find Least square estimates and root mean square error of parameter λ and T by using

extensive simulation technique.

METHOD-5.7.2: WEIGHTED LEAST SQUARES ESTIMATORS

The weighted least squares estimators can be obtained by minimizing

( )( )[ ]2

1 1=

+−=

n

iii n

iyGwP (5.7.14)

with respect to the unknown parameter, where

( )( )[ ]( ) ( )

( ) .,....,3,2,1;1i-ni

21n=

yGv

1 = w

2

ii nin =

+++

Therefore, in case of shifted exponential distribution the weighted least squares of λ and

T, say, WLSEλ̂ and WLSET̂ can be obtained by minimizing P with respect to λ and T,

86

( )( )[ ]{ }2

1 1exp1,

=

+−−−−=

n

iii n

iTxwPOr λ (5.7.15)



( )( )[ ]{ } ( )( ) ( )( )[ ] 0exp1

exp121

=−−−

+−−−−=

∂∂

=

TxTxn

iTxwPii

n

iii λλ

λ (5.7.16)

Let,

( )( )[ ] ( ) ( )( ) ( )( )[ ]TxTxuTxu iii −−−−=

∂∂

−−= λλ

λ expexp

Then (5.7.16) becomes

( )( )uTxn

iuwPi

n

ii −

+−−=

∂∂

=1 112

λ (5.7.17)

Now differentiating (5.7.17) with respect to λ, we have

( )( )

λλ ∂∂−

+−−=

∂∂

=

uTxn

iuwPi

n

ii

12

2

121(2

(5.7.18)


∂u, we have

( )( )[ ] ( )( )[ ] ( )( )[ ]=

++−−+−−−−=

∂∂ n

iiiii n

iTxTxTxwP1

2

2

2

1exp21(exp2 λλ

λ (5.7.19)

This is non-linear equation in x (i).



( )( )[ ] ( )( )[ ]{ } 01

exp1exp21

=

++−−+−−−=

∂∂

=

n

iiii n

iTxTxwTP λλ

(5.7.20)

Let,

( )( )[ ] ( )( )[ ]TxTuTxu ii −−=

∂∂

−−= λλλ expexp

=

+++−=

∂∂ n

ii n

iuuwTP

1 112 λ (5.7.21)

Differentiating (5.7.21) with respect to T, we have

87

=

+++−

∂∂=

∂∂ n

ii n

iuTuw

TP

1

22

2

1212 λ

(5.7.22)

Solving (5.7.22) by substituting values of u and Tu

∂∂

, we have,

( )( )[ ] ( )( )[ ]{ }=

++−−+−−−=

∂∂ n

iiii n

iTxTxwT

P1

22

2

1exp21exp2 λλλ

(5.7.23)


Let,

( )( )[ ] ( ) ( )( ) ( )( )[ ],expexp TxTxuTxu iii −−−−=∂∂

−−= λλ

λ then

=

+++−=

∂∂ n

ii n

iuuwTP

1 112 λ

(5.7.24)

Differentiating (5.7.24) with respect to λ,

=

+

+

∂∂+

+

∂∂+

+

∂∂−=

∂∂∂ n

ii n

iuuuuuuuwT

P1

22

122

λλ

λλ

λλ

λ (5.7.25)


∂u, so as to get,

( )( )[ ]( )( ) ( )( ) ( )( )[ ]

( )( )[ ] ( )( )=

++

+−−−−+

−−−−−−−−=

∂∂∂ n

i ii

iii

ii

ni

niTxTx

TxTxTxTxw

TP

1

2

11exp

exp21

exp2λλ

λλλλ

λ (5.7.26)


In case of SE distribution, if the shape parameter λ and T are unknown, then, the WLSE’s

of λ and T, say, WLSEλ̂ and WLSET̂ respectively, can be obtained by solving the non

linear equations (5.7.19) (5.7.23) and (5.7.26). One can use Newton-Raphson method to

solve the non linear equations (5.7.19) (5.7.23) and (5.7.26). Moreover one can find

Weighted Least square estimates and root mean square error of parameter λ and T by

using extensive simulation technique.

