Estimation of the entropy with generalized type I hybrid censored Weibull data

Estimation of the entropy with generalized type I hybrid censored Weibull data

Kyeongjun Lee ¹

1 Division of Mathematics and Big Data Science, Daegu University

Received 23 April 2020, revised 4 May 2020, accepted 12 May 2020

Abstract

In this paper, the estimation of the entropy of a Weibull distribution based on the generalized type I hybrid censoring scheme (GeHyC) has been considered. The maximum likelihood estimator (MLE) and approximate MLE are provided. The Bayes estimators for the entropy of the Weibull distribution based on the symmetric and asymmetric loss functions, such as the squared error (SqEL) balanced SqEL (BSqEL), linex (LinL), balanced LinL (BLinL), general entropy loss (GeEL) and balanced GeEL (BGeEL) functions, are provided. The Bayes estimators cannot be obtained explicitly, and Lindley’s approximation (LinA) is used to obtain the Bayes estimators. Simulation results are performed to see the effectiveness of the different estimators. Also, a real data set has been analyzed for illustrative purposes.

Keywords: Approximate maximum likelihood estimator, Bayes estimation, generalized type I hybrid censoring, Weibull data.

1. Introduction

The Weibull distribution (WeiD) has many real life application in testing lifetime data.

Since WeiD was originated with the paper by Weibull in 1935, it has been studied by many authors. The the probability density function (pdf) of the random variable X having the WeiD is given by

g(x; α, λ) = αλx ^α−1 exp (−λx ^α ) , x > 0, α > 0, λ > 0, (1.1) where λ and α are the scale and shape parameters, respectively.

In many life testing and reliability studies, it is well known that the lifetimes of test units may not be recorded exactly. There are also situations wherein the removal of units prior to failure is pre-planned in order to reduce the cost or time associated with testing.

The most common censoring schemes are type I and type II censoring. Epstein (1954) considered a hybrid censoring scheme (HyCS) in which the testing lifetime is terminated at a T ^∗ = min{X _r:n , T }, where r = 1, 2, · · · , n and T are fixed in advance. Recently, it

1

Assistant professor, Division of Mathematics and Big Data Science, Daegu University, Gyeongsan

38453, Korea. E-mail: indra [email protected]

has been studied by many authors - Bhattacharya et al. (2014), Childs et al. (2014) and Dey and Pradhan (2014). However, limitation of the HyCS is that few failures may occur before terminaton time. Because, under HyCS the testing lifetime will be no more than T ^∗ . Accuracy of maximum likelihood estimator (MLE) for a parameter of a distribution will be extremely low or MLE may not be computed. Therefore, Chandrasekar et al. (2004) propose GeHyC. GeHyC gives a assurance in terms of the number of failures (k) as well as time to terminate the test. GeHyC is designed to overcome the limitation in the HyCS.

Entropy defined in information theorem by Shannon (1948) depend on probability and statisics. The differential entropy of WeiD simplifies to

H(g) = 1 − log  αλ



+ γ

 1 − 1

α



, (1.2)

where γ is the Euler–Mascheroni constant. It is seen that a very sharply peaked distribution has very low entropy, whereas if the probability is spread out, the entropy is much higher.

In this sense, H(g) is a measure of uncertainty associated with g. Furthermore, as H(g) increases, g(x) approaches uniformity. Thus, entropy can be viewed as measuring the uni- formity of a distribution (Cho et al., 2015). Entroy estimation for various distributions has been studied by many authors. Baratpour et al. (2007) considered lower/upper bounds for entropy of upper record values. Also they provideded the entropy of upper record values.

Kang et al. (2012) considered the entropy estimators of a double exponential distribution based on multiply type II censoring. Cho et al. (2014) provided the entropy estimators of a Rayleigh distriution based on doubly-generalized type II HyCS. Cho et al. (2015) provided the entropy estimators of a WeiD based on generalized progressive HyCS.

In this paper, we consider entroy estimation for WeiD under GeHyC. In Section 2, MLE of entropy of WeiD for GeHyC is presented. Also, we obtained approximate MLE of entropy using Taylor series expansion. Moreover, approximate explicit expressions for Bayes entropy estimators are derived by using the LinA. In Section 3, a real life data set has been analyzed.

To know the performance of proposed entropy estimates based on GeHyC, a numerical study is also presented. Finally, Section 4 concludes.

2. Estimations

2.1. Maximum likelihood estimation

Suppose that n identical units are placed on a test and it has WeiD. Based on the observed failures as discussed in the previous section, the likelihood functions can be written as

L (α, λ) = n!

