Confidence intervals for the stress-strength reliability of the generalized exponential distribution

Confidence intervals for the stress-strength reliability of the generalized exponential distribution

Hong Kyung Pak ¹ · Sang Gil Kang ² · Woo Dong Lee ³

1 Department of Medical Information Technology and Data Science, Daegu Haany University

2 Department of Computer and Data Information, Sangji University

3 Department of Cosmeceutical Engineering, Daegu Haany University

Received 20 August 2018, revised 13 September 2018, accepted 17 September 2018

Abstract

In this article, two likelihood-based methods are developed to obtain the confidence intervals for the stress-strength reliability of the generalized exponential distribution.

When the parameter of interest is the stress-strength reliability of the generalized exponential distribution, we construct the confidence intervals of the stress-strength reliability based on the signed log-likelihood ratio statistic and the modified signed log-likelihood ratio statistic for the parameter of interest. Simulations are performed to show the accuracy of the proposed method in terms of estimated coverage probability and bilateral symmetry of the confidence intervals when the sample size is small. And an example is provided.

Keywords: Confidence interval, generalized exponential distribution, likelihood-based inference, modified signed log-likelihood ratio statistic, signed log-likelihood ratio statis- tic, stress-strength reliability.

1. Introduction

Kundu and Gupta (1999) introduced the one parameter generalized exponential (GE) distribution as an alternative to the gamma or Weibull distribution for fitting lifetime data.

This GE distribution is very flexible to modeling a lifetime data in place of the gamma or Weibull distributions (see also Gupta and Kundu, 2001; Kundu and Gupta, 2007; Raqab, 2002; Raqab and Ahsanullah, 2001). Specially, the GE distribution has many properties that are quite similar to those of the gamma distribution. But this GE distribution has the cumulative distribution function and the failure rate in an explicit form. And the distribution of the transformed random sample distributes as the χ ² distribution.

1

Professor, Department of Medical Information Technology and Data Science, Daegu Haany University, Kyungsan 38610, Korea.

2

Professor, Department of Computer and Data Information, Sangji University, Wonju 26339, Korea.

3

Corresponding author: Professor, Department of Cosmeceutical Engineering, Daegu Haany University,

Kyungsan 38610, Korea. E-mail : [email protected]

Since the failure rate of the GE is increasing, constant or decreasing according to the value of the shape parameter, it is a versatile distribution to analyze a lifetime data. Gupta and Kundu (2002) and Kundu and Gupta (2007) studied the statistical inference about the GE distribution. They developed the maximum likelihood estimator (MLE) and proved the sampling distributions of the estimators. They also developed the confidence intervals of the parameters based on the large sample properties of the MLE.

The statistical inference related with this stress-strength reliability is very common. In statistical reliability analysis, a system fails when the stress exceeds the strength. In other words, the system fails if and only if at any time the stress is greater than the strength. This stress-strength reliability is a measure of the system performance and has a meaning in the context of the reliability engineering. There are many studies related with statistical inference of the stress-strength reliability under various distributions such as bivariate exponential distribution (Awad et al., 1981), bivariate normal distribution (Gupta and Gupta, 1990) and exponential distribution (Tong, 1974, 1977). Kundu and Gupta (2005) studied the estimation of the stress-strength reliability for the GE distribution. They developed the MLE, uniformly minimum variance unbiased estimator and Bayes estimator of the stress-strength reliability.

Wong and Wu (2009) studied the interval estimation of the stress-strength reliability of GE using lower record data. They developed a likelihood-based analysis using Lagrange multiplier technique. When the parameter of interest is given, we use the transformation of a nuisance parameter which is orthogonal to the parameter of interest. This transformation simplifies the development of confidence intervals based on the likelihood-based statistics. In this sense, the approach is different to this paper. Lee et al. (2018) developed the likelihood based inference when the parameter of interest is the ratio of the shape parameters of the GE distributions.

