Optimal design of step stress partially accelerated life test under progressive type-II censored data with random removal for Frechet distribution Текст научной статьи по специальности «Математика»

S.Shahab, S.Anwar, A. Islam- ESTIMATION AND OPTIMAL DESIGN OF CINSTANT STRESS PARTIALLY ACCELERATED LIFE TEST FOR GOMPERTZ DISTRIBUTION WITH TYPE I CENSORING

OPTIMAL DESIGN OF STEP STRESS PARTIALLY ACCELERATED LIFE TEST UNDER PROGRESSIVE TYPE-II CENSORED DATA WITH RANDOM REMOVAL FOR FRECHET DISTRIBUTION

Sana Shahab, Sadia Anwar, Arif Ul Islam •

Department of Statistics & Operations Research (Aligarh Muslim University, Aligarh-202002, India)

e-mail: sana.shahb@gmail.com

ABSTRACT

In this article, progressive censoring and step stress partially accelerated life test are combined to develop a step-stress PALT with Progressively type-II Censored Data with the random removal. The removals from the test are assumed to have binomial distribution and uniform distribution and the life time of the testing products are considered to follow Frechet distribution. The parameters are estimated by using the maximum likelihood method and asymptotic confidence interval estimates of the model parameters are also evaluated by using Fisher information matrix. Statistically optimal PALT plans are developed such that the Generalized Asymptotic Variance (GAV) of the Maximum Likelihood Estimators (MLEs) of the model parameters at design stress is minimized. At the end, simulation study is performed to illustrate the statistical properties of the parameters.

KEYWORDS: Partially Accelerated Life Tests; Binomial Removal; Uniform Removal; Progressive Censoring; Maximum Likelihood Estimator; Generalized Asymptotic Variance

1 INTRODUCTION

When the product of high reliability is tested, the result of the some commonly used life test gives no or very few failures by the end of the test. In these types of the testing, the accelerated life testing (ALT) is used to obtain failures quickly. In such cases the testing is done at higher than usual use conditions. Three types of testing such as constant-stress, step-stress and progressive-stress are commonly used. In ALT, the mathematical model relating the lifetime of the unit and the stress is known or can be assumed. For detailed study of ALT see Nelson [1]. So as to, ALT data cannot be extrapolated to normal use condition. So, in such cases, partially accelerated life testing (PALT) is a more appropriate test to be used to estimate the statistical model parameters. Ismail et al. [2] introduced the Optimum Simple Time-Step Stress Plans for Partially Accelerated Life Testing with Censoring.

In many life tests, the experiment does not observe the failure times of all components. In such cases, the censored sampling arises. The most common censoring schemes are type-I censoring and type-II censoring. These two censoring schemes do not allow for units to be removed from the test at the points other than the final termination point. Moreover, there are some cases in which components are lost or removed from the test before failure. This would lead to progressive censoring. For progressive censoring see Balakrishnan and Aggarwala [3] and Balakrishnan [4]. Under the progressive type II censoring scheme, the experimenter puts n components on test at time zero. The first failure is observed at Yx and then Rx of surviving components is randomly selected and removed. When the second failure occurs at time Y2, R2 of surviving components is randomly

selected and removed and when (m-1)th failure is observed at the time, Rm_x of the surviving units are randomly selected and removed from the experiment, the experiment terminates when the

m-1

mth failure component is observed at and R = n - m R all removed. In this censoring

!=1

scheme R, R,............, R are all prefixed. However, in some practical experiments, these numbers

cannot be pre-fixed and they occur at random. Inference based on progressively Type II censored data is discussed by many authors. Yuen and Tse [5] considered the estimation problem for Weibull distribution under progressive Censoring with random removals. Yang et al. [6] statistically analyzed the Weibull Distributed Lifetime Data under Type-II Progressive Censoring with Binomial Removals. Wu [7] used progressively Type-II censored data with uniform removals to estimate the parameters of Pareto distribution. Ismail et al. [8] introduced the Optimal Design of Step-Stress Life Test with Progressively type-II Censored Exponential Data with binomial removals. Bander [9] estimated the maximum likelihood for Generalized Pareto Distribution under Progressive Censoring with Binomial Removals. Chang et al. [10] studied the progressive censoring with Random Removals for the Burr Type XII Distribution.

