Reliability Estimation for the Distribution of a k-Unit Parallel System with Rayleigh  Distribution as the Component Life Distribution

K. G. Potdar; D. T. Shirke

doi:10.17577/IJERTV2IS80806

Volume 02, Issue 08 (August 2013)

Reliability Estimation for the Distribution of a k-Unit Parallel System with Rayleigh Distribution as the Component Life Distribution

DOI : 10.17577/IJERTV2IS80806

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Authors : K. G. Potdar, D. T. Shirke
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Volume & Issue : Volume 02, Issue 08 (August 2013)
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Reliability Estimation for the Distribution of a k-Unit Parallel System with Rayleigh Distribution as the Component Life Distribution

K. G. Potdar

Department of Statistics, Ajara Mahavidyalaya, Ajara, Dist Kolhapur, India-416505,

D. T. Shirke

Department of Statistics, Shivaji University, Kolhapur, India-416004,

Abstract- A k-unit parallel system having component lifetime distribution to be Rayleigh is considered. Based on the progressively Type-II censored sample, the maximum likelihood estimator (MLE) of the scale parameter of the Rayleigh distribution is derived and is used to estimate reliability function. EM algorithm is used to obtain MLE. Asymptotic confidence interval for the scale parameter and reliability function is constructed. Confidence interval based on the log- transformed MLE is also constructed. Simulation study is conducted to investigate performance of estimates and confidence intervals. An example with real data is presented for illustration.

Index Terms- Progressively Type-II censoring, Rayleigh distribution, EM algorithm, MLE, Reliability, confidence interval, coverage probability.

AMS subject classification 62N01, 62N02, 62N05.

ACRONYMS

CI Confidence Interval

CDF Cumulative Density Function EM Expectation Maximization MLE Maximum Likelihood Estimate MSE Mean Square Error

PDF Probability Density Function

X(i) Order Statistics from a progressively censored sample of size m.

Ri Number of surviving units withdrawn from the experiment after ith failure.

L Log-likelihood function of all n observations. L() Likelihood function of observed data.

Lc Log-likelihood function of complete data. Ix() Observed Fisher information.

Iw() Complete Fisher information. Iw|x() Missing information.

R(t) Reliability of system at time t.

In a life testing experiment, censoring is common practice because of various restrictions on data collection such as time limit, cost etc. In such a situation we remove some observations in the experiment. Such data is called censored data. Censoring is broadly classified into two types; Type-I and Type-II censoring.

Type-I censoring is related with time. In this type, an experiment continues up to a pre-determined time T. Units having failure time after time T are not observed. Here, lifetime will be known exactly only if it is less than T. For example, suppose n units are put on test, but decision is made to terminate the test after time

T. In this experiment life times will be known exactly only for those units that fail before time T. In Type-I censoring, the number of exact life times observed is random.

NOTATIONS

n Sample size.

m Number of observed failures in a Type-II censored sample.

Scale parameter of the Rayleigh distribution.

MLE of .

g(y) PDF of the Rayleigh distribution. G(y) CDF of the Rayleigh distribution.

f(x) PDF of the life distribution of k unit parallel system.

F(x) CDF of the life distribution of k unit parallel system.

Type-II censoring is related with number of failures, that is, experiment continues up to the pre- determined number of failures. For example, in life testing experiment, suppose n units are put on test, but instead of continuing until all n units have failed; the test is terminated at the time of the mth (m n) unit failure. In case of Type-II censoring, the number of exact life times observed is fixed.

Based on the time epochs for removals of units from the experiment, the censoring is further classified in to two types, such as conventional censoring and progressive censoring. In conventional Type-I and Type- II censoring, units may not be removed before terminal point. In progressive censoring scheme, units may be

removed at different stages rather than at terminal point. Progressive censoring scheme is applied in both Type-I and Type-II censoring schemes.

=

=

In progressive Type-I censoring scheme, m censoring times T1, T2,, Tm and R1, R2,, Rm are fixed in advance. Since 1 = . At time T1 remove R1 units, at time T2 remove R2 units and so on. The experiment terminates at time Tm with Rm units still surviving. In progressive Type-II censoring scheme, suppose n units are put on test. The number m and R1, R2,.,Rm are fixed prior to the test. At the first failure R1 units are removed randomly from remaining n-1 units.

At the second failure R2 units are removed randomly from remaining n-2-R1 units and so on. At the mth failure all remaining Rm units are removed. Here, we observe failure time of m units and remaining n-m units

are removed at different stages of experiment. In conventional Type-II censoring scheme R1=R2== Rm-1 = 0 and Rm= n-m. In this paper, we consider the progressive Type-II censoring scheme.

Progressive Type-II censoring scheme for various lifetime distributions has been discussed by number of researchers. Cohen [1] studied MLE of the parameters of exponential and normal distribution for progressively Type-II censored samples. Mann ([2], [3]) considered Weibull distribution with progressive censoring. Balkrishnan ([4], [5] and [6]) discussed inference for the scaled half-logistic, Gaussian and extreme value distributions under progressive Type-II censoring scheme respectively. Ng [7] studied parameter estimation for modified Weibull distribution under progressively Type-II censored samples. Balkrishnan and Aggarwala [8] gave details about progressive Type-II censoring scheme.

Balkrishnan [9] studied various distributions and inferential methods for progressively censored data. Pradhan [10] considered point and interval estimation of a k-unit parallel system based on progressive Type-II censoring scheme with exponential distribution as the component life distribution. Chein and Balkrishnan [11] discussed consistency and asymptotic normality of MLE based on progressive Type-II censored samples.