In this paper, we focus on constructing the confidence intervals of the stress-strength reli- ability of the GE distribution. Generally, to construct a confidence interval of a parameter, the exact distribution of the estimator is necessary. When the exact distribution of the estimator is hard to obtain, we must resort to an approximation method like the Central Limit Theorem. However, even though the approximation method is developed, the cover- age probability of the confidence interval may be inaccurate and the confidence interval is quite asymmetric when the sample size is small. Cox and Hinkley (1974) introduced the signed log-likelihood ratio (SLR) statistic. This SLR statistic converges in distribution to the standard normal distribution when the sample size is large. But when the sample size is small, this statistic is quite inaccurate. The modified signed log-likelihood ratio (MSLR) statistic of Barndorff-Nielsen (1986, 1991) is highly accurate in the sense that it converges to the standard normal distribution very fast. So, the MSLR statistic is accurate when the sample size is small. However, the computation of the MSLR statistic is difficult.

The main purpose of this paper is to develop the confidence intervals of the stress-strength

reliability based on SLR and MSLR. Specially, we use an orthogonal transformation to

simplify computation. We want to show that the proposed confidence intervals are working

very well even with a small sample size. This paper is arranged as follows. In Section 2, the

confidence intervals based on the exact statistics, the SLR statistic and the MSLR statistic

will be developed. In Section 3, some simulations are performed to show the performances

of the proposed methods. An example using simulated data will be given. Section 4 presents

the conclusions of this paper.

2. Development of confidence intervals based on likelihood-based methods

The probability density function of the GE distribution with shape parameter λ is given by

f (x|λ) = λe ^−x (1 − e ^−x ) ^λ−1 , x > 0, (2.1) where λ > 0. We denote this distribution as GE(λ). The distribution function of GE(λ) is

F (x|λ) = (1 − e ^−x ) ^λ , x > 0, λ > 0.

Let X be a stress of a system whose distribution is GE(α) and Y be a strength with GE(β). Assume that X and Y are independent. Then the stress-strength reliability ψ is given by

ψ = P (X < Y ) = Z ∞

0

Z y 0

αe ^−x (1 − e ^−x ) ^α−1 βe ^−y (1 − e ^−y ) ^β−1 dxdy

= β

α + β . (2.2)

ψ is the probability of functioning of an item or a system which is subject to stress X.

Let X _i , i = 1, · · · , m denote random samples of stress from GE(α) and Y _j , j = 1, 2, · · · , n denote random sample of strength from GE(β). Then the likelihood function of α and β given observation x = (x _i , y _j ), i = 1, 2, · · · , m, j = 1, 2, · · · , n is given by

L(α, β|x) = α ^m e ^{− P}

^mi=1

^x

ⁱ

m

Y

i=1

(1 − e ^−x

ⁱ

) ^α−1 β ⁿ e ^{− P}

^nj=1

^y

^j

n

Y

j=1

(1 − e ^−y

^j

) ^β−1

= α ^m β ⁿ e ^−t

¹

^−t

²

m

Y

i=1

(1 − e ^−x

ⁱ

) ^α−1

n

Y

j=1

(1 − e ^−y

^j

) ^β−1 , (2.3)

where t ₁ = P m

i=1 x _i and t ₂ = P n j=1 y _j .

From the likelihood function (2.3), the MLE of α and β are calculated by α = − b ^m _t

3

and β = − b _t ⁿ

4

, where t 3 = P m

i=1 log (1 − e ^−x

ⁱ

) and t 4 = P n

j=1 log (1 − e ^−y

^j

), respectively.

By the invariance property of the MLE, the MLE of ψ is given by

ψ = b β b α + b b β =

 1 + α b

β b

 −1

.

Following Kundu and Gupta (2005) or Wong and Wu (2009), since −2αt 3 ∼ χ ² (2m) and

−2βt 4 ∼ χ ² (2n), where χ ² (ν) denotes a χ ² distribution with degree of freedom ν, and these two statistics are stochastically independent,

α b β

β α b = 1 − ψ ψ

ψ b

1 − b ψ ∼ F (2m, 2n), (2.4)

where F (ν ₁ , ν ₂ ) denotes a F distribution with degrees of freedom ν ₁ and ν ₂ . Then, the exact 100(1 − γ)% confidence interval for ψ is given by





1 1 + ^{1− b ψ}

ψ b F (1 − γ/2; 2m, 2n)

, 1

1 + ^{1− b ψ}

ψ b F (γ/2; 2m, 2n)



 , (2.5)

where F (γ; ν ₁ , ν ₂ ) is the 100γ% th percentile of F distribution with degrees of freedom ν ₁ and ν ₂ .

