# Estimation of Sine Inverse Exponential Model under Censored Schemes

Authors: M. Shrahili; I. Elbatal; Waleed Almutiry; Mohammed Elgarhy
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we introduce a new one-parameter model, which is named sine inverted exponential (SIE) distribution. The SIE distribution is a new extension of the inverse exponential (IE) distribution. The SIE distribution aims to provide the SIE model for data-fitting purposes. The SIE distribution is more flexible than the inverted exponential (IE) model, and it has many applications in physics, medicine, engineering, nanophysics, and nanoscience. The density function (PDFu) of SIE distribution can be unimodel shape and right skewed shape. The hazard rate function (HRFu) of SIE distribution can be J-shaped and increasing shaped. We investigated some fundamental statistical properties such as quantile function (QFu), moments (Mo), moment generating function (MGFu), incomplete moments (ICMo), conditional moments (CMo), and the SIE distribution parameters were estimated using the maximum likelihood (ML) method for estimation under censored samples (CS). Finally, the numerical results were investigated to evaluate the flexibility of the new model. The SIE distribution and the IE distribution are compared using two real datasets. The numerical results show the superiority of the SIE distribution.


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## Body

## 1. Introduction

In the recent years, inverse and half-inverse problems are studied in general operator theory [1–3], numerous authors have attracted the attention of generated families of distributions such as Kumaraswamy-G by [4], sine generated (S-G) by [5], Kumaraswamy Type-I half-logistic-G [6], Weibull-G [7], odd Fréchet -G by [8], the Burr type X-G by [9], Kumaraswamy Kumaraswamy-G [10], truncated Cauchy power-G by [11], generalized odd half-Cauchy-G by [12], and among others.The cumulative distribution function (CDFu) and PDFu of S-G are(1)Fx;ξ=sinπ2Gx;ξ,x∈R,(2)fx;ξ=π2gx;ξcosπ2Gx;ξ,x∈R.Lettinggx;ξandGx;ξ, the PDFu and CDFu of IE distribution, it has the following form:(3)gx=θx−2e−θ/x,θ>0,x>0,(4)Gx=e−θ/x,θ>0,x>0.The main idea for this paper was to introduce a new one-parameter model that is more flexible than the IE model by using the S-G family. The new model is called the SIE model. The SIE model is more flexible than the IE model and it has many applications in physics, medicine, nanophysics, and nanoscience [13–16]. This manuscript is arranged as follows. Section 2 presents materials and methods. In Section 3, statistical inference of the SIE model under the censored scheme is studied. Section 4 presents results and discussion. At the end of article, conclusions are discussed.


## 2. Materials and Methods

### 2.1. The New SIE Model

Letting random variableX to have SIE distribution, then the CDFu, PDFu, survival function (SFu), and HRFu of X are(5)Fx=sinπ2e−θ/x,x>0,θ>0,(6)fx=πθ2x2e−θ/xcosπ2e−θ/x,x>0,θ>0,(7)Rx=1−sinπ2e−θ/x,(8)hx=πθ/2x2e−θ/xcosπ/2e−θ/x1−sinπ/2e−θ/x,where θ is a scale parameter.Figures1–3 show the plots of PDFu, CDFu, and HRFu of the SIE model. The PDFu of the SIE model can be right skewed and unimodal shaped, while the HRFu of the SIE model can be increasing or J-shaped.Figure 1
Plots of PDFu for the SIE model.Figure 2
Plots of CDFu for the SIE model.Figure 3
Plots of HRFu for the SIE model.


### 2.2. Quantile and Median

If X∼ SIE, then the QFu of SIE is as follows:(9)Qu=θlnπ2Arcsinu−1,and by taking u = 0.5, we get the median (M) as M=θln3−1.MacGillivray’s skewness function is defined in [17] as(10)MGS=Qu+Q1−u−2MQu−Q1−u.Figure4 plots MGS for all values of the parameter θ.Figure 4
Plots of MGS for the SIE model.