(n − J )! α ^J λ ^J

J

Y

i=1

x ^α−1 _i:n exp

"

−λ

J

X

i=1

x ^α _i:n + (n − J )U ^α

!#

, (2.1)

where J = k and U = x k:n if T < x k:n , J = d 1 and U = T if x k:n < x d

:n < T < x d

+1:n <

x r:n , and J = r and U = x r:n if x r:n < T . Here, J means the number of failures and U

means the terminated time. On differentiating the log-likelihood function with respect to α

and λ, we obtain the two equations

∂logL

∂α = J

α − (n − J ) λU ^α logU − λ

J

X

i=1

x ^α _i:n logx i:n +

J

X

i=1

logx i:n = 0,

∂logL

∂λ = J

λ − (n − J ) U ^α −

J

X

i=1

x ^α _i:n = 0.

It is easy to obtain the MLE of λ, say ˆ λ(α), for the fixed α to be

ˆ λ(α) = J

(n − J ) U ^α + P J

i=1 x ^α _i:n (2.2)

and the MLE of α can be obtained by solving the following equation J

α +

J

X

i=1

logx i:n − ˆ λ(α)

" _J X

i=1

x ^α _i:n logx i:n + (n − J ) U ^α logU

#

= 0. (2.3)

Eq (2.3) can be written as

m(α) = α, (2.4)

where

m(α) = J

ˆ λ(α) h P J

i=1 x ^α _i:n logx _i:n + (n − J ) U ^α logU i

− P J

i=1 logx _i:n .

We use the algorithm introduced by Kundu (2007) to solve for Eq (2.4). We note that the MLE of the entropy of the WeiD based on GeHyC is obtained as

H = 1 − log ˆ  ˆ αˆ λ



+ γ

 1 − 1

ˆ α



. (2.5)

2.2. Approximate MLE

Because the log-likelihood equations cannot be solved explicitly, it will be desirable to consider an approximation to the likelihood equations that will provide explicit estimators of α and λ. In this section, therefore, we provide the approximate MLE which have explicit form. First of all, let y i:n = logx i:n . Also, let z i:n = (y i:n − µ)/σ and V = (U − µ)/σ where σ = 1/α and µ = −1/αlogλ. Then, the Eq (2.1) can be obtained as

L(µ, σ) = n!

(n − J )!

1 σ ^J

J

Y

i=1

f (z _i:n )[1 − F (V )] ^n−J , (2.6) where F (z) = 1−exp(−e ^z ), f (z) = exp(z −e ^z ). On differentiating the log-likelihood function with respect to µ and σ, we obtain the two estimating equations

∂logL

∂µ = − 1 σ

J

X

i=1

f ⁰ (z i:n ) f (z i:n ) + 1

σ (n − J ) f (V )

1 − F (V ) = 0, (2.7)

∂logL

∂σ = − J σ − 1

σ

J

X

i=1

f ⁰ (z _i:n )

f (z i:n ) z _i:n + 1

σ (n − J ) f (V )

1 − F (V ) V = 0. (2.8)

Because the Eqs (2.7) and (2.8) cannot be solved explicitly, we expand the function f ⁰ (z _i:n )/f (z _i:n ) and f (V )/[1 − F (V )] in Taylor series expansion as follows. Let p _i = i/(n + 1) and q i = 1 − p i for i = 1, 2, · · · , n. Also, let p J

= (p J + p J +1 )/2 and q J

= 1 − p J

for T < x r:n . Then we expand the function f ⁰ (z i:n )/f (z i:n ) in Taylor series expansion around the points ξ i = log(−logq i ). Also, we expand the function f (V )/[1 − F (V )] in Taylor series expansion around the points ξ J = log(−logq J ) if U 6= T . If U = T , we expand the function f (V )/[1 − F (V )] in Taylor series expansion around the points ξ J

= log(−logq J

). We can approximate the functions by

f ⁰ (z i:n )

f (z _i:n ) ' δ _i − β _i z _i:n , f (V )

1 − F (V ) ' 1 − δ _J + β _J z _{J :n} (2.9) where δ i = 1 + logq i [1 − log(−logq i )], β i = −logq i . By substituting Eqs (2.9) into Eqs (2.7) and (2.8), we obtain

∂logL

∂µ ' −

J

X

i=1

(δ i − β i z i:n ) + (n − J )(1 − δ J + β J z J :n ) = 0, (2.10)

∂logL

∂σ ' −J −

J

X

i=1

(δ _i − β _i z _i:n ² ) + (n − J )(1 − δ _J + β _J z _{J :n} )z _{J :n} = 0. (2.11) From Eq (2.10), we obtain as