2 THE MODEL AND TEST METHOD

2.1 The Frechet Distribution

case of the generalized extreme value

The Frechet distribution is a special

distribution. The generalized extreme value (GEV) distribution is continuous probability distributions developed within extreme value the Gumbel, Frechet and Weibull families also known as type I, II and III

a family of theory to combine extreme value

distributions. The lifetimes of the test items are assumed to follow a Frechet distribution. The probability density function (pdf) of the Gompertz distribution is given by

f (t) = ada t-a-1 exp

(1)

And the cumulative distribution function is given by

F (t) = exp

,JH r

The survival function of the Frechet distribution is given by

(2)

F (t ) = 1 - exp

f r^-a\

—

2.2 Assumptions

n identical and independent units are put on the life used condition and the lifetime of each testing unit follows Frechet distribution.

The test is terminated at the mth failure, where m is prefixed (m < n).

Each of the n units is first run under normal use condition. If it does not fail or remove from the test by a pre-specified time t , it is put under accelerated condition.

■

■ At the ith failure a random number of the surviving units, R,i = 1,2,......m -1, are randomly

selected and removed from the test. Finally, at the mth failure the remaining surviving units

m-1

R = n - m R are all removed from the test and the test is terminated.

i=1

■ The lifetime, say Y, of a unit under SS-PALT can be written as

_fT if T >t

Y = {t + (T-i)/p if T <t (3)

where T is the lifetime of the unit under normal use condition, t is the stress change time and ft is the acceleration factor; ft > 1. Therefore, the pdf of Y can be written as in the following form

Therefore probability density function (pdf) of Y can be written as 0 y < 0

f(y) = \f1 (y) 0 < y <T

f2 (y) y >T

f (y )=

a6ay ~a-1 exp

( / \-a\ {$)

adap(t + p{y-t)) a 1 exp

■ + p( y-t)

9

y < 0

0 < y <t y>t

(4)

0

F (y ) =

0

exp exp

i f I y 1

v

f

0

J

t + p( y-t)

e

y < 0

0 <y <t y >t

(5)

3 MAXIMUM LIKELIHOOD ESTIMATION

3.1 Parameter Estimation with the Binomial Removals

The number of units removed from the test at each failure time follows a binomial distribution and any individual unit being removed is independent of the others but with the same

f ,-1 \

probability p. That is, R1 ~ bino(n - m,p) and for i = 2, 3,......, m -1, Ri ~ bino

E rj, P

n - m - E r

v j=1 J

and r =

rm = n - m - ri - r2 - •

— r

' m-1 •

Let (y,r,51;,52,),i = 1,2,.......m denote the observation obtained form a progressively type-II

censored sample with random removals in a step-stress PALT. Here y^ < y^ <..........< y^.

Thus for the progressive censoring with the pre determined number of the removals

R = (R = r,.........., R-i = rm-i), the conditional likelihood function of the observations

y = {(y, r, ^i;, 52,), i = 1,2,........m.}can be defined as follow

L(yi;a,p,0,51,,5, | R = r) = ft{[/1 (yt )(F(y,))r J" f (y,)(F2(y,))r f2i) (6)

i=1

L ( y;0,a, ßAi Sn/R = r ) = ^

i=1

a6ay— exp

/ \-a

y±_ e

-a

1 - exp

f , -,-a\\r

fyi 1

e

JJ

aeaß{z + ß{y-T)) a1 exp

-a r, ^-ar

e

exp

v v

T + ß(y. -t)

e

JJ

(7)