Iliopoulos and Balkrishnan [12] studied likelihood

Dempster et al. [17] introduced EM algorithm. They presented maximum likelihood estimation for incomplete data. Mclachlan and Krishnan [18] gave more details about EM algorithm. Little and Rubin [19] discussed EM algorithm for exponential family of distributions. Pradhan and Kundu [20] used EM algorithm to estimate parameters of generalized exponential distribution under Type-II censoring scheme. Ng et al. [21] used EM algorithm to estimate parameters of lognormal and Weibull distributions under Type-II censoring scheme. In this article, we used EM algorithm for estimation of the parameters of a k-unit parallel system based on progressive Type-II censoring scheme when component lifetime follows Rayleigh distribution with scale parameter .

=

=

Parameter estimation is based on the life times of the system. We assumed that n items put on test and failure times of 1 = items are censored. Lifetimes of these censored items are unknown. We consider this data as missing and used EM algorithm to

compute MLE. Louis [22] presented technique for computing observed Fisher information within EM algorithm framework. We used this technique to obtain observed Fisher information. Asymptotic normal distribution of MLE is used to construct CI for the parameter.

Maximum likelihood method is used to estimate the parameter and reliability function of the Rayleigh distribution. This estimation procedure is discussed in Sectio 2. By using asymptotic normality of the MLE, we derived CI and coverage probability is computed in Section 3. To investigate performance of procedure, simulation study has been made in Section 4. Results of simulation study are discussed in Section 5. In Section 6 proposed estimation methods are illustrated for real data set. Conclusions are presented in Section 7.

Consider Rayleigh distribution with scale parameter . The PDF g y and CDF G y are respectively given by,

inference for Laplace distribution based on progressive Type-II censored samples. Krishna and Malik [13] discussed reliability estimation in Maxwell distribution

g y =

2y e

2

y 2

y 0, > 0

based on progressively Type-II censord data. Recently,

G y = 1 e

y 0, > 0

Potdar and Shirke [14] studied inference for the scale parameter of lifetime distribution for k-unit parallel system based on progressively censored data. Potdar and Shirke [15] discussed estimation for the distribution of a k-unit parallel system with exponential distribution as the component life distribution based on Type-II

Consider k unit parallel system with independent and identically distributed components. Let Y1, Y2, , Yk be the lifetimes, where Yi is the lifetime of the ith component with Yi ~ g y . Life time of the system X= max(Y1, Y2, . , Yk). The CDF of X is

x 2 k

progressively censored data. Potdar and Shirke [16] studied Inference for the parameters generalized inverted family of distributions.

F x = 1 e

x 0, > 0. (1)

and the PDF of X is

2

m

2 k1 2 0

2kx x

x

+ 3 Ri a xi, k,

f x =

2 e

1 e

x 0, > 0. (2)

i=1

Suppose n systems of k-unit parallel systems are under test and we observe lifetimes of m systems under progressive Type-II censoring. Let (R , R ,.., R )

2 k 1 m

3 Ri b xi, k, 0 . 6

i=1

where a xi, k, 0 = E z2 zij > xi

1 2 m ij

is a progressive censoring scheme. The likelihood

2kz z 2

z 2

k1

function for the observed data is

2 e

1 e

m = z2 dz .

L = C f(x

) 1 F x

Ri , (3)

x 2 k

i

i=1

i

m1

i

j

xi 1 1 e

zij 2

z2e

where C = n n j Ri .

and b xi, k, 0 = E ij

zij 2

zij > xi

m

2kx

j=1

i

x 2

i=1

1 e

L = C

i=1

i e

2

2 z 2

2kz z 2

2 e

z 2

1 e

k1

x i 2

Ã— 1 e

k1

= z e

z 2

x 2 k

dz.

x 2 k Ri

xi 1 e

1 1 e i

dL

(i)

We have to solve equation c = 0 to obtain 1

Ã— 1 1 e

. (4)

d

as the solution. But this equation does not have solution

Suppose x(1), x(2),., x(m) is the observed data and z , z , . , z is the censored data. We note that z

in the closed form. Therefore we used Newton-Raphson method and compute 1. By using 1, we

1 2 m

th i computed a x , k, 1 and b x , k, 1 . This ends the M

is a vector with Ri elements. That is at the i failure, we i i

remove Ri (i=1,2,.,m) systems. Observations on the removed systems are not available. The censored data Z= (z1, z2, . , zm) can be considered as missing data. X= (x(1), x(2), , x(m)) is observed data. W=(X, Z) is the complete data set. Then complete log likelihood function is

Lc = nlog 2 + nlog k 2nlog

step. We continue this procedure until convergence took place.

In Newton Raphson-method, we need to choose initial value of . We used least square estimate. Ng [7] discussed estimation of model parameters of modified Weibull distribution based on progressively Type-II censored data where the empirical distribution function is

m

+ log xi

m

2

1 x2

computed as (see Meeker and Escober [23])

i

i=1

m

i

xi 2

F x

i

= 1 (1 p j).

j=1

+ k 1 log 1 e

i=1

with

m Ri

1

p = 1

, for j = 1,2, , m.

+ log zij

n

i=1 j=1

2 zi2j

i=1 j=1

j j

k=2

Rk1

j + 1

m Ri

zij 2

The estimate of the parameters can be obtained by least

+ k 1 log 1 e . (5)

i=1 j=1

square fit of simple linear regression.

yi=x(i) with = -1/.