88

5.8 L-MOMENT ESTIMATORS

In this section we propose a method for estimating the unknown parameters of a SE

distribution based on the linear combinations of order statistics, see, for example, David

(1981), Hosking (1990), or David and Nagaraja (2003). The estimators obtained by this

method are popularly known as L-moment estimators (LME's). The LME's are analogous

to the conventional moment estimators but can be estimated by linear combinations of

order statistics, i.e., by L-statistics.

The LME's have theoretical advantages over conventional moments of being more robust

in the presence of outliers in the data. It is observed that LME's are less subject to bias in

estimation and sometimes more accurate in small samples than even the MLE's. First, we

discuss the case how to obtain the LME's when both the parameters of shifted

exponential distribution are unknown. If x(1) < x(2) <.... < x(n) denotes the ordered sample,

then using the same notation as Hosking (1990), First, we discuss the case how to obtain

the LME of shifted exponential distribution when both the parameters are unknown.

L-moments can be expressed as certain linear combinations of probability weighted

moments (PWMS). Let x1, x2,..., xn be identically independently distributed random

variables each with pdf )(xf and cdf )(xF and quantile function )(1 xF − ,then PWMS are

defined as ( ){ } ( )dxxfxFxFr

r −= )(1β ,where r = 0,1,2,3,----. Here we consider only

first two L-moments (λi = 1, 2) associated with x that can be expressed as 01 βλ = and

012 2 ββλ −= . Quantile function ( ) is required to be a strictly increasing monotone

function. This requirement implies that an inverse function ( -1) exists. As such, the cdf

can be expressed as ( ( )) = ( ) = , and subsequently differentiating this cdf with

respect to will yield the parametric form of the pdf for ( ) as ( ( )) =1/ ’( ).

Where ~ (0, 1) with pdf and cdf as 1 and , respectively. For, Shifted Exponential

distribution, ( )( ) ( )( ) TxFxTxxF +−−=−−−=λ

λ 1lnexp1)( ; here ( ) ( ) uuFuqF ==)(

89

and quantile function ( ) ( ) ( ) ( ) ( ) 1;;1ln1 ==+−−== − xfuxFTuxFuqλ

on Substituting )(xF ,

( )xF 1− and ( )xf in ( ){ } ( )dxxfxFxFr

r −= )(1β ,so as to get,

( );

1ln1

0

+−−= duuTu r

r λβ

Let, 01;10;11 ====−=−==− xuxudxduxuxu

( ) ( ) ( )

( ) ( )

−+−−=

−

+−=−−

+−=

1

0

1

0

1

0

0

1

11ln1

1ln

1ln

dxxTdxxx

dxxTxdxxTx

rrr

rr

rr

λβ

λβ

λβ

(5.8.1)

Where, ( ) −−=1

0

1ln1 dxxxA r

λ and ( ) −=

1

0

1 dxxTB r

( ) ;11

0 −= dxxTB r

Let, 01,10;1 ====−==− zxzxdzdxzx

( )11

1

0

1

0

10

1 +

+

−= +

rT

rzTdzzTdzzTB

rrr

(5.8.2)

( ) ;1ln1 1

0 −−= dxxxA r

λ

Let, 01,0;1 ==−∞===== txtxdtedxextnx tt

( ) ( ) ( ) ( ) ( ) ( ) dtetetr

etr

erttedteetA rtrtttttrt ∞−

+

−+−−−−−+

−

+−−=−−=

01432

1

0

132

11

1

λλLet, 00;; ==∞=−∞=−=−= ytytdydtyt

( ) ( ) ( ) dyeyeyr

eyr

eyeydteetA yrryyyytrt ∞

+−−−−−−−−−−

−+−−−−−+

−

+−=−−=

0

!12412312212121

0

132

11

1

λλ

( ) ( )

