1

Test for homogeneity of lifetimes of several systems under generalized inverted scale family of

distributions based on Type II censored sampling design

Shanubhogue, A1 and Raykundaliya, D.P

1,*.

1Department of Ststistics, Sardar Patel University, Vallbh Vidyangar- 388120, Gujrat-India

*corresponding author: dp_raykundaliya@spuvvn.edu, +919428435134

Abstract:

In this paper we consider generalized inverted scale family of distributions (GIFD) as

probability models for m independent systems of different make and study the problem of

simultaneous estimation of the unknown parameters under an extension of Type II censoring

scheme. We consider Maximum likelihood estimation (MLEs) of the scale and shape parameters.

Asymptotic confidence intervals for �+1 parameters based on the MLEs are also constructed.

As a special case, we consider generalized inverted exponential distribution (GIED) and

conducted simulation study to investigate performance of the estimates and confidence intervals

under type II censoring. Further, likelihood ratio test is used to test for homogeneity of several

scale parameters and the cost effectiveness of the Type II censoring scheme in planning of

experiment for the study of several populations is done using realistic cost function. We also

study effect of censoring on duration of experiment, total time on experiment, cost of experiment

and on estimates of parameters through simulation.

Keyword: Type II censoring scheme, generalized inverted exponential family of distribution

(GIED), Newton-Raphson method, ML estimation, Monte-Carlo Simulation

technique, likelihood ratio test, total time on test, cost function, proportion of

censoring

Introduction:

The industrial revolution onward man has become more and more dependent on

“Machine” or “System”. The failure of the machine stops the whole system abruptly. Therefore,

it is important to know among the several machines during the same work, which machine

should be faithful or reliable for their daily usage and having minimum maintenance but not

unduly expensive. This requires extensive results from reliability and life tastings. In life testing

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experiment, the objective is to obtain the estimates of mean time to failure, hazard rate and

reliability at a given point etc. In case of a study of systems from different brands performing

identical task the problems are: (i) simultaneous estimation of the parameters of the assumed

probability models for the life times and (ii) obtaining tests for the homogeneity of life time

distributions or equality of certain parameter of the life time distributions. As continuation of

life testing experiments till all the units put on test fail may lead to not termination of experiment

in a manageable duration of time and increased experimental cost, one has to resort to some kind

censoring scheme leading to enough data information in reasonable duration of time. Often,

Type II censoring is the most commonly used sampling plan. In statistics literature we find lots

of research papers which use the plan for various lifetime models such as normal, exponential

and Weibull. Few noteworthy research papers to cite are Gupta (1952), Cohen (1965), Mann et

al. (1974), Lawless (1982), Sinha (1986) and Hossain et al. (2003) .

In manufacturing setting, the problem of comparing effectiveness of products is

important. In this case after placing several independent samples of units manufactured by

several processes, the reliability engineer would like to make early and efficient decision on the

effectiveness of the products under the life test in terms of mean time to failure, standard hazard

rate functions, reliability function etc. Balakrishnan and Ng (2006) extensively study the

problem of comparing two populations in terms of stochastic ordering. Sharafi et al. (2013)

study a distribution free test for comparison of hazard rates of two distributions under Type II

censoring. Shafay et al. (2013) discuss the Bayesian inference on jointly Type II censored sample

for two exponential populations. Srivastava (1987) discusses the method of generalized Type II

censoring for comparison of life times of several systems under Weibull distribution. Further,

experimenters are also concerned with the cost of experiments, total time under experiments and

duration of experiment for better planning of the experiment. There are several costs associated

with the conduct of the experiment. In this pursuit, Srivastava (1989) studies a realistic cost

function for several systems put on tests under life experiment for the generalized Weibull

distribution.

The scale family of distributions play very important role in lifetime data analysis.

Exponential distribution, Rayleigh distribution and Weibull distribution are such distributions

which are widely used by researchers in reliability studies. Recently, Potdar and Shirke (2013)

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proposes a generalized inverted scale family of distributions (GIFD) and shows that generalized

inverted exponential distribution (GIED) (for detail one can refer; Abouammoh and Alshingti

(2009)), generalized inverted-half logistic distribution (GIHD) and generalized inverted Rayleigh

distribution (GIRD) are the members of GIFD. Further they obtain ML estimates of parameters

of GIFD and obtain confidence intervals for the parameters involved in the distributions and

discuss the inferential procedures for GIHD.

In this paper we discuss inferential problems on lifetimes of several systems under the

generalized inverted scale family of distributions based on a generalization of Type II censored

scheme for several samples. We further, study the reliability characteristics of the distributions.

Under generalized Type II censoring design we put ‘�’ units, for each ‘�’ types of systems,

simultaneously on test and continue the experiment till �∗ failures in each system are observed

i.e. the total number of units put on test are ‘��’ and the total number of failures we observe at

the end of experiment is � = ��∗. Here we assume that lifetimes of systems follow

generalized inverted scale family of distributions with shape parameter � and scale

parameters; � = 1,2, … ,�. The observed failure times are denoted by ��; � = 1,2, … , �∗; � =1,2, … ,�. Thus the experimental data under this scheme are(�, �, ��; � = 1,2, … , �∗; � =1,2, … ,�). The organization of whole paper is as below.

In Section 2, we give the probability density function, the survival function and the hazard

rate of the generalized inverted scale family of distributions (GIFD), and develop the likelihood

for Type II censored sampling design under GIFD. In Section 3, we derive the expressions for

maximum likelihood equations for the parameters and their asymptotic variance-covariance

matrix for both cases of known and unknown shape parameters. Further, we obtain asymptotic

confidence interval for �+ 1 parameters of the distributions. Section 4 discusses likelihood

ratio test for simultaneous testing of homogeneity of scale parameters when the shape parameter

is known and when it is unknown. In Section 5, we consider GIED as a special case and obtain

ML estimates of the parameters, confidence intervals for the parameters, reliability

characteristics of the distribution, and efficiency measures like mean square error (MSE) and

standard error(SE) for the parameters through Monte-Carlo simulations. Further the likelihood

ratio test for simultaneous testing of homogeneity of scale parameters, when the shape parameter

is known, is discussed and the cut-off points for the test statistics are also obtained through

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Monte-Carlo simulation. In Section 6, we discuss realistic cost function for the problem of

planning such experiments. We further study the effect of proportion of censoring on cost

function. Some concluding remarks are given in Section 7.

