Estimation of half triangle distribution under generalized progressive hybrid censored sample

Estimation of half triangle distribution under generalized progressive hybrid censored sample

Ho-Yong Kim ¹ · Suk-Bok Kang ²

12 Department of Statistics, Yeungnam University

Received 16 March 2018, revised 2 May 2018, accepted 8 May 2018

Abstract

Using some Taylor series expansions, we obtain maximum likelihood estimator (MLE) and some approximate maximum likelihood estimators (AMLEs) of the scale parameter in half triangle distribution based on generalized progressive hybrid censored samples. We also obtain some estimators of the relibility function by using the proposed estimators of the scale parameter of the half triangle distribution. Finally, Monte Carlo simulations are used to assess the validity of the proposed estimators. The proposed AMLEs are obtained explicitly with closed form and more efficient than the MLE of the scale parameter.

Keywords: Approximate maximum likelihood estimator, generalized progressive hybrid censored sample, half triangle distribution, reliability.

1. Introduction

The random variable X with a half triangle distribution has a probability density function (pdf)

f (x; θ) = 2 θ

 1 − x

θ



, 0 < x < θ (1.1)

and cumulative distribution function (cdf) F (x; θ) = 1 − 

1 − x θ

 2

, 0 < x < θ. (1.2)

A triangle distribution was applied to a kernel function in non-parametric density esti- mation. Some properties of the triangular distribution was studied by Balakrishnan and Nevzorov (2003). Han and Kang (2008) derived MLE and AMLEs of the scale parameter in the half triangle distribution under progressively type-II censored samples. Kang (2007) proposed some explicit estimators of the location parameter in the half triangle distribution based on multiply type II censored samples. Seo and Kang (2014) dealt with the problem of

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]

predicting censored data in a half triangle distribution with an unknown parameter based on progressively type-II censored samples. Han and Kang (2012) provided the MLE and AMLEs for the scale parameter in exponentiated half triangle distribution based on hybrid censored data. Kang et al. (2009) dealt with estimation based on type-I hybrid censord sam- ples from the half triangle distribution. Lee et al. (2016) examined statistical inferences of a scale parameter of the half triangle distribution under progressively type-I interval censored samples. Seo and Kang (2017) proposed a new method for the half logistic distribution based on a pivotal quantity from the progressively type-II censoring scheme.

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

They estimated the entropy of the two-parameter Weibull distribution based on generalized progressive censored samples. Cho et al. (2015b) derived the exact distribution of the MLE and confidence interval (CI) for the parameter of the exponential distribution under gener- alized progressive hybrid censoring scheme. Ahmed Elshahhat (2017) provided parameters estimation for the exponentiated Weibull distribution based on generalized progressive hy- brid censoring schemes. Lee (2017) obtained the MLE and Bayes estimator of the entropy of the inverse Weibull distribution under generalized progressive hybrid censoring scheme.

Nowadays, progressive hybrid censoring scheme has been used in the field of life testing problem and survival analysis but it had limitation. It cannot be reflected when few failures generate before pre-fixed time T . The generalized progressive hybrid censoring scheme can be described as follows. The integers k, m ∈ {1, 2, · · · , n} are pre-fixed such as k < m. We denote progressive type II censored sample by X 1:m:n , X 2:m:n , · · · , X m:m:n . 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. This scheme is guaranted a minimum k failures, that is X k:m:n . At the moment of the first failure, R 1 of the remaining units are randomly removed. Likewise at the moment of the second failure, R 2 of the remaining units are randomly removed and so on. This procedure continues until, immediately following the terminated time T ^∗ = max{X _k:m:n , min{X _m:m:n , T }}. All the remaining units are removed from the experiment.

Let D denote the number of observed failures up to time T .

In this paper, we propose MLE and AMLEs of the scale parameter θ in half triangle distribution based on a generalized progressive hybrid censoring scheme. First, the scale parameter of half triangle distribution is estimated by AMLEs using some kinds of Taylor series expansions, and then the MLE is calculated using these AMLEs as initial values. We also compare the suggested estimators by using the mean squared error (MSE) through Monte Carlo simulation from various censored samples.