The number of units removed at each failure time follows a binomial distribution such that

p(r = ri ) =

C n - m^

v ri J

Pr (1 - P)n

And for i=2, 3,......,m-1

PR = rI\RI-1 = r-1,........., R = r1 ) =

-1

n - m -y r

j=1

j

Pri (1 - p)n-m-Eri

where 0 < r < n - m - (r + r2 +........+ r_i). Furthermore, suppose that Ri is independent of Yi for

all i. Then the joint likelihood function can be found as

L(y,; a, p, 0, p, 61,, 52,) = L (y,; a, p, 0, p, 5,, 52, | R = r)p(R, p) (8)

where P (R, p) is the joint probability distribution of R = (rj, f\,...rm ) and is given by

m

S

r

P (R p) = P (Rm-1 = rm-1, Rm-2 = r^- R1 = r1)

= P (Rm-1 = rm-1/ Rm-2 = rm-2,-R1 = r1 )x P (Rm-2 = rm-2l Rm-3 = rm-3,-R1 = r1) Xp(Rm-3 = rm-31Rm-4 = rm-2,-R1 = r1 )x -P(R2 = r2/R1 = r1)P(R1 = r1)

(n - m) pZ51 i (1 - p)(m-1)(n-m)-^^(m-,)r

(n - m-h,=1 rijh,=1 ri!

(9)

Now by substituting L(y; a, P, 0, p, , d2j | R = r) and P(R, p) from the equation (7) and (9) in (8) we get the likelihood function

L (y,0, P, p,d d ) = n

i=1

a0 y;- exp

1 - exp

v v

JJ

aQa P(t + P(-T))-a-1exp

0

exp

v v

t + p( y-t)

0

vaY\r

JJ

(10)

( n - m )!

( m-1 1 m-1

n -m -h r- ! n r!

m-1

phr (1 -p)(m-1)(n-m)-h(m-i

v

i=1 J i=1

The log-likelihood of the above equation is given by

log L =

m„ mu mu / ,

m loga+ma log 0 - (a +1) h] log y-- h (y- /0)-a + ri h log (1 - exp (- (y-/

i=1 i=1 i=1 v

m {

m

i=1

log P-(a +1)2: log (T + P( y-T)))-h: rT + P( y-t)^

i=1

r f

t + p( yi-t) 0

+ri h log i=1

1 - exp

v v

0

+

JJ

m-1

(m-1)(n-m)- h (m-i)r

i=1

+ log C1 +hrilog p

i=1

log (1 - p )

The maximum likelihood estimators of P and 0 can be derived directly by maximizing the equation (7) instead of (10) because P(R, p) does not involves the parameter P and 0 . Similarly the binomial parameterp does not depend on L(y; a, P, 0, p, 8U, 62; | R = r), hence the MLE ofp can be

m

d

found by maximizing P(R; p) directly. Thus, the maximum likelihood estimates (MLEs), of p and 0 can be found by solving the following equations:

( r

-a ( yi 1

ry exp 1 11

aiogZ = ma-a0a-i ¿-a ^a-1 y_ 50 p y y

0

J

-a

-(y1

J

1 - exp ri (T+p(y, -t)) a exP

'"a '"a

a0a-1 y (t + P( y,-T))-a+a0a-1 y

( f ni (11)

' T + P(y,-t) ^

e

,=1

i=1

1 - exp

-a

fT + p( y,

e

= ma. -(a + 1)y y'-T x+a0ay-y^T

dP P V 'y T + p(y,-t) y (T + p(y,-t))

a+1

f

i (yi- T)(T + p(yi- T)) a 1 exp

-a0ay-i=1

t + p( y,-t)

0

v

1 - exp

a

^t + P( y,-t)^ 0

(12)

Independently, the MLE of the binomial parameter p can be obtained by solving the following equation:

m-1

5 iog L_m^ r±_ dp ,=1 p

(m- 1)(n-m)- y (m-i)r,

i=1

1 - p

(13)

Therefore we get p from equation (13)

m-1

Hr

i=1

P =

m-1 m-1

y r + (m -1) (n - m)- y (m - i) r

i=1 i=1

3.2 Estimation with the Uniform Removal

The number of units removed from the test at each failure time follows a uniform discrete

( i-1 1

distribution. That is, R ~ Unif (0,n-m) and for i = 2, 3,......, m -1, R ~ Unf 0,n -m-y r.

V j=1 J

and rm = n - m - r - r2 -.

