A New Two-Parameter Lifetime Model with Statistical Properties and
Applications
NAJWAN ALSADAT
Department of Quantitative Analysis, College of Business Administration, King Saud University,
P.O. Box 2455, Riyadh, 11451,
SAUDI ARABIA
Abstract: - A new lifetime distribution called the truncated Cauchy power length-biased exponential (TCP-
LBEX) distribution that extends the length-biased (LBEX) model is investigated. The statistical properties of
the TCP-LBEX model including the quantile function, incomplete moment, moment, and entropy are derived.
The method of maximum likelihood estimation was used to estimate the parameters of the TCP-LBEX. Monto
Carlo simulations are used to assess the behavior of parameters. Finally, we demonstrate applications of two
real-world data sets to show the flexibility and potentiality of the proposed model.
Key-Words: - Truncated Cauchy power family; Length-biased exponential; moments; Rényi Entropy;
Maximum likelihood estimation.
Received: July 20, 2022. Revised: January 19, 2023. Accepted: February 15, 2023. Published: March 29, 2023.
1 Introduction
Some scientists have recently proposed strategies
for incorporating probability models. This
parameter addition phenomenon generates more
classes of distributions, which are useful for
modeling datasets in engineering science, biological
science, economics, medicine, income, physics, and
environmental sciences. Several G-class of
distributions are the T-X class, [1], the odd Fréchet-
G class, [2], Kumaraswamy-G class, [3], odd
Dagum-G class, [4], weighted exponential-G class,
[5], alpha power class, [6], weighted exponentiated-
G class, [7], Marshall-Olkin alpha-power-G class,
[8], truncated inverted Kumaraswamy-G class, [9],
transmuted geometric-G class, [10], complementary
generalized transmuted Poisson-G class, [11], the
exponentiated odd log-logistic-G class, [12], Type-
II half logistic-G class, [13], Topp-Leone-G class,
[14], truncated Cauchy power Weibull-G class,
[15], Lomax-G class, [16], Type-I half logistic Burr
X-G class, [17], type I general exponential-G class,
[18], sine Topp-Leone-G class, [19], generalized
odd Weibull class, [20], a new power Topp-Leone-
G class, [21], odd power Lindley-G class, [22], the
transmuted transmuted-G class, [23], exponentiated
version of the M class, [24], the transmuted
Gompertz-G class, [25], transmuted odd Fréchet-G
class, [26], transmuted odd Lindley-G class, [27],
Topp Leone odd Lindley-G class, [28], Topp-Leone
odd log-logistic class, [29], odd Perks-G class, [30]
and Kumaraswamy transmuted-G class, [31],
among others.
Recently, [32], discussed the TCP-G class of
distributions. The distribution function (cdf) of the
TCP-G class is
where and the cdf of a baseline with
parameter vector is denoted by . The
corresponding probability density function (pdf)
and hazard rate function (hrf) respectively are
and
Relevant research has been provided depending on
the TCP-G family, for example, TCP-inverted
Topp–Leone model, [33], TCP-inverse exponential
model, [34], TCP-Lomax model, [35], TCP-
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Weibull-G class, [36], and TCP odd Frechet-G
class, [37].
[38], proposed the LBEX by assigning a weight to
the exponential (E) model using the concept
proposed by [39]. They investigated that the LBEX
model is more flexible than the E model. The cdf
and pdf of LBEX distribution are provided below
and
where is a scale parameter. An extension of
the LBEX model is proposed, and this extension is
constructed utilizing the TCP-G class and LBEX
model. This extension is termed the TCP-LBEX
distribution.
The remainder of this article is outlined as follows.
In Section 2, the cdf and pdf of the proposed
distribution are presented and an expansion of the
TCP-LBEX pdf is derived. The basic properties of
the distribution, including the quantile function,
moments, incomplete and conditional moments, and
entropy, are presented in Section 3. In Section 4,
the parameter estimation employing the maximum
likelihood estimation (MLE) approach is discussed
and Monto Carlo simulations are utilized to study
the behavior of the parameters. In Section 5, the
TCP-LBEX model is performed on two real-world
data sets to examine its feasibility using some
information criterion (INC) of the goodness of fit,
like; the Akaike INC (K1), Bayesian INC (K2),
consistent Akaike INC (K3), Hannan–Quinn INC
(K4), Cramѐr–Von Mises (K5), Anderson–Darling
(K6), Kolmogorov–Smirnov (K7) statistics and p-
value (K8). Lastly, Section 6 presents the
conclusions.