2. Maximum likelihood estimation

We obtain MLE for the scale parameter θ of half triangle distribution with pdf (1.1) based on generalized progressive hybrid censored samples. For a generalized progressive hybrid censored samples, we have one of the following three forms;

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.

Given a generalized progressive hybrid censored sample, the likelihood functions can be

Figure 2.1 Schematic picture of generalized progressive hybrid censoring scheme

written;

L i (θ) = K i V

Y

j=1

f (x j ; θ) [1 − F (x j ; θ)] ^R

w i (θ),

where i = 1, 2, 3 for Cases I, II and III, respectively, V 1 = k, V 2 = D, V 3 = m, w i (θ) = 1, for i=1,3, w 2 (θ) = [1 − F (T ; θ)] ^R

, R _k ^∗ = n − k − P k−1

j=1 R j , R ^∗ _D+1 = n − D − P D

j=1 R j and K i = Q V

j=1

P m

k=j (R k + 1).

Additionally, the corresponding log likelihood functions are:

l(θ) ∝ −J lnθ +

J

X

j=1

(2R _j + 1)log 

1 − x j:m:n

θ



+ W (θ), (2.1)

where J = k, W (θ) = 0 and R j = n − k = P k−1

j=1 R j for Case I, J = D and W (θ) = 2R ^∗ _D+1 ln 1 − ^T _θ
for Case II and J = m and W (θ) = 0 for Case III.

Differentiating (2.1) with θ, we obtain

∂l(θ)

∂θ = − J θ +

J

X

j=1

(2R _j + 1)

x

θ

1 − ^x

_θ + W

(θ) = 0, (2.2)

where W ⁰ (θ) = 0 for Cases I and III and W ⁰ (θ) = 2R ^∗ _D+1

T θ2

1−

for Case II

The random variable Z = ^X _θ has a standard half triangle distribution with pdf and cdf;

f (z) = 2(1 − z)

and

F (z) = 1 − (1 − z) ² .

Because the log-likelihood equations cannot be solved for θ to closed form explicitly, we must apply some numerical method. We used the Newton-Raphson method to obtain the maximum likelihood estimates.

3. Approximate maximum likelihood estimation

The approximate maximum likelihood estimating method was developed by Balakrishnan (1989) for the solved explicit estimators of the scale parameter in Rayleigh distribution.

Kang and Han (2015) obtained AMLEs of the shape parameter and scale parameter in the inverse Weibull distribution based on multiply type-II censored sample. They also proposed a graphical method for goodness-of-fit test by using AMLEs. Seo and Kang (2015) provided a simpler estimation equation than the likelihood equation for estimating the scale parameter of the half logistic distribution based on progressively type II censored samples. Because log-likelihood equations cannot be solved exlicit solutions, we consider an approximation to the likelihood equation which provide us with explicit estimator for θ. Let

ξ i:m:n = F ⁻¹ (p i:m:n ) = 1 − √ q i:m:n , ξ D = F ⁻¹ (p D ) = 1 − √

q D

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

We consider first two terms in Taylor series expansions of Z j:m:n /(1 − Z j:m:n ) and 1/(1 − Z j:m:n ). First, we can approximate the following functions

Z j:m:n

1 − Z j:m:n

≈ − ξ _j:m:n ² q i:m:n

+ 1

q i:m:n

Z j:m:n , (3.1)

Z _T 1 − Z T

≈ − ξ _D ² q D

+ 1 q D

Z _T . (3.2)

By substituting the equations (3.1) and (3.2) into the equation (2.2), we can obtain the following equation;

∂l(θ)

∂θ ≈ − 1 θ h

J +

J

X

j=1

(2R j + 1) − ξ _j:m:n ² q i:m:n

+ 1

q i:m:n

Z j:m:n

!

+ W (θ ⁽¹⁾ ) i

= 0, (3.3)

where W (θ ⁽¹⁾ ) = 0 for Cases I and III, and W (θ ⁽¹⁾ ) = 2R ^∗ _D+1 

− ^ξ _q

D

+ _q ¹

D

T 

for Case II.

