Estimation for a half-triangular distribution based on unied hybrid censored sample

Estimation for a half-triangular distribution based on unified hybrid censored sample

Young Eun Jeon ¹ · Suk-Bok Kang ²

12 Department of Statistics, Yeungnam University

Received 29 October 2018, revised 18 November 2018, accepted 19 November 2018

Abstract

In this paper, we derive some estimators of the scale parameter of half-triangular distribution based on unified hybrid censored samples. First, we obtain the maximum likelihood estimator (MLE) of the scale parameter. However, the MLE can’t be obtained explicitly with closed form because of the nonlinearity of the likelihood equation. So, we propose some approximate maximum likelihood estimators (AMLEs) using two different types of Taylor series expansions for some nonlinear functions of the likelihood equation.

Finally, we compare the proposed estimators in the sense of the mean squared error (MSE) through Monte Carlo simulation. From the simulation results, we can show that the proposed AMLEs are usually more efficient than the MLE.

Keywords: Approximate maximum likelihood estimator, half-triangular distribution, maximum likelihood estimator, unified hybrid censoring

1. Introduction

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

f (x; θ) = 2 θ

 1 − x

θ



, 0 < x < θ, (1.1)

and

F (x; θ) = 1 −  1 − x

θ

 ²

, 0 < x < θ. (1.2)

Triangular distribution was applied to a kernel function in nonparametric density estima- tion. Johnson (1997) studied the possibility of using the more intuitively obvious triangular distribution as a proxy for the beta distribution. Some properties of the triangular distribu- tion were studied by Balakrishnan and Nevzorov (2003). Kang (2007) proposed some explicit

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]

estimators for the half-triangular distribution based on multiply type-II censored samples.

Lee et al. (2008) proposed some approximate maximum likelihood estimators (AMLEs) of the scale parameter in a triangular distribution based on multiply type-II censored samples.

Han and Kang (2008) proposed the maximum likelihood estimator (MLE) and AMLE for the half-triangular distribution based on progressively type-II censored samples. Kang et al.

(2009) studied the AMLE of the scale parameter of the half-triangular distribution based on type-I hybrid censored samples. Kim et al. (2011) proposed Bayesian estimators of shape parameter and reliability function in the exponentiated half-triangular distribution based on type-I hybrid censored data. Kim and Kang (2018) obtained the MLE and some AMLEs of the scale parameter in half triangle distribution based on generalized progressive hybrid censored samples. Lee (2017) considered the MLE and Bayes estimation of the entropy of a inverse Weibull distribution under generalized progressive hybrid censoring scheme.

Type-I censoring occurs when the experiment is terminated at predetermined time and uses the observed samples up to that time. Type-II censoring occurs when the experiment is terminated as soon as samples are observed until predetermined observation number. This two type censorings were combined by Childs et al. (2003) and combined censoring was called hybrid censoring. Hybrid censoring also has two types of type-I hybrid censoring and type-II hybrid censoring. Two hybrid censorings have two disadvantages that the lifetime of experi- mental object was observed very shorts (type-I hybrid censoring) and the experimental time was very longs (type-II hybrid censoring). To resolve this problems, Chandrasekar et al.

(2004) proposed generalized hybrid censoring that combines type-I hybrid censoring and type-II hybrid censoring. However, generalized hybrid censoring also has two types (general- ized type-I hybrid and generalized type-II hybrid). So, Balalkrishnan et al. (2008) proposed unified hybrid censoring. They studied exact likelihood inference based on an unified hybrid censored sample from the exponential distribution. Habibirad (2011) discussed an EM algo- rithm for estimating the parameters of the generalized exponential distribution under unified hybrid censored data. Panahi and Sayyareh (2016) proposed estimation and prediction for Burr type XII distribution based on unified hybrid-censored. Ateya (2017) studied estima- tion under inverse Weibull distribution based on unified hybrid censored scheme. Gwag and Lee (2018) proposed some AMLEs of the scale parameter for the half logistic distribution under unified hybrid censored samples.