−+−−−−+

−

+−=

∞ ∞ ∞ ∞+−−−−−−−−−−

0 0 0 0

11241231221212 132

1 dyeydyeyr

dyeyr

dyeyrdyeyA yrryyyy

λ

( ) ( ) ( ) ( )( )

( )( )

+−Γ−+−−−−−+

−Γ

−

−Γ

+

−Γ−

−Γ=

22222 1

21

4

2

33

2

22

2

1

21

rrr

rA r

λ

90

( )( )( )

=Γ+

−+−−−−−+

−

+−Γ= 12;

1

11

16

1

39

1

241

22r

rrrA r

λ (5.8.3)

Adding (5.8.2) and (5.8.3), we have,

( )

1

11

1 1

12

1

++

−

−=

+

=

−

rT

kkr

r

k

k

r λβ

Let, r = 0 then,

( )1;

1

1

1

01

110

1

12

1

0 >+==+

−

−=

=

−

λλ

λβλ

β TTk

kk

k

Let, r = 1 then

( ) ( )

( )

( )

( ) 24

3

24

11

1

22

1

11

1

0

11

1

2

1

11

12

12

2

0

2

12

1

1

TTTTk

k

k

k

+=+

−=+

−

+

−

=+

−

−=

−

=

−

λλλλβ

1;2

11

2

31

24

322 2012 >=−−+=

+−

+=−= λ

λλλλλλββλ TTTT

If x(1) < x(2) <.... < x(n) denotes the ordered sample, then using the same notation as

Hosking (1990), we obtain the first and second sample L-moments as

( ) 1;11

11 >+==

=λ

λ

n

ii Tx

nl

(5.8.4)

( ) ( ) ( ) 1;2

11

1

21

12 >=−−

−=

=

λλ

lxinn

ln

ii

(5.8.5)

and the first two population L-moments are,

( ) 1;1

01 >+=== λλ

βλ TxE (5.8.6)

1;2

12 >= λ

λλ

(5.8.7)

respectively. Now to obtain the LME's of the λ and T unknown parameters, we need to

equate the sample L-moments with the population L-moments. Therefore, the LME's can

91

be obtained by dividing (5.8.4) by (5.8.5). Then, the LME's of λ and T, say, LMEλ̂ and

LMET̂ , respectively, can be obtained by solving the two equations (5.8.4) and (5.8.5).

Moreover one can find L-moment estimates and root mean square error of parameter λ

and T by using extensive simulation technique.

( ) 1;1ˆ1;

111

11 >−=>+==

=

λλ

λλ

lTTxn

l LME

n

ii

(5.8.8)

( ) ( ) ( ) 1;2

1ˆ1;2

11

1

22

21

12 >=>=−−

−=

=

ll

lxinn

l LME

n

ii λλ

λ (5.8.9)

5.9 NUMERICAL EXAMPLES AND DISCUSSIONS:

In this section we present results of some numerical experiments to compare the

performance of the different estimators proposed in the previous sections. We perform

extensive Monte Carlo simulations to compare the performance of the different

estimators, mainly with respect to their biases and root mean squared errors (RMSE's) for

different sample sizes. Note that, we consider sample size n = 10, 20, 30 for fixed λ = 2

when T = 3.We compute the RMSE's of estimators over 1000 replications.