### 2.3. Moments

Theorem 1.
LettingX be a r.v. from the SIE model, then its rth Mo is(11)θr′=∑i=0∞θr−1ΛΓ1−r2i+11−r.Proof.
LetX be a r. v. with pdf equation (6). The rth Mo of SIE distribution is computed as follows:(12)θr′=∫0∞xrfxdx=πθ2∫0∞xr−2e−θ/xcosπ2e−θ/xdx.
By inserting the expansioncosGx=∑i=0∞−1i/2i!Gx2i,n to the previous equation, then(13)θr′=∑i=0∞θ−1i2i!π22i+1∫0∞xr−δ−1e−2i+1θ/xdx.
The last equation can be rewritten as follows:(14)θr′=∑i=0∞Λ∫0∞xr−2e−2i+1θ/xdx,where Λ=∑i=0∞θ−1i/2i!π/22i+1.
Lety=θ/x; then,(15)θr′=∑i=0∞θr−1Λ∫0∞y−re−2i+1ydy.
Then,(16)θr′=∑i=0∞θr−1ΛΓ1−r2i+11−r.
The MGFu ofX is(17)MXt=∑r=0∞trr!θr′=∑r,i=0∞trθr−1ΛΓ1−rr!2i+11−r.
The ICMo, denoted byφst, of the SIE distribution is(18)φst=∫0txsfxdx=πθ2∫0txs−2e−θ/xcosπ2e−θ/xdx.
By using equation (18), φst will be as(19)φst=∑i=0∞θr−1Λγ1−r,θ/t2i+11−r,where γs,t=∫0tys−1e−ydy is the lower ICGFu where ICGFu is incomplete gamma function.
The CMo, denoted byτst, of the SIE distribution is(20)τst=∫t∞xsfxdx=πθ2∫t∞xs−2e−θ/xcosπ2e−θ/xdx.
By using equation (20), τst will be as given(21)τst=∑i=0∞θr−1ΛΓ1−r,θ/t2i+11−r,where Γs,t=∫t∞ys−1e−ydy is the upper ICGFu.


### 2.4. Order Statistics

LetX1,X2,…,Xn be random sample from the SIE distribution with order statistics (OS) X1,X2,…,Xn. The pdf of Tm of OS is(22)fXmx=n!m−1!n−m!Fm−1xfx1−Fxn−m.The pdf ofXm can be expressed as(23)fXmx=n!πθ2m−1!n−m!x−2e−θ/xsinm−1π2e−θ/xcosπ2e−θ/x1−sinπ2e−θ/xn−m.Specially, the pdfs of the lowest and greatest OS can be computed as(24)fX1x=nπθ2x−2e−θ/xcosπ2e−θ/x1−sinπ2e−θ/xn−1,(25)fXnx=nπθ2x−2e−θ/xsinn−1π2e−θ/xcosπ2e−θ/x.


## 2.1. The New SIE Model

Letting random variableX to have SIE distribution, then the CDFu, PDFu, survival function (SFu), and HRFu of X are(5)Fx=sinπ2e−θ/x,x>0,θ>0,(6)fx=πθ2x2e−θ/xcosπ2e−θ/x,x>0,θ>0,(7)Rx=1−sinπ2e−θ/x,(8)hx=πθ/2x2e−θ/xcosπ/2e−θ/x1−sinπ/2e−θ/x,where θ is a scale parameter.Figures1–3 show the plots of PDFu, CDFu, and HRFu of the SIE model. The PDFu of the SIE model can be right skewed and unimodal shaped, while the HRFu of the SIE model can be increasing or J-shaped.Figure 1
Plots of PDFu for the SIE model.Figure 2
Plots of CDFu for the SIE model.Figure 3
Plots of HRFu for the SIE model.


## 2.2. Quantile and Median

If X∼ SIE, then the QFu of SIE is as follows:(9)Qu=θlnπ2Arcsinu−1,and by taking u = 0.5, we get the median (M) as M=θln3−1.MacGillivray’s skewness function is defined in [17] as(10)MGS=Qu+Q1−u−2MQu−Q1−u.Figure4 plots MGS for all values of the parameter θ.Figure 4
Plots of MGS for the SIE model.


## 2.3. Moments

Theorem 1.
LettingX be a r.v. from the SIE model, then its rth Mo is(11)θr′=∑i=0∞θr−1ΛΓ1−r2i+11−r.Proof.
LetX be a r. v. with pdf equation (6). The rth Mo of SIE distribution is computed as follows:(12)θr′=∫0∞xrfxdx=πθ2∫0∞xr−2e−θ/xcosπ2e−θ/xdx.
By inserting the expansioncosGx=∑i=0∞−1i/2i!Gx2i,n to the previous equation, then(13)θr′=∑i=0∞θ−1i2i!π22i+1∫0∞xr−δ−1e−2i+1θ/xdx.
The last equation can be rewritten as follows:(14)θr′=∑i=0∞Λ∫0∞xr−2e−2i+1θ/xdx,where Λ=∑i=0∞θ−1i/2i!π/22i+1.
Lety=θ/x; then,(15)θr′=∑i=0∞θr−1Λ∫0∞y−re−2i+1ydy.
Then,(16)θr′=∑i=0∞θr−1ΛΓ1−r2i+11−r.
The MGFu ofX is(17)MXt=∑r=0∞trr!θr′=∑r,i=0∞trθr−1ΛΓ1−rr!2i+11−r.
The ICMo, denoted byφst, of the SIE distribution is(18)φst=∫0txsfxdx=πθ2∫0txs−2e−θ/xcosπ2e−θ/xdx.
By using equation (18), φst will be as(19)φst=∑i=0∞θr−1Λγ1−r,θ/t2i+11−r,where γs,t=∫0tys−1e−ydy is the lower ICGFu where ICGFu is incomplete gamma function.
The CMo, denoted byτst, of the SIE distribution is(20)τst=∫t∞xsfxdx=πθ2∫t∞xs−2e−θ/xcosπ2e−θ/xdx.
By using equation (20), τst will be as given(21)τst=∑i=0∞θr−1ΛΓ1−r,θ/t2i+11−r,where Γs,t=∫t∞ys−1e−ydy is the upper ICGFu.