In this paper, we deal with estimators of the scale parameters of the half-triangualr dis- tribution under unified hybrid censoring. In section 2, unified hybrid censoring is explained simply. In section 3, the maximum likelihood estimatior is obtained for the scale parameter of half-triangular distribution. However, the MLE can’t be obtained explicitly with closed form. In section 4, we propose AMLEs which have some closed forms using Taylor series expansion to estimate the scale parameter of the half-triangular distribution. Finally, we will compare the proposed estimators and MLE in the sense of the mean squared error (MSE).

2. Unified hybrid censoring

Unified hybrid censoring will be described before estimating the scale parameters of the

half-triangular distribution. The lifetime of experimental objects is observed gradually and

that times are expressed by X _1:n , X _2:n , . . . , X _n:n . Let k be the number of experimental

objects that should be observed at least, T ₁ be the experiment end prediction time, and T ₂

be the extended experiment end prediction time. Let D _j , j = 1, 2 be the number of observed objects until T _j , j = 1, 2 and the predetermined observed number r. That is, it is assumed that k < r < n. We need to determined somethings before experiment is conducted, for example, k, r, T 1 and so on. In accordance with terminate point and observed number, unified hybrid censoring is divided into six cases.

Figure 1. Unified hybrid censoring scheme

Case 1 : 0 < X _k:n < X _r:n < T ₁ < T ₂ , the experiment ends at the point of T ₁ .

Case 2 : 0 < X k:n < T 1 < X r:n < T 2 , the experiment ends at the point of X r:n .

Case 3 : 0 < X k:n < T 1 < T 2 < X r:n , the experiment ends at the point of T 2 .

Case 4 : 0 < T 1 < X k:n < X r:n < T 2 , the experiment ends at the point of X r:n .

Case 5 : 0 < T 1 < X k:n < T 2 < X r:n , the experiment ends at the point of T 2 .

Case 6 : 0 < T 1 < T 2 < X k:n < X r:n , the experiment ends at the point of X k:n .

In a word, when the k-th observation occurs before T ₁ , the experiment terminates at min(max{X _r:n , T ₁ }, T ₂ ) and when the k-th observation occurs between T ₁ and T ₂ , the ex- periment terminates at min(X r:n , T 2 ). If it occurs after T 2 , the experiment terminates at X k:n . For six cases, the likelihood functions based on unified hybrid censored sample are

L 1 (θ) = n!

(n − D 1 )!

D

₁

Y

i=1

f (x i:n )[1 − F (T 1 )] ^n−D

¹

, D 1 = r, . . . , n,

L j (θ) = n!

(n − r)!

r

Y

i=1

f (x i:n )[1 − F (x r:n )] ^n−r , j = 2, 4,

L _l (θ) = n!

(n − D ₂ )!

D

₂

Y

i=1

f (x _i:n )[1 − F (T ₂ )] ^n−D

²

, D ₂ = k, . . . , r − 1; l = 3, 5,

L 6 (θ) = n!

(n − k)!

k

Y

i=1

f (x i:n )[1 − F (x k:n )] ^n−k . (2.1)

Although L 2 and L 4 , L 3 and L 5 have different range, it is unified into one because the equation is the same.

3. Maximum likelihood estimation

The random variable Z = X/θ has a standard half-triangular distribution with pdf and cdf.

f (z) = 2(1 − z), 0 < z < 1,

F (z) = 1 − (1 − z) ² , 0 < z < 1. (3.1) Using the likelihood functions (2.1) in the previous section, we can obtain the log-likelihood functions as follows.

ln L 1 (θ) = K 1 +

D

1

X

i=1

ln f (z i:n ) + (n − D 1 ) ln [1 − F (z T

1

)] − D 1 ln θ, D 1 = r, . . . , n,

ln L _j (θ) = K ₂ +

r

X

i=1

ln f (z _i:n ) + (n − r) ln [1 − F (z _r:n )] − r ln θ, j = 2, 4,

ln L l (θ) = K 3 +

D

₂

X

i=1

ln f (z i:n ) + (n − D 2 ) ln [1 − F (z T

₂

)] − D 2 ln θ, D 2 = k, . . . , r − 1; l = 3, 5, ln L ₆ (θ) = K ₄ +

k

X

i=1

ln f (z _i:n ) + (n − k) ln [1 − F (z _k:n )] − k ln θ, (3.2)

where Z _T

_j

= T _j /θ, j = 1, 2 and K ₁ , K ₂ , K ₃ , K ₄ are constant. On differentiating the log- likelihood functions with respect to θ in turn and equation to zero, we obtain the likelihood equations as

