Estimation for the half-logistic distribution based on generalized progressive hybrid censoring

Estimation for the half-logistic distribution based on generalized progressive hybrid censoring

Sung-Ok Lee ¹ · Suk-Bok Kang ²

12 Department of Statistics, Yeungnam University

Received 8 May 2018, revised 8 June 2018, accepted 11 June 2018

Abstract

We obtain maximum likelihood estimator (MLE) and approximate maximum like- lihood estimators (AMLEs) of the scale parameter in half-logistic distribution under generalized progressive hybrid censored samples. We also obtain a Bayes estimator of the scale parameter under squared error loss function. Finally, we examine the validity of the proposed estimators using simulated data and real data. AMLEs are obtained explicitly with closed form and AMLEs are more efficient than MLE. Bayes estimator of the scale parameter is more efficient than MLE and AMLEs.

Keywords: Approximate maximum likelihood estimators, generalized progressive hybrid censored sample, half-logistic distribution, maximum likelihood estimator.

1. Introduction

The probability density function f (x) and the cumulative distribution function F (x) of a random variable X having a half-logistic distribution with the scale parameter σ are given by;

f (x; σ) = 2 exp − _σ ^x

σ 1 + exp − ^x _σ
 ² , x ≥ 0, σ > 0 (1.1) and

F (x; σ) = 1 − exp − ^x _σ

1 + exp − ^x _σ
, x ≥ 0, σ > 0. (1.2) Several authors have studied about inference for the half-logistic distribution. Balakrishnan (1985) founded a few relapse relations with regard to the moments of the half-logistic dis- tribution. He also founded product moments of order statistics. Kang et al. (2008) obtained MLE and AMLEs of the scale parameter in the half-logistic distribution under progressive

1

Graduate student, Department of Statistics, Yeungnam University, Gyeongsan 38541, Korea.

2

Corresponding author: Professor, Department of Statistics, Yeungnam University, Gyeongsan 38541,

Korea. E-mail: [email protected]

Type-II censored samples, and compared mean squared error (MSE) of the suggested estima- tors. Kang et al. (2009) derived MLE and AMLEs of the scale parameters in half logistic dis- tribution under the proposed double hybrid censored samples. Kang and Seo (2011) derived the MLE and AMLEs of the scale parameter for the exponentiated half-logistic distribution under progressively Type-II censored samples. Kang et al. (2013) derived MLEs and Bayes estimators of the unknown parameters in an exponentiated half-logistic distribution under a progressively Type-II censored samples. They obtained approximate confidence intervals (CIs) using the MLEs and credible intervals for Bayes estimators. Kang and Seo (2014a) ob- tained some entropy estimators by using Bayes estimators of an unknown parameter in the generalized half-logistic distribution under Type-II censored samples. Kang and Seo (2014b) explained some Bayesian estimation and prediction problems for the exponentiated half lo- gistic distribution under lower record values. Kang and Han (2015) suggested AMLEs of the shape parameter and scale parameter in the inverse Weibull distribution based on multiply Type-II censored samples. They proposed a graphical method for goodness-of-fit test under multiply Type-II censoring scheme by using AMLEs. Seo and Kang (2015a) derived some moment estimators and MLEs of unknown parameters in the exponentiated half-logistic distribution. They obtained entropy estimator and approximate CIs for parameter. Seo and Kang (2015b) proposed new method based on a pivotal quantity for estimating the scale parameter of the half-logistic distribution based on progressively Type-II censored samples.

Seo and Kang (2015c) derived the entropy of the half-logistic distribution with the scale parameter under Type-II censoring schemes. Seo and Kang (2016) suggested more efficient approach methods for estimating shape parameters in the exponentiated half-logistic distri- bution under record values. Seo and Kang (2017) derived unbiased estimators of location parameter and scale parameter in half-logistic distribution based on progressively Type-II censoring schemes using new method of pivotal quantities.

A generalized progressive hybrid censored scheme was introduced by Cho et al. (2015).