2. Generalized inverted scale family of distributions (GIFD) and likelihood function for

type II censoring design

Consider an item whose life time is denoted by�. The random variable � is said to have

two -parameter generalized inverted scale family of distributions, if its distribution function is

given by

��(�; , �) = �(� ≤ �) = 1 − �� � !"#$%, � ≥ 0; �, > 0 (2.1)

where �(�) is a parameter free survival function. If � has the distribution function (2.1), then

the corresponding density function, reliability function and hazard function are respectively

given by

)(�; , �) = %!"* ℎ � !"# �� � !"#$%, , � ≥ 0; �, > 0 (2.2)

��(�; , �) = �� � !"#$% (2.3)

and

-�(�; , �) = ./(")0/(") = 123*4� 523#�6� 523#$175�6� 523#$1 = %!"* 84� 523#6� 523#9 . (2.4)

The marginal likelihood function for �-th type of system under type II censoring based on

observing G*failures from� units is

: = ;!(;,=∗)!∏ )(��)?�@A (�=∗)B;,=∗=∗�C . (2.5)

Therefore, the joint likelihood of the whole experiment is

: = ∏ :DC = ∏ E ;!(;,=∗)!∏ )(��)?�@A (�=∗)B;,=∗=∗�C FDC . (2.6)

Substituting the density function in (2.2) and survival function in (2.3) in (2.6) we get

: = ∏ G ;!(;,=∗)!∏ %!H"IH* ℎ J !H"IHK L� J !H"IHKM%, �� � !H"N∗H#$%(;,=∗)=∗�C ODC . (2.7)

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3. Maximum likelihood estimation and confidence interval

The log likelihood function of (2.7) is

P = �PQ � ;!(;,=∗)!# +��∗PQ� − �∗∑ PQ − 2DC ∑ ∑ ln �� +=∗�C DC ∑ ∑ ln Lℎ J !H"IHKM=∗�C DC

+(� − 1)∑ ∑ ln L� J !H"IHKM + �(� − �∗)∑ PQ �� � !H"N∗H#$DC =∗�C DC (3.1)

Differentiating (3.1) with respect to � and (� = 1,2,⋯ ,�) we get

VWV% = D=∗% +∑ ∑ ln L� J !H"IHKM + (� − �∗)∑ PQ �� � !H"N∗H#$DC =∗�C DC (3.2)

VWV!H = − =∗!H − !H*∑ X "IH4′Y 52H3IHZ4Y 52H3IHZ[=∗�C − (%, )!H* ∑ X "IH

6′Y 52H3IHZ6Y 52H3IHZ[ − %(;,=∗)"N∗H!H*6′J 52H3N∗HK6J 52H3N∗HK

=∗�C (3.3)

for, � = 1,2,⋯ ,�

Equating equation (3.2) to zero we have,

�\ = − D=∗∑ ∑ ]^86Y 52H3IHZ9_(;,=∗)∑ W`L6J 52H3N∗HKMaHb5N∗Ib5aHb5 (3.4)

The estimates of parameters (� = 1,2,⋯ ,�) can be obtained in two different cases: (i) shape

parameter � is known and (ii) shape parameter � is unknown.

3.1 Maximum likelihood estimation when shape parameter c is known

The solutions of ML equations (3.3) cannot be obtained explicitly. Hence, we use

Newton-Raphson method for the data (�, �, ��; � = 1,2, … , �∗; � = 1,2, … ,�). The method

requires some initial values of ( , d, e, … , D). The initial values are obtained using the least

square method as suggested in Ng (2005). The method uses the empirical distribution function

for �-th population as given below

�fg��h = 1 − ∏ (1 − ik)�kC

with

ik = ;,k_ , for l = 1,2… , �;� = 1,2… , �∗; � = 1,2, … ,�

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The parameters’s are estimated by fitting the linear regression through the method of least

square. Here we consider

m� = �� , where

m� = 675n�57opHg3I75h#5/1r�57opHg3Ih#5/1* s � = 1,2… , �∗; � = 1,2, … ,� ,

and �t is such that

�f(�t) = 0

The least square estimates of ; � = 1,2, … ,� are given by

ut = ∑ vIH"IHN∗Ib5∑ "IH*N∗Ib5 ; � = 1,2, … . ,� (3.5)

These values are used as initial solution for the ML estimation by the Newton-Raphson method.

The MLEs of reliability (�(�); � = 1,2, …�) and hazard rate (-(�); � = 1,2, … ,�) can be

evaluated using invariance property of MLEs as

�p(�) = �� � !p"H#$% (3.6)

-g�; u, �h = %!p"* w4J 52p3HK6J 52p3HKx for � = 1,2, … ,� (3.7)

3.1.1 2 Observed Fisher information matrix under the design

The Fisher information matrix is

y = �gzkh# ,

where

zk = −{ J V*WV!HV!|K for �, l = 1,2, … ,�.

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Now for � = l, V*WV!H* = =∗!H* + ∑ "IH* !H}~�

�4Y 52H3IHZ4′′Y 52H3IHZ,�4′Y 52H3IHZ�*_d!H"IH4Y 52H3IHZ4′Y 52H3IHZ

84Y 52H3IHZ9* ���=∗�C

+(� − 1)∑ "IH* !H}~��6Y 52H3IHZ6′′Y 52H3IHZ,�6′Y 52H3IHZ�

*_d!H"IH6Y 52H3IHZ6′Y 52H3IHZ86Y 52H3IHZ9* ��

�=∗�C

+ %(;,=∗)"N∗H* !H} �6J 52H3N∗HK6′′J 52H3N∗HK,Y6′J 52H3N∗HKZ*_d!H"IH6J 52H3N∗HK6′J 52H3N∗HKL6J 52H3N∗HKM* � for (� = 1,2,⋯ ,�)

(3.8)

where, ℎ′(∙) = VV!H ℎ(∙), �′(∙) = VV!H�(∙), ℎ′′(∙) = V*V!H* ℎ(∙), �′′(∙) = V*V!H*�(∙) for (� = 1,2,⋯ ,�).

Since systems are independently functioning in the experimental set up, we have

V*WV!HV!| = 0 ∀l ≠ � = 1,2, …� (3.9)

Theorem 3.1: Let assume that all the usual regularity conditions needed for the asymptotic

distribution of mle hold true. Then, the limiting distribution of u , as � → ∞ such that =∗;

remains constant, is �-variate normal with mean vector and dispersion matrix y, .

3.2 Maximum likelihood estimation when shape parameter c is unknown

The solutions of ML equations 3.2 and 3.3 cannot be obtained explicitly. Hence, we use Newton-

Raphson iterative method for the data (�, �, ��; � = 1,2, … , �∗; � = 1,2, … ,�). The initial

values of the parameters of ; � = 1,2, … ,� are obtained through the method of least square as

discussed in Section 3.1 and initial value � is obtained by using equation (3.4). These initial

values are used in the Newton-Raphson method to obtain the MLEs of (�, ). The MLE of

reliability �(�)and hazard rate -(�), � = 1,2, … ,� can be evaluated using invariance

property of MLEs as

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�p(�) = �� � !p"H#$%�

(3.10)

-�g�; u, �\h = %�!p"* w4J 52p3HK6J 52p3HKx for � = 1,2, … ,� (3.11)

3.2.1 Observed Fisher information matrix under the design

The Fisher information matrix is

� = −{ wV*WV%* ∆′∆ yx y = �gzkh# ,

where

zk = V*WV!HV!| for �, l = 1,2, … ,�

and

∆′ = � V*WV%V!5 , V*WV%V!* , … , V*WV%V!a#.