∂ ln L 1 (θ)

∂θ = − 1 θ

"

D 1 − (n − D 1 ) 2

1 − z _T

₁

z T

₁

−

D

₁

X

i=1

1 1 − z _i:n z i:n

#

= 0,

∂ ln L _j (θ)

∂θ = − 1 θ

"

r − (n − r) 2 1 − z r:n

z _r:n −

r

X

i=1

1 1 − z i:n

z _i:n

#

= 0, j = 2, 4,

∂ ln L l (θ)

∂θ = − 1 θ

"

D 2 − (n − D 2 ) 2 1 − z T

₂

z T

₂

−

D

₂

X

i=1

1 1 − z i:n

z i:n

#

= 0, l = 3, 5, (3.3)

∂ ln L ₆ (θ)

∂θ = − 1 θ

"

k − (n − k) 2 1 − z k:n

z _k:n −

k

X

i=1

1 1 − z i:n

z _i:n

#

= 0.

We can find the MLE of θ as values ˆ θ by solving the equations (3.3). However, since the equations (3.3) can not be solved explicitly for θ, the MLE for θ is obtained using the Newton-Rapson method.

4. Approximate maximum likelihood estimation

In the previous section, the log-likelihood functions are differentiated with respect to θ to obtain the maximum likelihood estimator. However, the obtained equations are not linear equations and it is difficult to obtain exact solution. Thus, we obtain AMLE using Taylor series expansion. Let,

ξ i:n = F ⁻¹ (p i:n ) = 1 − √ q i:n ,

ξ D

^∗_j

= F ⁻¹ (p D

^∗_j

) = 1 − pq D

^∗_j

, j = 1, 2, where

p i:n = i

(n + 1) , q i:n = 1 − p i:n , p _D

^∗

j

= (p _D

_j

+ p _D

_j

₊₁ ) 2 , q _D

^∗

j

= 1 − p _D

^∗

j

, j = 1, 2.

We consider two types in Taylor series expansions of _1−z ¹

i:n

and _1−z ^z

^i:n

i:n

. First, we can approximate the functions by

1 1 − z i:n

≈ α 1i + β 1i z i:n , 1

1 − z _T

_j

≈ α 1D

^∗_j

+ β 1D

^∗_j

z T

_j

, j = 1, 2, (4.1)

where

α 1i = 1 − 2ξ i:n

q _i:n , β 1i = 1 q _i:n , α 1D

_j^∗

= 1 − 2ξ D

^∗_j

q D

_j^∗

, β 1D

^∗_j

= 1 q D

_j^∗

, j = 1, 2.

The equations (4.1) are substituted into the equation (3.3) and then the following equations are obtained.

∂ ln L 1 (θ)

∂θ = − 1 θ

"

D 1 − 2(n − D 1 )(α 1D

^∗₁

+ β 1D

^∗₁

z T

₁

)z T

₁

−

D

₁

X

i=1

(α 1i + β 1i z i:n )z i:n

#

= 0,

∂ ln L _j (θ)

∂θ = − 1 θ

"

r − 2(n − r)(α _1r + β _1r z _r:n )z _r:n −

r

X

i=1

(α _1i + β _1i z _i:n )z _i:n

#

= 0, j = 2, 4,

∂ ln L l (θ)

∂θ = − 1 θ

"

D 2 − 2(n − D 2 )(α 1D

^∗₂

+ β 1D

^∗₂

z T

₂

)z T

₂

−

D

₂

X

i=1

(α 1i + β 1i z i:n )z i:n

#

= 0, l = 3, 5,

∂ ln L ₆ (θ)

∂θ = − 1 θ

"

k − 2(n − k)(α _1k + β _1k z _k:n )z _k:n −

k

X

i=1

(α _1i + β _1i z _i:n )z _i:n

#

= 0. (4.2)

Therefore, we can obtain the following AMLEs.