From the equation (3.3), we can obtain an estimator of θ as follows;

θ ˜ 1 = B 1

A ₁ , (3.4)

where

A 1 =

J

X

j=1

(2R j + 1) X j:m:n

q j:m:n

+ 2R ^∗ _D+1 T q D

, (3.5)

B ₁ = J +

J

X

j=1

(2R _j + 1) ξ _j:m:n ²

q _j:m:n + 2R ^∗ _D+1 ξ _D ²

q _D , (3.6)

where 2R ^∗ _{D+1 q} ^T

D

and 2R ^∗ _D+1 ^ξ _q

D

are 0 for Cases I and III.

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

1 1 − Z j:m:n

≈ (1 − 2ξ j:m:n ) q i:m:n

+ 1

q i:m:n

Z j:m:n , (3.7)

1 1 − Z T

≈ (1 − 2ξ D ) q D

+ 1 q D

Z T . (3.8)

By substituting the equations (3.7) and (3.8) into the equation (2.2), we can obtain

∂l(θ)

∂θ ≈ − 1 θ h

J −

J

X

j=1

(2R j + 1) ·  (1 − 2ξ _j:m:n ) q i:m:n

+ 1

q i:m:n

Z j:m:n



− W (θ ⁽²⁾ ) i

= 0, (3.9)

where W (θ ⁽²⁾ ) = 0 for Cases I and III, and W (θ ⁽²⁾ ) = 2R ^∗ _D+1  _(1−2ξ

D

) q

+ _q ¹

D

T 

for Case II.

The equation (3.9) is a quadratic equation in θ, with its roots given by θ ˜ 2 = −A 2 ± pA ² ₂ − 4J B 2

2J , (3.10)

where

A 2 = −

J

X

j=1

(2R j + 1) 1 − 2ξ j:m:n

q _j:m:n X j:m:n − 2R ^∗ _D+1 1 − 2ξ D

q _D T, (3.11)

B 2 = −

J

X

j=1

(2R j + 1) 1 q j:m:n

X _j:m:n ² − 2R ^∗ _D+1 1 q D

T ² , (3.12)

where −2R _D+1 ^{∗ 1−2ξ} _q

D

T and −2R ^∗ _{D+1 q} ¹

D

T ² are 0 for Cases I and III.

Because B 2 < 0 and J > 0, only one root is admissible, and hence another AMLE of θ is given by

θ ˜ ₂ = −A 2 + pA ² ₂ − 4J B 2

2J . (3.13)

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

Z j:m:n

1 − Z _j:m:n ≈ ξ ³ _j:m:n

(p1 − p _j:m:n ) ³ + (1 − 3ξ j:m:n )

(p1 − p _j:m:n ) ³ Z j:m:n + ξ ³ _j:m:n

(p1 − p _j:m:n ) ³ Z _j:m:n ² , (3.14) Z T

1 − Z _T ≈ ξ _D ³ ( √

1 − p D ) ³ + (1 − 3ξ D ) ( √

1 − p D ) ³ Z _T + ξ _D ³ ( √

1 − p D ) ³ Z _T ² . (3.15)

By substituting the equations (3.14) and (3.15) into the equation (2.2), we can obtain the following equation;

∂l(θ)

∂θ ≈ − 1 θ h

J −

J

X

j=1

(2R j + 1)  ξ _j:m:n ³

(p1 − p j:m:n ) ³ + (1 − 3ξ j:m:n ) (p1 − p j:m:n ) ³ Z j:m:n

+ ξ _j:m:n ³

(p1 − p j:m:n ) ³ Z _j:m:n ² 

+ W (θ ⁽³⁾ ) i

= 0, (3.16)

where W (θ ⁽³⁾ ) = 0 for Cases I, III, and W (θ ⁽³⁾ ) = − P J

j=1 (2R ^∗ _D+1 )  _ξ

( √

1−p

)

+ ₍ ^{(1−3ξ √} _1−p

⁾

D

)

T + ₍ ^√ _1−p ^ξ

D

)

Z _T ² 

for Case II.