V*WV%* = −D=∗%* (3.12)

V*WV%V!H = − !H*∑ X "IH6′Y 52H3IHZ6Y 52H3IHZ[ − (;,=∗)"N∗H!H*

6′J 52H3N∗HK6J 52H3N∗HK=∗�C For (� = 1,2,⋯ ,�) (3.13)

V*WV!H* = =∗!H* + ∑ "IH* !H}~��4Y 52H3IHZ4′′Y 52H3IHZ,�4′Y 52H3IHZ�

*_d!H"IH4Y 52H3IHZ4′Y 52H3IHZ84Y 52H3IHZ9* ��

�=∗�C

+(� − 1)∑ "IH* !H}~��6Y 52H3IHZ6′′Y 52H3IHZ,�6′Y 52H3IHZ�

*_d!H"IH6Y 52H3IHZ6′Y 52H3IHZ86Y 52H3IHZ9* ��

�=∗�C

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+ %(;,=∗)"N∗H* !H} �6J 52H3N∗HK6′′J 52H3N∗HK,Y6′J 52H3N∗HKZ*_d!H"IH6J 52H3N∗HK6′J 52H3N∗HKL6J 52H3N∗HKM* � for � = 1,2, … ,�

(3.14)

where, ℎ′(∙) = VV!H ℎ(∙), �′(∙) = VV!H�(∙), ℎ′′(∙) = V*V!H* ℎ(∙), �′′(∙) = V*V!H*�(∙) for � = 1,2, … ,�.

Since the systems are independently functioning in the experimental set up, we have

V*WV!HV!| = 0 ∀l ≠ � = 1,2, …�. (3.15)

Theorem 3.2: Let assume that all the usual regularity conditions needed for the asymptotic

distribution of mle hold true. Then the limiting distribution of (�\, u), as � → ∞ such that =∗;

remains constant, is(� + 1)-variate normal with mean vector (�, ) and dispersion matrix �,

under regularity conditions.

3.3 Confidence interval

Assuming asymptotic normal distribution for the MLEs, CIs for (�, , d, e, … , D) are

constructed. Let (�\, u , ud, ue, … , uD) are MLE of (�, , d, e, … , D) respectively. Let �\d(�\) and �\dguh; � = 1,2, … ,� is the estimated variance of �\and u; � = 1,2, … ,� respectively.

Then, 100(1 − �)% asymptotic CI for � and : � = 1,2, … ,� are respectively given by

J�\ − �� d� ��\d(�\), �\ + �� d� ��\d(�\)K and Yu − �� d� ��\dguh, u + �� d� ��\dguhZ For � = 1,2, … ,� (3.16)

where �� d� is the upper 100(1 − �)"4 percentile of standard normal distribution.

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4. Testing of hypotheses

The proposed design will have significance only when we are able to ascertain that the m

type of systems are not all have identical life time. This can be done by developing ANOVA

approach for the proposed design. However, we will utilize likelihood approach to develop a test.

The testing of hypothesis problem is to test

�t: = d = ⋯ = D = against � : ≠ kforatleastonepair(�, l), � ≠ l. (4.1)

As we are considering maximum likelihood estimation, the use likelihood ratio test is

much convenient. The test statistic is

¤¥¦ = max%,! :(�, , �)max%,! :(�, , �) The test based on −2PQ(¤¥¦) rejects H0 in support of H1 if it is larger than upper �thcut point of

chi-square distribution (m-1) degrees of freedom.

4.1 Computation of likelihood under H0

The log likelihood equation (3.1) can be written under to as

P = �PQ � ;!(;,=∗)!# + ��∗ ln � − ��∗ ln − 2∑ ∑ ln �� + ∑ ∑ ln ℎ J !"IHK=∗�C DC =∗�C DC

+(� − 1)∑ ∑ ln� J !"IHK=∗�C DC + �(� − �∗) ∑ ln� � !"N∗H#DC (4.2)

Differentiate (4.2) with respect to � and we have

VWV% = D=∗% +∑ ∑ ln� J !"IHK=∗�C DC + (� − �∗)∑ ln� � !"N∗H#DC (4.3)

VWV! = −D=∗! − !*∑ ∑ n "IH ∙ 4′Y

523IHZ4Y 523IHZs=∗�C DC − (%, )!* ∑ ∑ n "IH ∙ 6′Y523IHZ6Y 523IHZs=∗�C DC − %(;,=∗)!* ∑ w "N∗H

6′J 523N∗HK6J 523N∗HKxDC

(4.4)

Differentiate (4.3) and (4.4) with respect to � and we have

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V*WV%* = −D=∗%* (4.5)

V*WV%V! = − !*∑ n "IH ∙ �′Y523IHZ6Y 523IHZs − (;,=∗)!* ∑ w "N∗H

�′J 523N∗HK6J 523N∗HKxDC =∗�C (4.6)

V*WV!* = D=∗!* + !}∑ ∑ "IH* ª««

¬4Y 523IHZ4′′Y 523IHZ,�4′Y 523IHZ�*_d!"IH4Y 523IHZ4′Y 523IHZ

�4Y 523IHZ�* ­®®

=∗�C DC

+ (%, )!} ∑ ∑ "IH* ª««¬6Y 523IHZ�′′Y 523IHZ,��′Y 523IHZ�

*_d!"IH6Y 523IHZ�′Y 523IHZ�6Y 523IHZ�

* ­®®=∗�C DC

+ %(;,=∗)!} ∑ "N∗H* °6J 523N∗HK�′′J 523N∗HK,Y6′J 523N∗HKZ*_d!"IH6J 523N∗HK�′J 523N∗HK

Y6J 523N∗HKZ* ±DC (4.7)

Since the likelihood equations 4.3 and 4.4 are not mathematically tractable, we use the Newton-

Raphson method to obtain the estimates of parameters �²Q³. 5. Application of generalized inverted exponential distribution (GIED)

Consider a member of generalized Inverted family of distribution, namely Generalized

Inverted Exponential distribution suggested by Abouammoh and Alshingti (2009). Let, � be

generalized Inverted Exponential random variable. The cdf and pdf of � are respectively,

��(�) = 1 − L1 − ´�, 523#M% � ≥ 0, �, > 0 (5.1)

and

)�(�) = %!"* ´�, 523# L1 − ´�, 523#M%, � ≥ 0, �, > 0 (5.2)

Here � and are scale and shape parameters respectively.