Case 1 : ˆ θ 1 = −B 1 + pB ² ₁ − 4D 1 C 1

2D ₁ ,

Case 2, 4 : ˆ θ 1 = −B 2 + pB ² ₂ − 4rC 2

2r ,

Case 3, 5 : ˆ θ ₁ = −B ₃ + pB ² ₃ − 4D ₂ C ₃ 2D 2

,

Case 6 : ˆ θ ₁ = −B 4 + pB ² ₄ − 4kC 4

2k , (4.3)

where

B 1 = −2(n − D 1 )α 1D

^∗₁

T 1 −

D

₁

X

i=1

α 1i X i:n , C 1 = −2(n − D 1 )β 1D

₁^∗

T ₁ ² −

D

₁

X

i=1

β 1i X _i:n ² ,

B 2 = −2(n − r)α 1r X r:n −

r

X

i=1

α 1i X i:n , C 2 = −2(n − r)β 1r X _r:n ² −

r

X

i=1

β 1i X _i:n ² ,

B ₃ = −2(n − d ₂ )α _1D

^∗

2

T ₁ −

D

2

X

i=1

α _1i X _i:n , C ₃ = −2(n − D ₂ )β _1D

^∗

2

T ₂ ² −

D

2

X

i=1

β _1i X _i:n ² ,

B 4 = −2(n − k)α 1k X k:n −

k

X

i=1

α 1i X i:n , C 4 = −2(n − k)β 1k X _k:n ² −

k

X

i=1

β 1i X _i:n ² .

Since β 1i > 0, C 1 < 0, C 2 < 0, C 3 < 0, C 4 < 0, the estimator ˆ θ 1 is always positive.

Second, we can approximate the functions by z _i:n

1 − z i:n

≈ α 2i + β 2i z i:n , z T

j

1 − z T

_j

≈ α 2D

^∗_j

+ β 2D

^∗_j

z T

j

, j = 1, 2, (4.4)

where

α _2i = −ξ _i:n ² q i:n

, β _2i = 1 q i:n

,

α 2D

_j^∗

=

−ξ _D ²

∗ j

q D

_j^∗

, β 2D

^∗_j

= 1 q D

^∗_j

, j = 1, 2.

The equations (4.4) are substituted into the equation (3.3) and then we can obtain another estimators as follows.

Case 1 : ˆ θ 2 = 2(n − D 1 )β 2D

^∗₁

T 1 + P D

₁

i=1 β 2i X i:n

D 1 − 2(n − D 1 )α 2D

^∗₁

− P D

₁

i=1 α 2i

,

Case 2, 4 : ˆ θ 2 = 2(n − r)β _2r X _r:n + P r

i=1 β _2i X _i:n r − 2(n − r)α 2r − P r

i=1 α 2i

,

Case 3, 5 : ˆ θ ₂ = 2(n − D ₂ )β _2D

^∗

2

T ₂ + P D

2

i=1 β _2i X _i:n D 2 − 2(n − D 2 )α 2D

^∗₂

− P D

₂

i=1 α 2i

, (4.5)

Case 6 : ˆ θ 2 = 2(n − k)β 2k X k:n + P k

i=1 β 2i X i:n

k − 2(n − k)α _2k − P k

i=1 α _2i .

The AMLE ˆ θ ₂ is always positive value since α _2i < 0 and β _2i > 0.

5. Simulation results

In order to compare the MLE obtained in Section 3 and the AMLEs using the Taylor series expansion obtained in Section 4, the biases and MSEs are obtained using Monte Carlo simulations. The simulation procedure is repeated 10,000 times for the sample sizes n = 20 and 40, θ = 1, and various censored schemes.

From Table 5.1, it can be seen that MSE decreases when the sample size n is generally

increased. When the sample size n is fixed, the MSE decreases as the predetermined ob-

servation number r increases. When the sample size n and the predetermined observation

number r are fixed, MSE decreases as the number k of experimental objects that should

be observed at least increases. Moreover, MSE decreases as the observation end point T ₁

and the extended observation end point T 2 increase. Usually AMLEs ˆ θ i (i = 1, 2) are more

efficient than the MLE ˆ θ. Among the AMLEs, the AMLE ˆ θ 1 is generally more efficient than

the AMLE ˆ θ 2 . The MLE ˆ θ can’t be obtained explicitly with closed form. So we should be

used the Newton-Rapson method. But the AMLEs have some closed forms and the MSEs

of the AMLEs are usually smaller than those of the MLE. Therefore, the proposed AMLEs

are simple and usually have good performance.