ˆ

µ = A ₁ − B 1 σ, ˆ (2.12)

where A 1 =

P

i=1

β

y

+β

(n−J )y

P

i=1

β

+β

(n−J ) , B 1 =

P

i=1

δ

−(n−J )(1−δ

) P

i=1

β

+β

(n−J ) . From Eq (2.11), we obtain ˆ σ as follow

ˆ

σ = −F ₁ + pF ₁ ² + 4J G ₁

2J , (2.13)

where F 1 =

J

X

i=1

δ i (y i:n − A 1 ) − (n − J )(1 − δ J )(y J :n − A 1 ) − 2B 1 J

X

i=1

β i (y i:n − A I )

− 2(n − J )β J B ₁ (y _{J :n} − A 1 ), G 1 =

J

X

i=1

β i (y i:n − A 1 ) ² + (n − J )β J (y J :n − A 1 ) ² .

If U = T , expanding f (V )/[1 − F (V )] around the point ξ J

, and following the same procedure as above, we obtain

ˆ

µ = A ₂ − B ₂ σ, ˆ (2.14)

where

A 2 = P J

i=1 β i y i:n + β J

(n − J )logT P J

i=1 β i + β J

(n − J ) , B 2 = P J

i=1 δ i − (n − J )(1 − δ J

:n ) P J

i=1 β i + β J

(n − J ) ,

δ J

= 1 + logq J

[1 − log(−logq J

)], β J

= −logq J

.

Also, we obtain

ˆ

σ = −F 2 + pF ₂ ² + 4J G ₂

2J , (2.15)

where

F 2 =

J

X

i=1

δ i (y i:n − A 2 ) − (n − J )(1 − δ J

)(logT − A 2 ) − 2B 2 J

X

i=1

β i (y i:n − A 2 )

− 2(n − J )β J

B 2 (logT − A 2 ), G ₂ =

J

X

i=1

β _i (y _i:n − A ₂ ) ² + (n − J )β _J

(logT − A ₂ ) ² .

Since G 1 > 0 and G 2 > 0, the estimator ˆ σ is always positive root. Hence, get the approximate MLEs of the α and λ under GeHyC as follows

ˆ α A = 1

ˆ

σ and ˆ λ A = exp(− ˆ α A µ). ˆ

With α and λ replaced by the ˆ α _A and ˆ λ _A , in Eq (1.2), the entropy estimator of the WeiD based on GeHyC are obtained as:

H ˆ A = 1 − log

 ˆ α A λ ˆ

1 αAˆ

A

 + γ

 1 − 1

ˆ α _A



. (2.16)

2.3. Bayes estimation

In this subsection, we consider the Bayes estimates for entropy. For estimating the entropy, squared error loss (SqEL), Linex loss (LinL) and general entropy loss (GeEL) functions have been taken into consideration. Moreover, we consider the Bayes estimates under balanced squared error loss (BSqEL), balanced Linex loss (BLinL) and balanced general entropy loss (BGeEL) functions.

In WeiD, the λ has a conjugate gamma prior, when the α is known. However, when the α and λ are unknown, both the parameters do not have conjugate priors. Therefore, we assume that α and λ have a gamma prior with the parameters (a ₁ , b ₁ ) and (a ₂ , b ₂ ), respectively. Also, we assume that the priors of α and λ are independent. Then, the posterior distribution of α and λ, given data X X X is written as

π(α, λ|X X X) ∝ α ^{J +b}

⁻¹ λ ^{J +b}

⁻¹

J

Y

i=1

x ^α−1 _i:n e ^−λ [ ^a

+ P

i=1

x

+(n−J )U

] −a

α ,

where X X X = (x 1:n , x 2:n , · · · , x J :n ). Now, we derive the Bayes entropy estimates under the SqEL, LinL and GeEL. Bayes entropy estimates under the SqEL is derived as,

H ˜ _S = 1 K

Z ∞ 0

Z ∞ 0

H(g)α ^{J +b}

⁻¹ λ ^{J +b}

⁻¹

J

Y

i=1

x ^α−1 _i:n e ^−λ [ ^a

+ P

i=1

x

+(n−J )U

] ^−a

α dαdλ.