1 1

In order to obtain MLE of , we use EM algorithm due to Dempster et al. [17]. For the E step in EM algorithm we take Expectation of Zij. The derivative of Lc with respect to is taken for the M step, where

y = ln 1 F k x i1 +F k x i , for i=1,2,..,m.

i

2

F x 0 = 0.

m 2 xi 2

i

i

dLc = 2n + 2 x2 2 k 1 xi e

The least square estimates of is given by

3

3

3

3

d

i=1

i=1 1 e

xi 2

m x2
xi 2

= i=1 (i) . (7)

6 k 1

x2 e

m

i=1

x 2

i

x(i)yi

E

4

i=1

i . (9) 1 e

We used as an initial value of in Newton-

Raphson method. Reliability function at time t is

R t = P X > = 1 F t .

Missing information is given by

m

t 2 k

I = R I i

R t = 1 1 e

t 0, > 0.

W|X

i=1

m

i W|X

Ri

d2log f Z X ,

ij i

The Maximum likelihood estimate of R(t) is

= EZ|X

d2 .

(10)

R t = 1 1 e

t 2 k

t 0, > 0.

Consider

i=1 j=1

We compute observed Fisher information using

f zij ;

f Z X , =

the idea of missing information principle of Louis [22].

z|X

ij i

1 F xi;

2kz

zij 2

k1

The observed information

ij e

2

1 e

= complete information missing information

Ix() = Iw() Iw|x(), (8)

where,

Let,

=

1 1 e

.

xi 2 k

Complete information = Iw

= E d2 L ,

2

d

log f = log fz|X Zij Xi, .

where, L is the log likelihood function of all n observations.

Therefore,

zij 2

logf = log 2 + log k + log z 2 log

Now,

ij

zij 2

xi 2 k

L = nlog 2 + nlog k

n n

+ k 1 log 1 e

log 1 1 e .

2nlog + log x 1 x2

zij 2

i 2 i

dlogf

2 2z2

2 k 1 z2e

n i=1

xi 2

i=1 = +

d

ij

3

ij

zij 2

+ k 1 log 1 e .

i=1

3 1 e

x 2

x 2 k1

and

2 i

i

n

dL 2n 2

2 k 1 n

x2e

xi 2

2kxi e

1 e

.

d = +

x2

i

3

i .

xi 2

3

xi 2 k

3 1 1 e

i=1

i=1 1 e

d2 L

n

2n 6 2

4 k 1 n

x4 e

xi 2

and

2

d2 = 2 4 xi

6

2 2

zij

2 4

i

i

i=1

i=1 1 e i

d2logf

2 6zij

4 k 1 zij e

i

i

n

n

x 2

2

=

d2 2 4

6

zij 2 2

+ 6 k 1 xi e .

1 e

i

i

4 x 2

zij 2

i=1 1 e

6 k 1 z2e

ij

ij

+

Complete information is given by,

zij 2

2n Iw = 2 +

n

i

i

6 E x2

4

4 1 e

xi 2
2 k2

xi 2

4 k 1

i=1

n

xi 2

x4e

4kx4 e

i

1 e

x

1 ke

2 k

+ E i

6 i

6 xi 2 2

1 1 e

i=1 1 e

xi 2

i

4k2x4 e2

xi 2 2(k1)

1 e

2 = 1

n

I n

is the estimated variance of n .

+ 2

6

xi 2 k

1 1 e

Therefore, 100(1-)% asymptotic CI for is given by

n /2 2 n , n + /2 2 n , (12)

2 xi 2

xi 2 k1

i

6kx2 e 1 e

+ .

where /2 is the upper 100(/2)th percentile of

i

x 2 k

6 1 1 e

standard normal distribution.

Meeker and Escober [23] reported that the

Hence Missing information is

m

asymptotic CI for based on ln( n ) has better coverage probability. An approximate 100(1-)% CI for ln() is

W|X

IW|X = Ri I i

ln n 2 ln n ,

i=1

R

2 , (13)

m i d2log f Zij Xi ,

ln(

) + 2 ln( )

= EZ|X i=1 j=1

d2 ,

n /2 n

m Ri

where 2 ln n is the estimated variance of

= 2 n m + 6 E z2

ln n which is approximately obtained by

2 4

i

m R

i=1 j=1

ij

zij 2

2 ln n

2

n

n 2

Hence, an approximate 100(1-

4 k 1

z4e

)% CI for is

+ 6

E ij

zij 2 2

2

i=1 j=1

1 e

/2 n

6 k 1 m Ri

zij 2

z2e

n e

n

, n e

n

. (14)

4

E

i=1 j=1

ij

1 e

zij 2

4. Simulation Study

R xi 2

xi 2 k2

xi 2

m i x4 e

1 e

1 ke

+ 4k i

6

i=1 j=1

xi 2 k

1 1 e

A simulation study is carried out to study the performance of MLE by considering bias and MSE for various progressively Type-II censoring scheme.

m Ri

4 2 xi 2

x 2 2 k1

i

Approximate CIs based on MLE and log-transformed

MLE are compared through their coverage probability.

4k2

xi e

1 e

Balkrishnan and Sandhu [24] presented algorithm for

6 2 k 2

sample generation from progressively Type-II censored

i=1 j=1

1 1 e

xi

scheme. Using this algorithm, we generate samples from

m Ri

2 xi 2

x 2 k1

i

the distribution of a k-unit parallel system with Rayleigh

distribution as the component life distribution.

6k

4

xi e 1 e

x 2 k

. (11)

Algorithm

Generate i.i.d. observations (W1, W2, , Wm) from

i=1 j=1

1 1 e

i

U(0, 1).
For (R1, R2, . , Rm) censoring scheme, set Ei= 1/

Using expressions in equations (10) and (11), we

(i + Rm+Rm-1+. + Rm-i+1) for i=1,2,.,m.
Set Vi= W Ei for i=1,2, .. , m.

obtained Fisher information.
Set Ui=

i

1 Vm

Vm-1

.. V

m-i+1

) for i=1,2, . , m.