We consider the estimation of λ and T, when λ and T is unknown. The estimates and

root mean square error of MLE, reliability, hazard rate of parameter λ and T can be

obtained from (5.2.4), (5.2.5), (5.2.9), (5.3.1), (5.3.5) (5.4.1) and (5.4.5). Similarly,

method of moment, L-moment estimates and root mean square error of parameters λ and

T can be obtained from (5.5.3), (5.5.4), (5.8.8) and (5.8.9). Moreover, percentile, least

squares, weighted least squares estimates and root mean square error of parameters λ and

T can be obtained by solving (5.6.8), (5.6.13), (5.6.18), (5.7.6), (5.7.10), (5.7.13),

(5.7.19), (5.7.23) and (5.7.26) respectively, with respect to λ and T. The results are

reported in Table 5.9.1 and 5.9.2. It is observed in Table 5.9.2 that most of the estimators

usually over estimate λ and T, as far as PCE, LSE and WLSE concern, which extremely

over estimate λ while for parameter T is quite closer to estimate value of T. The Rmse's

and Biases are also quite close to the MLE, MME and LME. In the context of

92

computational complexities, MLE is easiest to compute. It do not involve any non-linear

equation while solving, where as the PCE, LSE and WLSE involve non-linear equations

and they need to be calculated by some iterative processes.

From Table-5.9.1 it is clear that for increasing values of sample size ( )tR ˆ , ML ( )tR ˆ , and

ML ( ),t̂h ( )th ˆ remains constant, while Rmse ( )tR ˆ and Rmse ( )th ˆ

decreases. For, inverse

exponential distribution, Cumulative density function is defined as

( )[ ] 0>,T>;exp1)( λλ xTxxF −−−= , Therefore,( )( ) TxFx +−−=

λ1ln

, which is

useful for simulation. Table 5.9.1: Simulated values of biases and Rmse's of estimators of t = 5, 2=λ when

3=T .

n ( )tR ˆ ( )th ˆ ( )( )tRML ˆ RMSE’s ( )( )tR ˆ ( )( )thML ˆ RMSE’s ( )( )th ˆ

10 0.01831 2 0.01894 0.02725 2.5121 0.8477

20 0.01831 2 0.01835 0.01795 2.2413 0.5151

30 0.01831 2 0.01805 0.01417 2.1682 0.4029

93

Table 5.9.2: Simulated values of biases and Rmse's of estimators of 2=λ when

3=T .

n Methods λ̂ Bias ( )λ̂ RMSE ( )λ̂ T̂ Bias ( )T̂ RMSE ( )T̂

10 MLE 2.5121 0.5121 0.8477 3.0475 0.0475 0.0471

MOM 2.7272 0.7272 1.4200 3.0589 0.0589 0.1323

PCE 529.5668 527.5668 748.0989 8.7066 5.7066 8.0315

LSE 856.8664 854.8664 3560.683 24.7996 21.7996 183.5345

WLSE 689.6171 687.6171 1138.325 16.8320 13.8320 128.6947

LME 2.3353 0.3353 1.0410 2.9979 -0.0021 0.0999

20 MLE 2.2413 0.2413 0.5151 3.0239 0.0239 0.02412

MOM 2.3128 0.3128 0.8250 3.0334 0.0334 0.0971

PCE 615.2094 613.2094 737.4605 10.005 7.005 10.7279

LSE 1876.512 1874.512 3989.643 6.1703 3.1703 5.1075

WLSE 993.47 991.47 991.47 9.5805 6.5805 6.5805

LME 2.1755 0.1755 0.6326 3.0005 0.0005 0.06603

30 MLE 2.1682 0.1682 0.4029 3.0165 0.0165 0.01619

MOM 2.2484 0.2484 0.6134 3.0217 0.0217 0.08046

PCE 939.6682 937.6682 1191.527 11.8204 8.8204 10.2521

LSE 3744.838 3742.838 3821.506 11.03024 8.0302 8.03397

WLSE 870.5641 868.5641 1158.479 11.0519 8.0519 11.7960

LME 2.1169 0.1169 0.4809 2.9994 -0.0006 0.05273

94

Remark: Here, RMSE’s and Bias are obtained from,

λλ −= ˆBias and .1000

ˆ'

21000

1

=

−=

i

isRMSEλλ

and TTBias −= ˆ and .1000

ˆ'

21000

1

=

−=

i

i TTsRMSE

5.10 COMPARISONS OF DIFFERENT METHODS USING GRAPH

For a quick understanding, the relative biases and the relative RMSE’s of the different

estimators of the parameters λ and T is presented in figure-5.10.1, 5.10.2, 5.10.3 and

5.10.4 with sample sizes 10, 20, and 30.