The reliability function of GIED (, �) is given by

�(�) = L1 − e�7523#M% , � ≥ 0, �, > 0 (5.3)

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The failure rate function of GIED (, �) is given by

-�(�) = .(")¦(") = %!"* µJ7523K ,µJ7523K , � ≥ 0, �, > 0 (5.4)

Since the mean of the distribution has no closed form and it is finite only if� > 1 , we shall

consider here the median time to system failure (¶·�¸�) which is given by

¶·�¸� = n− !W`8 ,(t.¹)519s, �, > 0 (5.5)

It is clearly seen that the distribution belongs to GIFD if we take

� � !"# = L1 − ´�, 523#M (5.6)

and

ℎ � !"# = ´�, 523# (5.7)

and some other results

ℎ′ � !"# = ··! ℎ � !"# = ··! ´�, 523# = −´�, 523#, ℎ′′ � !"# = ´�, 523# �′ � !"# = ´�, 523#, �′′ � !"# = −´�, 523# (5.8)

Substituting the equations 5.6 to 5.8 in equations 3.1 to 3.13 we obtain the log likelihood

function, maximum likelihood equations for parameters, Fisher information of parameters for

generalized inverted exponential distribution. Details are omitted to avoid repetitions.

5.1 Algorithm, Numerical Exploration and Conclusions

In this Section, a Monte Carlo simulation study is conducted to compare the performance

of the estimates. Maximum likelihood estimates are obtained for observations generated through

the Type II censoring design when numbers of systems to be compared are 2 and 3 for known as

well as unknown shape parameter. All calculations are performed with the aid of R- language

version- R.2.12.0. The simulation study is conducted for both known as well as unknown shape

parameter.

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5.2.1 Known Shape Parameter

In this section, we carryout simulation study for two sets of parameter values � = 2, � = 2.5, = 1.5, d = 1.3 and for� = 3, � = 2.5, = 1.5, d = 1.3, e = 1.4,

and for different values of u and �∗. Here we keep total number of failures in the whole

experiment � = ��∗ fixed. We simulate 1000 samples for each case. The simulation results are

summarized in Table 1 and Table 2. To carry out our objective we proceed through following

algorithm.

Step 1: Taking � = 2, � = 2.5, = 1.5, d = 1.3 we generate � random numbers from �½{¾(�, , d, … , D) for each type of systems. The same is repeated for the

parameters� = 3, � = 2.5, = 1.5, d = 1.3, e = 1.4.

Step 2: Generate �∗ Type II censored observations for each type of systems. The generated �∗ failure times are (� , �d, … , �=∗); � = 1,2, … ,� for each type of systems.

Step 3: Obtain initial estimate of parameters ; � = 1,2, … ,� using the least method discussed

in Section 3.1 i.e. evaluate initial estimates using formulae

ut = ∑ vIH"IHN∗Ib5∑ "IH*N∗Ib5 ; � = 1,2, … . ,�.

Step 4: Obtain initial value for sample information matrix ¿pusing the value obtained in Step 3

and also obtain the score vector ¸ ′ = � VWV!5 , VWV!* , … , VWV!a#. Step 5: Use Newton-Raphson iterative method

uÀÁÂ = uÃW· + ¿p, �uÃW·# ∗ ¸.

Step 6: Repeat the Step 5 until the ∑ ÄuÀÁÂ − uÃW·ÄDC < Æ where Ç is very small predefined

quantity.

Step 7: Repeat the procedures in Steps 1 to Step 6 for Q = 1000 times and obtain the following

quantities.

(a) {y = ∑ !pH|È|b5;

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(b) Mean Squared Error, ¶¸{ = ∑ (!pH|,!H)*È|b5 ` where; � = 1,2, … ,� the values of

parameters given in Step 1.(c) Average of variance-covariance matrices computed for different simulated samples, say y∗, .(d) Evaluate Reliability functions �pk(�) and hazard rater�k(�); � = 1,2, … ,�; l =1,2, … , n using equations 5.3 and 5.4 for each simulated sample and obtain the average

of all these values.

(e) Obtain MSE(Reliability) = ∑ (0pH|("H),0H("H))*È|b5 ^ and MSE (hazard rate)=

∑ (ÌH|("H),ÌH("H))*Í|b5 `

respectively at median given in equation (5.5).

Step 8: Obtain Standard Error (SE) of estimates by taking square root of diagonal

elementsy∗, .

Table 1

Maximum likelihood estimate of parameters, reliability and hazard rates and their efficiency measures � = 2,� = 2.5, = 1.5, d = 1.3²Q³� = (0.4700,0.5424), �(�) = (0.5,0.5), -(�) = (2.4099,2.0886) � �∗ Ò dÒ � Ò(� ) �dÒ(�d) r �(� ) r\(�d)

12

06

{y 1.5200 1.3068 0.5059 0.5117 2.3851 2.0486 ¶¸{ 0.0917 0.0746 0.0120 0.0129 0.1731 0.1396 ¸{ 0.3164 0.2727 - - - -

24

12

{y 1.5134 1.3107 0.5019 0.5030 2.4016 2.0774 ¶¸{ 0.0450 0.0318 0.0062 0.0069 0.0875 0.0733 ¸{ 0.2200 0.1907 - - - -

36

16

{y 1.5011 1.3007 0.5045 0.5048 2.3924 2.0723 ¶¸{ 0.0320 0.0250 0.0044 0.0047 0.0619 0.0494 ¸{ 0.1774 0.1538 - - - -

48

24

{y 1.4993 1.3011 0.5039 0.5029 2.3949 2.0788 ¶¸{ 0.0235 0.0164 0.0033 0.0031 0.0464 0.0324 ¸{ 0.1532 0.1328 - - - -

60

30

{y 1.4966 1.3071 0.5042 0.5000 2.3939 2.0881 ¶¸{ 0.0187 0.0152 0.0026 0.0028 0.0367 0.0292 ¸{ 0.1365 0.1193 - - - -

72

36

{y 1.5014 1.3045 0.5019 0.5006 2.4025 2.0862 ¶¸{ 0.0158 0.0126 0.0022 0.0024 0.0305 0.0245 ¸{ 0.1249 0.1086 - - - -

84

42

{y 1.4972 1.3007 0.5031 0.5017 2.3983 2.0830 ¶¸{ 0.0128 0.0093 0.0018 0.0017 0.0249 0.0185 ¸{ 0.1120 0.0950 - - - -

96

48

{y 1.5049 1.3036 0.4999 0.5002 2.4098 2.0877 ¶¸{ 0.0116 0.0085 0.0016 0.0015 0.0225 0.0164 ¸{ 0.1083 0.0938 - - - -

15

Table 2

Maximum likelihood estimate of parameters, reliability and hazard rates and their efficiency measures