To develop the likelihood-based inference for ψ, let ω = α ^m β ⁿ be a nuisance parameter.

This transformation plays an important role in developing highly accurate methods. By this transformation, the parameter of interest ψ and the nuisance parameter ω is orthogonal, and thus the constraint MLE of ω is obtained in an explicit form and the computation of MSLR is simplified. As mentioned in the introduction, Wong and Wu (2009) also developed MSLR in this model, but they did not consider transformation of nuisance parameter. Though their results were also compatible to our results, their approach is different from ours. We can simplify the computation of MSLR by the orthogonal transformation of parameters.

Let θ = (ψ, ω). Under this reparametrization, the log-likelihood of θ = (ψ, ω) is given by

l ≡ l(θ) = l(ψ, ω) = log(ω) − t 1 − t 2 − t 3 − t 4 + ω

^N1

t 3 ( 1 − ψ ψ )

n N

+ t 4 ( 1 − ψ ψ )

−

^m_N

! , (2.6)

where N = m + n. Also, the MLE of ω is given by

ω = b α b ^m β b ⁿ .

The first partial derivatives of the log-likelihood (2.6) with respect to ψ and ω are given by

l ψ; ≡ ∂l

∂ψ = ω

^N1

ψ ⁻² N

 1 − ψ ψ

 −

^m_N



m ψ

1 − ψ t 4 − nt 3



, (2.7)

l ω; ≡ ∂l

∂ω = 1

ω + ω

^N1

⁻¹ N t 3

 1 − ψ ψ



_Nⁿ

+ t 4

 ψ

1 − ψ



^m_N

!

. (2.8)

When the parameter of interest ψ is fixed, the constrained MLE of ω is

ω b ψ = − N

( ^1−ψ _ψ )

^Nn

t 3 + ( ^1−ψ _ψ ) ⁻

^mN

t 4

! ^N

, (2.9)

and let b θ ψ = (ψ, ω b ψ ).

2.1. Confidence interval based on the signed log-likelihood ratio statistic The SLR statistic r is given by

r ≡ r(ψ) = sgn( b ψ − ψ) n

2[l(b θ) − l(b θ _ψ )] o

¹₂

, (2.10)

where sgn(a) is 1 if a > 0, 0 if a = 0 and -1 if a < 0. This statistic r converges in distribution to the standard normal distribution when the sample size n → ∞

Using r, the approximate 100(1 − γ)% confidence interval for ψ can be obtained from n

ψ; |r(ψ)| ≤ z

^γ

2

o

, (2.11)

where z

^γ

2

is the 100(1 − ^γ ₂ )% th percentile of the standard normal distribution.

It is well known that this statistic r distributes as the standard normal distribution with rate O _p (n ⁻

¹²

) (Cox and Hinkley, 1974). When the sample size n is small, the approxima- tion to the standard normal distribution is not accurate. This fact may cause problems in constructing confidence interval about ψ with small sample size.

2.2. Confidence interval based on the modified signed log-likelihood ratio statis- tic

There are various methods to improve the accuracy of the approximation to normality of the SLR statistic. The MSLR statistic of Barndorff-Nielsen (1986, 1991) is an excellent method to improve the accuracy of the approximation. Based on r, the MSLR statistic r ^∗ has the form

r ^∗ ≡ r ^∗ (ψ) = r(ψ) + r(ψ) ⁻¹ log  u(ψ) r(ψ)

 .

This statistic approximately distributes as the standard normal distribution to the order of O _p (n ⁻

³²

). Therefore, r ^∗ is more accurate than r when the sample size is small.