3. Confidence Intervals

Then (U1, U2, , Um) is the progressively Type-II censored sample from U(0,1).

For given values of the parameter , set

i

Using property of asymptotic normality of MLE we construct CI for . Let is the MLE of and

= 2 log 1 (U )1/k 1/2, for i=1, , m. (15) Then (x(1), x(2), , x(m)) is the required

progressively Type-II censored sample from the

distribution of a k-unit parallel system with Rayleigh distribution as the component life distribution. For

Table 1.

Progressively Type-II censored schemes used for simulation study

n	m	Scheme no.	Scheme
5	2	[1]	(3,0)
		[2]	(0,3)
		[3]	(1,2)
		[4]	(2,1)
15	5	[5]	(10, 4*0)
		[6]	(4*0, 10)
		[7]	(2,2,2,2,2)
15	10	[8]	(5,9*0)
		[9]	(9*0,5)
		[10]	(3,2, 8*0)
20	10	[11]	(10,9*0)
		[12]	(9*0,10)
25	10	[13]	(15,9*0)
		[14]	(9*0,15)
		[15]	(5,5,5,7*0)
25	15	[16]	(10,14*0)
		[17]	(14*0,10)
30	10	[18]	(20, 9*0)
		[19]	(9*0,20)
30	15	[20]	(15, 14*0)
		[21]	(14*0,15)
		[22]	(5,5,5,12*0)
30	20	[23]	(10, 19*0)
		[24]	(19*0,10)
		[25]	(0,5,5,17*0)
50	20	[26]	(30,19*0)
		[27]	(19*0,30)
50	35	[28]	(15,34*0)
		[29]	(34*0,15)
		[30]	(5,5,5,32*0)

In Table 1, scheme (a, b) stands for R1= a and R2=b. Similar meaning holds for schemes described through completely specified vector, while scheme (10, 4*0) means R1=10 and rest four Ris are zero. i.e. R2=R3=R4=R5=0.

Simulation was carried out for 3-unit parallel system and 5-unit parallel system (i.e. k=3, k=5) with =1. EM algorithm and Newton-Raphson method are used to compute MLE. For each particular progressive censoring scheme, 10,000 sets of observations were generated. The bias, the MSE and the coverage probability for the corresponding approximate CIs for are displayed in the Tables 2 and 3 for k=3 and k=5 respectively.Further, reliability estimate for different time period t= 1 given in Tables 4 and 5 for k=3 and k=5 respectively.

simulation study we consider 30 different progressively Type-II censored schemes as mentioned in Table I.

Table 2.

Bias, MSE and Coverage Probability for k=3 and =1

Schem e No.	Bas	MSE	Confidence level (MLE) 90% 95%		Confidence level (log MLE) 90% 95%
[1]	-0.0195	0.0406	0.8653	0.9075	0.8806	0.9323
[2]	-0.0196	0.0351	0.8683	0.9118	0.8865	0.9376
[3]	-0.0210	0.0359	0.8675	0.9080	0.8831	0.9335
[4]	-0.0222	0.0377	0.8651	0.9092	0.8832	0.9350
[5]	-0.0082	0.0174	0.8861	0.9327	0.8919	0.9432
[6]	-0.0079	0.0133	0.8829	0.9324	0.8893	0.9414
[7]	-0.0083	0.0143	0.8847	0.9297	0.8919	0.9414
[8]	-0.00274	0.0096	0.8951	0.9425	0.8961	0.9482
[9]	-0.0023	0.0082	0.8945	0.9421	0.8944	0.9497
[10]	-0.0036	0.0096	0.8929	0.9416	0.8966	0.9457
[11]	-0.0039	0.0093	0.8927	0.9422	0.8977	0.9513
[12]	-0.0027	0.0076	0.8908	0.9407	0.8949	0.9462
[13]	-0.0051	0.0093	0.8897	0.9408	0.8959	0.9455
[14]	-0.0035	0.0067	0.8990	0.9460	0.9052	0.9505
[15]	-0.0037	0.0087	0.8945	0.9412	0.8990	0.9464
[16]	-0.0024	0.0063	0.9017	0.9456	0.9025	0.9515
[17]	-0.00157	0.0053	0.8943	0.9430	0.8963	0.9471
[18]	-0.0038	0.0091	0.8927	0.9431	0.8990	0.9492
[19]	-0.0045	0.0066	0.8919	0.9437	0.8977	0.9453
[20]	-0.00314	0.0064	0.8952	0.9433	0.8963	0.9468
[21]	-0.0039	0.0048	0.9022	0.9496	0.9039	0.9534
[22]	-0.0031	0.0061	0.8952	0.9425	0.8981	0.9471
[23]	-0.0018	0.0049	0.8939	0.9423	0.8951	0.9457
[24]	-0.00096	0.0041	0.8988	0.9463	0.9008	0.9490
[25]	-0.0016	0.0047	0.8985	0.9451	0.9026	0.9482
[26]	-0.0013	0.0048	0.8960	0.9451	0.8966	0.9489
[27]	-0.0016	0.0033	0.8981	0.9449	0.8999	0.9498
[28]	-0.00001	0.0028	0.9002	0.9498	0.9021	0.9509
[29]	-0.00034	0.0024	0.8971	0.9475	0.8961	0.9485
[30]	-0.0004	0.0027	0.9022	0.9525	0.9019	0.9537

From Table II and III we observe the following for the MLE of the scale parameter.