Table 5.1 MSEs (biases) of the estimators of θ

n r k T

1

T

2

θ ˆ θ ˆ

1

θ ˆ

2

20 18

14

0.3 0.6

0.0335(-0.0035) 0.0328(-0.0057) 0.0329(-0.0095) 11 0.0367(-0.0005) 0.0359(-0.015) 0.0360(-0.0053) 8 0.0370(-0.0004) 0.0360(-0.0022) 0.0361(-0.0060)

16

14 0.0372(-0.0178) 0.0367(-0.0195) 0.0368(-0.0230) 12 0.0396(-0.0157) 0.0393(-0.0176) 0.0394(-0.0210) 11 0.0399(-0.0205) 0.0395(-0.0222) 0.0396(-0.0256)

13 11 0.0440(-0.0221) 0.0430(-0.0228) 0.0433(-0.0253)

6 0.0445(-0.0224) 0.0439(-0.0231) 0.0438(-0.0256)

18 14

0.4 0.6

0.0324(-0.0059) 0.0324(-0.0087) 0.0325(-0.0124)

11 0.0349(0.0030) 0.0347(0.0066) 0.0348(-0.0030)

8 0.0361(-0.0060) 0.0348(-0.0072) 0.0350(-0.0108)

16

14 0.0375(-0.0150) 0.0356(-0.0190) 0.0357(-0.0193) 12 0.0376(-0.0127) 0.0363(-0.0135) 0.0364(-0.0168) 11 0.0382(-0.0083) 0.0384(-0.0104) 0.0384(-0.0136)

13 11 0.0380(0.0008) 0.0376(0.0001) 0.0375(-0.0024)

6 0.0397(-0.0016) 0.0386(-0.0017) 0.0385(-0.0042)

18 14

0.4 0.7

0.0301(-0.0253) 0.0293(-0.0289) 0.0296(-0.0335) 11 0.0303(-0.0191) 0.0294(-0.0224) 0.0297(-0.0269) 8 0.0312(-0.0243) 0.0298(-0.0271) 0.0301(-0.0318)

16

14 0.0309(-0.0283) 0.0303(-0.0305) 0.0304(-0.0343) 12 0.0321(-0.0247) 0.0313(-0.0267) 0.0314(-0.0305) 11 0.0322(-0.0256) 0.0314(-0.0275) 0.0314(-0.0313)

13 11 0.0334(0.0025) 0.0321(0.0024) 0.0320(-0.0002)

6 0.0368(0.0005) 0.0357(0.0002) 0.0356(-0.0024)

40 38

34

0.3 0.6

0.0123(-0.0072) 0.0115(-0.0083) 0.0115(-0.0104) 32 0.0123(-0.0039) 0.0123(-0.0056) 0.0124(-0.0074) 30 0.0146(-0.0047) 0.0139(-0.0053) 0.0139(-0.0072)

35

32 0.0149(-0.0040) 0.0144(-0.0051) 0.0144(-0.0069) 29 0.0166(0.0056) 0.0162(-0.0065) 0.0163(-0.0083) 26 0.0176(-0.087) 0.0174(-0.0098) 0.0175(-0.0118)

32

30 0.0172(-0.191) 0.0172(-0.0202) 0.0172(-0.0219) 27 0.0189(-0.0107) 0.0189(-0.0130) 0.0189(-0.0134) 24 0.0195(-0.0062) 0.0195(-0.0073) 0.0195(-0.0089)

38 34

0.4 0.6

0.0118(-0.0058) 0.0112(-0.0071) 0.0112(-0.0093) 32 0.0122(-0.0049) 0.0122(-0.0066) 0.0124(-0.0085)

30 0.0140(0.0050) 0.0138(0.0039) 0.0138(0.0020)

35

32 0.0145(-0.0052) 0.0144(-0.0066) 0.0144(-0.0085) 29 0.0157(-0.0042) 0.0157(-0.0055) 0.0157(-0.0073) 26 0.0167(0.0002) 0.0166(-0.0033) 0.0166(-0.0028)

32

30 0.0166(-0.0133) 0.0165(-0.0144) 0.0165(-0.0161) 27 0.0173(-0.0115) 0.0172(-0.0103) 0.0172(-0.0142) 24 0.0182(-0.0109) 0.0180(-0.0118) 0.0180(-0.0135)