2 The New TCP-LBEX Model
By substituting (4) in (2), the TCP-LBEX cdf of
random variable X is obtained as
The corresponding pdf is
The reliability function (sf) of the TCP-LBEX
distribution is provided below
For the TCP-LBEX distribution, the hazard rate
function (hrf) is expressed as follows:
The reversed hazard rate (RHR) function of the
TCP-LBEX distribution is provided below
The cumulative hazard rate (CHR) function of the
TCP-LBEX distribution is provided below
Figure 1 presents the pdf of the TCP-LBEX model
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Fig. 1: cdf of the TCP-LBEX model.
Figure 2 presents the cdf of the TCP-LBEX
model.
Fig. 2: cdf of the TCP-LBEX model.
Fig. 3: hrf of the TCP-LBEX model.
Fig. 4: sf of the TCP-LBEX model.
3 Statistical Properties of the TCP-
LBEX Model
In this section, the statistical properties of the TCP-
LBEX model, such as the useful expansion,
quantiles, moments, generating functions,
incomplete moments, and entropy, are discussed.
3.1 Useful expansion
In this subsection, the expansion of the TCP-LBEX
pdf is established. Using the binomial series
expansion,
which holds for , and is a positive real
non-integer. Further, the following relation is used:
where is any positive real noninteger. By
substituting (8) in (7), the TCP-LBEX pdf becomes
Further, by substituting (9) in (10) and performing
some algebraic manipulations, the TCP-LBEX pdf
can be written as
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where
.
3.2 Quantiles
Quantiles are fundamental for estimating and
simulating the distribution parameters. The quantile
function of can be obtained by inverting (6) as
We cannot have closed form for Equation (12) but
we can solve it numerically.
3.3 Moments
For with a pdf (11), the moment of is
By substituting
and some algebraic
manipulations, the moment achieves the
following form:
The mg function of can be derived from
(11) as follows:
The numerical values of specific parameters of the
first four ordinary moments variance (
skewness (S), kurtosis (K), and coefficient of
variation (CV) of the TCP-LBEX model are
mentioned in Table 1.
From the data presented in Table 1, as the value of
increases, the values of moments decrease for
constant . In contrast, the values of S, K, and
CV decrease.
3.4 Conditional Moments
The upper incomplete moment of the TCP-
LBEX distribution can be given by
where
is the upper
incomplete gamma function. Similarly, the
lower incomplete moment of the distribution is
expressed as
where
is the lower
incomplete gamma function.
Table 1. Some numerical results of moments for the
TCP-LBEX model at = 0.5
CV
K
S
0.5
54
0.6
33
1.1
06
2.5
79
0.3
27
2.1
06
9.4
81
1.0
32
0.
5
0.8
66
1.1
59
2.1
43
5.1
01
0.4
09
1.6
46
7.1
97
0.7
39
1.
0
1.0
76
1.5
99
3.1
04
7.5
45
0.4
41
1.4
79
6.5
3
0.6
17
1.
5
1.2
33
1.9
78
3.9
97
9.9
08
0.4
57
1.3
93
6.2
28
0.5
48
2.
0
1.3
59
2.3
12
4.8
32
12.
195
0.4
66
1.3
42
6.0
61
0.5
02
2.
5
1.4
2.6
5.6
14.
0.4
1.3
5.9
0.4
3.
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63
1
17
408
7
09
59
69
0
1.5
52
2.8
81
6.3
59
16.
555
0.4
73
1.2
86
5.8
91
0.4
43
3.
5
1.6
29
3.1
3
7.0
63
18.
64
0.4
75
1.2
69
5.8
43
0.4
23
4.
0
1.6
98
3.3
59
7.7
34
20.
668
0.4
76
1.2
56
5.8
09
0.4
06
4.
5
1.7
6
3.5
73
8.3
75
22.
642
0.4
77
1.2
46
5.7
83
0.3
92
5.
0
1.8
16
3.7
73
8.9
9
24.
568
0.4
77
1.2
38
5.7
64
0.3
8
5.
5
1.8
67
3.9
62
9.5
81
26.
447
0.4
77
1.2
32
5.7
49
0.3
7
6.
0
1.9
14
4.1
4
10.
15
28.
282
0.4
77
1.2
26
5.7
37
0.3
61
6.
5
1.9
58
4.3
08
10.
7
30.
078
0.4
76
1.2
22
5.7
27
0.3
53
7.
0
1.9
98
4.4
69
11.
231
31.
836
0.4
76
1.2
19
5.7
2
0.3
45
7.
5
2.0
36
4.6
22
11.
746
33.
558
0.4
76
1.2
16
5.7
14
0.3
39
8.
0
2.0
72
4.7
68
12.
245
35.