. - rm-1 .Such that,

m

PR = '1 ) =-~T

n-m +1 And for i=2, 3 ...m-1.

PR = 'i\R-1 = 1,........., R = ' ) =

i-1

n-m- ^ ' +1

j=i

where p(r), the joint probability distribution of R = (ri, ri,...rm ) and is given by

p(r = r ) =-1m-i--(14)

n-m- ^ r +1

i=1

where 0 < ' < n-m-(' + ^ +........+ ^),i = 1,2,..............m-1.

1

It is clear that P(R=r) does not depend on the parameters ß and 9 and, hence the maximum likelihood estimators can be derived directly by maximizing the equations (7) and then solving the equations (11)

4 FISHER INFORMATION MATRIX & ASYMPTOTIC CONFIDENCE INTERVAL

The asymptotic variance-covariance matrix of the ML estimators of the parameters can be approximated by numerically inverting the Fisher-information matrix F and The Fisher information matrix is obtained by taking the negative second partial derivatives of the log-likelihood function and for the binomial removal it can be written

F =

a2/ a2/ a2 /

ae2 aeaß aeap

a2/ a2/ a2 /

aßae aß2 aßap

a2/ a2/ a/

apae apaß ap2

And for the uniform removal, fisher information matrix can be written as

F =

a2/ a2/

ae2 aeaß

a2/ a2/

aßae aß2

Elements of Fisher Information matrix are

•^2 1 T m m

^ -ma-a(a "OE^-2 +a(a-1)ea-2 E (r+ß(y,— t))"

se2 e i=1

—a

Wi exp

+ aE" i=1

r , x-a

y_i e

i=1

a-2 ,.-a

(a- 1)ea-2 -y"

Wi 2a exp

1 - exp

f , ^-a\

y_i e

j

-a2e2a-2 EE-i=1

-2

1 - exp

f s N-a^

y± e

j.

ri (T+ß(yi -t)) aexp

+a

2:-i=1

a

^T + ß( y-r)Va

e

(a -1) ea-2 + ae2a-2 (t + ß (y - t))"

1 - exp

T + ß( yi-t)

v-a^

e

ri (T + ß(yi -t)) 2a 1exp

-a2e2a—2 E-i=1

a

2

fT + ß( y,

e

1-exp

a

fT + ß(y, -t)Va

e

2

S log L ma

( yi-t)"

2 =-^f + (a + 1)E 2

Sß2 ß2 V ^ [T + ß( y-t)] 2

-a(a + 1)eaEE

( yi-T

i=1 (T + ß(yi -t))

a+2

f

ri (yi -t)2 (T + ß(yi -t)) a 2exp

-aeaE-

T + ß( y/-t)

e

-(a + 1) + aea

-a Y

fT + ß( yi Va

e

i=1

1 - exp

T + ß( yi-t)

-a

e

ri (yi- t)2 (t+ß (yi- t)) 2a 2 exp

-aE-

-a

-2

fT + ß( y-T)Va

e

s2 log L _ m ri

T~2 = E 2 SP i=1 p

i=1

m-1

1 - exp

r T + ß(yi-t^ e

(m - 1)(n - m )- E (m - i ) ri i=1

( p—1)2

m

m

m

2

m

m

2

2 2 m

ajogz = ajogz = aV_1 z, ( _T))-a-1 (^ _r)

dedp apae ZV n '

i=1

r (y _r)(r + p(y _r))_a_ 1 6a_1 exp

a2 Z" . =1

e

a

1+ea(r+p( y _r))"

1_exp

+p( y _r) e

a

ne2a^ (y _r)(r + ^(y _r))_2a_1 exp

+ a2

_2

a

i =1

1_exp

a

-ft) ,

e

a 2ln L a 2ln L a 2ln L a 2ln L

apae aep dpdp dfidp

= 0

The variance covariance and covariance matrix of the parameter for the binomial removal can be written

Z =

a2/ a2/ a2/

ap2 apae apap

a2/ a2/ a2/

aeap ae2 aeap

a2/ a2/ a/

apap apae ap2

AVar(p) ACov(pe) ACov(pp) ACov(ep) AVar (e) ACov(epJ ACov(pp) ACov[pe) AVar(p)