The equation (3.16) is a quadratic equation in θ, with its roots given by θ ˜ 3 = −B 3 ± pB ₃ ² − 4A 3 C 3

2A 3

, (3.17)

where

A 3 =

J

X

j=1

(2R j + 1) (1 − p(1 − p _j:m:n )) ³

(p1 − q j:m:n ) ³ + 2R ^∗ _D+1 (1 − p(1 − p _D )) ³ ( √

1 − q _D ) ³ − J, (3.18)

B ₃ =

J

X

j=1

(2R _j + 1) 1 − 3(1 − p(1 − p j:m:n ))

(p1 − q j:m:n ) ³ X _j:m:n + 2R ^∗ _D+1 1 − 3(1 − p(1 − p D )) ( √

1 − q D ) ³ T, (3.19) C 3 =

J

X

j=1

(2R j + 1) 1

(p1 − q _j:m:n ) ³ X _j:m:n ² + 2R ^∗ _D+1 1 ( √

1 − q D ) ³ T ² , (3.20)

where 2R ^∗ _D+1 ⁽¹⁻

√

(1−p

))

( √

1−q

)

, 2R ^∗ _D+1 ¹⁻³⁽¹⁻

√

(1−p

)) ( √

1−q

)

T , and 2R ^∗ _{D+1 (} ^√ _1−q ¹

D

)

T ² are all 0 for Cases I, III.

Because A 3 is negative from the results of various simulations, only one root is admissi- ble and hence we propose another AMLE of θ as follows

θ ˜ 3 = −B 3 − pB ₃ ² − 4A 3 C ₃ 2A 3

. (3.21)

Now we consider a progressive type II censored sample scheme as special case of generalized

progressive hybrid censored sample.

Figure 3.1 Progressive type II censoring scheme

Case III of generalized progressive hybrid censoring is the same as progressive type-II censoring.

L III = K 3 m

Y

j=1

f (x j:m:n )[1 − F (x j:m:n )] ^R

.

Han and Kang (2008) derived MLE and AMLEs of scale parameter θ in a half triangle distribution based on progressive type II censored samples. AMLEs will be used as initial values in the Newton-Raphson method. Let ˆ θ i (i = 1, 2, 3) be the maximum likelihood estimates that are obtained by using ˜ θ i as the initial values in Newton-Raphson method, respectively. We compare proposed estimators based on progressive type II censored samples by means of MSE. These values are given in Table 3.1.

Table 3.1 The relative MSEs of the estimators of θ under progressive type II censoring scheme

n m Scheme θ ˜

θ ˜

θ ˜

θ ˆ

θ ˆ

θ ˆ

10 10 (10*0) 0.052075 0.044918 0.037404 0.045300 0.039601 0.034942 6 (3*0,2,2,0) 0.095282 0.090419 0.086915 0.092895 0.087105 0.084475 6 (2*0.4.3*0) 0.095980 0.084881 0.076632 0.087988 0.077964 0.072946 5 (5,4*0) 0.113378 0.096693 0.085544 0.100071 0.087317 0.081097 15 15 (15*0) 0.032322 0.028225 0.023040 0.028125 0.024861 0.021453 10 (5,9*0) 0.052933 0.045674 0.038265 0.046239 0.040413 0.035792 10 (4*0,3,3*0,2,0) 0.054973 0.051131 0.048332 0.052347 0.049003 0.046863 10 (0,3,6*0,2,0) 0.056819 0.050518 0.044936 0.052070 0.046404 0.042618 10 (2*0,1,0,2,0,2,3*0) 0.056919 0.050718 0.045214 0.052259 0.046663 0.042901 20 20 (20*0) 0.022787 0.020108 0.016236 0.019848 0.017727 0.015106 15 (3*0,2,4*0,3,6*0) 0.035365 0.031138 0.026398 0.031594 0.028035 0.024784 10 (5,2*0,5,6*0) 0.056308 0.049060 0.042280 0.050339 0.044140 0.039754 10 (2*0,1,0,2,0,2,2*0,5) 0.062499 0.062780 0.062944 0.062649 0.062874 0.062964 10 (2*0,3,0,2,0,2,2*0,3) 0.058440 0.058126 0.058214 0.058108 0.058161 0.058236 30 30 (30*0) 0.014056 0.012602 0.010076 0.012296 0.011163 0.009384 20 (3*0,5,*0,5,12*0) 0.024896 0.021937 0.018084 0.021931 0.019541 0.016889 20 (2*0,10,17*0) 0.023662 0.020848 0.016948 0.020679 0.018438 0.015789 20 (9*0,10,10*0) 0.026745 0.023720 0.020001 0.023917 0.021393 0.018779 15 (5,6*0,10,7*0) 0.038368 0.033971 0.029378 0.034761 0.030877 0.027620 15 (10,6*0,5,7*0) 0.036502 0.032149 0.027321 0.032673 0.028970 0.025654