The 100(1 − γ)% confidence interval for ψ can be developed from n

ψ; |r ^∗ (ψ)| ≤ z

^γ

2

o

. (2.12)

If the log-likelihood function l(θ) can be written as l(θ; b θ, a), where a is an ancillary statistic, then u(ψ) is given by

u(ψ) =  

 l _{;b θ} (b θ) − l _{;b θ} (b θ ψ ) l _{ω;b θ} (b θ ψ )    n

  j θθ (b θ)

      j ωω (b θ ψ )

  

o

¹₂

,

where the sample-space derivatives are defined as

l _{;b θ} (θ) = ∂l(θ; b θ, a)

∂ b θ , the mixed derivatives as

l _{ω;b θ} (θ) = ∂ ² l(θ; b θ, a)

∂ω∂ b θ = ∂

∂ω l _{;b θ} (θ),

j θθ (b θ) is the observed information matrix and j ωω (b θ ψ ) is the observed nuisance information

matrix. Except full-rank exponential family distributions and certain transformation models,

it is often difficult or impossible to compute the statistic u.

Since our model is a full-rank exponential model, the log-likelihood function (2.6) based on the data x is only related to the minimal sufficient statistic t = (t ₃ , t ₄ ). There is a one- to-one transformation between b θ and t, and the transformation Jacobian matrix is ∂ b θ/∂t.

The sample-space derivatives with respect to b θ in the formula for u can be computed based on the sample-space derivatives with respect to t. Since j θθ (b θ) = l _{θ;b θ} (b θ) (Barndorff-Nielsen and Cox, 1994) and by canceling the determinant of the transformation Jacobian matrix, the statistic u can be reduced to

u(ψ) =  

 l ;t (b θ) − l ;t (b θ ψ ) l ω;t (b θ ψ )    

  l θ;t (b θ)

  





    j θθ (b θ)

    

 j ωω (b θ ψ )   







1 2

,

where the sample-space derivatives l ;t (θ) = ∂l(θ; t)/∂t and the mixed derivatives l ω;t (θ) =

∂ ² l(θ; t)/∂ω∂t.

In our case, we have found that

l ψψ; = ∂ ² l

∂ψ ² = ω

^N1

N

 ψ

1 − ψ



^m_N

ψ ⁻³

 1

1 − ψ + 2

 

nt 3 − mt 4

ψ 1 − ψ

 ,

l ψω; = ∂ ² l

∂ψ∂ω = ω

^N1

⁻¹ ψ ⁻² N

 ψ

1 − ψ



^m_N

 m N

ψt 4

1 − ψ − nt 3

N

 ,

l _ωω; = ∂ ² l

∂ω ² = − 1

ω ² + ω

^N1

⁻² N

 1 N − 1



t ₃  1 − ψ ψ



_Nⁿ

+ t ₄

 ψ

1 − ψ



^m_N

! .

Then the observed information matrix j θθ (θ) is j θθ (θ) =

 −l ψψ; −l ψω;

−l ψω; −l ωω;

 , and j ωω (θ) = −l ωω; . The sample-space derivative is

l ;t = (l ;t

₃

l ;t

₄

) ^T =







 1−ψ ψ



_Nⁿ

ω

^N1

t 3

 ψ 1−ψ



^m_N

ω

^N1

t 4







T

,

and the mixed derivative is

l θ;t =

 l ψ;t

₃

l ψ;t

₄

l ω;t

₃

l ω;t

₄



=





− _{N ψ} ^nω

₂^N1

_t

3

 ψ 1−ψ



^m_N

mω

^N1

N ψ

²

t

3

 ψ 1−ψ



^m_N

+1 ω

N¹−1

(

^1−ψ_ψ

)

^Nn

N t

3

ω

N¹−1

(

_1−ψ^ψ

)

^mN

N t

4



 .

Now, we have developed the modified signed log-likelihood statistics r ^∗ when the parameter

of interest is ψ.

3. Numerical studies

In this section, through simulation study and real data analysis, we will investigate the performances of the proposed methods. We will compare the estimated coverage probabilities of the 100(1 − γ)% confidence intervals for ψ based on (2.5), (2.11) and (2.12). Specially, we are interested in the coverage probabilities and their bilateral symmetry, when the sample size is small.

For the confidence interval for ψ, we are interested in the upper or lower error probabilities which is the proportion of ψ that falls outside the upper or lower bound of the confidence interval. To estimate these probabilities, we generate 10,000 independent random samples of size (m, n) =(1,1), (1,2), (2,1), (2,2), (3,5), (5,3), (10,10), (15,20) and (20,20) from GE(α) and GE(β). And we assume α = 2; β =1, 2, 3, 4, 8, so ψ =, 1/3, 1/2, 3/5, 2/3, 4/5.