And for the uniform removal case it can be written as

Z =

a2 / a2/

ap2 apae

a2 / a2/

aeap ae2

AVar (p) ACov(pe) ACov (ep) AVar (e)_

The 100(1 -£)% asymptotic confidence interval for e, p and p can be written as

1 + z avar(e) p±Z ^AVar(p)

and

p + Z .. jAVar (p)

m

2

_1

2

2

2

5 OPTIMUM TEST PLAN

The present criterion by which one can choose the optimal value of t is based on the determinant of the Fisher's information matrix. Maximization of that that determinant is equivalent to minimization of the generalized asymptotic variance (GAV) of the MLE of the model parameters. The GAV is the reciprocal of the determinant of the Fisher's information matrix F that is

GAV = A

So, the optimal value of t is chosen in such a way that the determinant of the Fisher's information matrix F is maximized and then the GAV is minimized. This is called the D-optimality criterion.

6 SIMULATION STUDY

In order obtain MLEs of p, 0 and p and to study the properties of these estimates through Mean squared errors (MSEs),) and the coverage rate of asymptotic confidence intervals for different sample sizes, a simulation study is performed. Moreover, we will determine the optimal stress change time which minimizes the generalized asymptotic variance of the MLE of parameters. To perform the simulation study, we used the following steps

a) First specify the value of n and m.

b) The value of the parameters are chosen to be a = 2.87,0 = 3.02,p = 2.62,p = 0.67,t = 3.5.

c) Generate a random sample with size n and censoring size m with random removals, r, i = 1,2,.........m -1 from the random variable Y given by (4).

f i-1 ^ f i-1 ^

rj

d) Generate a group value Rt ~ bino n - m - y r, p and also Rt ~ Unif 0, n - m - y

V j=1 J V j=1

where, r = n - m - r - r -............- ^ i

5 m 12 m-1.

e) For different sample sizes n= 20, 60, 80,100 and 120, compute the ML estimates.

f) The mean squared error (MSE), the coverage rate of the 95% confidence interval of parameters and Bias are obtained associated with the MLE of the parameters, optimal value of t and also the Optimal GAV of the MLEs of the model parameters are obtained numerically for each sample size.

Table 1(i); Simulation study results with Binomial Removals for a = 2.87,0 = 3.02, p = 2.62, p = 0.67, t = 3.5.