Ó = Ô,c = Õ. Ö, ×Ø = Ø. Ö,×Õ = Ø. Ô, ×Ô = Ø. Ù,Ú = (Û. ÙÜÛÛ, Û. ÖÙÕÙ, Û. ÖÛÔÝ), �(�) = (Û. Ö, Û. Ö, Û. Ö), Þ(Ú) = (Õ. ÙÛßß, Õ. ÛààÝ, Õ. ÕÙßÔ) � �∗ Ò dÒ eÒ � Ò(� ) �dÒ(�d) �eÒ(�e) r �(� ) r\(�d) -e�(�e) 12 04

{y 1.5173 1.2979 1.4037 0.5077 0.5148 0.5019 2.3782 2.0380 2.2404 ¶¸{ 0.0974 0.0703 0.0849 0.0127 0.0126 0.0124 0.1838 0.1359 0.1555 ¸{ 0.3163 0.2706 0.2981 - - - - - -

24

08

{y 1.5011 1.3073 1.4105 0.5068 0.5040 0.5038 2.3836 2.0747 2.2350 ¶¸{ 0.0468 0.0350 0.0460 0.0065 0.0065 0.0071 0.0917 0.0682 0.0817 ¸{ 0.2183 0.1900 0.2054 - - - - - -

36

12

{y 1.5003 1.3088 1.3985 0.5047 0.5010 0.5051 2.3919 2.0848 2.2307 ¶¸{ 0.0309 0.0234 0.0254 0.0043 0.0044 0.0042 0.0604 0.0457 0.0506 ¸{ 0.1773 0.1547 0.1653 - - - - - -

48

16

{y 1.5018 1.3079 1.4025 0.5031 0.5003 0.5027 2.3978 2.0872 2.2395 ¶¸{ 0.0243 0.0180 0.0204 0.0034 0.0033 0.0033 0.0477 0.0345 0.0405 ¸{ 0.1534 0.1336 0.1433 - - - - - -

60

20

{y 1.4982 1.2989 1.4044 0.5037 0.5034 0.5009 2.3958 2.0774 2.2457 ¶¸{ 0.0193 0.0138 0.0150 0.0027 0.0026 0.0024 0.0378 0.0269 0.0295 ¸{ 0.1367 0.1185 0.1281 - - - - - -

72

24

{y 1.5039 1.3049 1.3981 0.5010 0.5005 0.5030 2.4058 2.0867 2.2383 ¶¸{ 0.0161 0.0127 0.0128 0.0022 0.0024 0.0021 0.0308 0.0246 0.0254 ¸{ 0.1252 0.1086 0.1163 - - - - - -

84

28

{y 1.5059 1.3017 1.3995 0.4997 0.5015 0.5022 2.4108 2.0834 2.2411 ¶¸{ 0.0126 0.0110 0.0115 0.0018 0.0021 0.0017 0.0246 0.0217 0.0228 ¸{ 0.1159 0.1002 0.1077 - - - - - -

96

32

{y 1.4928 1.3048 1.3961 0.5045 0.4998 0.5013 2.3930 2.0891 2.2447 ¶¸{ 0.0113 0.0090 0.0093 0.0016 0.0017 0.0015 0.0224 0.0174 0.0184 ¸{ 0.1065 0.0932 0.1000 - - - - - -

The conclusions for these studies are given below.

We observe that for the known shape parameter α, averages of estimated values of the

parameters are very close to their true values and the averages of mean square errors are relatively

small. Further, we observe that the estimates and standard/mean square errors are decreasing

functions of number of systems � of each type put on test.

5.2 Unknown shape parameter

Similar study, with not much change in the algorithm, one can make simulation study for

the case of unknown shape parameter. We make simulation studies for the set of parameter

values � = 2, � = 2.5, = 1.5, d = 1.3, � = (0.4700,0.5424) and � = 3, � = 2.5, = 1.5, d = 1.3, e = 1.4,� = (0.4700,0.5424,0.5036) by taking n=1000.

16

Table 3

Maximum likelihood estimate of parameters, reliability and hazard rates and their efficiency measures Ó = Õ,c = Õ. Ö, ×Ø = Ø. Ö, ×Õ = Ø. Ô, Ú = (Û. ÙÜÛÛ, Û. ÖÙÕÙ), �(�) = (Û. Ö, Û. Ö), Þ(Ú) = (Õ. ÙÛßß, Õ. ÛààÝ) � �∗ �\ Ò dÒ � Ò(� ) �dÒ(�d) r �(� ) r\(�d)

12

06

{y 7.6968 1.2929 1.3116 0.4602 0.4559 3.7798 3.3771 ¶¸{ 483.34 0.2609 0.2044 0.0267 0.0296 10.2981 10.1162 ¸{ 39.47 0.4694 0.4134 - - - -

24

12

{y 3.8356 1.3949 1.2183 0.4813 0.4744 2.9270 2.5705 ¶¸{ 8.7988 0.1435 0.0970 0.0110 0.0112 1.5460 1.2587 ¸{ 2.7442 0.3482 0.3023 - - - -

36

18

{y 3.2538 1.4270 1.2373 0.4866 0.4854 2.7328 2.3744 ¶¸{ 3.2557 0.0869 0.0624 0.0065 0.0070 0.7267 0.5515 ¸{ 1.5771 0.2859 0.2474 - - - -

48

24

{y 3.0131 1.4484 1.2575 0.4907 0.4894 2.6355 2.2883 ¶¸{ 1.6604 0.0678 0.0485 0.0047 0.0048 0.4568 0.3386 ¸{ 1.1334 0.2505 0.2170 - - - -

60

30

{y 2.8370 1.4723 1.2734 0.4927 0.4935 2.5645 2.2208 ¶¸{ 0.9501 0.0519 0.0376 0.0037 0.0038 0.3027 0.2347 ¸{ 0.8982 0.2270 0.1965 - - - -

72

36

{y 2.8082 1.4614 1.2727 0.4959 0.4926 2.5484 2.2211 ¶¸{ 0.8139 0.0477 0.0324 0.0033 0.0033 0.2687 0.2107 ¸{ 0.8030 0.2053 0.1787 - - - -

84

42

{y 2.7971 1.4622 1.2738 0.4943 0.4907 2.5464 2.2219 ¶¸{ 0.6105 0.0384 0.0271 0.0025 0.0029 0.2006 0.1695 ¸{ 0.7198 0.1896 0.1648 - - - -

96

48

{y 2.6923 1.4838 1.2859 0.4956 0.4959 2.5015 2.1663 ¶¸{ 0.4429 0.0306 0.0245 0.0023 0.0023 0.1701 0.1250 ¸{ 0.6354 0.1799 0.1562 - - - -

17

Table 4

Maximum likelihood estimate of parameters, reliability and hazard rates and their efficiency measures Ó = Ô,c = Õ. Ö, ×Ø = Ø. Ö, ×Õ = Ø.Ô, ×Ô = Ø. Ù,Ú = (Û. ÙÜÛÛ, Û. ÖÙÕÙ, Û. ÖÛÔÝ),�(�) = (Û. Ö, Û. Ö, Û. Ö), Þ(Ú) = (Õ. ÙÛßß, Õ. ÛààÝ, Õ. ÕÙßÔ) � �∗ �\ Ò dÒ eÒ � Ò(� ) �dÒ(�d) �eÒ(�e) r �(� ) r\(�d) -e�(�e)