Lee (2017) obtained the MLE and Bayes estimator of the entropy of the inverse Weibull dis- tribution under generalized progressive hybrid censoring scheme. Lately, progressive hybrid censoring schemes have become pretty popular in many reliability and a life-testing problem analysis. But the progressive hybrid censoring scheme has the limitation that it cannot be applied when quite a few failures happen before time T . So, generalized progressive hybrid censored scheme was developed. The generalized progressive hybrid censoring scheme can be explained as follows. The integers k, m ∈ {1, 2, · · · , n} are pre-fixed such as k < m. T is also pre-fixed time point satisfying T ∈ (0, ∞). As well as, the integers R 1 , R 2 , · · · , R m are pre-fixed satisfying P m

i=1 R i + m = n. In this case, at the moment of the first failure, R 1

of remaining units are randomly removed. Likewise at the moment of the second failure, R 2

of remaining units are randomly removed and so on. This procedure continues until, imme- diately following the terminated time T ^∗ = max{X k:m:n , min{X m:m:n , T }}, where X 1:m:n , X 2:m:n , · · · , X m:m:n denote such a generalized progressive hybrid censored sample. All the remaining units are removed from the experiment. From this scheme, a minimum kth failures of observed failures can be guaranteed.

In this paper, we propose the MLE and AMLEs of the scale parameter σ in half-logistic

distribution under a generalized progressive hybrid censoring scheme. First, the scale pa-

rameter of half-logistic distribution is estimated by AMLE method using two different kinds

of Taylor series expansions, and then we will use these AMLEs as initial values to obtain

the MLE. We also obtain Bayes estimator with conjugate prior under squared error loss

fuction. Finally, we compare the suggested estimators by using the MSE through Monte Carlo simulation from various censored schemes.

2. Maximum likelihood estimation

We consider MLE for the scale parameter σ of half-logistic distribution with pdf (1.1) based on generalized progressive hybrid censored samples. Let D denote the number of observed failures until time T . If T < X _k:m:n < X _m:m:n , we have Case I, then generalized progressive hybrid censored sample is (X 1:m:n , X 2:m:n , · · · , X k:m:n ). If X k:m:n < T < X m:m:n , we have Case II. We can get D, such as X D:m:n < T < X D+1:m:n , then generalized progressive hybrid censored sample is (X 1:m:n , X 2:m:n , · · · , X D:m:n ). If X k:m:n < X m:m:n < T , we have Case III, then generalized progressive hybrid censored sample is (X 1:m:n , X 2:m:n , · · · , X m:m:n ).

Therefore, we can have three different cases as follows;

· Case I : X 1:m:n , X 2:m:n , · · · , X k:m:n , if T < X k:m:n < X m:m:n

· Case II : X 1:m:n , · · · , X k:m:n , · · · , X D:m:n , if X k:m:n < T < X m:m:n

· Case III : X 1:m:n , · · · , X _k:m:n , · · · , X _m:m:n , if X _k:m:n < X _m:m:n < T

Figure 2.1 Graphical representation of generalized progressive hybrid censoring scheme

Therefore, the likelihood functions of σ are given by;

Case I : L I (σ) = K 1 k−1

Y

i=1

f (x i:m:n )[1 − F (x i:m:n )] ^R

f (x k:m:n )[1 − F (x k:m:n )] ^R

, (2.1)

Case II : L _II (σ) = K ₂

D

Y

i=1

f (x _i:m:n )[1 − F (x _i:m:n )] ^R

[1 − F (T )] ^R

, (2.2)

Case III : L III (σ) = K 3 m

Y

i=1

f (x i:m:n )[1 − F (x i:m:n )] ^R

, (2.3)

where

K 1 = [

k

Y

i=1 m

X

k=i

(R k + 1)], K 2 = [

D

Y

i=1 m

X

k=i

(R k + 1)], K 3 = [

m

Y

i=1 m

X

k=i

(R k + 1)],

R ^∗ _k = n − k −

k−1

X

j=1

R j and R ^∗ _D+1 = n − D −

D

X

j=1

R j .