The bias and MSE of the MLE decrease with increase in sample size n as well as with increase in the effective sample size m.

The bias and MSE of the MLE decrease as k (no. of units in parallel system) increases.

Table 3.

Bias, MSE and Coverage Probability for k=5 and =1

Table 4.

Bias, MSE and Coverage Probability of reliability estimate for k=3, t=1 and =1, R(t)=0.7474

Scheme No.	Bias	MSE	Confidence level (MLE) 90% 95%		Confidence level (log MLE) 90% 95%
[1]	-0.0160	0.0258	0.8733	0.9171	0.8831	0.9362
[2]	-0.0091	0.0217	0.8807	0.9290	0.8909	0.9407
[3]	-0.0122	0.0224	0.8788	0.9248	0.8879	0.9403
[4]	-0.0113	0.0237	0.8792	0.9256	0.8917	0.9367
[5]	-0.0046	0.0115	0.8879	0.9397	0.8941	0.9449
[6]	-0.0042	0.0078	0.8925	0.9418	0.8969	0.9482
[7]	-0.0044	0.0087	0.8918	0.9412	0.8932	0.9458
[8]	0.0009	0.0065	0.8985	0.9472	0.9002	0.9492
[9]	-0.0014	0.0053	0.8997	0.9460	0.8992	0.9492
[10]	0.00014	0.0064	0.8958	0.9442	0.8957	0.9475
[11]	-0.0019	0.0063	0.8962	0.9458	0.8978	0.9479
[12]	-0.0023	0.0046	0.8970	0.9450	0.8979	0.9477
[13]	-0.0017	0.0061	0.8971	0.9473	0.8991	0.9495
[14]	-0.0026	0.0042	0.8977	0.9456	0.8994	0.9504
[15]	-0.0017	0.0055	0.9009	0.9463	0.9006	0.9502
[16]	-0.0005	0.0043	0.8993	0.9465	0.8998	0.9491
[17]	-0.0007	0.0034	0.8958	0.9481	0.8978	0.9481
[18]	-0.0016	0.0061	0.8963	0.9438	0.8965	0.9490
[19]	-0.0016	0.0039	0.8944	0.9452	0.8969	0.9454
[20]	-0.0016	0.0042	0.9013	0.9490	0.9029	0.9507
[21]	-0.0013	0.0031	0.8962	0.9461	0.8965	0.9484
[22]	-0.0018	0.0040	0.8970	0.9440	0.8997	0.9478
[23]	-0.0001	0.0033	0.8968	0.9464	0.8978	0.9484
[24]	0.0006	0.0027	0.9026	0.9502	0.9045	0.9512
[25]	-0.00002	0.0031	0.9005	0.9524	0.9020	0.9529
[26]	-0.0014	0.0032	0.8973	0.9435	0.8958	0.9446
[27]	-0.0011	0.0021	0.8971	0.9448	0.8968	0.9460
[28]	-0.0008	0.0019	0.8980	0.9499	0.8989	0.9500
[29]	0.0004	0.0015	0.9047	0.9488	0.9032	0.9497
[30]	-0.0003	0.0019	0.9004	0.9463	0.9009	0.9489

Scheme No.	Bias	MSE	Confidence level (MLE) 90% 95%
[1]	-0.0648	0.0427	0.8258	0.8716
[2]	-0.0627	0.0374	0.8424	0.8867
[3]	-0.0620	0.0378	0.8390	0.8881
[4]	-0.0662	0.0395	0.8414	0.8880
[5]	-0.0302	0.0164	0.8702	0.9193
[6]	-0.0265	0.0125	0.8774	0.9299
[7]	-0.0267	0.0132	0.8780	0.9246
[8]	-0.0166	0.0085	0.8856	0.9342
[9]	-0.0143	0.0071	0.8878	0.9356
[10]	-0.0157	0.0083	0.8811	0.9310
[11]	-0.0174	0.0082	0.8867	0.9360
[12]	-0.0146	0.0066	0.8820	0.9335
[13]	-0.0168	0.0082	0.8819	0.9323
[14]	-0.0141	0.0060	0.8853	0.9356
[15]	-0.0170	0.0078	0.8872	0.9336
[16]	-0.0107	0.0054	0.8892	0.9361
[17]	-0.0095	0.0043	0.8937	0.9421
[18]	-0.0155	0.0077	0.8918	0.9422
[19]	-0.0116	0.0055	0.8873	0.9384
[20]	-0.0113	0.0055	0.8869	0.9348
[21]	-0.0084	0.0040	0.8922	0.9403
[22]	-0.0104	0.0051	0.8916	0.9407
[23]	-0.0086	0.0041	0.8868	0.9412
[24]	-0.0064	0.0034	0.8884	0.9386
[25]	-0.0084	0.0039	0.8901	0.9415
[26]	-0.0073	0.0039	0.8890	0.9408
[27]	-0.0058	0.0028	0.8929	0.9448
[28]	-0.0049	0.0022	0.8984	0.9472
[29]	-0.0040	0.0019	0.8964	0.9465
[30]	-0.0044	0.0022	0.8986	0.9453