24

08

{y 4.9530 1.3474 1.1634 1.2517 0.4468 0.4478 0.4477 3.4987 3.0095 3.2462 ¶¸{ 32.9732 0.1482 0.1087 0.1251 0.0199 0.0190 0.0192 5.9173 3.7950 4.1568 ¸{ 5.6021 0.3417 0.2946 0.3169 - - - - - -

36

12

{y 3.7464 1.3992 1.1971 1.2955 0.4617 0.4673 0.4660 3.0371 2.6188 2.8132 ¶¸{ 6.0686 0.0969 0.0683 0.0829 0.0110 0.0113 0.0102 1.5790 1.1968 1.2909 ¸{ 2.2504 0.2875 0.2451 0.2657 - - - - - -

48

16

{y 3.3883 1.4173 1.2222 1.3097 0.4696 0.4740 0.4761 2.8825 2.4756 2.6624 ¶¸{ 3.2677 0.0671 0.0550 0.0625 0.0079 0.0074 0.0077 0.9885 0.6801 0.8306 ¸{ 1.5922 0.2503 0.2165 0.2315 - - - - - -

60

20

{y 3.1061 1.4421 1.2493 1.3409 0.4804 0.4800 0.4824 2.7300 2.3677 2.5397 ¶¸{ 1.8389 0.0574 0.0416 0.0484 0.0056 0.0060 0.0058 0.6293 0.4656 0.5389 ¸{ 1.2094 0.2286 0.1979 0.2123 - - - - - -

72

24

{y 3.0412 1.4359 1.2477 1.3457 0.4828 0.4814 0.4851 2.6981 2.3424 2.5212 ¶¸{ 1.3220 0.0448 0.0336 0.0419 0.0047 0.0047 0.0047 0.4747 0.3555 0.4222 ¸{ 1.0363 0.2065 0.1795 0.1939 - - - - - -

84

28

{y 2.8649 1.4628 1.2744 1.3692 0.4894 0.4869 0.4874 2.5988 2.2585 2.4355 ¶¸{ 0.8869 0.0395 0.0311 0.0337 0.0037 0.0036 0.0041 0.3370 0.2440 0.3134 ¸{ 0.8748 0.1956 0.1706 0.1832 - - - - - -

96

32

{y 2.8502 1.4554 1.2650 1.3650 0.4908 0.4898 0.4883 2.5903 2.2454 2.4244 ¶¸{ 0.7773 0.0329 0.0260 0.0307 0.0035 0.0032 0.0032 0.3053 0.2125 0.2584 ¸{ 0.8057 0.1816 0.1579 0.1705 - - - - - -

In the presence of unknown shape parameter�, from Table 3 and Table 4 it is seen that

the MLEs of scale parameters; � = 1,2, … ,�, the reliability characteristics and hazard rate are

getting estimated closely to their true values. However the convergence rate is slow compared to

the convergence rate when shape parameter� is known. Perhaps, it may be the effect of estimate

of unknown shape parameter�. Further we can say, slightly large sample size is required than

what we consider for the estimates to reach their true values.

5.3 Testing of Hypothesis under GIED

The hypothesis given in equation 4.1 can be tested for GIED by simply substituting

equation (5.6-5.7) in (4.2-4.8) we have required equations. Details are omitted to avoid

repetitions. The likelihood equations are not mathematically tractable for known as well as

unknown shape parameter we use the Newton-Raphson method to obtain the estimate of

parameter . Here we deal with only known shape parameter.

18

We demonstrate the test procedure for � = 2 and � = 3 . We generate data under our

design for the parameter values under H1:� = 2.5, = 1.5, d = 1 and

H1:� = 2.5, = 1.9, d = 1.5, e = 1 respectively. Then carryout the test procedure as

suggested above. The procedure is repeated for the different choices of u and �∗. The results are

produced in the table 5 and table 6 respectively.

Table 5

Likelihood Ratio Test for testing áÛ: ×Ø = ×Õ = ×âãáØ: ×Ø ≠ ×Õ when c = Õ. Ö, ×Ø = Ø. Ö, ×Õ = Ø. Ô

ä å∗ ×p ×ØÒ ×ÕÒ ææáÛ ææáØ çÕ p-value

12 06 1.3358 1.6602 1.0435 19.89 21.17 2.58 0.1085

24 12 1.2360 1.4820 1.0064 55.40 57.20 3.62 0.0573

36 18 1.2892 1.5217 1.0778 107.65 109.76 4.22 0.0399

48 24 1.2828 1.4946 1.0838 146.05 148.55 4.99 0.0255

60 30 1.2782 1.5000 1.0698 188.27 191.75 6.95 0.0083

72 36 1.2719 1.4907 1.0679 258.68 262.65 7.95 0.0048

84 42 1.2668 1.5241 1.0299 295.13 301.63 13.00 0.0003

96 48 1.2579 1.5139 1.0208 348.38 355.89 15.02 0.0001

Table 6

Likelihood Ratio Test for testing áÛ: ×Ø = ×Õ = e = ×âãáØ: ×è ≠ ×é(è ≠ é = Ø, Õ, Ô) when c = Õ. Ö, ×Ø = Ø. ß, ×Õ = Ø. Ö, ×Ô = Ø

ä å∗ ×p ×ØÒ ×ÕÒ ×ÔÒ ææáÛ ææáØ çÕ p-value

24 08 1.5648 2.0149 1.6065 1.1429 58.28 61.84 7.13 0.0283

36 12 1.4565 1.9386 1.4481 1.4481 102.26 108.26 12.35 0.0021

48 16 1.6085 2.0605 1.6637 1.6637 152.59 159.92 14.68 0.0006

60 20 1.5570 1.9373 1.6765 1.6765 204.43 213.22 17.58 0.0001

72 24 1.5060 1.9868 1.5475 1.5475 253.28 266.30 26.04 0.0000

84 28 1.4195 1.9031 1.4848 1.4848 300.09 317.54 35.00 0.0000

96 32 1.4345 1.9097 1.4978 1.4978 356.75 376.75 40.14 0.0000

From the Table 5, we infer that the power the test is poor for small sizes. However as the

sample size becomes 36(total number failures observed irrespective type of systems) it exhibits

its power in identifying the alternative. It requires more sample size to detect small departure

from homogeneity. From Table 6, it can reveal that for comparing homogeneity of three systems,

as sample size becomes 24 it exhibits its power to identify magnitude of departure from

homogeneity. The consistency of the test is also inferred as sample size tends to 96 the p value

becomes almost zero up to three digits in both Table 5 and Table 6.