Binomial case 95% Confidence interval

n m coverage

t F 1

0 P p CP- CP- CPp cpp

on 9 2.983612 4.889763 U.897212 U.92U39 U.9U121 U.9U313 3.8746 1.336

2U 19 2.977351 4.886342 U.895871 U.92141 U.9U341 U.9U583 3.8786 1.465

9 2.953811 4.86783U U.847492 U.92156 U.9U547 U.9U876 3.7424 1.493

19 2.9U7351 4.858361 U.8U8313 U.92183 u.9u645 u.9u963 3.7413 1.5U2

An 29 2.895634 4.8889U7 U.8U4721 U.9219U U.9U673 U.9U991 3.74U9 1.573

6U 39 2.893631 4.68367U U.8UU838 U.92199 U.9U843 U.91234 3.7289 1.638

49 2.89U731 4.642846 U.799743 U.92213 u.9u863 U.91425 3.7263 1.693

59 2.865341 4.619843 U.795982 U.92254 U.91633 U.91473 3.7U84 1.699

9 2.862563 4.983741 U.769371 U.92261 U.9174U U.91533 3.7U52 1.712

19 2.86u726 4.738421 U.766932 U.92275 u.91834 U.91642 3.5566 1.734

29 2.846535 4.597361 u.759826 u.9328u u.91876 u.91735 3.55u3 1.782

sn 39 2.818732 4.55836 U.685821 U.93289 U.91899 U.91841 3.4371 1.791

8U 49 2.815721 4.387461 U.588763 U.93385 U.92934 U.91893 3.5778 1.832

59 2.687631 4.334524 U.559831 U.93481 U.92997 u.91934 3.U766 1.854

o9 2.665434 4.284712 u.53u841 u.93541 u.93u13 u.91953 3.U355 1.871

79 2.646213 4.U97361 U.5U8349 U.94753 U.93U84 U.91979 2.7UU9 1.889

9 2.619736 3.869763 U.487354 U.9513U U.93U99 U.91991 2.7987 1.92U

19 2.6U5531 3.898731 U.379421 u.95353 U.93194 U.92421 2.7354 1.943

29 2.576435 3.757365 U.339741 u.95365 u.93245 U.92632 2.7354 2.132

39 2.557261 3.728371 u.336821 u.95475 u.93385 U.92713 2.7U47 2.223

1 nn 49 2.397251 3.68651U U.3U9431 U.95573 U.93642 U.92795 2.7U28 2.264

1UU 59 2.3U736U 3.428761 U.3U4814 U.95752 U.93752 U.92846 2.7U11 2.349

o9 2.152841 3.u87361 U.233193 u.95883 U.9384U u.92896 2.7UU6 2.382

79 2.119423 2.787361 u.23u341 u.95992 U.94671 u.92888 2.6937 2.467

o9 2.U94381 2.629834 U.178287 u.96862 u.94689 U.92913 2.6795 2.484

99 2.UU7378 2.198347 U.145931 U.97432 U.94778 U.92999 2.6654 2.961

9 2.UU3841 2.UU7973 U.U89831 U.97652 U.95U32 U.93252 2.6473 2.999

19 1.997763 1.775983 u.u8972u u.97743 u.95u75 u.93419 2.5531 3.u12

29 1.947345 1.999631 u.u84566 u.97832 u.95174 u.93555 2.5139 3.u58

39 1.917371 1.929831 u.u59741 u.97865 u.95195 u.9386u 2.4961 3.184

49 1.886351 1.9U9832 U.U57631 u.97921 u.95348 U.94642 2.4741 3.452

i on 59 1.8U7363 1.9U5987 U.UU5574 U.98134 U.95534 U.94875 2.3961 3.872

12U 69 1.743251 1.889874 U.UU5174 U.98353 U.95613 U.95875 2.3756 3.891

79 1.797356 1.899642 u.uu3752 u.98463 u.95732 u.95999 2.3367 3.928

o9 1.586352 1.858943 u.uu1734 u.98561 u.95822 u.96641 2.32u5 3.963

99 1.559736 2.81874 u.uu1538 u.98673 u.95913 U.96831 2.1858 3.971

1U9 1.5372651 1.77321 U.UU9634 U.98751 u.95989 u.97654 2.U751 3.984

119 1.386345 1.79731 U.UU2752 U.98462 U.96143 U.97943 2.U356 3.991

Table 1(ii); Simulation study results with Binomial Removals for a = 2.87,0 = 3.02, p = 2.62, p = 0.67, t = 3.5.