We set the confidence levels as γ = 0.01 and 0.05. The simulation results for confidence intervals are given in Table 3.1 when γ = 0.05 and Table 3.2 when γ = 0.01. In these tables, F means the results based on (2.5), r based on (2.11) and r ^∗ based on (2.12). For each statistic, the result is more accurate if the first and the second columns are closer to γ/2 and the third column is closer to 1 − γ.

From Table 3.1 and Table 3.2, we can observe that when the sample size is small, r ^∗ is highly accurate even when m = 1 and n = 1. Also r ^∗ is symmetric in the sense that the first two columns are close to γ/2. The results of r ^∗ are almost the same as those of F . r does not achieve the target values even when the sample size is (m, n) =(5,3) or (3,5). These results are invariant with respect to the values of ψ.

As mentioned above r and r ^∗ converge to standard normal distribution with rate O p (n ^−1/2 ) and O p (n ^−3/2 ), respectively. We can see that r ^∗ converges to standard normal distribution when the sample size is small. And r ^∗ has bilateral symmetry of the confidence intervals.

Example The following data are mentioned in Lee et al. (2018). They generated artificial data sets from GE distributions with m=10, n=10 (N =20) and α = 1, β =2. Hence the true parameter ψ = 2/3. In that paper of Lee et al. (2018), their parameter of interest was the ratio of two shape parameters. Using the same data, we consider X as a distribution of stress and Y as a distribution of strength. Then the observations of stress X are 1.085, 0.239, 0.504, 0.382, 1.897, 1.541, 1.312, 3.768, 2.602, 0.004 and those of strength Y are 0.101, 0.818, 0.989, 1.452, 4.205, 0.955, 2.796, 3.287, 0.813, 0.634. From these data, the MLE of α is 0.964 and the MLE of β is 1.785. So, the MLE of ψ is 0.649. The 95% confidence interval for ψ are reported in Table 3.3. These confidence intervals are calculated based on (2.5), (2.11) and (2.12).

In this Table 3.3, we have found that the confidence interval based on r ^∗ is the same as

the interval based on the exact distribution. The length of the confidence interval based on r

is the shortest of them all, but as seen in simulation results, r does not achieve the nominal

level when the sample size is small.