246
0.4
75
1.2
13
5.7
09
0.3
33
8.
5
2.1
06
4.9
08
12.
729
36.
902
0.4
75
1.2
11
5.7
05
0.3
27
9.
0
2.1
37
5.0
43
13.
201
38.
528
0.4
74
1.2
09
5.7
02
0.3
22
9.
5
2.1
68
5.1
73
13.
66
40.
126
0.4
74
1.2
07
5.7
0.3
17
10
.0
2.1
96
5.2
98
14.
107
41.
696
0.4
73
1.2
06
5.6
98
0.3
13
10
.5
2.2
24
5.4
18
14.
543
43.
24
0.4
73
1.2
05
5.6
96
0.3
09
11
.0
2.2
5
5.5
35
14.
969
44.
759
0.4
72
1.2
04
5.6
95
0.3
05
11
.5
2.2
75
5.6
48
15.
385
46.
254
0.4
72
1.2
03
5.6
95
0.3
02
12
.0
2.2
99
5.7
57
15.
792
47.
727
0.4
71
1.2
02
5.6
94
0.2
99
12
.5
2.3
22
5.8
63
16.
19
49.
178
0.4
71
1.2
02
5.6
94
0.2
95
13
.0
2.3
44
5.9
67
16.
58
50.
608
0.4
7
1.2
01
5.6
94
0.2
93
13
.5
2.3
66
6.0
67
16.
962
52.
019
0.4
7
1.2
5.6
94
0.2
9
14
.0
3.5 Entropy
The Rényi entropy is defined using (:
Using (2.2), the following expression is obtained:
Using the same procedure employed for the useful
expansion (8) and by performing some
simplifications, the following expression is
obtained:
Where
Thus,
Some numerical results of for the TCP-LBEX
distribution for some choices of parameter
and are listed in Table 2.
Table 2. Some numerical results of Rényi entropy
for the TCP-LBEX model.
0.354
0.09
0.028
0.5
0.5
0.451
0.266
0.19
1.0
0.485
0.311
0.238
1.5
0.501
0.33
0.258
2.0
0.51
0.34
0.269
2.5
0.516
0.346
0.275
3.0
0.52
0.35
0.279
3.5
0.956
0.692
0.574
0.5
2.0
1.054
0.868
0.792
1.0
1.087
0.913
0.84
1.5
1.103
0.932
0.86
2.0
1.112
0.942
0.871
2.5
1.118
0.948
0.877
3.0
1.122
0.952
0.881
3.5
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The numerical values in Table 2 belong to [0.028,
1.122]; this indicates that and have an
important impact on the amount of information.
4 Parameter Estimation
The MLE technique is utilized in this section to
estimate the unknown parameters of the TCP-
LBEX distribution. Suppose that be an n-
th random sample from the specified distribution
(7). The TCP-LBEX distribution's log-likelihood
function (LLF) is supplied by
The LLF in equation (18) can be computed by
differentiating (18) regards to and:
and
respectively. The MLEs of the parameters and
are denoted by and and are obtained by
solving the above last system of equations (19) and
(20). It is commonly more convenient to employ
nonlinear optimization techniques, like the quasi-
Newton approach, to numerically optimize the
sample likelihood function.
4.1 Simulation Study
In this section, a simulation study has been
conducted to illustrate the MLEs of and for the
TCP-LBEX model. The estimates are assessed and
compared based on the root mean square errors
(RMSErs). For this purpose, the following
algorithm is adopted.
Step 1: A random sample of size n = 50, 100, 200,
300, 500, 700, and 1000 are generated from the
TCP-LBEX distribution.
Step 2: The parameter values are considered as
,
,, and
.
Step 3: For the selected values of parameters and
each sample of size n, the MLEs are calculated.
Step 4: Steps 1–3 are repeated, and N = 10000
times, representing various samples.
Step 5: The outcomes of the simulation study are
presented in Table 3 and Table 4.