n m Bias ^ Bias ^ Bias p

on 9 0.006691 0.009879 0.089431

20 19 0.006687 0.009773 0.067909

9 0.005982 0.005928 0.063989

19 0.005791 0.005721 0.047298

An 29 0.005194 0.004823 0.045901

60 39 0.003791 0.004594 0.028432

49 0.003913 0.003909 0.018931

59 0.003492 0.003791 0.015986

9 0.003389 0.003588 0.014982

19 0.002780 0.002791 0.011955

29 0.002678 0.002279 0.011577

sn 39 0.002569 0.002254 0.008793

80 49 0.002378 0.002093 0.003985

59 0.002354 0.002056 0.003416

69 0.002334 0.001973 0.001567

79 0.001682 0.001671 0.001391

9 0.001494 0.001498 0.001198

19 0.001475 0.001289 0.001203

29 0.000971 0.001182 0.001982

39 0.000849 0.000678 0.000689

1 nn 49 0.000692 0.000451 0.000486

100 59 0.000578 0.000381 0.000198

69 0.000387 0.000078 0.000139

79 0.000234 0.000029 0.000116

89 0.000209 7.35x10-5 0.000104

99 0.000209 3.74x10-5 0.000101

9 9.87x10-5 9.59x10-6 0.000094

19 9.31x10-5 9.38x10-6 0.000047

29 7.52x10-5 9.27x10-7 0.000029

39 4.39x10-5 5.62x10-7 0.000016

49 2.58x10-5 5.39x10-7 8.79x10-5

i on 59 5.81x10-7 2.76x10-7 5.91x10-5

120 69 4.79x10-7 5.73x10-8 4.13x10-5

79 3.35x10-7 4.79x10-8 9.95x10-6

89 2.79x10-7 3.79x10-8 8.88x10-6

99 2.12x10-7 1.87x10-8 6.83x10-6

109 2.07x10-7 8.67x10-9 4.67x10-6

119 6.87x10-8 5.19x10-9 1.39x10-6

Table 2; Simulation study results with uniform removals for a = 2.87,0 = 3.02, p = 2.62, p = 0.67 and r = 3.5