Let Z = ^X _σ and V = ^T _σ . The random variable Z has a standard half-logistic distribution with pdf and cdf;

f (z) = 2e ^−z

(1 + e ^−z ) ² , z ≥ 0 (2.4)

and

F (z) = 1 − e ^−z

1 + e ^−z , z ≥ 0. (2.5)

The functions of F (z), f (z), and f ⁰ (z) are satisfied as;

−f (z)

1 − F (z) = − 1 + F (z)

2 , −f ⁰ (z)

f (z) = F (z). (2.6)

Therefore, Cases I, II, and III can be merged and written as;

ln L(σ) ∝ −I ln σ +

I

X

i=1

ln f (z _i:m:n ) +

" _I X

i=1

R _i ln[1 − F (z _i:m:n )] + W (θ)

#

, (2.7) where

I = k, W (θ) = 0, R k = n − k −

k−1

X

j=1

R j for Case I,

I = D, W (θ) = R ^∗ _D+1 ln[1 − F (T )] for Case II,

I = m, W (θ) = 0 for Case III.

By differentiating, the corresponding log-likelihood equations can be written as;

∂ ln L

∂σ ∝ − 1 2σ

"

2I −

I

X

i=1

R i z i:m:n −

I

X

i=1

(R i + 2)F (z i:m:n )z i:m:n − W (θ ⁽¹⁾ )

#

= 0. (2.8)

Finally, we can obtain the MLE of σ as;

ˆ σ =

P I i=1

 R _i + (R _i + 2)F (Z _i:m:n ) 

Z _i:m:n + W (θ ⁽¹⁾ )

2I , (2.9)

where W (θ ⁽¹⁾ ) = 0 for Cases I, III and W (θ ⁽¹⁾ ) = 2R ^∗ _D+1 T (1 + F (V )) for Case II.

Because the log-likelihood equation cannot be solved for σ to closed form explicitly, we apply simplely iterative procedure from the equation. It was suggested by Kundu (2007).

Define ˆ σ = h(σ). Begin with an initial value of σ, from σ ⁽⁰⁾ . Then we can acquire such as σ ⁽¹⁾ = h(σ ⁽⁰⁾ ). As a mentioned above, we can conduct interatively and then obtain σ ⁽ⁿ⁺¹⁾ = h(σ ⁽ⁿ⁾ ). If |σ ⁽ⁿ⁺¹⁾ − σ ⁽ⁿ⁾ | < , for some pre-assigned permitted limit  > 0, stop the iterative procedure, and then we can obtain the MLE ˆ σ by σ ⁽ⁿ⁺¹⁾ .

3. Approximate maximum likelihood estimation

As mentioned earlier, the log-likelihood equations cannot be solved to closed form explicitly for σ. Therefore, it will be worth considering an approximate maximum likelihood estimation method to the likelihood equation in this section.

Let

ξ i:m:n = F ⁻¹ (p i:m:n ) = − ln

 q _i:m:n 1 + p i:m:n

 , ξ _D = F ⁻¹ (p _D ) = − ln

 q _D 1 + p D

 , where

p i:m:n = 1 −

m

Y

j=m−i+1

j + r _m−j+1 + · · · + r _m j + 1 + r m−j+1 + · · · + r m

, i = 1, · · · , m, q _i:m:n = 1 − p _i:m:n , and q _D = 1 − p _D .

First, we can approximate the following functions using Taylor series expansion as;

F (z i:m:n ) ' α 1i + β 1i z i:m:n , (3.1)

F (V ) ' α _1T + β _1T V, (3.2)

where

α 1i = p i:m:n + 1

2 q i:m:n (1 + p i:m:n ) ln  q i:m:n

1 + p _i:m:n

 , β 1i = 1

2 q i:m:n (1 + p i:m:n ), α 1T = p D + 1

2 q D (1 + p D ) ln  q D

1 + p D

 , β 1T = 1

2 q D (1 + p D ).

By substituting the equations (3.1) and (3.2) into the equation (2.8), we can obtain the following equation (3.3);

∂ ln L

∂σ ∝ − 1 2σ

h 2I −

I

X

i=1

R i z i:m:n −

I

X

i=1

(R i + 2)(α 1i + β 1i z i:m:n )z i:m:n − W (θ ⁽¹⁾ ) i

= 0, (3.3)

where W (θ ⁽¹⁾ ) = 0 for Cases I, III, and W (θ ⁽¹⁾ ) = R ^∗ _D+1 

V + V (α 1T + β 1T V ) 

for Case II.