19

6. The cost function

Suppose we start life testing experiment with � machines each from m makes or brands.

Further, we shall assume that cost of failure of a system is constant, say ê , irrespective of the

brand. Since the experiment is terminated after observing total G=m G* where G

* is a fixed

number failures being observed on each brand of systems. Then the total cost due to failure

is� = mê G∗ which is fixed and pre-planned. In this scheme there are � separate sub

experiments in the sense that the way we observe the machines of type � does not depend upon

the failures of machines of type �ì, for � ≠ �ì. Let �=∗; � = 1,2… ,� denote the time of failure of �∗th machine of the �th type. Define �Díî by �Díî = �²ï(�=∗ , �=∗d, … , �=∗D), (6.1)

So that �Díî , denotes the duration of experiment for which the whole experiment under Type II

censoring lasts. Under this experimental scheme, the component of cost from the view point of

length of the experiment, is êe�Díî. Now we shall consider the cost related with total experimental time for all the units put

under test. Supposeêd is the cost per unit time associated with testing time of a machine,

irrespective of the brand. Let � , �d, … , �=∗ be the failure times of �∗ systems out of � systems

of type�, � = 1,2, . . . . , �. Then the total time the on test for the ith

sub experiment is �� +(� − 1)(�d − � ) + (� − 2)(�e − �d) + ⋯+ (� − �∗ + 1)g�=∗ − �(=∗, )h. Hence, the total

machine time under Type II censoring is �d where

�d = ∑ � + �d +⋯ .+(� − �∗ + 1)�=∗C (6.2)

The contribution of this cost is �dêd.

The fourth component of the cost relates to the total number of machines used in the

whole experiment, which under Type II censoring equal to �� = ð. This cost deals with the

actual procurement, storage and handling of the machines. We shall assume thatñ(ð), an

increasing functions of ð, is the cost of putting N machines on experiment. In some cases, ñ(ð) may increase slower than ð or ñ may even be discontinuous.

Finally, let êt denote the overhead cost associated with conducting the experiment. We

shall assume that êt represents general costs such as those associated with planning the

experiment, administrative and consultancy costs, etc. However, we make the assumption that êtis independent ofð, the total number of machines used in the experiment; such costs are

already included in ñ(ð). With the above assumptions, the actual cost of the experiment under

20

Type II censoring design is ê = êt + ê � + êd�d + êe�Díî + ñ(ð) (6.3)

We carryout Monte-Carlo simulation study for the choice of cost êt = 100, ê = 5, êd = 10, êe = 10 and ñ(ð) = 0.5ð for the cost effectiveness of Type II censoring scheme with different

censoring proportions. We consider � = 2 and � = 3 and results are given in Table 7 and

Table 8 respectively. Further, we fix total number of failures � and change the number of units

to be tested at each sub experiment. The results are given in Table 9.

It can be seen from Table 7 and Table 8, total time under experiment, duration of test and

cost associated with experiments in Type II experiments are comparatively less than no

censoring for m=2 as well as m=3. Therefore, in terms of cost, Type II censoring is

recommended for comparison of several systems.

From Table 9 and Table 10 we can see that as proportion of censoring increases when

total number of failures is fixed, the total time on test also increases linearly and duration time of

the experiment decreases. The facts can be seen from Figure 1 and Figure 2. Further, from Table

11 we can observe that as proportion of censoring increases when number of units to be tested is

fixed, the total time on test, duration of test and the total cost of experiment are decreasing while

mean square error of estimates are increasing.

21

Table 7

Comparative Table for total time under experiment and duration of experiment

when no censoring and 1/2 Censoring Ó = Õ,c = Õ. Ö, ×Ø = Ø. Ö, ×Õ = Ø. Ô

No Censoring ½ Censoring (å/Ó) ä å òÕ òóôõ ö òÕ òóôõ ö

12 12 18.0732 2.9559 442.291 9.9583 0.5272 382.291

24 24 35.4111 3.5768 753.879 19.8305 0.5302 633.879

36 36 53.5503 4.3074 1074.577 29.9269 0.539 894.577

48 48 71.3685 4.8756 1390.441 39.833 0.5388 1150.441

60 60 89.5124 5.4885 1710.009 49.7208 0.5372 1410.009

72 72 107.6014 5.9442 2027.456 59.7131 0.538 1667.456

84 84 125.1195 6.209 2337.285 69.8531 0.5423 1917.285

96 96 143.2841 6.7334 2656.175 79.4327 0.5393 2176.175

Table 8

Comparative Table for total time under experiment and duration of experiment

when no censoring and 1/3 Censoring Ó = Ô,c = Õ. Ö, ×Ø = Ø. Ö, ×Õ = Ø. Ô, ×Ô = Ø. Ù

No Censoring 2/3 Censoring (G/m) ä å òÕ òóôõ ö òÕ òóôõ ö

12 12 26.6254 2.6323 590.577 12.1862 0.3952 303.78

24 24 53.7196 3.607 1069.266 24.4007 0.4008 509.386

36 36 80.7912 4.2592 1544.504 36.8109 0.3974 706.117

48 48 106.9302 5.1795 2013.097 48.9774 0.3999 903.276

60 60 134.4159 5.5035 2489.194 61.1037 0.4023 1105.037

72 72 161.7802 6.293 2968.732 73.2893 0.4041 1304.934

84 84 187.9244 6.6052 3431.296 85.653 0.4035 1506.565

96 96 215.6138 6.4423 3904.561 97.9054 0.4036 1707.09

22

Table 9

Total time under experiment, duration of experiment and cost of experiment for fixed failures Ó = Ô,å = ØÕ, c = Õ. Ö, ×Ø = Ø. Ö, ×Õ = Ø. Ô, ×Ô = Ø. Ù ä òÕ òóôõ ö ä òÕ òóôõ ö

4 8.718 1.574 228.92 10 11.0219 0.4387 249.606

5 8.6535 0.8871 222.906 11 11.6579 0.4155 257.234

6 8.9423 0.6879 225.302 12 12.1182 0.3897 263.079

7 9.4497 0.5848 230.845 13 12.7585 0.377 270.855

8 10.0057 0.5146 237.203 14 13.1883 0.3582 276.465

9 10.6023 0.4851 244.374 15 13.8141 0.3506 284.147

Table 10

Total time under experiment, duration of experiment and cost of experiment for fixed failures Ó = Ô,å = ÕÙ, c = Õ. Ö, ×Ø = Ø. Ö, ×Õ = Ø. Ô, ×Ô = Ø. Ù(÷ = Ø − åÓä : proportion of censoring)