Scheme No.	Bias	MSE	Confidence level (MLE) 90% 95%		Confidence level (log MLE) 90% 95%
[1]	-0.0160	0.0258	0.8733	0.9171	0.8831	0.9362
[2]	-0.0091	0.0217	0.8807	0.9290	0.8909	0.9407
[3]	-0.0122	0.0224	0.8788	0.9248	0.8879	0.9403
[4]	-0.0113	0.0237	0.8792	0.9256	0.8917	0.9367
[5]	-0.0046	0.0115	0.8879	0.9397	0.8941	0.9449
[6]	-0.0042	0.0078	0.8925	0.9418	0.8969	0.9482
[7]	-0.0044	0.0087	0.8918	0.9412	0.8932	0.9458
[8]	0.0009	0.0065	0.8985	0.9472	0.9002	0.9492
[9]	-0.0014	0.0053	0.8997	0.9460	0.8992	0.9492
[10]	0.00014	0.0064	0.8958	0.9442	0.8957	0.9475
[11]	-0.0019	0.0063	0.8962	0.9458	0.8978	0.9479
[12]	-0.0023	0.0046	0.8970	0.9450	0.8979	0.9477
[13]	-0.0017	0.0061	0.8971	0.9473	0.8991	0.9495
[14]	-0.0026	0.0042	0.8977	0.9456	0.8994	0.9504
[15]	-0.0017	0.0055	0.9009	0.9463	0.9006	0.9502
[16]	-0.0005	0.0043	0.8993	0.9465	0.8998	0.9491
[17]	-0.0007	0.0034	0.8958	0.9481	0.8978	0.9481
[18]	-0.0016	0.0061	0.8963	0.9438	0.8965	0.9490
[19]	-0.0016	0.0039	0.8944	0.9452	0.8969	0.9454
[20]	-0.0016	0.0042	0.9013	0.9490	0.9029	0.9507
[21]	-0.0013	0.0031	0.8962	0.9461	0.8965	0.9484
[22]	-0.0018	0.0040	0.8970	0.9440	0.8997	0.9478
[23]	-0.0001	0.0033	0.8968	0.9464	0.8978	0.9484
[24]	0.0006	0.0027	0.9026	0.9502	0.9045	0.9512
[25]	-0.00002	0.0031	0.9005	0.9524	0.9020	0.9529
[26]	-0.0014	0.0032	0.8973	0.9435	0.8958	0.9446
[27]	-0.0011	0.0021	0.8971	0.9448	0.8968	0.9460
[28]	-0.0008	0.0019	0.8980	0.9499	0.8989	0.9500
[29]	0.0004	0.0015	0.9047	0.9488	0.9032	0.9497
[30]	-0.0003	0.0019	0.9004	0.9463	0.9009	0.9489

Scheme No.	Bias	MSE	Confidence level (MLE) 90% 95%
[1]	-0.0648	0.0427	0.8258	0.8716
[2]	-0.0627	0.0374	0.8424	0.8867
[3]	-0.0620	0.0378	0.8390	0.8881
[4]	-0.0662	0.0395	0.8414	0.8880
[5]	-0.0302	0.0164	0.8702	0.9193
[6]	-0.0265	0.0125	0.8774	0.9299
[7]	-0.0267	0.0132	0.8780	0.9246
[8]	-0.0166	0.0085	0.8856	0.9342
[9]	-0.0143	0.0071	0.8878	0.9356
[10]	-0.0157	0.0083	0.8811	0.9310
[11]	-0.0174	0.0082	0.8867	0.9360
[12]	-0.0146	0.0066	0.8820	0.9335
[13]	-0.0168	0.0082	0.8819	0.9323
[14]	-0.0141	0.0060	0.8853	0.9356
[15]	-0.0170	0.0078	0.8872	0.9336
[16]	-0.0107	0.0054	0.8892	0.9361
[17]	-0.0095	0.0043	0.8937	0.9421
[18]	-0.0155	0.0077	0.8918	0.9422
[19]	-0.0116	0.0055	0.8873	0.9384
[20]	-0.0113	0.0055	0.8869	0.9348
[21]	-0.0084	0.0040	0.8922	0.9403
[22]	-0.0104	0.0051	0.8916	0.9407
[23]	-0.0086	0.0041	0.8868	0.9412
[24]	-0.0064	0.0034	0.8884	0.9386
[25]	-0.0084	0.0039	0.8901	0.9415
[26]	-0.0073	0.0039	0.8890	0.9408
[27]	-0.0058	0.0028	0.8929	0.9448
[28]	-0.0049	0.0022	0.8984	0.9472
[29]	-0.0040	0.0019	0.8964	0.9465
[30]	-0.0044	0.0022	0.8986	0.9453

Table 5. Bias, MSE and Coverage Probability of reliability estimate for k=5, t=1 and =1, R(t)=0.8991

Scheme No.	Bias	MSE	Confidence level (MLE) 90% 95%
[1]	-0.0512	0.0203	0.8512	0.8869
[2]	-0.0455	0.0169	0.8506	0.8911
[3]	-0.0447	0.0168	0.8550	0.8920
[4]	-0.0471	0.0185	0.8511	0.8920
[5]	-0.0226	0.0062	0.8791	0.9179
[6]	-0.0164	0.0040	0.8852	0.9279
[7]	-0.0169	0.0045	0.8826	0.9256
[8]	-0.0113	0.0029	0.8840	0.9260
[9]	-0.0095	0.0023	0.8917	0.9341
[10]	-0.024	0.0031	0.8800	0.9239
[11]	-0.0118	0.0028	0.8878	0.9287
[12]	-0.0082	0.0019	0.8895	0.9318
[13]	-0.0107	0.0028	0.8782	0.9237
[14]	-0.0086	0.0019	0.8870	0.9329
[15]	-0.0107	0.0025	0.8840	0.9281
[16]	-0.0078	0.0017	0.8933	0.9346
[17]	-0.0063	0.0013	0.8925	0.9366
[18]	-0.0112	0.0027	0.8936	0.9341
[19]	-0.0073	0.0016	0.8940	0.9370
[20]	-0.0072	0.0017	0.8894	0.9333
[21]	-0.0058	0.0013	0.8939	0.9380
[22]	-0.0080	0.0017	0.8900	0.9365
[23]	-0.0061	0.0013	0.8936	0.9399
[24]	-0.0047	0.0011	0.8875	0.9383
[25]	-0.0060	0.0013	0.8939	0.9404
[26]	-0.0056	0.0012	0.8939	0.9369
[27]	-0.0043	0.0008	0.8976	0.9457
[28]	-0.0031	0.0007	0.8970	0.9424
[29]	-0.0028	0.0006	0.8914	0.9415
[30]	-0.0039	0.0007	0.8961	0.9444