## 2.4. Order Statistics

LetX1,X2,…,Xn be random sample from the SIE distribution with order statistics (OS) X1,X2,…,Xn. The pdf of Tm of OS is(22)fXmx=n!m−1!n−m!Fm−1xfx1−Fxn−m.The pdf ofXm can be expressed as(23)fXmx=n!πθ2m−1!n−m!x−2e−θ/xsinm−1π2e−θ/xcosπ2e−θ/x1−sinπ2e−θ/xn−m.Specially, the pdfs of the lowest and greatest OS can be computed as(24)fX1x=nπθ2x−2e−θ/xcosπ2e−θ/x1−sinπ2e−θ/xn−1,(25)fXnx=nπθ2x−2e−θ/xsinn−1π2e−θ/xcosπ2e−θ/x.


## 3. Statistical Inference under Censored Samples

For different reasons, such as time constraints, money, or other resources, reliability or lifespan testing trials are typically censored. Generally speaking, there are two types of censorship schemes: Type-I and Type-II CS. Estimation using these two censoring techniques will be discussed in this section of the paper. If we use type-I censoring, we have a set time, sayX, but the amount of things that fail during the trial is completely random. Type-II censoring, on the contrary, is a process that continues until the stated number of failures is reached.


### 3.1. ML Estimation under Type-I Censor

Assume thatX1,X2,…,Xr be a type-I CS of size r obtained from lifetime testing experiment on k items whose lifetime follows the PDFu for SIE. The likelihood function (LLFu) of type-I CS is given as(26)L=n!n−r!1−Fx0n−r∏i=1rfxi.The log-LLFu corresponding to equation (26) is given by(27)lnL=lnn!n−r!+n−rln1−sinπ2e−θ/x0+rlnπ2+rlnθ−2∑i=1rlnxi−∑i=1rθxi+∑i=1rlncosπ2e−θ/xi.The ML equations for the SIE distribution are as follows:(28)∂lnL∂θ=n−rπe−θ/x0cosπ/2e−θ/x02x01−sinπ/2e−θ/x0+rθ−∑i=1rθxi+δπ2θ∑i=1rθxiδe−θ/xiδtanπ2e−θ/xiδ.Then, the ML estimators for the parameterθ are computed by putting ∂lnL/∂θ=0 and solving.


### 3.2. ML Estimation under Type-II Censor

LetX1,X2,…,Xr be a type-II CS of size r observed from lifetime testing experiment on k items whose lifetime has the PDFu for SIE. The LLFu of type-II CS is(29)L=n!n−r!1−Fxrn−r∏i=1rfxi.The log-LLFu corresponding to equation (29) is given by(30)lnL=lnn!n−r!+n−rln1−sinπ2e−θ/xr+rlnπ2+rlnθ−2∑i=1rlnxi−∑i=1rθxi+∑i=1rlncosπ2e−θ/xi.The ML equations for the SIE distribution are as follows:(31)∂lnL∂θ=n−rπe−θ/xrcosπ/2e−θ/xr2xr1−sinπ/2e−θ/xr+rθ−∑i=1rθxi+δπ2θ∑i=1rθxiδe−θ/xiδtanπ2e−θ/xiδ.Then, the ML estimators for the parameterθ is calculated by putting ∂lnL/∂θ=0 and solving.