For Cases I and III, we can obtain an estimator of σ as follows;

˜

σ ₁ = −B ₁ ± pB ₁ ² − 8IC ₁

4I , (3.4)

where

B 1 = −

I

X

i=1

[R i + (R i + 2)α 1i ]X i:m:n , C 1 = −

I

X

i=1

(R i + 2)β 1i X i:m:n 2 .

For Case II, we can obtain an estimator of σ as follows;

˜

σ ₁ = −B 2 ± pB ₂ ² − 8IC 2

4I , (3.5)

where

B 2 = −  X ^I

i=1

[R i + (R i + 2)α 1i ]X i:m:n + R ^∗ _D+1 T (1 + α 1T )  ,

C ₂ = −  X ^I

i=1

(R _i + 2)β _1i X _i:m:n ² + R ^∗ _D+1 T ² β _1T  .

Second, we can approximate the following functions using Taylor series expansion as;

F (z _i:m:n )z _i:m:n ' α 2i + β _2i z _i:m:n , (3.6)

F (V )V ' α 2T + β 2T V, (3.7)

where

α 2i = − 1

2 q i:m:n (1 + p i:m:n ) n

ln  1 + p i:m:n

q _i:m:n

o 2

, β 2i = 1

2 q i:m:n (1 + p i:m:n ) ln  1 + p i:m:n

q i:m:n



+ p i:m:n , α 2T = − 1

2 q D (1 + p D ) n

ln  1 + p D

q _D

o 2

, β 2T = 1

2 q D (1 + p D ) ln  1 + p D

q _D

 + p D .

By substituting the equations (3.6) and (3.7) into the equation (2.8), we can obtain the following equation (3.8);

∂ ln L

∂σ ∝ − 1 2σ

h 2I −

I

X

i=1

R _i z _i:m:n −

I

X

i=1

(R _i + 2)(α _2i + β _2i z _i:m:n ) − W (θ ⁽¹⁾ ) i

= 0, (3.8)

where W (θ ⁽¹⁾ ) = 0 for Cases I, III, and W (θ ⁽¹⁾ ) = R _D+1 ^∗ 

V + (α _2T + β _2T V ) 

for Case II.

For Cases I and III, we can obtain another estimator of σ as follows;

˜ σ 2 = E ₁

G 1

, (3.9)

where

G 1 = 2I −

I

X

i=1

(R i + 2)α 2i , E 1 =

I

X

i=1

[R i + (R i + 2)β 2i ]X i:m:n .

For Case II, we can obtain another estimator of σ as follows;

˜ σ ₂ = E 2

G ₂ , (3.10)

where

G ₂ = 2I −

I

X

i=1

(R _i + 2)α _2i − R ^∗ _D+1 α _2T ,

E 2 =

I

X

i=1

[R i + (R i + 2)β 2i ]X i:m:n + R ^∗ _D+1 T (1 + β 2T ).

4. Bayes estimation

In this section, we consider the conjugate prior for σ as

π(σ) ∝ σ ^−α−1 exp(−β/σ), σ > 0, (4.1) where α > 0 and β > 0. For β = 0, π(σ) reduced to a improper priors and results of this paper will hold for this non-informative prior as α = 2 and β = 1.

By combining likelihood function and the estimation (4.1), the posterior density function of σ is obtained as

π(σ|x) ∝ σ ^−(I+α+1) exp

"

− 1 σ

( _I X

i=1

(R i + 1)x i:m:n + b + W ⁽¹⁾ )#

×

I

Y

i=1

h 1 + exp 

− x _i:m:n σ

i ^−(R

+2)

W ⁽²⁾ , (4.2)

where W ⁽¹⁾ = R ^∗ _D+1 T for Case II, W ⁽¹⁾ = 0 for Cases I and III, and W ⁽²⁾ = 1 + exp − ^T _σ
 ^−R

for case II, W ⁽²⁾ = 1 for Cases I and III.

The Bayes estimator of σ under squared error loss function is the posterior mean. But the posterior mean do not simplified to nice closed forms. So, we use the Tierney and Kadane (1986) approximation method.