But, Bayes estimates in the above Eq cannot be solved explicitly, it will be desirable to consider an LinA that will provide explicit estimators of entropy. For the two parameter case (θ 1 , θ 2 ), the LinA can be written as

ˆ

g = g  ˆ θ ₁ , ˆ θ ₂ 

+ 0.5 (A + l ₃₀ B ₁₂ + l ₀₃ B ₂₁ + l ₂₁ C ₁₂ + l ₁₂ C ₂₁ ) + p ₁ A ₁₂ + p ₂ A ₂₁ , (2.17) where

A =

2

X

i=1 2

X

j=1

u _ij τ _ij , B _ij = (u _i τ _ii + u _j τ _ij )τ _ii , C _ij = 3u _i τ _ii τ _ij + u _j (τ _ii τ _jj + 2τ _ij ² ),

l ij = ∂ ^i+j l(θ 1 , θ 2 )

∂θ ⁱ ₁ ∂θ ₂ ^j , i + j = 3, i, j = 0, 1, 2, 3, p i = ∂p

∂θ i

, u i = ∂g

∂θ i

, u ij = ∂ ² g

∂θ i θ j

, p = logπ (θ 1 , θ 2 ) , A ij = u i τ ii + u j τ ji .

Here, l means the log-likelihood function. τ ij means the (i, j)th element of [−∂ ² l/∂θ ⁱ ₁ ∂θ ^j ₂ ] ⁻¹ . First of all, we compute ˆ g, we have

l ₃₀ = 2J α ³ − ˆ λ

" _J X

i=1

x ^α _i:n ^ˆ (logx _i:n ) ³ + (n − J )U ^{α ˆ} (logU ) ³

# ,

l 21 = −

" _J X

i=1

x ^α _i:n ^ˆ (logx i:n ) ² + (n − J )U ^{α ˆ} (logU ) ²

# ,

l 12 = ∂ ³ l

∂α∂λ ² = 0, l 03 = ∂ ³ l

∂λ ³ = 2J

λ ³ , p 1 = a 1 − 1

α − b 1 , p 2 = a 2 − 1 λ − b 2 . Also, we have

τ 11 = W

N ^∗ , τ 12 = τ 21 = −V

N ^∗ and τ 22 = S N ^∗ , where S = J/ ˆ α ² + ˆ λ h

P J

i=1 x ^α _i:n ^ˆ (logx i:n ) ² + (n − J )U ^{α ˆ} (logU ) ² i

, V = P J

i=1 x ^α _i:n ^ˆ (logx i:n ) + (n − J )U ^{α ˆ} (logU ), W = J/ ˆ λ ² and N ^∗ = SW − V ² .

Then, we compute the Bayes entropy estimates by using LinA. First of all, we compute Bayes entropy estimator against SqEL function. Then, we have

g(α, λ) = H(g) = 1 − log  αλ



+ γ

 1 − 1

α



, u 1 = γ α ² − 1

α + 1 α ² logλ, u 2 = − 1

αλ , u 11 = − 2γ α ³ + 1

α ² − 2

α ³ logλ, u 12 = u 21 = 1

α ² λ , u 22 = 1

αλ ² .

From Eq (2.17), the Bayes entropy estimator against SqEL is obtained as

H ˆ S = ˆ H(g) + 1 2N ^∗2

"

N ^∗ (u 11 W − 2u 12 V + u 22 S) + l 30 (u 1 W − u 2 V )W

+ l ₀₃ (u ₂ S − u ₁ V )S + l ₂₁ {−3u 1 W V + u ₂ (W S + 2V ² )}

#

+ 1

N ^∗ [p 1 (u 1 W − u 2 V ) + p 2 (u 2 S − u 1 V )] . (2.18) Also, we compute the Bayes entropy estimator against LinL, we have

H ˆ _L = − 1 h log

"

e ^{−h ˆ H(g)} + 1 2N ^∗2

"

N ^∗ (u ₁₁ W − 2u ₁₂ V + u ₂₂ S) + l ₃₀ (u ₁ W − u ₂ V )W

+ l 03 (u 2 S − u 1 V )S + l 21 {−3u 1 W V + u 2 (W S + 2V ² )}

#

+ 1

N ^∗ {p 1 (u 1 W − u 2 V ) + p 2 (u 2 S − u 1 V )}

#

, (2.19)

where

u 1 = −h  γ α ² − 1

α + 1 α ² logλ



e ^−hH(g) , u 2 = h

 1 αλ



e ^−hH(g) ,

u 11 = h

"

2γ α ³ − 1

α ² + 2

α ³ logλ + h  γ α ² − 1

α + 1 α ² logλ

 ² #

e ^−hH(g) ,

u ₁₂ = u ₂₁ = − h α ² λ



1 + h  γ

α − 1 + 1 α logλ



e ^−hH(g) , u 22 = −h

 1 αλ ²

 1 − c

α

 

e ^−hH(g) .