ä ÷ Ò dÒ eÒ � Ò(� ) �dÒ(�d) �eÒ(�e) r � (� ) r\(�d) -e�(�e) òÕ òóôõ ö

24 0.67 EV 1.5015 1.307 1.4182 0.5015 0.5048 0.5004 2.4031 2.0719 2.2465 24.2905 0.3987

502.892

MSE 0.0561 0.039 0.0454 0.0074 0.007 0.0068 0.1049 0.0745 0.0454 - - -

27 0.7 EV 1.5124 1.299 1.4118 0.5023 0.508 0.5025 2.4001 2.0614 2.2395 26.1149 0.3766

525.415

MSE 0.0462 0.0375 0.0416 0.0062 0.0069 0.0064 0.0879 0.0736 0.0792 - - -

30 0.73 EV 1.5011 1.3034 1.4169 0.506 0.5052 0.5 2.3866 2.0709 2.2486 27.7067 0.3537

545.604

MSE 0.0422 0.0326 0.0391 0.0058 0.0061 0.0059 0.0814 0.0641 0.0726 - - -

33 0.76 EV 1.5028 1.3214 1.3988 0.5054 0.4974 0.507 2.3887 2.0964 2.2241 29.3456 0.3339

566.295

MSE 0.0422 0.033 0.037 0.0058 0.0057 0.0059 0.0816 0.0608 0.0723 - - -

36 0.78 EV 1.5147 1.3058 1.4141 0.5004 0.5037 0.5006 2.4073 2.0757 2.2463 30.9336 0.3254

586.59

MSE 0.0395 0.0306 0.0366 0.0053 0.0056 0.0055 0.0745 0.0593 0.0665 - - -

39 0.79 EV 1.5053 1.3137 1.4065 0.5037 0.499 0.5028 2.3153 2.0909 2.2389 32.4637 0.3107

606.244

MSE 0.0374 0.0321 0.0308 0.0051 0.0056 0.0048 0.0716 0.0593 0.0585 - - -

42 0.81 EV 1.504 1.3115 1.4113 0.5042 0.5003 0.5013 2.3935 2.087 2.2442 34.0454 0.3009

626.463

MSE 0.0374 0.0206 0.0328 0.0051 0.0046 0.005 0.0715 0.0485 0.0614 - - -

45 0.82 EV 1.511 1.312 1.4141 0.5009 0.5 0.4999 2.4058 2.0879 2.2501 35.5085 0.2934

645.519

MSE 0.0333 0.0256 0.0299 0.0044 0.0046 0.0046 0.0627 0.0486 0.0563 - - -

48 0.83 EV 1.5126 1.3138 1.4092 0.5006 0.4992 0.5016 2.4068 2.0905 2.243 37.0428 0.2854

665.282

MSE 0.0353 0.0255 0.0305 0.0048 0.0046 0.0046 0.0668 0.0481 0.0502 - - -

23

Fig. 1:÷ vs total time under test

Fig. 2:÷ vs duration of test

Table 11

Maximum likelihood estimate of parameters, reliability and hazard rates and their efficiency measures, total

time under experiment, duration of experiment and cost of experiment for fixed failures Ó = Ô, ä = ÕÙ, c = Õ. Ö, ×Ø = Ø. Ö, ×Õ = Ø. Ô, ×Ô = Ø. Ù(÷ = Ø − å(CÓå∗)Óä : proportion of censoring)

å∗ ÷ Ò dÒ eÒ � Ò(� ) �dÒ(�d) �eÒ(�e) r �(� ) r\(�d) -e�(�e) òÕ òóôõ ö

2 0.92 EV 1.5465 1.3527 1.4369 0.4997 0.4918 0.5015 2.4098 2.1139 2.2425 14.5499 0.2195

343.69

MSE 0.1046 0.0768 0.0999 0.0120 0.0118 0.0106 0.1718 0.1266 0.1661 - - -

3 0.88 EV 1.5281 1.3185 1.4488 0.5014 0.5030 0.4932 2.4030 2.0776 2.2716 16.7091 0.2546

395.64

MSE 0.0817 0.0559 0.0977 0.0101 0.0010 0.0106 0.1446 0.1022 0.1313 - - -

4 0.83 EV 1.5199 1.3141 1.4306 0.5026 0.5045 0.4979 2.3987 2.0730 2.2716 18.4367 0.2820

443.19

MSE 0.0678 0.0527 0.0601 0.0088 0.0093 0.0086 0.1250 0.0994 0.1060 - - -

6 0.75 EV 1.5297 1.3126 1.4111 0.4983 0.5035 0.5045 2.4154 2.0761 2.2321 21.5628 0.3410

535.04

MSE 0.0635 0.0446 0.522 0.0079 0..80 0.0080 0.1114 0.0849 0.0985 - - -

8 0.67 EV 1.5040 1.3133 1.4128 0.5064 0.5023 0.5027 2.3846 2.0801 2.2387 24.3820 0.3986

623.81

MSE 0.0522 0.0403 0.0452 0.0071 0.0071 0.0073 0.1009 0.0757 0.0848 - - -

9 0.63 EV 1.5078 1.3057 1.4107 0.5042 0.5053 0.5032 2.3933 2.0703 2.2371 25.7072 0.4278

667.35

MSE 0.0501 0.0403 0.0452 0.0068 0.0071 0.0072 0.0967 0.0754 0.0841 - - -

10 0.58 EV 1.4993 1.3025 1.3971 0.5077 0.5063 0.5089 2.3798 2.0673 2.2171 27.1351 0.4611

711.96

MSE 0.0489 0.0359 0.0438 0.0068 0.0066 0.0071 0.0962 0.0738 0.0821 - - -

12 0.50 EV 1.5033 1.3120 1.4000 0.5060 0.5017 0.5069 2.3865 2.0825 2.2240 29.9872 0.5369

801.24

MSE 0.0471 0.0340 0.0396 0.0065 0.0060 0.0063 0.0920 0.0636 0.0778 - - -

24

7. Concluding remarks

In this article we obtain maximum likelihood estimates of parameters of m lifetimes

distributions using a generalization of Type II censoring scheme, proposed by Srivastava (1989),

under the assumption of generalized inverted scale family of distributions. We consider

generalized inverted family of exponential distribution (GIED) as an example for the more

general procedure we discuss. Then we obtain the estimates of parameters involved in the

distributions, reliability and hazard rate at a given point. The efficiency measures like MSE and

SE are simulated. The likelihood ration test is discussed to test homogeneity of lifetimes of

several brands. Finally we have studied the cost effectiveness of the Type II censored sampling

scheme, through Monte-Carlo simulation. Expressions given in this article can also be used for

generalized inverted half-logistic distribution and generalized inverted Rayleigh distribution etc.

Note: We have written a subroutine to compute the test statistics in R language. The source code

is available on request.

Acknowledgement:

The authors would like to thank the referee for their valuable comments and suggestions which

helped us to improve the earlier version of this paper. The second author would like to thank

University Grant Commission (UGC) for financial support. This research is funded by

University Grant Commission (UGC) for Minor Research Project (F.37-549/2009(SR)).

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