38 34

0.4 0.7

0.0115(-0.0070) 0.0110(-0.0094) 0.0111(-0.0119) 32 0.0117(-0.0020) 0.0117(-0.0048) 0.0119(-0.0073) 30 0.0123(-0.0024) 0.0121(-0.0049) 0.0122(-0.0073)

35

32 0.0137(-0.0129) 0.0135(-0.0148) 0.0136(-0.0170) 29 0.0138(-0.0142) 0.0136(-0.0160) 0.0137(-0.0182) 26 0.0143(-0.0134) 0.0141(-0.0152) 0.0142(-0.0173)

32

30 0.0140(-0.0163) 0.0140(-0.0176) 0.0140(-0.0194)

27 0.0155(-0.0194) 0.0155(-0.0206) 0.0155(-0.0224)

24 0.0161(-0.0234) 0.0156(-0.0242) 0.0156(-0.0260)

References

Ateya, S. F. (2017). Estimation under inverse Weibull distribution based on Balakrishnan’s unified hybrid censored scheme. Communications in Statistics - Simulation and Computation, 46, 3645-3666.

Balakrishnan, N. and Nevzorov, V. B. (2003). A primer on statistical distributions, John Wiley & Sons New York.

Balakrishnan, N., Rasouli, A. and Sanjari Farsipour, N. (2008). Exact likelihood inference based on an unified hybrid censored sample from the exponential distribution. Journal of Statistical Computation and Simulation, 78, 475-488.

Chandrasekar, B., Childs, A. and Balakrishnan, N. (2004). Exact likelihood inference for the exponential distribution under generalized type-I and type-II hybrid censoring. Naval Research Logistics, 51, 994- 1004.

Childs, A., Chandrasekar, B. and Balakrishnan, N. (2003). Exact likelihood inference based on type-I and type-II hybrid censored samples from the exponential distribution. Annals of the Institute of Statistical Mathematics, 55, 319-330.

Gwag, J. and Lee, K. (2018). Estimation of the scale parameter of the half logistic distribution under unified hybrid censored sample. Journal of the Korean Data & Information Science Society , 29, 13-25.

Habibirad, A. (2011). An EM algorithm for estimating the parameters of the generalized exponential dis- tribution under unified hybrid censored data. Journal of Statistical Research of Iran, 8, 149-162.

Han, J. T. and Kang, S. B. (2008). Estimation for the half-triangle distribution based on progressively type-II censored samples. Journal of the Korean Date & Information Science Society, 19, 951-957.

Johnson, D. (1997). The triangular distribution as a proxy for the beta distribution in risk analysis. The Statistician, 46, 387-398.

Kang, S. B. (2007). Estimation in a half-triangle distribution based on multiply type-II censored samples.

Journal of the Korean Date & Information Science Society, 18, 793-801.

Kang, S. B., Cho, Y. S. and Han, J. T. (2009). Estimation for the half triangle distribution based on type-I hybrid censored samples. Journal of the Korean Date & Information Science Society, 20, 961-969.

Kang, S. B. and Seo, J. I. (2011). Estimation in an exponentiated half logistic distribution under progres- sively type-II censoring. Communications for Statistical Applications and Methods, 18, 657-666.

Kim, H. Y. and Kang, S. B. (2018). Estimation of half triangle distribution under generalized progressive hybrid censored sample. Journal of the Korean Date & Information Science Society, 29, 839-850.

Kim, Y. K., Kang, S. B. and Seo, J. I. (2011). Bayesian estimations on the exponentiated half triangle distribution under type-I hybrid censoring. Journal of the Korean Date & Information Science Society, 22, 565-574.

Lee, K. (2017). Estimation of entropy of the inverse weibull distribution under generalized progressive hybrid censored data. Journal of the Korean Data & Information Science Society , 28, 659-668.

Lee, W. J., Han, J. T. and Kang, S. B. (2008). Estimation for a triangular distribution based on multiply type-II censored samples. Journal of the Korean Date & Information Science Society, 19, 319-330.

Panahi, H. and Sayyareh, A. (2016). Estimation and prediction for a unified hybrid-censored Burr type XII

distribution. Journal of Statistical Computation and Simulation, 86, 55-73.