4. Estimation of the reliability

The reliability function of the half triangle distribution is

R(t) = 1 − F (t) = P [X > t] =

 1 − t

θ

 ²

, t > 0.

We can use the proposed estimators of the scale parameter θ to estimate the reliability function. The estimators of the reliability function are as follows.

R ˆ _i (t) =

 1 − t

θ ˆ i

 ²

, i = 1, 2, 3 and R ˜ _i (t) =

 1 − t

θ ˜ _i

 ²

, i = 1, 2, 3.

The relative MSEs of the proposed estimators of the reliability function R(t) are given in Figure 4.1 for size n = 20, m = 20(20 ∗ 0) and m = 10(2 ∗ 0, 1, 0, 2, 0, 2, 2 ∗ 0, 5).

[n = 20, m = 20(20 ∗ 0)], [n = 20, m = 10(2 ∗ 0, 1, 0, 2, 0, 2, 2 ∗ 0, 5)]

Figure 4.1 The relative MSEs of ˆ R

(t) and ˜ R

(t)

5. The simulation results

We obtain MLE and AMLEs of the scale parameter θ, and reliability function of half triangle distribution based on generalized progressive hybrid type II censored samples. We compare the suggested estimators in the sense of MSE using Monte Carlo simulation for various kinds of censoring schemes. From Table 3.1 and Figure 4.1, we obtain the following results. In most cases, ˜ θ 3 is more efficient than the other estimators, MSE decreases as m, n increases. ˆ θ 3 and ˜ θ 3 are generally more efficient than ˆ θ 1 , ˜ θ 1 , ˆ θ 2 , and ˜ θ 2 in the sense of MSE.

The biases and MSEs of proposed estimators for different n, m, k, T based on generalized progressive hybrid censored sample are given in Table 5.1. We observe that MSEs decrease as n increases. For fixed n, m, T, MSEs decrease as k increases and for fixed n, k, T, MSEs decrease as m increases for all schemes. In most cases, ˜ θ _i , i = 1, 2, 3 are more efficient than θ ˆ _i . ˜ θ ₁ is generally more efficient than the other estimators.

Bayes and interval estimations for the scale parameter θ based on generalized progressive hybrid type-II censored sample are not be considered in this paper. Therefore, in the future study, Bayes and interval estimations will be needed.

Table 5.1 MSEs (biases) of the estimators of θ under generalized progressive hybrid censoring scheme

n m k T Scheme θ ˆ θ ˜

20 15 2 0.1 (0*14,5) θ ˆ

0.39974(0.19543) θ ˜

0.39830(0.19382)

θ ˆ

0.39974(0.19544) θ ˜

0.39968(0.19533)

θ ˆ

0.39974(0.19544) θ ˜

0.39976(0.19546)

(0, 2, 0*11, 1,2) θ ˆ

0.39688(0.16385) θ ˜

0.39534(0.16245)

θ ˆ

0.39689(0.16386) θ ˜

0.39683(0.16379)

θ ˆ

0.39689(0.16386) θ ˜

0.39691(0.16388)

0.3 (0*14, 5) θ ˆ

0.10650(0.05795) θ ˜

0.10632(0.05617)

θ ˆ

0.10650(0.05795) θ ˜

0.10653(0.05762)

θ ˆ

0.10650(0.05795) θ ˜

0.10651(0.05799)

(0,2, 0*11, 1, 2) θ ˆ

0.10000(0.04178) θ ˜

0.10001(0.03981)

θ ˆ

0.10000(0.04180) θ ˜

0.10011(0.04139)

θ ˆ

0.10000(0.04180) θ ˜

0.10000(0.04181)