Finally, we compute the Bayes entropy estimator against GeEL, we have H ˆ E =

"

H(g) ˆ ^−q + 1 2N ^∗2

"

N ^∗ (u 11 W − 2u 12 V + u 22 S) + l 30 (u 1 W − u 2 V )W

+ l ₀₃ (u ₂ S − u ₁ V )S + l ₂₁ {−3u 1 W V + u ₂ (W S + 2V ² )}

#

+ 1

N ^∗ {p ₁ (u ₁ W − u ₂ V ) + p ₂ (u ₂ S − u ₁ V )}

# ^−1/q

, (2.20)

where

u ₁ = −q  γ α ² − 1

α + 1 α ² logλ



H(g) ^−q−1 , u ₂ = q

αλ H(g) ^−q−1 , u 11 = q

"

H(g)  2γ α ³ − 1

α ² + 2 α ³ logλ



+ (q + 1)  γ α ² − 1

α + 1 α ² logλ

 2 #

H(g) ^−q−2 ,

u 12 = u 21 = − q α ² λ



H(g) + (q + 1)  γ

α − 1 + 1 α logλ



H(g) ^−q−2 , u 22 = q

αλ ²

 q + 1

α − H(g)



H(g) ^−q−2 .

Also, we compute the Bayes estimates against BSqEL, BLinL and BGeELs. Using Eqs (2.18–2.20), the Bayes entropy estimates against BSqEL, BLinL and BGeELs are obtained as

H ˆ BS = w ˆ H + (1 − w) ˆ H S , ˆ H BL = − 1 h log h

we ^{−h ˆ H} + (1 − w)e ^{−h ˆ H}

i , H ˆ BE = h

w ˆ H ^−q + (1 − w) ˆ H _E ^−q i −1/q

. (2.21)

3. Real data and simulation results

3.1. Real data analysis

To illustrate the proposed methods, we analyze a real life data set from Linhart and Zucchini (1986). The following real life data set denote broken times of the air conditioning system of an airplane. The ordered broken times are as follows: 1, 3, 5, 7, 11, 11, 11, 12, 14, 14, 14, 16, 16, 20, 21, 23, 42, 47, 52, 62, 71, 71, 87, 90, 95, 120, 120, 225, 246, 261.

Before progressing further, first we perform a goodness-of-fit test by using the Anderson- Darling. p-value of Anderson-Darling test is .159. Therefore, the Anderson-Darling test con- clude that the data are coming from the WeiD. All the hyperparameters equal to 0.0001 for Bayes estimates. We compute the MLE, approximate MLE and the Bayes entropy estimates.

In this real life data, we created a new data set by GeHyC. We assumed that r = 25. Also, we take case I (k = 20 and T = 50), case II (k = 15 and T = 50) and case III (k = 15 and T = 100). Table 3.1 shows the estimates of entropy based on GeHyC.

Table 3.1 Estimation of entropy for example

Case w H ˆ H ˆ

H ˆ

H ˆ

H ˆ

h = −2 h = 2 h = 4 q = −2 q = 2 q = 4 I 4.9998 4.9835 5.0572 5.1168 4.9831 4.9219 5.0631 5.3693 3.6963

0.3 5.0400 5.0817 4.9881 4.9452 5.0441 5.2585 4.0874

0.5 5.0285 5.0583 4.9914 4.9608 5.0314 5.1845 4.3481

0.7 5.0170 5.0349 4.9947 4.9764 5.0188 5.1106 4.6087

II 4.9729 4.9558 5.0389 5.1068 4.9519 4.8825 5.0459 5.4315 3.6182

0.3 5.0191 5.0666 4.9582 4.9096 5.0240 5.2939 4.0246

0.5 5.0059 5.0399 4.9624 4.9277 5.0094 5.2022 4.2956

0.7 4.9927 5.0131 4.9666 4.9458 4.9948 5.1105 4.5665

III 4.9649 4.9501 5.0050 5.0462 4.9570 4.9143 5.0089 5.1917 4.1253

0.3 4.9930 5.0218 4.9594 4.9295 4.9957 5.1237 4.3772

0.5 4.9850 5.0056 4.9610 4.9396 4.9869 5.0783 4.5451

0.7 4.9770 4.9893 4.9626 4.9497 4.9781 5.0330 4.7130

3.2. Simulation results

Table 3.2 The biases (within bracket) and MSEs of proposed entropy estimators

T r k w Hˆ HˆA HˆS

HˆL HˆE

h = −0.5 h = 0.5 h = 2.0 q = −0.5 q = 0.5 q = 2.0

1.0 32 22 .0549(-.0068) .0552(-.0087) .0572(-.0046) .0597(.0047) .0548(-.0142) .0528(-.0232) .0576(-.0026) .0542(-.0192) .0509(-.0302) .3 .0565(-.0053) .0582(.0012) .0548(-.0120) .0533(-.0183) .0568(-.0038) .0544(-.0155) .0520(-.0232) .5 .0560(-.0057) .0572(-.0011) .0548(-.0105) .0537(-.0150) .0562(-.0047) .0545(-.0130) .0527(-.0185) .7 .0555(-.0061) .0563(-.0034) .0548(-.0090) .0542(-.0117) .0557(-.0055) .0546(-.0105) .0535(-.0138) 26 .0451(-.0287) .0454(-.0308) .0467(-.0275) .0479(-.0196) .0456(-.0354) .0448(-.0431) .0469(-.0256) .0456(-.0398) .0444(-.0482)