### 3.3. Simulation Outcomes

Numerical outcomes are given in this section to examine how the estimators behave in cases of full, TIC, and TIIC estimations. With the help of Mathematica 9, you can compute the mean square errors (A1) as well as the lower limit (L1) and upper bound (U1) of the confidence interval, as well as the average length (AvLe) for 90 percent and 95 percent. The following is a description of how the following algorithm works:(i)
SIE distribution generates 5000 random samples with sizen = 30, 50, 100, 200, 300, 500, 1000, and 2000.(ii)
True parameterθ values are taken.(iii)
The termination time is set toT = 1.5 in the event of TIC and 3 in the absence of TIC. Three levels of censorship are chosen: r = 80%, r = 90% (TIIC), and r = 100% (complete sample).(iv)
The ML estimates,A1, L1, U1, and AvLe for various parameter values are computed.(v)
Tables1–6 include numerical outputs based on completeness and TIIC, while Tables 7–10 contain TIC-based simulation findings.Table 1
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 0.8 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.6394
0.0336
0.4777
0.8010
0.3233
0.4467
0.8320
0.3853
50
0.6410
0.0327
0.5152
0.7667
0.2515
0.4911
0.7908
0.2997
100
0.6454
0.0295
0.5474
0.7235
0.1761
0.5305
0.7403
0.2098
200
0.6519
0.0294
0.5700
0.6939
0.1238
0.5582
0.7057
0.1476
300
0.6530
0.0286
0.5824
0.6836
0.1013
0.5727
0.6933
0.1207
500
0.6630
0.0284
0.5938
0.6722
0.0785
0.5863
0.6797
0.0935
1000
0.6720
0.0282
0.6043
0.6597
0.0554
0.5990
0.6650
0.0660
2000
0.6813
0.0276
0.6118
0.6509
0.0391
0.6080
0.6546
0.0466Table 2
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 0.9 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.7151
0.0161
0.5501
0.9000
0.3500
0.5165
0.9336
0.4170
50
0.7203
0.0125
0.5856
0.8550
0.2694
0.5598
0.8808
0.3210
100
0.7270
0.0099
0.6222
0.8119
0.1897
0.6040
0.8301
0.2260
200
0.7344
0.0090
0.6476
0.7813
0.1337
0.6348
0.7941
0.1592
300
0.7359
0.0085
0.6594
0.7685
0.1090
0.6490
0.7789
0.1299
500
0.7446
0.0079
0.6723
0.7569
0.0845
0.6642
0.7650
0.1007
1000
0.7448
0.0076
0.6849
0.7447
0.0598
0.6791
0.7504
0.0713
2000
0.7536
0.0056
0.6925
0.7347
0.0422
0.6885
0.7388
0.0503Table 3
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 1 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.8200
0.0129
0.6229
0.9972
0.3743
0.5871
1.0330
0.4459
50
0.8157
0.0073
0.6616
0.9498
0.2882
0.6341
0.9774
0.3434
100
0.7893
0.0040
0.6983
0.9003
0.2021
0.6789
0.9197
0.2408
200
0.7965
0.0017
0.7271
0.8699
0.1428
0.7134
0.8835
0.1701
300
0.8024
0.0013
0.7438
0.8610
0.1171
0.7326
0.8722
0.1396
500
0.7986
0.0008
0.7544
0.8448
0.0904
0.7457
0.8535
0.1077
1000
0.8011
0.0004
0.7690
0.8331
0.0641
0.7629
0.8392
0.0763
2000
0.8005
0.0002
0.7778
0.8231
0.0453
0.7735
0.8274
0.0539Table 4
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 0.8 and θ=1.5.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.6394
0.0336
0.4777
0.8010
0.3233
0.4467
0.8320
0.3853
50
0.6410
0.0327
0.5152
0.7667
0.2515
0.4911
0.7908
0.2997
100
0.6454
0.0295
0.5474
0.7235
0.1761
0.5305
0.7403
0.2098
200
0.6519
0.0294
0.5700
0.6939
0.1238
0.5582
0.7057
0.1476
300
0.6530
0.0286
0.5824
0.6836
0.1013
0.5727
0.6933
0.1207
500
0.6630
0.0284
0.5938
0.6722
0.0785
0.5863
0.6797
0.0935
1000
0.6720
0.0282
0.6043
0.6597
0.0554
0.5990
0.6650
0.0660
2000
0.6813
0.0276
0.6118
0.6509
0.0391
0.6080
0.6546
0.0466Table 5
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 0.9 and θ=1.5.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.7151
0.0161
0.5501
0.9000
0.3500
0.5165
0.9336
0.4170
50
0.7203
0.0125
0.5856
0.8550
0.2694
0.5598
0.8808
0.3210
100
0.7270
0.0099
0.6222
0.8119
0.1897
0.6040
0.8301
0.2260
200
0.7344
0.0090
0.6476
0.7813
0.1337
0.6348
0.7941
0.1592
300
0.7359
0.0085
0.6594
0.7685
0.1090
0.6490
0.7789
0.1299
500
0.7446
0.0079
0.6723
0.7569
0.0845
0.6642
0.7650
0.1007
1000
0.7448
0.0076
0.6849
0.7447
0.0598
0.6791
0.7504
0.0713
2000
0.7536
0.0056
0.6925
0.7347
0.0422
0.6885
0.7388
0.0503Table 6
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 1 and θ=1.5.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.8200
0.0129
0.6229
0.9972
0.3743
0.5871
1.0330
0.4459
50
0.8157
0.0073
0.6616
0.9498
0.2882
0.6341
0.9774
0.3434
100
0.7893
0.0040
0.6983
0.9003
0.2021
0.6789
0.9197
0.2408
200
0.7965
0.0017
0.7271
0.8699
0.1428
0.7134
0.8835
0.1701
300
0.8024
0.0013
0.7438
0.8610
0.1171
0.7326
0.8722
0.1396
500
0.7986
0.0008
0.7544
0.8448
0.0904
0.7457
0.8535
0.1077
1000
0.8011
0.0004
0.7690
0.8331
0.0641
0.7629
0.8392
0.0763
2000
0.8005
0.0002
0.7778
0.8231
0.0453
0.7735
0.8274
0.0539Table 7
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIC for T = 1.5 and θ=0.2.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.1671
0.0040
0.1252
0.2090
0.0838
0.1172
0.2170
0.0998
50
0.1672
0.0033
0.1350
0.1995
0.0645
0.1288
0.2056
0.0769
100
0.1674
0.0027
0.1444
0.1898
0.0455
0.1400
0.1942
0.0542
200
0.1685
0.0018
0.1533
0.1857
0.0324
0.1502
0.1888
0.0386
300
0.1688
0.0017
0.1557
0.1820
0.0263
0.1531
0.1845
0.0314
500
0.1696
0.0012
0.1575
0.1778
0.0203
0.1555
0.1797
0.0242
1000
0.1714
0.0010
0.1641
0.1787
0.0146
0.1627
0.1801
0.0174
2000
0.1719
0.0008
0.1667
0.1771
0.0103
0.1658
0.1781
0.0123Table 8
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIC for T = 3 and θ=0.2.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.1898
0.0051
0.1447
0.2349
0.0902
0.1360
0.2435
0.1075
50
0.1913
0.0032
0.1591
0.2295
0.0704
0.1523
0.2362
0.0839
100
0.1923
0.0024
0.1668
0.2159
0.0491
0.1621
0.2206
0.0585
200
0.1925
0.0021
0.1760
0.2110
0.0350
0.1727
0.2143
0.0417
300
0.1931
0.0019
0.1788
0.2073
0.0285
0.1761
0.2100
0.0340
500
0.1939
0.0016
0.1828
0.2050
0.0222
0.1807
0.2071
0.0264
1000
0.1940
0.0016
0.1828
0.2050
0.0222
0.1807
0.2071
0.0264
2000
0.1942
0.0012
0.1864
0.2020
0.0157
0.1849
0.2035
0.0186Table 9
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIC for T = 1.5 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.6316
0.0335
0.4710
0.7921
0.3211
0.4403
0.8229
0.3826
50
0.6323
0.0314
0.5081
0.7565
0.2484
0.4843
0.7803
0.2959
100
0.6382
0.0311
0.5408
0.7156
0.1748
0.5241
0.7324
0.2083
200
0.6403
0.0306
0.5684
0.6922
0.1238
0.5565
0.7040
0.1475
300
0.6481
0.0302
0.5777
0.6784
0.1007
0.5680
0.6881
0.1200
500
0.6494
0.0294
0.5903
0.6685
0.0782
0.5828
0.6760
0.0931
1000
0.6578
0.0288
0.6003
0.6554
0.0551
0.5950
0.6607
0.0657
2000
0.6583
0.0276
0.6088
0.6478
0.0390
0.6050
0.6515
0.0465Table 10
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIC for T = 3 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.7492
0.0117
0.5783
0.9400
0.3617
0.5437
0.9747
0.4310
50
0.7575
0.0080
0.6099
0.8852
0.2753
0.5835
0.9115
0.3280
100
0.7639
0.0063
0.6470
0.8409
0.1939
0.6284
0.8594
0.2310
200
0.7646
0.0045
0.6760
0.8133
0.1373
0.6629
0.8264
0.1635
300
0.7738
0.0041
0.6878
0.7998
0.1120
0.6771
0.8105
0.1334
500
0.7748
0.0040
0.6878
0.7998
0.1120
0.6771
0.8105
0.1334
1000
0.7807
0.0039
0.6994
0.7860
0.0866
0.6911
0.7943
0.1031
2000
0.7820
0.0037
0.7115
0.7726
0.0611
0.7056
0.7785
0.0729From Tables1–10, we can note that when n increases, the MLE and AvLe are decreased.