Figure-5.10.1 and 5.10.2 show the average relative biases and RMSE’s of the different

estimator of λ with different sample sizes 10, 20 and 30.

Figure-5.10.3 and 5.10.4 show the average relative biases and RMSE’s of the different

estimator of T with different sample sizes 10, 20 and 30.

0500

1000150020002500300035004000

10 20 30

MLE

MOM

PCE

LSE

WLSE

LME

Figure:5.10.1

n------->

Relative b

aises

0500

10001500200025003000350040004500

10 20 30

MLE

MOM

PCE

LSE

WLSE

LME

Figure:5.10.2

n--------->

Relative

Rm

se's

95

5.11 CONCLUSIONS

In this section we present results of some numerical experiments to compare the

performance of the different estimators proposed in the previous sections. We perform

extensive Monte Carlo simulations to compare the performance of the different

estimators, mainly with respect to their biases and root mean squared errors (RMSE's) for

different sample sizes. Note that, we consider sample size n = 10, 20, 30 for fixed λ = 2

when T = 3. We compute the Rmse's of estimators over 1000 replications.

From Table:5.9.1 we conclude that for unknown shape parameter λ and scale parameter

T, as far as reliability as well as ML reliability is concern, reliability remains same for

different sample size considered. As far as hazard rate is concern, hazard rate remains

same for different sample size considered while ML hazard rate decreases for different

sample size considered. While RMSE’s of reliability and hazard rate decreases for

different sample size considered here.

From Table:5.9.2 comparing all the methods, we conclude that for unknown shape

parameter λ and scale parameter T, LME, MLE and MME perform best for different

values of λ, T and n considered, while LSE do not perform well for different values of λ,

T and n considered. As far as PCE, LSE and WLSE concern, parameter λ is extremely

over estimate while T is under estimate as compare to λ all the time for different values of

n considered. As far as LSE concern, parameter λ is extremely over estimate when

sample size is 30 while T is under estimate as compare to λ all the time for different

values of n considered. As far as PCE, LSE and WLSE are concern, estimate of

-5

0

5

10

15

20

25

10 20 30

MLE

MOM

PCE

LSE

WLSE

LME

Figure:5.10.3

n-------->

Relative b

aises

020406080

100120140160180200

10 20 30

MLE

MOM

PCE

LSE

WLSE

LME

Figure:5.10.4

n-------->R

elative Rm

se's

96

parameter T is similar when sample size is 30 while λ is extremely over estimate. As n

increases, the estimate of λ decreases and its biased increases for PCE, LSE and WLSE

while estimate of λ increases and its biased decreases for MLE, MOM and LME, as far as

estimate of T is concern, estimate of T decreases and its biased increases for MLE, MOM

and LME while for PCE estimate of T increases and its biased decreases. While LSE and

WLSE decreases and its biased increases for sample size 20 again LSE and WLSE

increases and its biased decreases when sample size 30 computationally, the MLE

involve only one dimensional optimization whereas the rest of the estimators involve two

dimensional optimization. From the graph (figure-5.9.1, 5.9.2, 5.9.3 and 5.9.4), we

conclude that for unknown shape parameter λ and scale parameter T, LME, MLE and

MME perform best for different values of λ, T and n considered, while LSE do not

perform well for different values of λ, T and n considered. For a quick understanding, the

relative biases and the relative Rmse’s of the different estimators of the parameters λ and

T is presented in figure-5.9.1, 5.9.2, 5.9.3 and 5.9.4 with sample sizes 10, 20, and 30.

chapter: 5 shifted exponential distribution: different …€¦ · shifted exponential...

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