3 0.1 (0*14,5) θ ˆ

0.21837(0.99186) θ ˜

0.21812(0.98161)

θ ˆ

0.21837(0.99187) θ ˜

0.21837(0.99111)

θ ˆ

0.21837(0.99187) θ ˜

0.21837(0.99203)

(0, 2, 0*11, 1,2) θ ˆ

0.17638(0.03368) θ ˜

0.17583(0.03268)

θ ˆ

0.17638(0.03368) θ ˜

0.17634(0.03361)

θ ˆ

0.17638(0.03368) θ ˜

0.17637(0.03368)

0.3 (0*14, 5) θ ˆ

0.10650(0.05795) θ ˜

0.10632(0.05617)

θ ˆ

0.10650(0.05795) θ ˜

0.10653(0.05762)

θ ˆ

0.10650(0.05795) θ ˜

0.10651(0.05799)

(0,2, 0*11, 1, 2) θ ˆ

0.10002(0.04178) θ ˜

0.10001(0.03981)

θ ˆ

0.10002(0.04178) θ ˜

0.10012(0.04139)

θ ˆ

0.10002(0.04178) θ ˜

0.10003(0.04181)

Table 5.1 Continued

n m k T Scheme θ ˆ θ ˜

20 12 2 0.1 (0*11,8) θ ˆ

0.40200(0.20825) θ ˜

0.40044(0.20660)

θ ˆ

0.40201(0.20826) θ ˜

0.40194(0.20815)

θ ˆ

0.40201(0.20826) θ ˜

0.40202(0.20828)

(0,2,2,0*8,4) θ ˆ

0.40979(0.13918) θ ˜

0.40826(0.13750)

θ ˆ

0.40979(0.13919) θ ˜

0.40975(0.13909)

θ ˆ

0.40979(0.13919) θ ˜

0.40982(0.13921)

0.3 (0*11,8) θ ˆ

0.14535(0.06757) θ ˜

0.14511(0.06575)

θ ˆ

0.14535(0.06758) θ ˜

0.14538(0.06725)

θ ˆ

0.14535(0.06758) θ ˜

0.14537(0.06760)

(0,2,2,0*8,4) θ ˆ

0.14784(0.03378) θ ˜

0.14786(0.03072)

θ ˆ

0.14783(0.03381) θ ˜

0.14801(0.03308)

θ ˆ

0.14782(0.03381) θ ˜

0.14787(0.03377)

3 0.1 (0*11,8) θ ˆ

0.23187(0.11638) θ ˜

0.23157(0.11536)

θ ˆ

0.23187(0.11639) θ ˜

0.23185(0.11631)

θ ˆ

0.23187(0.11639) θ ˜

0.23186(0.11640)

(0,2,2,0*8,4) θ ˆ

0.20153(0.01478) θ ˜

0.20103(0.01355)

θ ˆ

0.20153(0.01478) θ ˜

0.20150(0.01468)

θ ˆ

0.20153(0.01478) θ ˜

0.20152(0.01478)

0.3 (0*11,8) θ ˆ

0.14535(0.06757) θ ˜

0.14511(0.06575)

θ ˆ

0.14535(0.06758) θ ˜

0.14538(0.06725)

θ ˆ

0.14535(0.06758) θ ˜

0.14537(0.06760)

(0,2,2,0*8,4) θ ˆ

0.14784(0.03378) θ ˜

0.14786(0.03072)

θ ˆ

0.14783(0.03381) θ ˜

0.14801(0.03308)

θ ˆ

0.14782(0.03381) θ ˜

0.14787(0.03377)

40 35 8 0.1 (0*34,5) θ ˆ

0.08417(0.01659) θ ˜

0.08414(0.01626)

θ ˆ

0.08417(0.01659) θ ˜

0.08417(0.01657)

θ ˆ

0.08417(0.01659) θ ˜

0.08416(0.01659)

(0,2,0*31,1,2) θ ˆ

0.07956(0.01603) θ ˜

0.07950(0.01565)

θ ˆ

0.07956(0.01603) θ ˜

0.07956(0.01600)

θ ˆ

0.07956(0.01603) θ ˜

0.07956(0.01603)