Table 3.1 The coverage probabilities of the 95% confidence intervals for ψ

m n F r r

^∗

1 1 .0254 .0262 .9484 .0409 .0383 .9208 .0238 .0252 .9510

1 2 .0253 .0254 .9493 .0438 .0267 .9295 .0242 .0250 .9508

2 1 .0250 .0251 .9499 .0258 .0459 .9283 .0246 .0247 .9507

ψ = 1/3 2 2 .0256 .0254 .9490 .0314 .0318 .9368 .0253 .0253 .9494

3 5 .0233 .0288 .9479 .0304 .0287 .9409 .0232 .0287 .9481

5 3 .0247 .0241 .9512 .0244 .0313 .9443 .0242 .0240 .9518

10 10 .0249 .0240 .9511 .0265 .0255 .9480 .0249 .0240 .9511

15 20 .0254 .0243 .9503 .0274 .0242 .9484 .0254 .0243 .9503

20 20 .0263 .0228 .9509 .0272 .0235 .9493 .0264 .0228 .9508

1 1 .0259 .0216 .9525 .0393 .0331 .9276 .0238 .0207 .9555

1 2 .0249 .0221 .9530 .0430 .0228 .9342 .0239 .0219 .9542

2 1 .0269 .0257 .9474 .0277 .0464 .9259 .0263 .0250 .9487

2 2 .0264 .0265 .9471 .0341 .0330 .9329 .0262 .0262 .9476

ψ = 1/2 3 5 .0270 .0213 .9517 .0347 .0212 .9441 .0269 .0212 .9519

5 3 .0259 .0247 .9494 .0259 .0333 .9408 .0259 .0247 .9494

10 10 .0266 .0261 .9473 .0280 .0275 .9445 .0266 .0261 .9473

15 20 .0241 .0255 .9504 .0258 .0255 .9487 .0241 .0256 .9503

20 20 .0255 .0264 .9481 .0260 .0267 .9473 .0255 .0264 .9481

1 1 .0216 .0225 .9559 .0336 .0340 .9324 .0205 .0213 .9582

1 2 .0223 .0247 .9530 .0427 .0256 .9317 .0214 .0242 .9544

2 1 .0242 .0253 .9505 .0250 .0472 .9278 .0236 .0242 .9522

2 2 .0246 .0250 .9504 .0309 .0313 .9378 .0240 .0245 .9515

ψ = 3/5 3 5 .0268 .0253 .9479 .0351 .0253 .9396 .0268 .0253 .9479

5 3 .0262 .0277 .9461 .0261 .0356 .9383 .0259 .0277 .9464

10 10 .0263 .0262 .9475 .0275 .0274 .9451 .0263 .0262 .9475

15 20 .0255 .0267 .9478 .0271 .0264 .9465 .0255 .0268 .9477

20 20 .0245 .0227 .9528 .0254 .0234 .9512 .0245 .0227 .9528

1 1 .0256 .0229 .9515 .0376 .0346 .9278 .0244 .0216 .9540

1 2 .0251 .0239 .9510 .0457 .0250 .9293 .0248 .0232 .9520

2 1 .0248 .0275 .9477 .0256 .0450 .9294 .0239 .0269 .9492

2 2 .0256 .0228 .9516 .0321 .0287 .9392 .0251 .0226 .9523

ψ = 2/3 3 5 .0262 .0258 .9480 .0357 .0258 .9385 .0261 .0257 .9482

5 3 .0254 .0256 .9490 .0254 .0316 .9430 .0253 .0256 .9491

10 10 .0235 .0268 .9497 .0252 .0285 .9463 .0235 .0268 .9497

15 20 .0234 .0249 .9517 .0244 .0248 .9508 .0234 .0249 .9517

20 20 .0260 .0229 .9511 .0266 .0239 .9495 .0260 .0229 .9511

1 1 .0234 .0258 .9508 .0364 .0391 .9245 .0219 .0246 .9535

1 2 .0252 .0236 .9512 .0470 .0243 .9287 .0239 .0228 .9533

2 1 .0214 .0263 .9523 .0226 .0464 .9310 .0206 .0255 .9539

2 2 .0254 .0255 .9491 .0317 .0325 .9358 .0250 .0254 .9496

ψ = 4/5 3 5 .0227 .0266 .9507 .0300 .0266 .9434 .0227 .0266 .9507

5 3 .0259 .0266 .9475 .0257 .0344 .9399 .0257 .0265 .9478

10 10 .0233 .0248 .9519 .0240 .0263 .9497 .0233 .0248 .9519

15 20 .0258 .0241 .9501 .0278 .0240 .9482 .0258 .0241 .9501

20 20 .0229 .0252 .9519 .0237 .0259 .9504 .0229 .0252 .9519

Table 3.2 The coverage probabilities of the 99% confidence intervals for ψ

m n F r r

^∗