## 3.1. ML Estimation under Type-I Censor

Assume thatX1,X2,…,Xr be a type-I CS of size r obtained from lifetime testing experiment on k items whose lifetime follows the PDFu for SIE. The likelihood function (LLFu) of type-I CS is given as(26)L=n!n−r!1−Fx0n−r∏i=1rfxi.The log-LLFu corresponding to equation (26) is given by(27)lnL=lnn!n−r!+n−rln1−sinπ2e−θ/x0+rlnπ2+rlnθ−2∑i=1rlnxi−∑i=1rθxi+∑i=1rlncosπ2e−θ/xi.The ML equations for the SIE distribution are as follows:(28)∂lnL∂θ=n−rπe−θ/x0cosπ/2e−θ/x02x01−sinπ/2e−θ/x0+rθ−∑i=1rθxi+δπ2θ∑i=1rθxiδe−θ/xiδtanπ2e−θ/xiδ.Then, the ML estimators for the parameterθ are computed by putting ∂lnL/∂θ=0 and solving.


## 3.2. ML Estimation under Type-II Censor

LetX1,X2,…,Xr be a type-II CS of size r observed from lifetime testing experiment on k items whose lifetime has the PDFu for SIE. The LLFu of type-II CS is(29)L=n!n−r!1−Fxrn−r∏i=1rfxi.The log-LLFu corresponding to equation (29) is given by(30)lnL=lnn!n−r!+n−rln1−sinπ2e−θ/xr+rlnπ2+rlnθ−2∑i=1rlnxi−∑i=1rθxi+∑i=1rlncosπ2e−θ/xi.The ML equations for the SIE distribution are as follows:(31)∂lnL∂θ=n−rπe−θ/xrcosπ/2e−θ/xr2xr1−sinπ/2e−θ/xr+rθ−∑i=1rθxi+δπ2θ∑i=1rθxiδe−θ/xiδtanπ2e−θ/xiδ.Then, the ML estimators for the parameterθ is calculated by putting ∂lnL/∂θ=0 and solving.