Let L(σ; x) be the likelihood functin of σ based on the n observations and π(σ|x) denote the posterior distribution of σ. Then the posterior mean of a function g(σ) can be written as

E(g(σ)|X) = Z

g(σ)π(σ|x)dσ = R e ^nl

^(σ) dσ

R e ^nl(σ) dσ , (4.3)

where

l(σ) = 1

n ln π(σ|x) and l ^∗ (σ) = l(σ) + 1

n ln g(σ).

Tierney and Kadane (1986) showed that the equation (4.3) can be approximated as ;

E(g(σ)|X) =  ψ ^∗ ψ

 ^1/2

exp(n{l ^∗ (ˆ σ ^∗ ) − l(ˆ σ)}) =  ψ ^∗ ψ

 ^1/2 g(ˆ σ ^∗ )π(ˆ σ ^∗ |x)

π(ˆ σ|x) , (4.4) where ˆ σ ^∗ and ˆ σ maximize l ^∗ (σ) and l(σ), respectively and ψ ^∗ and ψ are minus the inverse of the second derivatives of l ^∗ (σ) and l(σ) at ˆ σ ^∗ and ˆ σ, respectively.

In this case,

l(σ) = 1

n −(I + α + 1) ln σ − 1 σ

" _I X

i=1

(R i + 1)x i:m:n + b + W ⁽¹⁾

#

−

I

X

i=1

(R _i + 2) ln(1 + exp(−x _i:m:n /σ)) + ln W ⁽²⁾

!

, (4.5)

l ^∗ (σ) = l(σ) + 1

n ln σ. (4.6)

By substituting the equations (4.5) and (4.6) into the equation (4.4), the Bayes estimator ˆ

σ _B of a function g(σ) = σ under squared error loss function is obtained by

ˆ

σ B =  |ψ ^∗ |

|ψ|

 ^1/2  ˆ σ ^I+α+1 σ ˆ ^{∗ I+α}

 exp

" _I X

i=1

(R i + 1) + b + W ⁽¹⁾

#  1 ˆ σ − 1

ˆ σ ^∗

 !

×

I

Y

i=1

 1 + exp(−x _i:m:n /ˆ σ) 1 + exp(−x i:m:n /ˆ σ ^∗ )

 ^R

+2

W ⁽³⁾ , (4.7)

where W ⁽³⁾ =  1+exp(−T /ˆ σ) 1+exp(−T /ˆ σ

)

 ^R

for Case II, W ⁽³⁾ = 1 for Cases I and III.

5. Simulation results and illustration example

5.1. Simulated results

In this section, we simulate biases and MSEs of the proposed estimators for different n, m, k , T and progressive Type-II censored sampling schemes that were introduced by Balakrishnan and Aggarwala (2000) by Monte Carlo method (based on 10,000 Monte Carlo runs). Such as n = 20, 30, m = 26, 22, 18, 14, k = 3, 4, 6, 8, T = 0.5, 1, and (0 ∗ 17, 2) or (0 ∗ 2, 1, 0 ∗ 5, 1, 0 ∗ 9) and so on. These values are given in Table 5.2.

AMLEs of σ are obtained explicitly with closed form and AMLEs of σ are more efficient than MLE. AMLE ˜ σ ₂ is more efficient than AMLE ˜ σ ₁ . Bayes estimator of σ is more efficient than MLE and AMLEs.

For fixed n, k, T , MSEs decrease as m increases with all scheme. For fixed n, m, k, MSEs decrease as T increases with all scheme. For fixed T, k, m, MSEs decrease as n increases with all scheme. For fixed n, m, T , MSEs decrease as k increases with all scheme. Hence, the results from the suggested estimators can be efficiently used in the part of various statistic areas such as inference, modeling, and life tests.

5.2. Real data

In this subsection, we use the following real data suggested by Lawless (1982). He suggested a particular form of electrical insulation that be taken to a continuously increasing voltage stress. A particular form means successive failure times. This data already was applied by Balakrishnan and Wong (1991).