0.3 (0*34,5) θ ˆ

0.04319(0.03118) θ ˜

0.04317(0.03033)

θ ˆ

0.04319(0.03118) θ ˜

0.04320(0.03102)

θ ˆ

0.04319(0.03118) θ ˜

0.04320(0.03119)

(0,2,0*31,1,2) θ ˆ

0.04871(0.02593) θ ˜

0.04869(0.02498)

θ ˆ

0.04871(0.02593) θ ˜

0.04873(0.02575)

θ ˆ

0.04871(0.02593) θ ˜

0.04871(0.02594)

12 0.1 (0*34,5) θ ˆ

0.06428(-0.01547) θ ˜

0.06424(-0.01592)

θ ˆ

0.06428(-0.01547) θ ˜

0.06427(-0.01552)

θ ˆ

0.06428(-0.01547) θ ˜

0.06428(-0.01548)

(0,2,0*31,1,2) θ ˆ

0.06016(0.00019) θ ˜

0.06000(0.01565)

θ ˆ

0.06016(0.00019) θ ˜

0.06009(-0.00039)

θ ˆ

0.06016(0.00019) θ ˜

0.06015(0.00012)

0.3 (0*34,5) θ ˆ

0.04425(0.04283) θ ˜

0.04420(0.04197)

θ ˆ

0.04425(0.04283) θ ˜

0.04426(0.04267)

θ ˆ

0.04425(0.04283) θ ˜

0.04425(0.04284)

(0,2,0*31,1,2) θ ˆ

0.04470(0.03894) θ ˜

0.04466(0.03804)

θ ˆ

0.04470(0.03894) θ ˜

0.04471(0.03878)

θ ˆ

0.04470(0.03894) θ ˜

0.04471(0.03895)

Table 5.1 Continued

n m k T Scheme θ ˆ θ ˜

40 31 8 0.1 (0*31,9) θ ˆ

0.08470(0.01634) θ ˜

0.08467(0.01600) θ ˆ

0.08470(0.01634) θ ˜

0.08470(0.01631) θ ˆ

0.08470(0.01634) θ ˜

0.08470(0.01634) (0,2,2,0*27,5) θ ˆ

0.08555(-0.00944) θ ˜

0.08543(-0.00996) θ ˆ

0.08555(-0.00944) θ ˜

0.08554(-0.00949) θ ˆ

0.08555(-0.00944) θ ˜

0.08555(-0.00945) 0.3 (0*31,9) θ ˆ

0.04688(0.04302) θ ˜

0.04683(0.04215) θ ˆ

0.04688(0.04303) θ ˜

0.04689(0.04287) θ ˆ

0.04688(0.04303) θ ˜

0.04688(0.04304) (0,2,2,0*27,5) θ ˆ

0.04872(0.03504) θ ˜

0.04871(0.03406) θ ˆ

0.04872(0.03504) θ ˜

0.04873(0.03485) θ ˆ

0.04872(0.03504) θ ˜

0.04872(0.03504) 12 0.1 (0*31,9) θ ˆ

0.06712(0.00265) θ ˜

0.06705(0.00221) θ ˆ

0.06712(0.00266) θ ˜

0.06711(0.00261) θ ˆ

0.06712(0.00266) θ ˜

0.06711(0.00265) (0,2,2,0*27,5) θ ˆ

0.06508(-0.01841) θ ˜

0.06499(-0.01920) θ ˆ

0.06508(-0.01841) θ ˜

0.06506(-0.01852) θ ˆ

0.06508(-0.01841) θ ˜

0.06507(-0.01842) 0.3 (0*31,9) θ ˆ

0.04578(0.04258) θ ˜

0.04573(0.04171) θ ˆ

0.04578(0.04258) θ ˜

0.04578(0.04242) θ ˆ

0.04578(0.04258) θ ˜

0.04578(0.04259) (0,2,2,0*27,5) θ ˆ

0.04622(0.03381) θ ˜

0.04620(0.03283) θ ˆ

0.04622(0.03382) θ ˜

0.04624(0.03363) θ ˆ

0.04622(0.03382) θ ˜

0.04622(0.03382)

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