.3 .0462(-.0278) .0470(-.0223) .0455(-.0334) .0449(-.0388) .0464(-.0265) .0455(-.0365) .0445(-.0424) .5 .0459(-.0281) .0465(-.0242) .0454(-.0321) .0449(-.0359) .0460(-.0271) .0453(-.0342) .0446(-.0385) .7 .0456(-.0283) .0459(-.0260) .0453(-.0307) .0450(-.0330) .0457(-.0278) .0453(-.0320) .0448(-.0346) 36 22 .0547(-.0066) .0550(-.0085) .0570(-.0044) .0595(.0049) .0546(-.0140) .0526(-.0230) .0575(-.0024) .0540(-.0190) .0508(-.0300)

.3 .0563(-.0051) .0580(.0014) .0546(-.0118) .0532(-.0181) .0566(-.0036) .0542(-.0153) .0518(-.0230) .5 .0558(-.0055) .0570(-.0009) .0546(-.0103) .0536(-.0148) .0561(-.0045) .0543(-.0128) .0525(-.0183) .7 .0554(-.0060) .0561(-.0032) .0547(-.0088) .0540(-.0115) .0555(-.0053) .0545(-.0103) .0534(-.0136) 26 .0450(-.0285) .0452(-.0307) .0465(-.0273) .0477(-.0194) .0455(-.0353) .0446(-.0429) .0468(-.0254) .0455(-.0396) .0442(-.0480)

.3 .0461(-.0276) .0469(-.0222) .0453(-.0332) .0447(-.0386) .0462(-.0263) .0453(-.0363) .0443(-.0422) .5 .0457(-.0279) .0463(-.0240) .0452(-.0319) .0447(-.0357) .0459(-.0270) .0452(-.0341) .0445(-.0383) .7 .0454(-.0281) .0457(-.0258) .0451(-.0305) .0448(-.0328) .0455(-.0276) .0451(-.0318) .0446(-.0344) 1.5 32 22 .0445(-.0222) .0447(-.0262) .0462(-.0209) .0476(-.0142) .0449(-.0278) .0438(-.0343) .0465(-.0193) .0447(-.0313) .0430(-.0387)

.3 .0457(-.0213) .0466(-.0166) .0448(-.0261) .0440(-.0307) .0459(-.0202) .0446(-.0286) .0434(-.0338) .5 .0453(-.0216) .0460(-.0182) .0447(-.0250) .0441(-.0283) .0455(-.0208) .0446(-.0268) .0437(-.0305) .7 .0450(-.0218) .0454(-.0198) .0446(-.0239) .0442(-.0259) .0451(-.0214) .0445(-.0250) .0440(-.0272) 26 .0434(-.0233) .0437(-.0272) .0451(-.0220) .0463(-.0154) .0439(-.0288) .0429(-.0353) .0454(-.0204) .0437(-.0323) .0423(-.0396)

.3 .0446(-.0224) .0454(-.0178) .0437(-.0271) .0430(-.0317) .0448(-.0212) .0436(-.0296) .0426(-.0347) .5 .0442(-.0227) .0448(-.0193) .0436(-.0260) .0431(-.0293) .0444(-.0218) .0435(-.0278) .0428(-.0314) .7 .0439(-.0229) .0442(-.0209) .0435(-.0250) .0432(-.0269) .0440(-.0224) .0435(-.0260) .0430(-.0282) 36 22 .0387(-.0100) .0392(-.0145) .0403(-.0085) .0418(-.0019) .0389(-.0153) .0377(-.0218) .0407(-.0069) .0386(-.0188) .0368(-.0262)

.3 .0398(-.0090) .0408(-.0043) .0388(-.0137) .0380(-.0183) .0401(-.0078) .0386(-.0161) .0373(-.0213) .5 .0395(-.0093) .0402(-.0059) .0388(-.0127) .0382(-.0159) .0397(-.0085) .0386(-.0144) .0377(-.0181) .7 .0932(-.0096) .0396(-.0076) .0388(-.0116) .0384(-.0136) .0393(-.0091) .0387(-.0127) .0381(-.0149) 26 .0377(-.0111) .0381(-.0155) .0392(-.0096) .0406(-.0030) .0379(-.0163) .0368(-.0228) .0395(-.0080) .0376(-.0198) .0361(-.0270)