## 3.3. Simulation Outcomes

Numerical outcomes are given in this section to examine how the estimators behave in cases of full, TIC, and TIIC estimations. With the help of Mathematica 9, you can compute the mean square errors (A1) as well as the lower limit (L1) and upper bound (U1) of the confidence interval, as well as the average length (AvLe) for 90 percent and 95 percent. The following is a description of how the following algorithm works:(i)
SIE distribution generates 5000 random samples with sizen = 30, 50, 100, 200, 300, 500, 1000, and 2000.(ii)
True parameterθ values are taken.(iii)
The termination time is set toT = 1.5 in the event of TIC and 3 in the absence of TIC. Three levels of censorship are chosen: r = 80%, r = 90% (TIIC), and r = 100% (complete sample).(iv)
The ML estimates,A1, L1, U1, and AvLe for various parameter values are computed.(v)
Tables1–6 include numerical outputs based on completeness and TIIC, while Tables 7–10 contain TIC-based simulation findings.Table 1
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 0.8 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.6394
0.0336
0.4777
0.8010
0.3233
0.4467
0.8320
0.3853
50
0.6410
0.0327
0.5152
0.7667
0.2515
0.4911
0.7908
0.2997
100
0.6454
0.0295
0.5474
0.7235
0.1761
0.5305
0.7403
0.2098
200
0.6519
0.0294
0.5700
0.6939
0.1238
0.5582
0.7057
0.1476
300
0.6530
0.0286
0.5824
0.6836
0.1013
0.5727
0.6933
0.1207
500
0.6630
0.0284
0.5938
0.6722
0.0785
0.5863
0.6797
0.0935
1000
0.6720
0.0282
0.6043
0.6597
0.0554
0.5990
0.6650
0.0660
2000
0.6813
0.0276
0.6118
0.6509
0.0391
0.6080
0.6546
0.0466Table 2
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 0.9 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.7151
0.0161
0.5501
0.9000
0.3500
0.5165
0.9336
0.4170
50
0.7203
0.0125
0.5856
0.8550
0.2694
0.5598
0.8808
0.3210
100
0.7270
0.0099
0.6222
0.8119
0.1897
0.6040
0.8301
0.2260
200
0.7344
0.0090
0.6476
0.7813
0.1337
0.6348
0.7941
0.1592
300
0.7359
0.0085
0.6594
0.7685
0.1090
0.6490
0.7789
0.1299
500
0.7446
0.0079
0.6723
0.7569
0.0845
0.6642
0.7650
0.1007
1000
0.7448
0.0076
0.6849
0.7447
0.0598
0.6791
0.7504
0.0713
2000
0.7536
0.0056
0.6925
0.7347
0.0422
0.6885
0.7388
0.0503Table 3
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 1 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.8200
0.0129
0.6229
0.9972
0.3743
0.5871
1.0330
0.4459
50
0.8157
0.0073
0.6616
0.9498
0.2882
0.6341
0.9774
0.3434
100
0.7893
0.0040
0.6983
0.9003
0.2021
0.6789
0.9197
0.2408
200
0.7965
0.0017
0.7271
0.8699
0.1428
0.7134
0.8835
0.1701
300
0.8024
0.0013
0.7438
0.8610
0.1171
0.7326
0.8722
0.1396
500
0.7986
0.0008
0.7544
0.8448
0.0904
0.7457
0.8535
0.1077
1000
0.8011
0.0004
0.7690
0.8331
0.0641
0.7629
0.8392
0.0763
2000
0.8005
0.0002
0.7778
0.8231
0.0453
0.7735
0.8274
0.0539Table 4
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 0.8 and θ=1.5.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.6394
0.0336
0.4777
0.8010
0.3233
0.4467
0.8320
0.3853
50
0.6410
0.0327
0.5152
0.7667
0.2515
0.4911
0.7908
0.2997
100
0.6454
0.0295
0.5474
0.7235
0.1761
0.5305
0.7403
0.2098
200
0.6519
0.0294
0.5700
0.6939
0.1238
0.5582
0.7057
0.1476
300
0.6530
0.0286
0.5824
0.6836
0.1013
0.5727
0.6933
0.1207
500
0.6630
0.0284
0.5938
0.6722
0.0785
0.5863
0.6797
0.0935
1000
0.6720
0.0282
0.6043
0.6597
0.0554
0.5990
0.6650
0.0660
2000
0.6813
0.0276
0.6118
0.6509
0.0391
0.6080
0.6546
0.0466Table 5
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 0.9 and θ=1.5.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.7151
0.0161
0.5501
0.9000
0.3500
0.5165
0.9336
0.4170