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Table 3. MLEs and RMSEr of the TCP-LBEX
distribution for set 1 and set 2
n
set 1
set 2
MLE
RMSEr
MLE
RMSEr
50
0.519876
0.010675
0.549776
0.015231
0.583619
0.096384
1.763210
0.652452
100
0.519020
0.006504
0.526846
0.006895
0.580228
0.050699
1.731120
0.467523
200
0.507657
0.002476
0.512182
0.002960
0.532262
0.013490
1.621920
0.109372
300
0.506596
0.001206
0.501286
0.001231
0.504893
0.006243
1.561980
0.063621
500
0.503032
0.000785
0.501934
0.000863
0.510978
0.003014
1.488500
0.028265
700
0.503911
0.000366
0.504906
0.000489
0.505968
0.001960
1.527870
0.024482
1000
0.502257
0.000234
0.500685
0.000212
0.499607
0.001162
1.505420
0.007254
Table 4. MLEs and RMSEr of the TCP-LBEX
distribution for set 3 and set 4
n
set 3
set 4
MLE
RMSEr
MLE
RMSEr
50
0.536645
0.013190
1.504310
0.032153
2.312980
1.042390
1.507250
0.037610
100
0.514007
0.004442
1.518710
0.010780
2.160900
0.424585
1.521170
0.013241
200
0.506586
0.003029
1.505840
0.003223
2.009150
0.119023
1.505360
0.003831
300
0.502757
0.001286
1.501350
0.001183
2.000800
0.090311
1.500930
0.001369
500
0.505382
0.000974
1.503320
0.000541
2.071520
0.064444
1.503850
0.000738
700
0.502108
0.000509
1.502030
0.000213
2.036330
0.042179
1.501890
0.000286
1000
0.500067
0.000247
1.498890
0.000053
2.019420
0.016566
1.498660
0.000069
From the above Table 3 and Table 4, the RMSErs
of the TCP-LBEX model decrease when n increase.
5 Applications
In this section, we used two real-world data sets to
compare the TCP-LBEX model to several other
known competing models like; the LBEX model,
Burr X-EX (BrXEX) model, Marshall–Olkin EX
(MOEX) model, Kumaraswamy Marshall–Olkin
EX (KMOEX) model, Kumaraswamy EX (KEX)
model, beta EX (BEX) model, generalized
Marshall–Olkin EX (GMOEX) model, EX model
and Marshall–Olkin Kumaraswamy EX (MOKEX)
model) in this part to highlight its usefulness in data
modeling. To estimate the parameters of the
competing models, the MLE approach is employed.
To choose the optimal model, the K1, K2, K3, K4,
K5, K6, K7, and K8model selection criteria and
goodness of fit tests are utilized.
The first Data set: This data set contains the
survival periods (in days) of 72 guinea pigs infected
with virulent tubercle bacilli, [40]. For the first data
set, MLEs and standard errors (SErs) are computed.
Table 5 shows the numerical results of MLEs and
the SErs for all competitive models for the first data
set. Based on the numerical numbers in Table 6 and
the information in Figure 3, we can conclude that
the TCP-LBEX model provides the best fit for the
first data set because the TCP-LBEX model has the
lowest numerical value in K1, K2, K3, K4, K5, K6,
K7 but has the largest value in K8.
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Table 5. MLEs and SErs for the first data set.
Mo
dels
MLEs
SErs
TCP
-
LBE
X
0.7
81
1.66
5
0.0
92
0.30
1
LBE
X
0.9
25
2
0.0
76
8
BrX
EX
0.4
75
0.20
55
0.0
60
0.01
2
MO
EX
8.7
78
1.37
9
3.5
55
0.19
3
KM
OEX
(,
δ, γ,
)
0.3
73
1
3.47
82
3.3
06
3
0.2
99
0
0.1
35
8
0.86
2
0.7
81
1.1
13
KEX
(δ,
γ,
)
3.3
04
1
1.10
02
1.0
37
1
1.1
06
1
0.76
42
0.6
14
1
BEX
(δ,
γ,
β)
0.8
07
3
3.46
12
1.3
31
1
0.6
96
1
1.00
32
0.8
55
1
GM
OEX
(λ,
α,
)
0.1
78
9
47.6
350
4.4
65
2
0.0
70
2
44.9
011
1.3
27
0
EX
0.5
40
0.0
63
MO
KEX
(,
δ, γ,
)
0.0
08
1
2.71
62
1.9
86
1
0.0
99
2
0.0
02
1
1.31
58
0.7
83
9
0.0
48
1
Table 6. Numerical values of K1, K2, K3, K4,
K5, K6, K7, and K8 for the first data set.
Mod
els
K1
K2
K3
K4
K5
K6
K7
K8
TCP-
LBEX
191.
60
191.
31
191.
77
193.
41
0.
08
0.
48
0.
09
(0.6
35)
LBEX
210.
40
212.
68
210.
45
211.
30
0.
25
1.
52
0.
14
(0.1
30)
BrXE
X
235.
30
239.
90
235.
50
237.
10
0.
52
2.
90
0.
22
(0.0
02)
MOE
X
210.
36
214.
92
210.
53
212.
16
0.
17
1.
18
0.
10
(0.4
30)
KM
OEX
207.
82
216.
94
208.
42
211.
42
0.