.3 .0387(-.0101) .0396(-.0054) .0378(-.0148) .0370(-.0193) .0389(-.0089) .0376(-.0172) .0365(-.0222) .5 .0384(-.0104) .0391(-.0071) .0378(-.0137) .0372(-.0170) .0386(-.0096) .0376(-.0154) .0368(-.0191) .7 .0381(-.0107) .0385(-.0087) .0377(-.0127) .0374(-.0146) .0382(-.0102) .0376(-.0137) .0371(-.0159)

n = 60

1.0 46 34 .0340(-.0051) .0341(-.0062) .0349(-.0047) .0359(.0006) .0340(-.0100) .0332(-.0152) .0351(-.0032) .0338(-.0127) .0329(-.0183) .3 .0346(-.0048) .0353(-.0011) .0340(-.0085) .0334(-.0122) .0348(-.0038) .0339(-.0104) .0332(-.0144) .5 .0344(-.0049) .0349(-.0023) .0340(-.0076) .0336(-.0102) .0346(-.0042) .0339(-.0089) .0334(-.0117) .7 .0343(-.0050) .0345(-.0034) .0340(-.0066) .0337(-.0082) .0343(-.0046) .0339(-.0074) .0336(-.0091) 38 .0305(-.0171) .0306(-.0183) .0312(-.0169) .0318(-.0121) .0307(-.0217) .0302(-.0264) .0314(-.0155) .0307(-.0243) .0302(-.0292)

.3 .0310(-.0169) .0314(-.0136) .0306(-.0203) .0303(-.0236) .0311(-.0159) .0306(-.0221) .0302(-.0199) .5 .0309(-.0170) .0312(-.0146) .0306(-.0194) .0303(-.0217) .0309(-.0163) .0306(-.0207) .0303(-.0232) .7 .0307(-.0170) .0309(-.0156) .0306(-.0185) .0304(-.0199) .0308(-.0166) .0306(-.0192) .0304(-.0207) 54 34 .0337(-.0048) .0338(-.0059) .0346(-.0043) .0355(.0010) .0337(-.0097) .0329(-.0148) .0348(-.0028) .0335(-.0124) .0326(-.0180)

.3 .0343(-.0045) .0350(-.0008) .0337(-.0082) .0331(-.0118) .0345(-.0034) .0336(-.0101) .0328(-.0140) .5 .0341(-.0046) .0346(-.0019) .0337(-.0072) .0332(-.0098) .0342(-.0038) .0336(-.0086) .0331(-.0114) .7 .0339(-.0047) .0342(-.0031) .0337(-.0063) .0334(-.0078) .0340(-.0042) .0336(-.0071) .0333(-.0088) 38 .0302(-.0167) .0303(-.0179) .0309(-.0165) .0315(-.0117) .0304(-.0214) .0299(-.0261) .0311(-.0151) .0304(-.0239) .0298(-.0289)

.3 .0307(-.0166) .0311(-.0132) .0303(-.0200) .0300(-.0233) .0308(-.0156) .0303(-.0218) .0299(-.0252) .5 .0305(-.0166) .0308(-.0142) .0303(-.0190) .0300(-.0214) .0306(-.0159) .0303(-.0203) .0300(-.0228) .7 .0304(-.0167) .0306(-.0152) .0302(-.0181) .0301(-.0195) .0304(-.0162) .0302(-.0189) .0300(-.0204) 1.5 46 34 .0287(-.0224) .0288(-.0247) .0294(-.0222) .0299(-.0182) .0290(-.0262) .0286(-.0300) .0296(-.0210) .0290(-.0282) .0286(-.0323)