50
0.7203
0.0125
0.5856
0.8550
0.2694
0.5598
0.8808
0.3210
100
0.7270
0.0099
0.6222
0.8119
0.1897
0.6040
0.8301
0.2260
200
0.7344
0.0090
0.6476
0.7813
0.1337
0.6348
0.7941
0.1592
300
0.7359
0.0085
0.6594
0.7685
0.1090
0.6490
0.7789
0.1299
500
0.7446
0.0079
0.6723
0.7569
0.0845
0.6642
0.7650
0.1007
1000
0.7448
0.0076
0.6849
0.7447
0.0598
0.6791
0.7504
0.0713
2000
0.7536
0.0056
0.6925
0.7347
0.0422
0.6885
0.7388
0.0503Table 6
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIIC for xr = 1 and θ=1.5.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.8200
0.0129
0.6229
0.9972
0.3743
0.5871
1.0330
0.4459
50
0.8157
0.0073
0.6616
0.9498
0.2882
0.6341
0.9774
0.3434
100
0.7893
0.0040
0.6983
0.9003
0.2021
0.6789
0.9197
0.2408
200
0.7965
0.0017
0.7271
0.8699
0.1428
0.7134
0.8835
0.1701
300
0.8024
0.0013
0.7438
0.8610
0.1171
0.7326
0.8722
0.1396
500
0.7986
0.0008
0.7544
0.8448
0.0904
0.7457
0.8535
0.1077
1000
0.8011
0.0004
0.7690
0.8331
0.0641
0.7629
0.8392
0.0763
2000
0.8005
0.0002
0.7778
0.8231
0.0453
0.7735
0.8274
0.0539Table 7
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIC for T = 1.5 and θ=0.2.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.1671
0.0040
0.1252
0.2090
0.0838
0.1172
0.2170
0.0998
50
0.1672
0.0033
0.1350
0.1995
0.0645
0.1288
0.2056
0.0769
100
0.1674
0.0027
0.1444
0.1898
0.0455
0.1400
0.1942
0.0542
200
0.1685
0.0018
0.1533
0.1857
0.0324
0.1502
0.1888
0.0386
300
0.1688
0.0017
0.1557
0.1820
0.0263
0.1531
0.1845
0.0314
500
0.1696
0.0012
0.1575
0.1778
0.0203
0.1555
0.1797
0.0242
1000
0.1714
0.0010
0.1641
0.1787
0.0146
0.1627
0.1801
0.0174
2000
0.1719
0.0008
0.1667
0.1771
0.0103
0.1658
0.1781
0.0123Table 8
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIC for T = 3 and θ=0.2.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.1898
0.0051
0.1447
0.2349
0.0902
0.1360
0.2435
0.1075
50
0.1913
0.0032
0.1591
0.2295
0.0704
0.1523
0.2362
0.0839
100
0.1923
0.0024
0.1668
0.2159
0.0491
0.1621
0.2206
0.0585
200
0.1925
0.0021
0.1760
0.2110
0.0350
0.1727
0.2143
0.0417
300
0.1931
0.0019
0.1788
0.2073
0.0285
0.1761
0.2100
0.0340
500
0.1939
0.0016
0.1828
0.2050
0.0222
0.1807
0.2071
0.0264
1000
0.1940
0.0016
0.1828
0.2050
0.0222
0.1807
0.2071
0.0264
2000
0.1942
0.0012
0.1864
0.2020
0.0157
0.1849
0.2035
0.0186Table 9
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIC for T = 1.5 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.6316
0.0335
0.4710
0.7921
0.3211
0.4403
0.8229
0.3826
50
0.6323
0.0314
0.5081
0.7565
0.2484
0.4843
0.7803
0.2959
100
0.6382
0.0311
0.5408
0.7156
0.1748
0.5241
0.7324
0.2083
200
0.6403
0.0306
0.5684
0.6922
0.1238
0.5565
0.7040
0.1475
300
0.6481
0.0302
0.5777
0.6784
0.1007
0.5680
0.6881
0.1200
500
0.6494
0.0294
0.5903
0.6685
0.0782
0.5828
0.6760
0.0931
1000
0.6578
0.0288
0.6003
0.6554
0.0551
0.5950
0.6607
0.0657
2000
0.6583
0.0276
0.6088
0.6478
0.0390
0.6050
0.6515
0.0465Table 10
MLEs, estimates,A1, L1, U1, and AvLe of the SIE model under TIC for T = 3 and θ=0.8.
n
MLE
A1
90%
95%
L1
U1
AvLe
L1
U1
AvLe
30
0.7492
0.0117
0.5783
0.9400
0.3617
0.5437
0.9747
0.4310
50
0.7575
0.0080
0.6099
0.8852
0.2753
0.5835
0.9115
0.3280
100
0.7639
0.0063
0.6470
0.8409
0.1939
0.6284
0.8594
0.2310
200
0.7646
0.0045
0.6760
0.8133
0.1373
0.6629
0.8264
0.1635
300
0.7738
0.0041
0.6878
0.7998
0.1120
0.6771
0.8105
0.1334
500
0.7748
0.0040
0.6878
0.7998
0.1120
0.6771
0.8105
0.1334
1000
0.7807
0.0039
0.6994
0.7860
0.0866
0.6911
0.7943
0.1031
2000
0.7820
0.0037
0.7115
0.7726
0.0611
0.7056
0.7785
0.0729From Tables1–10, we can note that when n increases, the MLE and AvLe are decreased.