Table 5.1 A particular form of electrical insulation for increasing voltage stress 12.3 21.8 24.4 28.6 43.2 46.9 70.7 75.3 95.5 98.1 138.6 151.9

We can get MLE ˆ σ=41.4194, AMLEs ˜ σ 1 =41.4291 and ˜ σ 2 =41.2023, and Bayes estimate ˆ

σ B =41.5159 for complete data. For this example, we can apply n = 12, m = 10 (0 ∗ 9, 2), T =

1, k = 4. Then we can get MLE ˆ σ=43.4116, AMLEs ˜ σ 1 =43.42258 and ˜ σ 2 =43.1539, and Bayes

estimate ˆ σ B =43.3463.

Table 5.2 The relative MSEs (biases) of the estimators of σ

n m k T Scheme σ ˆ σ ˜

σ ˜

ˆ σ

20 18 3 0.5 (0*17, 2) 0.21029(0.1244) 0.21028(0.1245) 0.20879(0.1207) 0.15146(0.1017)

(0*2, 1, 0*5, 1, 0*9) 0.21470(0.1087) 0.21468(0.1087) 0.21317(0.1050) 0.16267(0.0989)

(0*8, 1, 1, 0*8) 0.21072(0.1241) 0.21070(0.1241) 0.20922(0.1203) 0.15189(0.1013)

1 (0*17, 2) 0.13631(0.0693) 0.13625(0.0707) 0.13310(0.0606) 0.11067(0.0778)

(0*2, 1, 0*5, 1, 0*9) 0.14180(0.0403) 0.14167(0.0407) 0.14080(0.0353) 0.11553(0.0447)

(0*8, 1, 1, 0*8) 0.14821(0.0435) 0.14802(0.0441) 0.14733(0.0384) 0.11297(0.0406)

4 0.5 (0*17, 2) 0.14755(0.0747) 0.14755(0.0748) 0.14681(0.0713) 0.11345(0.0825)

(0*2, 1, 0*5, 1, 0*9) 0.13208(0.0396) 0.13208(0.0397) 0.13127(0.0362) 0.11195(0.0545)

(0*8, 1, 1, 0*8) 0.14798(0.0743) 0.14797(0.0744) 0.14724(0.0709) 0.11511(0.0799)

1 (0*17, 2) 0.12553(0.0666) 0.12546(0.0680) 0.12281(0.0580) 0.10517(0.0819)

(0*2, 1, 0*5, 1, 0*9) 0.12452(0.0357) 0.12438(0.0361) 0.12383(0.0307) 0.11553(0.0501)

(0*8, 1, 1, 0*8) 0.13744(0.0408) 0.13724(0.0414) 0.13670(0.0357) 0.12983(0.0537)

14 3 0.5 (0*13, 6) 0.22354(0.1137) 0.22353(0.1138) 0.22217(0.1101) 0.14675(0.0977)

(0*10, 1,2, 0*1, 3) 0.22359(0.1137) 0.22358(0.1138) 0.22222(0.1101) 0.14675(0.0977)

(0*9, 1, 2, 1, 2, 0*1) 0.21029(0.1244) 0.21028(0.1245) 0.20879(0.1207) 0.14681(0.0977)

1 (0*13, 6) 0.14343(0.0723) 0.14340(0.0727) 0.14203(0.0675) 0.11781(0.0792)

(0*10, 1,2, 0*1, 3) 0.15040(0.0631) 0.15021(0.0636) 0.14927(0.0581) 0.12786(0.0692)

(0*9, 1, 2, 1, 2, 0*1) 0.15795(0.0508) 0.15756(0.0515) 0.15724(0.0454) 0.12616(0.0421)

4 0.5 (0*13, 6) 0.15477(0.0645) 0.15477(0.0646) 0.15403(0.0612) 0.12716(0.1000)

(0*10, 1,2, 0*1, 3) 0.15482(0.0645) 0.15482(0.0645) 0.15408(0.0611) 0.12717(0.1000)

(0*9, 1, 2, 1, 2, 0*1) 0.15495(0.0643) 0.15494(0.0644) 0.15421(0.0610) 0.12761(0.1004)