.3 .0292(-.0222) .0295(-.0195) .0289(-.0250) .0286(-.0277) .0293(-.0214) .0289(-.0265) .0286(-.0293) .5 .0291(-.0223) .0293(-.0203) .0288(-.0243) .0286(-.0262) .0291(-.0217) .0288(-.0253) .0286(-.0273) .7 .0289(-.0223) .0291(-.0212) .0288(-.0235) .0287(-.0247) .0290(-.0220) .0288(-.0241) .0286(-.0254) 38 .0286(-.0225) .0287(-.0248) .0293(-.0223) .0298(-.0183) .0289(-.0262) .0285(-.0301) .0295(-.0211) .0289(-.0283) .0285(-.0324) .3 .0291(-.0223) .0294(-.0196) .0288(-.0251) .0285(-.0278) .0292(-.0215) .0288(-.0266) .0285(-.0294) .5 .0290(-.0224) .0292(-.0204) .0288(-.0244) .0286(-.0263) .0290(-.0218) .0287(-.0254) .0285(-.0274) .7 .0288(-.0224) .0290(-.0213) .0287(-.0236) .0286(-.0248) .0289(-.0221) .0287(-.0242) .0285(-.0255) 54 34 .0242(-.0088) .0244(-.0115) .0249(-.0084) .0254(-.0045) .0243(-.0123) .0238(-.0161) .0250(-.0072) .0242(-.0142) .0237(-.0182) .3 .0247(-.0085) .0250(-.0058) .0243(-.0112) .0239(-.0139) .0248(-.0077) .0242(-.0126) .0238(-.0154) .5 .0245(-.0086) .0248(-.0066) .0243(-.0105) .0240(-.0124) .0246(-.0080) .0242(-.0115) .0239(-.0135) .7 .0244(-.0087) .0246(-.0075) .0242(-.0098) .0241(-.0110) .0244(-.0083) .0242(-.0104) .0240(-.0116) 38 .0241(-.0089) .0243(-.0116) .0248(-.0084) .0253(-.0046) .0242(-.0123) .0238(-.0161) .0249(-.0073) .0241(-.0143) .0236(-.0183)

.3 .0246(-.0086) .0249(-.0059) .0242(-.0113) .0238(-.0140) .0247(-.0078) .0241(-.0127) .0237(-.0155) .5 .0244(-.0087) .0247(-.0067) .0242(-.0106) .0239(-.0125) .0245(-.0081) .0241(-.0116) .0238(-.0136) .7 .0243(-.0088) .0245(-.0076) .0241(-.0099) .0240(-.0111) .0243(-.0084) .0241(-.0105) .0239(-.0117)

To know the performance of proposed estimators of entropy based on GeHyC, we perform a numerical study. Comparison between proposed estimators is made with respect to their mean square error (MSE). The MSE values of proposed estimators are computed by using Monte Carlo simulations. We replicated 10,000 times GeHyC data of size n drawn from an WeiD. Without loss of generality, the true value of α and λ is taken as 1. All the hyperpa- rameters equal to 0.0001, for Bayes estimation. In each case, the Bayes entroy estimators against the LinL, GeEL, BLinL, BGeEL are obtained for distinct values of h and q, namely -0.5, 0.5 and 2. Also, the Bayes entroy estimators against the balanced loss functions are obtained for distinct values of w, namely 0.3, 0.5 and 0.7. And, various schemes have been taken into consideration to compute MSEs and biases of proposed estimators, MSEs and biases are presented in Table 3.2. We have presented MSEs and biases of the numerical MLE and approximate MLE in the fifth and sixth column of the table. Uniformly, seventh - thir- teenth columns present four values. In each cell, the values show the biases (within bracket) and MSEs. The first value represents the MSEs and biases of Bayes entropy estimates by using the LinA. The last three values represents the MSEs and biases of entropy of Bayes entropy estimates against balanced loss functions.

In general, it can be seen that the MSEs of all estimates reduce with an increasing in sample size n. For fixed sample size n, the MSEs of all estimates reduce with an increasing in number r. For fixed sample size n and r, the MSEs of all estimates reduce with an increasing in time T . For fixed sample size n, r and time T , the MSEs of all estimates reduce with an increasing in guarantee sample size k. It can be seen that the MLE and approximate MLE behave quite similarly in terms of MSEs. Also, it can be ssen that Bayes estimators are better than the MLE and approximate MLE in terms of MSEs. Also, the problem of selecting a suitable loss function is concerned, it can be seen that GeEL emerges as the best loss function. For Bayes entropy estimation, the choice h = 4 and q = 4 seem to be a reasonable choice against LinL and GeEL. Also, the choice w = 0.3 seems to be a reasonable choice against BLinL and BGeEL, while w = 0.7 seems to be a reasonable choice against BSqEL.

4. Conclusions

In this paper, we considered entropy estimators for the WeiD based on GeHyC. This

paper obtained numerical MLE, approximate MLE and Bayes entropy estimators in the

WeiD based on GeHyC and compared these estimates in terms of their MSE by using Monte

Carlo simulation. Bayes entropy estimators are provided against different loss functions, and

it can be seen that the Bayes entropy estimator against the GeEL works better than other

estimators in this situation. Due to the censoring of large observations due to GeHyC, the

bias appears to be mostly negative. Therefore, additional research is needed to reduce the

bias.

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