11
0.
61
0.
09
(0.5
30)
KEX
209.
42
216.
24
209.
77
212.
12
0.
11
0.
74
0.
09
(0.5
00)
BEX
207.
38
214.
22
207.
73
210.
08
0.
15
0.
98
0.
11
(0.3
40)
GM
OEX
210.
54
217.
38
210.
89
213.
24
0.
16
1.
02
0.
09
(0.5
10)
EX
234.
63
236.
91
234.
68
235.
54
1.
25
6.
53
0.
27
(0.0
60)
MO
KEX
209.
44
218.
56
210.
04
213.
04
0.
12
0.
79
0.
10
(0.4
40)
Fig. 3: Fitted cdf, pdf, and pp plots for the first data
set
The second data set: This data collection contains
information from 20 individuals and consists of
histories pertaining to relief periods (in minutes) for
patients who have taken an analgesic, [41]. Table 7
shows the numerical results of MLEs and the SErs
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for all competitive models for the second data set.
Based on the numerical numbers in Table 8 and the
information in Figure 4, we can conclude that the
TCP-LBEX provides the best fit for the second data
set because the TCP-LBEX model has the lowest
numerical value in K1, K2, K3, K4, K5, K6, K7 but
has the largest value in K8.
Table 7. MLEs and SErs for the second data set.
Mo
dels
MLEs
SErs
TCP
-
LBE
X
0.41
3
11.7
14
0.07
7
6.93
9
LBE
X
0.95
02
0.15
01
BrX
EX
1.16
35
0.32
07
0.33
0
0.03
0
MO
EX
54.4
74
2.31
6
35.5
82
0.37
4
KM
OEX
(,
δ, γ,
)
8.86
79
34.8
258
0.2
989
4.8
988
9.14
59
22.3
119
0.2
387
3.1
757
KEX
(δ,
γ,
)
83.7
558
0.56
79
3.3
329
42.3
612
0.32
61
1.1
880
BEX
(δ,
γ, β)
81.6
333
0.54
21
3.5
142
120.
410
0.32
72
1.4
101
GM
OEX
(λ,
α,
)
0.51
92
89.4
623
3.1
691
0.25
61
66.2
782
0.7
721
EX
0.52
6
0.11
7
MO
KEX
(,
δ, γ,
)
0.13
33
33.2
322
0.5
711
1.6
691
0.33
20
57.8
371
0.7
211
1.8
141
Table 8. Numerical values of K1, K2, K3, K4, K5,
K6, K7, and K8 for the second data set.
Mod
els
K1
K2
K3
K4
K5
K6
K7
K8
TCP-
LBEX
36.
43
35.
03
37.
14
36.
82
0.
06
0.
35
0.
13
(0.9
0)
LBEX
54.
32
55.
31
54.
54
54.
50
0.
53
2.
76
0.
32
(0.0
7)
BrXE
X
48.
10
50.
10
48.
80
48.
50
0.
24
1.
39
0.
25
(0.1
7)
MO
EX
43.
51
45.
51
44.
22
43.
90
0.
14
0.
80
0.
18
(0.5
5)
KM
OEX
42.
80
46.
84
45.
55
43.
60
0.
19
0.
08
0.
15
(0.8
6)
KEX
41.
78
44.
75
43.
28,
42.
32
0.
07
0.
45
0.
14
(0.8
6)
BEX
43.
48
46.
45
44.
98
44.
02
0.
12
0.
70
0.
16
(0.8
0)
GM
OEX
42.
75
45.
74
44.
25
43.
34
0.
08
0.
51
0.
15
(0.7
8)
EX
67.
67
68.
67
67.
89
67.
87
0.
96
4.
60
0.
44
(0.0
04)
MO
KEX
41.
58
45.
54
44.
25
42.
30
0.
11
0.
60
0.
14
(0.8
7)
Fig. 4: Fitted cdf, pdf, and pp plots for the second
data set
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6 Conclusion
A new two-parameter model named the TCP-LBEX
model was proposed. Some of the statistical
properties of the TCP-LBEX model were
investigated. The maximum likelihood estimator of
the TCP-LBEX model was derived. Monto Carlo
simulations are used to assess the behavior of
parameters. Using two real data sets, the proposed
model achieved better goodness of fit than some of
the other competitive models. The limitation of our
work is that we only used the complete samples and
maximum likelihood method to estimate the
parameters of the suggested model. For future
directions, the other authors can estimate the
parameters of the suggested model using different
methods of estimation and utilizing different
censored schemes.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Najwan Alsadat has written, reviewed, and actively
participated in all the publication stages of this
manuscript.
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Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
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