1 (0*13, 6) 0.12925(0.0688) 0.12921(0.0691) 0.12798(0.0640) 0.11664(0.0747)

(0*10, 1,2, 0*1, 3) 0.13622(0.0596) 0.13602(0.0601) 0.13523(0.0545) 0.12252(0.0597)

(0*9, 1, 2, 1, 2, 0*1) 0.14377(0.0472) 0.14338(0.0479) 0.14319(0.0418) 0.11931(0.0409)

30 26 6 0.5 (0*25, 4) 0.10542(0.0561) 0.10542(0.0561) 0.10505(0.0539) 0.09663(0.0725)

(0*9, 2, 0*9, 2, 0*6) 0.10807(0.0528) 0.10806(0.0528) 0.10773(0.0506) 0.09762(0.0655)

(0*15, 2, 2, 0*9) 0.10547(0.0561) 0.10547(0.0561) 0.10510(0.0539) 0.09663(0.0725)

1 (0*25, 4) 0.07751(0.0491) 0.07749(0.0493) 0.07695(0.0459) 0.07896(0.0673)

(0*9, 2, 0*9, 2, 0*6) 0.07779(0.0042) 0.07772(0.0004) 0.07755(0.0008) 0.06941(0.0099)

(0*15, 2, 2, 0*9) 0.08512(0.0371) 0.08497(0.0374) 0.08481(0.0336) 0.08056(0.0611)

8 0.5 (0*25, 4) 0.08322(0.0148) 0.08322(0.0149) 0.08293(0.0127) 0.07508(0.0336)

(0*9, 2, 0*9, 2, 0*6) 0.08586(0.0115) 0.08585(0.0116) 0.08561(0.0093) 0.07938(0.0364)

(0*15, 2, 2, 0*9) 0.08326(0.1244) 0.08326(0.0148) 0.08298(0.0126) 0.07508(0.0336)

1 (0*25, 4) 0.06928(0.0454) 0.06926(0.0456) 0.06880(0.0422) 0.07715(0.0641)

(0*9, 2, 0*9, 2, 0*6) 0.06956(0.0006) 0.06949(0.0009) 0.06939(0.0028) 0.06895(0.0104)

(0*15, 2, 2, 0*9) 0.07689(0.0334) 0.07674(0.0338) 0.07666(0.0299) 0.08081(0.0617)

22 6 0.5 (0*21, 8) 0.11016(0.0678) 0.11016(0.0679) 0.10971(0.0656) 0.09947(0.0759)

(0*18, 4, 0*2, 4) 0.11016(0.0678) 0.11016(0.0679) 0.10971(0.0656) 0.09947(0.0759)

(0*15, 1, 3, 1, 3, 0*3) 0.11016(0.0678) 0.11016(0.0679) 0.10971(0.0656) 0.09949(0.0759)

1 (0*21, 8) 0.08286(0.0571) 0.08284(0.0574) 0.08229(0.0539) 0.06816(0.0530)

(0*18, 4, 0*2, 4) 0.08447(0.0553) 0.08440(0.0555) 0.08398(0.0520) 0.06994(0.0508)

(0*15, 1, 3, 1, 3, 0*3) 0.09038(0.0468) 0.09018(0.0472) 0.09016(0.0432) 0.07824(0.0395)

8 0.5 (0*21, 8) 0.08432(0.0255) 0.08432(0.0255) 0.08402(0.0233) 0.07367(0.0454)

(0*18, 4, 0*2, 4) 0.08204(0.0190) 0.08204(0.0191) 0.08175(0.0168) 0.07367(0.0454)

(0*15, 1, 3, 1, 3, 0*3) 0.08345(0.0151) 0.08346(0.0151) 0.08307(0.0127) 0.07367(0.0454)

1 (0*21, 8) 0.07345(0.0531) 0.07342(0.0533) 0.07295(0.0499) 0.06762(0.0525)

(0*18, 4, 0*2, 4) 0.07505(0.0512) 0.07498(0.0515) 0.07464(0.0480) 0.06940(0.0503)

(0*15, 1, 3, 1, 3, 0*3) 0.07505(0.0512) 0.07498(0.0515) 0.07464(0.0480) 0.07791(0.0398)

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