MLE 95%Confidence

n m Interval coverage Bias- Bias- * T

0 h CP- CP- P 0 P I f

on 9 3.94710 4.98997 0.91018 0.87329 0.097631 0.009959 5.9829 1.009

20 19 3.92741 4.98975 0.91317 0.87440 0.093859 0.009936 5.9693 1.119

9 3.91931 4.99742 0.91712 0.87489 0.073185 0.009368 5.8746 1.239

19 3.91673 4.95836 0.91738 0.87511 0.068166 0.008489 5.7837 1.265

An 29 3.91391 4.98890 0.91740 0.87546 0.057217 0.008299 5.4728 1.363

60 39 3.91832 4.99888 0.91765 0.87599 0.043608 0.006943 5.4643 1.371

49 3.67721 4.99817 0.91785 0.87632 0.041735 0.006509 5.4489 1.398

59 3.65360 4.98736 0.91889 0.87790 0.030360 0.006297 5.4098 1.403

9 3.48721 4.95742 0.91940 0.87793 0.018429 0.006098 5.4071 1.412

19 3.46831 4.92646 0.91985 0.87888 0.009588 0.006024 5.3064 1.425

29 3.47974 4.90896 0.91990 0.89999 0.005981 0.005949 5.1984 1.451

sn 39 3.29346 4.90693 0.91998 0.91354 0.005945 0.005439 5.1697 1.463

80 49 3.25312 4.78931 0.92011 0.91616 0.005674 0.005190 5.0983 1.470

59 3.21038 4.75726 0.92042 0.91659 0.005395 0.004987 5.0582 1.623

69 3.09531 4.73842 0.92086 0.91923 0.005194 0.004956 5.0193 1.674

79 3.09836 4.73571 0.92090 0.91987 0.004004 0.004547 4.9875 1.680

9 3.05647 4.59831 0.92119 0.92156 0.003598 0.003757 4.8572 1.731

19 3.05563 4.56828 0.92187 0.92181 0.003283 0.002899 4.6365 2.643

29 3.03844 4.29784 0.92319 0.92615 0.003093 0.002875 4.5324 2.684

39 3.03573 4.27641 0.92355 0.90842 0.003062 0.002598 4.5084 2.299

1 nn 49 3.01963 4.25989 0.92488 0.92476 0.000999 0.002429 4.3948 2.384

100 59 2.98450 4.23791 0.92556 0.92589 0.000739 0.002125 4.2874 2.715

69 2.95741 4.09912 0.92580 0.92757 0.000721 0.001356 4.2683 2.764

79 2.93474 4.06983 0.92666 0.92783 0.000699 0.000896 4.2543 2.754

89 2.91093 4.06728 0.92691 0.92791 0.000570 0.000597 4.1974 2.794

99 2.90983 4.01734 0.92921 0.92798 0.000398 0.000496 4.0746 2.790

9 2.90657 3.93837 0.92957 0.92799 0.000096 0.000063 3.9973 2.917

19 2.90633 3.90973 0.93421 0.93523 0.000068 0.000039 3.8374 2.932

29 2.90435 3.68347 0.93511 0.92645 0.000036 0.000019 3.6467 2.938

39 2.90313 3.65531 0.92585 0.92685 0.000019 0.000013 3.5621 2.972

49 2.86414 3.48327 0.92881 0.92985 9.96x10-5 9.02x10-5 3.4788 2.977

1 on 59 2.84531 3.45177 0.93183 0.93145 9.28x10-7 8.84x10-5 3.1845 2.999

120 69 2.84313 3.42641 0.93371 0.93351 9.09x10-7 8.36x10-5 3.1477 3.031

79 2.75443 3.28421 0.93612 0.93831 8.45x10-7 7.79x10-5 3.1248 3.074

89 2.73249 3.26947 0.93722 0.935145 8.33x10-7 7.34x10-5 3.0983 3.187

99 2.71734 3.24874 0.94513 0.93690 7.84x10-7 6.78x10-5 3.0387 3.191

109 2.59931 3.19834 0.94721 0.93800 6.69x10-7 6.34x10-5 3.0276 3.284

119 2.51677 3.15893 0.94882 0.93641 4.05x10-7 5.99x10-6 2.9975 3.291

7 CONCLUSION

This paper considers the SS-PALT under type-II progressive censoring with Binomial and uniform removals assuming frechet distribution. Comparison between both removal are shown. The Newton-Raphson method is applied to obtain the optimal stress-change time t* which minimizes the GAV.

The numerical study for obtaining the optimum plan for binomial removal is tabulated in table 1 for different sample size and table 2 describes uniform removal for possible values of scale and shape parameters. From the above results it is easy to find that for the fixed values of the parameters, the error and optimal time decrease with increasing sample size n.

Performance of testing plans and model assumptions are usually evaluated by the properties of the maximum likelihood estimates of model parameters. Hence from the numerical result we can conclude that estimates of binomial are more stable with relatively small error with increasing sample size. Therefore, the test design obtained here is robust design and work well for binomial removal.

As a future work, this study can be extended to explore the situation under type-I progressive censoring

REFERENCES

1. Nelson, W., 199U. Accelerated Life Testing: Statistical Models, Data Analysis and Test Plans. John Wiley and Sons, New York.

2. Aly, H.M. and Ismail, A.A., 2UU8. Optimum Simple Time-Step Stress Plans for Partially Accelerated Life Testing with Censoring, Far East Journal of Theoretical Statistics. Vol. 24, no.2, pp.175 - 2UU.

3. Balakrishnan, N. and Aggarwala, R., 2UUU. Progressive Censoring: Theory, Methods and Applications. Birkhauser, Boston, Mass, USA.

4. Balakrishnan, N., 2UU7. Progressive censoring methodology: an appraisal, Test, vol. 16, no. 2, pp. 211-259.

5. Yuen, H. K., and Tse, S. K., 1996. Parameters Estimation for Weibull distributed Lifetime under progressive Censoring with random removals. Journal of Statistical Computation and Simulation, vol. 55, pp. 57-71.

6. Tse, S.K., Yang, C. and. Yuen, H.K., 2UUU. Statistical Analysis of Weibull Distributed Lifetime Data under Type-II Progressive Censoring with Binomial Removals. Journal of Applied Statistics, vol. 27, no. 8, pp. 1U33-1U43.

7. Wu, S.J., 2UU3. Estimation for the Two Parameter Pareto Distribution under Progressive Censoring with Uniform Removals. Journal of Statistical Computation and Simulation, vol. 73, no. 2, pp. 125-134.

8. Ismail, A. and Sarhan, A. M., 2UU9. Optimal Design of Step-Stress Life Test with Progressively type-II Censored Exponential Data. International Mathematical Forum, vol. 4, no. 40, 1963 - 1976.

9. Bander Al-Zahrani, 2012. Maximum Likelihood Estimation for Generalized Pareto Distribution under Progressive Censoring with Binomial Removals. Open Journal of Statistics, 2, pp.420-423.

10. Wu, S.J., Chen, Y. J. and Chang, C.T., 2UU7. Statistical Inference Based on Progressively Censored Samples with Random Removals from the Burr Type XII Distribution. Journal of Statistical Computation and Simulation, vol. 77, no. 1, pp. 19-27.