The bias and MSE of the MLE decrease with increase in sample size n as well as with increase in effective sample size m. The bias and MSE of the MLE decrease as k (no. of units in parallel system) increases. The MSE of the MLE is relatively small for conventional Type-II censoring scheme as compared with progressive Type-II censoring scheme. There is negligible difference between bias in case of conventional Type-II censoring scheme and progressive Type-II censoring scheme. Coverage probability in case of progressive Type-II censoring scheme are better than confidence levels in case of conventional Type-II censoring scheme. Same trend is observed in confidence levels for the log-transformed MLE. Coverage probability for k=5 is better than coverage probability for k=3 in case of small sample size. Same trend is observed in coverage probability for the log-transformed MLE. There is negligible difference in coverage probability for different k for large sample size

in both MLE and log-transformed MLE case. Coverage probability using log-transformed MLE are close to nominal levels as compared to the coverage probability of MLE for small size, while for large sample size both are similar.

The bias and MSE of the MLE of Reliability decrease with increase in sample size n as well as with increase in the effective sample size m. The bias and MSE of the MLE of Reliability decrease as k (no. of units in parallel system) increases. There is negligible difference between the bias and MSE of the MLE of Reliability in case of conventional Type-II censoring scheme and progressive Type-II censoring scheme. For small sample size, coverage probability in case of conventional Type-II censoring are better than coverage probability in case of progressive Type-II censoring, whereas for large sample size coverage probability in case of progressive Type-II censoring are better.

Consider the following data which represent the number of revolutions to failure (in hundreds of millions) for each of 23 ball bearings given by Lieblein and Zelen

[25] (also given by Lawless [26]).

0.1788, 0.2892, 0.33, 0.4152, 0.4212, 0.4560, 0.4840,

0.5184, 0.5196, 0.5412, 0.5556, 0.6780, 0.6864, 0.6864,

0.6888, 0.8412, 0.9312, 0.9864, 105.12, 105.84, 127.92,

128.04, 173.40

According to Raqab and Madi [27], scale parameter Rayleigh distribution satisfactory fit to this data. We consider this data as outcome for life time of two unit parallel systems and three unit parallel systems. Reliability estimate for k=2, 3 and different time periods (t=0.5, 1, 1.5, 2) is given in Tables 6.

k	n	m	Scheme	MLE	t=0.5	R(t)^ t=1 t=1.5		t=2
2	11	7	(1,1,1,1,3*0)	0.6141	0.7651	0.1361	0.0051	0
			(4,6*0)	0.7332	0.8617	0.2871	0.0302	0.0012
			(6*0,4)	0.7155	0.8507	0.2635	0.0245	0.0008
			(3*0,1,1,1,1)	0.7606	0.8769	0.3236	0.0405	0.0020
		11	(11*0)	0.7380	0.8645	0.2935	0.0319	0.0013
3	7	4	(1,1,1,0)	0.7064	0.9388	0.3523	0.0327	0.0010
			(3,3*0)	0.8906	0.9802	0.6321	0.1657	0.0192
			(3*0,3)	0.7451	0.9523	0.418	0.0512	0.0022
		7	(7*0)	0.7894	0.9639	0.4898	0.0789	0.0049

k	n	m	Scheme	MLE	t=0.5	R(t)^ t=1 t=1.5		t=2
2	11	7	(1,1,1,1,3*0)	0.6141	0.7651	0.1361	0.0051	0
			(4,6*0)	0.7332	0.8617	0.2871	0.0302	0.0012
			(6*0,4)	0.7155	0.8507	0.2635	0.0245	0.0008
			(3*0,1,1,1,1)	0.7606	0.8769	0.3236	0.0405	0.0020
		11	(11*0)	0.7380	0.8645	0.2935	0.0319	0.0013
3	7	4	(1,1,1,0)	0.7064	0.9388	0.3523	0.0327	0.0010
			(33*0)	0.8906	0.9802	0.6321	0.1657	0.0192
			(3*0,3)	0.7451	0.9523	0.418	0.0512	0.0022
		7	(7*0)	0.7894	0.9639	0.4898	0.0789	0.0049

Table 6. Reliability Estimate for k=2, 3

From Table 6 we observe the following.

Reliability estimate increase with increase in effective sample size m. Reliability estimate increase as k (no. of units in parallel system) increases. Reliability estimate in case of progressive Type-II censoring scheme are better than reliability estimate in case of conventional Type-II censoring scheme.

Conclusion

The study reveals that for small sample size n and the smallest effective sample size m, EM algorithm method works well. Overall both conventional Type-II censoring scheme and progressive Type-II censoring scheme give better results. According to MSE, conventional Type-II censoring method are superior to the progressive Type-II censoring method, while CIs perform well in case of progressive Type-II censoring methods. In this study both conventional and progressive censoring methods give better performance. In many situations units are removed or lost from the experiment before the completion. For example individuals from clinical trial may drop from the experiment. In such situations we have

no alternative but to use progressive censoring method and analyze the data accordingly.