1 1 .0044 .0055 .9901 .0083 .0099 .9818 .0040 .0051 .9909 1 2 .0045 .0055 .9900 .0110 .0060 .9830 .0041 .0053 .9906 2 1 .0047 .0056 .9897 .0052 .0118 .9830 .0047 .0053 .9900 2 2 .0046 .0051 .9903 .0065 .0075 .9860 .0046 .0051 .9903 ψ = 1/3 3 5 .0045 .0052 .9903 .0056 .0054 .9890 .0045 .0052 .9903 5 3 .0056 .0054 .9890 .0059 .0076 .9865 .0056 .0054 .9890 10 10 .0058 .0043 .9899 .0060 .0045 .9895 .0058 .0043 .9899 15 20 .0060 .0051 .9889 .0066 .0051 .9883 .0060 .0051 .9889 20 20 .0051 .0039 .9910 .0052 .0040 .9908 .0052 .0039 .9909 1 1 .0056 .0044 .9900 .0107 .0077 .9816 .0055 .0041 .9904 1 2 .0048 .0046 .9906 .0105 .0052 .9843 .0048 .0043 .9909 2 1 .0051 .0049 .9900 .0058 .0103 .9839 .0047 .0047 .9906 2 2 .0058 .0053 .9889 .0081 .0083 .9836 .0056 .0053 .9891 ψ = 1/2 3 5 .0061 .0034 .9905 .0081 .0034 .9885 .0061 .0034 .9905 5 3 .0054 .0040 .9906 .0056 .0057 .9887 .0053 .0040 .9907 10 10 .0064 .0051 .9885 .0070 .0055 .9875 .0064 .0051 .9885 15 20 .0032 .0047 .9921 .0034 .0047 .9919 .0032 .0048 .9920 20 20 .0052 .0064 .9884 .0056 .0066 .9878 .0052 .0064 .9884 1 1 .0037 .0049 .9914 .0074 .0096 .9830 .0033 .0043 .9924 1 2 .0041 .0040 .9919 .0082 .0047 .9871 .0040 .0039 .9921 2 1 .0047 .0050 .9903 .0055 .0113 .9832 .0046 .0049 .9905 2 2 .0053 .0062 .9885 .0077 .0081 .9842 .0050 .0062 .9888 ψ = 3/5 3 5 .0047 .0051 .9902 .0068 .0055 .9877 .0046 .0051 .9903 5 3 .0052 .0053 .9895 .0055 .0070 .9875 .0052 .0053 .9895 10 10 .0044 .0047 .9909 .0049 .0050 .9901 .0044 .0047 .9909 15 20 .0046 .0063 .9891 .0050 .0064 .9886 .0046 .0064 .9890 20 20 .0043 .0047 .9910 .0049 .0051 .9900 .0043 .0047 .9910 1 1 .0071 .0036 .9893 .0108 .0078 .9814 .0066 .0032 .9902 1 2 .0047 .0044 .9909 .0109 .0051 .9840 .0046 .0043 .9911 2 1 .0048 .0054 .9898 .0054 .0124 .9822 .0045 .0052 .9903 2 2 .0057 .0041 .9902 .0078 .0063 .9859 .0056 .0041 .9903 ψ = 2/3 3 5 .0054 .0048 .9898 .0072 .0051 .9877 .0054 .0047 .9899 5 3 .0054 .0047 .9899 .0057 .0073 .9870 .0052 .0047 .9901 10 10 .0045 .0052 .9903 .0050 .0057 .9893 .0045 .0052 .9903 15 20 .0055 .0044 .9901 .0059 .0044 .9897 .0055 .0044 .9901 20 20 .0046 .0040 .9914 .0054 .0041 .9905 .0046 .0040 .9914 1 1 .0042 .0055 .9903 .0080 .0108 .9812 .0040 .0052 .9908 1 2 .0044 .0038 .9918 .0103 .0044 .9853 .0043 .0037 .9920 2 1 .0044 .0051 .9905 .0049 .0125 .9826 .0043 .0049 .9908 2 2 .0044 .0055 .9901 .0070 .0074 .9856 .0044 .0055 .9901 ψ = 4/5 3 5 .0043 .0052 .9905 .0065 .0055 .9880 .0042 .0050 .9908 5 3 .0061 .0043 .9896 .0064 .0080 .9856 .0061 .0043 .9896 10 10 .0047 .0056 .9897 .0049 .0061 .9890 .0047 .0056 .9897 15 20 .0052 .0047 .9901 .0059 .0047 .9894 .0052 .0047 .9901 20 20 .0039 .0051 .9910 .0039 .0056 .9905 .0039 .0051 .9910

Table 3.3 The 95% confidence intervals of example data lower upper

F 0.429 0.820

r 0.432 0.819

r

^∗

0.429 0.820

4. Concluding remarks

In this paper, we have considered the confidence interval estimation of the strength- reliability of the GE distribution. Specially, we developed the confidence intervals based the SLR statistic and the MSLR statistic. Using the orthogonal transformation to the pa- rameter of interest, we have found the constraint MLE of nuisance parameter in an explicit form and used it to develop the SLR and MSLR statistics. Through numerical studies, we can see that r ^∗ is accurate and gives desirable results even in a small sample size.

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