## 4. Results and Discussion

### 4.1. Application

In this section, two real datasets are analyzed to explain the benefit of the SIE model compared to the IE model. To compare the competitive models, we suggested some information criterion (ICr) as minus two log-LLFu (D1), Akaike ICr (D2), the correct Akaike ICr (D3), Bayesian ICr (D4), and Hannan–Quinn ICr (D5).The first data are known as ball bearing data, and it represents the number of rotations before ball bearing failure obtained [18]. The second data set consists of 100 observations of breaking stress of carbon fibres (in Gba) given by [19]. According to Tables 11 and 12, our new model is better suited than the IE model and has the lowest values for all statistics.Table 11
MLEs,D1, D2, D3, D4, and D5 for ball bearing data.
Models
MLE and SE
D1
D2
D3
D4
D5
SIE
73.586 (11.48)
243.07
245.07
245.252
244.45
245.382
IE
20.403 (2.55)
253.927
255.927
256.109
255.307
256.24Table 12
MLEs,D1, D2, D3, D4, and D5 for carbon fibres’ data.
Models
MLE and SE
D1
D2
D3
D4
D5
SIE
2.817 (11.48)
348.384
350.384
350.404
350.384
351.438
IE
2.135 (2.55)
397.956
399.956
399.977
399.956
401.011


### 4.2. Discussion

From the modelling to ball bearing and carbon fibres datasets, we see that the SIE model provides the greatest fit for the both datasets. The numerical values in Tables11 and 12 are proposed; the both datasets supported the superiority of the SIE model.


## 4.1. Application

In this section, two real datasets are analyzed to explain the benefit of the SIE model compared to the IE model. To compare the competitive models, we suggested some information criterion (ICr) as minus two log-LLFu (D1), Akaike ICr (D2), the correct Akaike ICr (D3), Bayesian ICr (D4), and Hannan–Quinn ICr (D5).The first data are known as ball bearing data, and it represents the number of rotations before ball bearing failure obtained [18]. The second data set consists of 100 observations of breaking stress of carbon fibres (in Gba) given by [19]. According to Tables 11 and 12, our new model is better suited than the IE model and has the lowest values for all statistics.Table 11
MLEs,D1, D2, D3, D4, and D5 for ball bearing data.
Models
MLE and SE
D1
D2
D3
D4
D5
SIE
73.586 (11.48)
243.07
245.07
245.252
244.45
245.382
IE
20.403 (2.55)
253.927
255.927
256.109
255.307
256.24Table 12
MLEs,D1, D2, D3, D4, and D5 for carbon fibres’ data.
Models
MLE and SE
D1
D2
D3
D4
D5
SIE
2.817 (11.48)
348.384
350.384
350.404
350.384
351.438
IE
2.135 (2.55)
397.956
399.956
399.977
399.956
401.011


## 4.2. Discussion

From the modelling to ball bearing and carbon fibres datasets, we see that the SIE model provides the greatest fit for the both datasets. The numerical values in Tables11 and 12 are proposed; the both datasets supported the superiority of the SIE model.


## 5. Conclusion

In this study, we proposed a new one-parameter model, which is called the SIE model. Some basic statistical properties of the SIE model are also proposed. Estimation of the SIE parameter was assessed by using the ML method of estimation censored and complete samples. Application to carbon fibres datasets were used to explain the importance of SIE model against the IE model. The SIE model as we see is very flexible and simple model, so many authors can use it in the future articles. The authors can use Bayesian estimation under complete and various censored schemes to estimate its parameters. Also, ranked set sampling papers can apply the new model. Also, the authors which are interested in distribution theory can generalize more extensions of this model by many different ways.



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*Source: 7330385-2021-10-21.xml*