Cohen, A. C., 1963. Progressively censored samples in life testing, Technometrics, 5, 327-329.
Mann, N. R.,1969. Exact three-order-statistic confidence bounds on reliable life for a Weibull model with progressive censoring, Journal of the American Statistical Association, 64, 306-315.
Mann, N. R., 1971. Best linear invariant estimation for Weibull parameters under progressive censoring, Technometrics, 13, 521-533.
Balkrishnan N. and Asgharzadeh A. , 2005. Inference for the Scaled Half-Logistic Distribution based on progressively Type-II censored samples , Communication in Statistics Theory and Methods, 34, 73-87.
Balakrishnan, N., Kannan, N., Lin, C. T. and Ng, H.

K. T., 2003. Point and interval estimation for Gaussian distribution, based on Progressively Type- II Censored samples, IEEE Transactions on Reliability, 52(1), 90-95. (2)
Balakrishnan, N., Kannan, N., Lin, C. T. and Wu, S.
1. S. , 2004. Inference for the extreme value distribution under Progressive Type-II Censoring, Journal of Statistical Computation and Simulation, 74, 25-45.
Ng, H. K. T., 2005. Parameter estimation for a modified Weibull distribution, for progressively Type-II censored samples, IEEE transactions on Reliability, 54, 374-380.
Balakrishnan, N. and Aggarwala, R., 2000. Progressive Censoring: Theory, Methods and Applications, Birkhauser, Boston.
Balakrishnan, N., 2007. Progressive Censoring Methodology: An Appraisal. Test, 16(2), 211-259.
Pradhan, B., 2007. Point and Interval Estimation for the Lifetime Distribution of a k-Unit Parallel System Based on Progressively Type-II Censored Data, Economic Quality Control, 22, 2, 175-186.
Chein, T. L. and Balkrishnan N., 2011. Asymptotic properties of maximum likelihood estimators based on progressively Type-II censoring, Metrika, 74, 3, 349-360
Iliopoulous, G. and Balkrishnan L., 2011. Exact Likelihood Inference for Laplace Distribution based on Type-II censored samples, Journal of Statistical Planning and Inference , 141, 1224-1239.
Krishna H. and Malik M., 2012. Reliability estimation in Maxwell distribution with progressively censored data. J. Stat. Comput. Simul. 82: 623-641.
Potdar K. G. and Shirke D. T., 2012. Inference for the scale parameter of lifetime distribution of k-unit parallel system based on progressively censored data. J. Stat. Comput. Simul. Online published http://dx.doi.org/10.1080/00949655. 2012. 700314:1-15
Potdar K. G. and Shirke D. T., 2012. Inference for the distribution of a k-unit parallel system with exponential distribution as the component life distribution based on Type- II progressively censored sample. Int. J. Agricult. Stat. Sci. 8(2): 503-517.
Potdar K. G. and Shirke D. T., 2013. Inference for the parameters generalized inverted family of distributions. ProbStat Forum. 6: 18-28.
Dempster, A. P. , Laird, N. M. and Rubin, D. B., 1977. Maximum Likelihood from Incomplete Data via the EM algorithm, Journal of the Royal Statistical Society, B, 39, 1-38.
Mclachlan, G. J. and Krishnan, T., 1997. The EM Algorithm and Extensions, John Wiley and sons.
Little, Roderick J. A. and Rubin, Donald B., 2002. Statistical analysis with missing data, Wiley Series in Probability and Statistics.
Pradhan, B. and Kundu, D., 2009. On Progressively Censored Generalized Exponential Distribution, Test, 18, 497-515.
Ng, H. K. T., Chan, P.S. and Balkrishnan N., 2002. Estimation of parameters from progressively censored data using EM algorithm, Computational Statistics and Data Analysis, 39, 371-386.
Louis, T.A., 1982. Finding the observed information matrix using the EM algorithm, Journal of Royal Statistical Society, B, 44, 226-233
Meeker, W. Q. and Escober, L.A., 1998. Statistical Methods for Reliability Data, John Wiley & Sons, New York.
Balakrishnan, N. and Sandhu, R. A., 1995. A simple simulation algorithm for generating progressive Type-II censored samples, The American Statistician, 49, 229-230.
Leibein, J. and Zelen, M. (1956). Statistical Investigation of the Fatigue Life of Deep-Groove Ball Bearings, Journal of Research of the National Bureau of Standards, 57, 273-316.
Lawless, J. F. (1982). Statistical Models and Methods of Lifetime Data, John Wiley and Sons.
Raqab, M. Z. and Madi, M. T. (2002). Bayesian Prediction of the Total Time on Test Using Doubly Censored Rayleigh data, Journal of Statistical Computation and Simulation, 72, 781-789.

G. Potdar is an Assistant Professor of Statistics at the Ajara Mahavidyalaya, Ajara. He received the B.Sc. in 1990, M.Sc. in 1992 and M.Phil. in 2008 in Statistics from Shivaji University, Kolhapur, Maharashtra. His area of interest includes reliability theory and inference.

D. T. Shirke is Professor of Statistics at the Shivaji University, Kolhapur. He received the B.Sc. in 1985, M.Sc. in 1987 and Ph.D. in Statistics in 1993 from Shivaji University, Kolhapur, Maharashtra. His areas of interest include reliability theory, inference, industrial statistics, survival analysis and data mining.

Reliability Estimation for the Distribution of a k-Unit Parallel System with Rayleigh Distribution as the Component Life Distribution

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