# A New Lifetime Distribution and Its Power Transformation

Authors: Ammar M. Sarhan; Lotfi Tadj; David C. Hamilton
Journal: Journal of Probability and Statistics
 (2014)

---

## Abstract

New one-parameter and two-parameter distributions are introduced in this paper. The failure rate of the one-parameter distribution is unimodal (upside-down bathtub), while the failure rate of the two-parameter distribution can be decreasing, increasing, unimodal, increasing-decreasing-increasing, or decreasing-increasing-decreasing, depending on the values of its two parameters. The two-parameter distribution is derived from the one-parameter distribution by using a power transformation. We discuss some properties of these two distributions, such as the behavior of the failure rate function, the probability density function, the moments, skewness, and kurtosis, and limiting distributions of order statistics. Maximum likelihood estimation for the two-parameter model using complete samples is investigated. Different algorithms for generating random samples from the two new models are given. Applications to real data are discussed and compared with the fit attained by some one- and two-parameter distributions. Finally, a simulation study is carried out to investigate the mean square error of the maximum likelihood estimators, the coverage probability, and the width of the confidence intervals of the unknown parameters.


---

## Body

## 1. Introduction

Lindley [1] proposed a one-parameter distribution, now known as the Lindley distribution, with the following probability density function (pdf):
(1)
f
(
t
)
=
β
2
β
+
1
(
1
+
t
)
e
-
β
t
,
t
>
0
,
β
>
0
.
The failure rate function of the Lindley distribution is always increasing. The properties of the Lindley distribution are studied in detail by Ghitany et al. [2]. There are situations in which the Lindley distribution may not be suitable from a theoretical or applied point of view, Ghitany et al. [3]. For this reason, Ghitany et al. [3] used a power transformation, X
=
T
1
/
α, to introduce the power Lindley distribution which is a more flexible distribution. The pdf of PL(
α
,
β
) is
(2)
f
(
x
)
=
α
β
2
β
+
1
(
1
+
x
α
)
e
-
β
x
α
,
x
>
0
,
α
,
β
>
0
.
Ghitany et al. [3] showed that the hazard function of PL(
α
,
β
) can be increasing, decreasing, and decreasing-increasing-decreasing depending on the values of the parameters. They also discussed some of the statistical properties of the distribution and used the maximum likelihood method to estimate its two unknown parameters and applied it to a real data set. In spite of the flexibility of the PL(
α
,
β
) to fit some real data sets, it fails to fit some other data sets.The main aim of this paper is to introduce two new distributions. The first is a one-parameter distribution which is similar to the Lindley distribution and the second is the power transformation of the one-parameter distribution. We refer to these two distributions asN
(
β
)and TN(
α
,
β
)respectively. The hazard function of N
(
β
)is only unimodal, while the hazard function of TN(
α
,
β
)can be decreasing, increasing, unimodal, decreasing-increasing-decreasing, or increasing-decreasing-increasing depending on the values of its two parameters. The variety of shapes of the hazard function of the TN(
α
,
β
)enables it to be a good model to fit different data sets.The rest of the paper is organized as follows. Section2 introduces the new one-parameter distribution and some of its characteristics are discussed in Section 3. Section 4 presents the transformation of the new distribution, TN(
α
,
β
). Different characteristics of TN(
α
,
β
), such as the hazard function, quantiles, random sample generation, moments, and order statistics distributions, are discussed in Section 5. Section 6 discusses the maximum likelihood estimate of the two parameters of TN(
α
,
β
). Applications of the two models are presented in Section 6. Monte Carlo Simulation study is carried out in Section 7 to examine the accuracy of the maximum likelihood estimators of the TN(
α
,
β
)parameters as well as the coverage probability and average width of the confidence intervals for the parameters. Finally, Section 8 concludes this paper.


## 2. The New Distribution

Consider the random variableTwhose pdf is given by
(3)
f
(
t
)
=
β
1
+
β
[
β
+
(
1
+
2
β
t
)
e
-
β
t
]
e
-
β
t
,
t
≥
0
,
β
>
0
.
The survival function (sf) ofTis given by
(4)
S
(
t
)
=
1
1
+
β
[
β
+
(
1
+
β
t
)
e
-
β
t
]
e
-
β
t
,
t
≥
0
,
β
>
0
,
while its hazard rate function is given by
(5)
h
(
t
)
=
β
[
β
+
(
1
+
2
β
x
)
e
-
β
t
]
β
+
(
1
+
β
t
)
e
-
β
t
,
t
≥
0
,
β
>
0
.
For simplicity, from now on, we refer to this distribution as N
(
β
).Interpretation. There are two different interpretations ofN
(
β
)as follows.(1)
The pdff
(
t
)is a mixture density of two mixture components. One follows Exp(
β
) and the other is the lifetime of a two independent component series system with Exp
(
β
) and G
(
2
,
β
) and mixture weights a
1
=
β
/
(
1
+
β
)anda
2
=
1
/
(
1
+
β
), respectively. This means that N
(
β
) can be expressed in terms ofExp
(
β
)andG
(
2
,
β
)asN
(
β
)
=
a
1
Exp
(
β
)
+
a
2
min
⁡
{
Exp
(
β
)
,
G
(
2
,
β
)
}.
(2)
The random variableTcan be described as a mixture of three components: Exp
(
β
), Exp
(
2
β
), and a  G
(
2,2
β
) with mixture weights b
1
=
β
/
(
1
+
β
)
,
b
2
=
1
/
2
(
1
+
β
)andb
3
=
1
/
2
(
1
+
β
), respectively. This means thatN
(
β
)can be expressed in terms ofExp
(
β
),Exp
(
2
β
)andG
(
2,2
β
)asN
(
β
)
=
b
1
Exp
(
β
)
+
b
2
Exp
(
2
β
)
+
b
3
G
(
2,2
β
). Some characteristics ofN
(
β
)are derived in the next section.


## 3. Characteristics ofN
(
β
)

In this section, algorithms are described to obtain quantiles ofN
(
β
)and to generate samples fromN
(
β
). Also, the moment generating function and the moments of this distribution are derived.


### 3.1. Quantiles

The100
qth quantile, q
∈
(
0,1
), can be derived as follows.(1)
Letq
∈
(
0,1
);
(2)
Solve the following equation numerically inu
∈
(
0,1
)
:
(6)
[
β
+
(
1
-
ln
⁡
u
)
u
]
u
-
(
1
+
β
)
(
1
-
q
)
=
0
.
(3)
The100
qth quantile is t
q
=
-
ln
⁡
u
/
β.


### 3.2. Random Sample Generation

We provide below three equivalent algorithms to generate a random variate fromN
(
β
).Algorithm 1.
(1) Generateu
~
U
(
0,1
).
(2) Solve numerically the following equation inv
∈
(
0,1
):[
β
+
(
1
-
ln
⁡
v
)
v
]
v
-
(
1
+
β
)
u
=
0
.
(3) SetT
=
-
ln
⁡
v
/
β.Algorithm 2.
(1) Generate I from the set{
1,2
}such thatP
(
I
=
j
)
=
a
j,  j
=
1,2. (a)
IfI
=
1, sety
1
=
-
ln
⁡
u
/
β, whereu
~
U
(
0,1
).
(b)
IfI
=
2, sety
2
=
min
⁡
{
-
ln
⁡
u
1
/
β
,
-
ln
⁡
u
2
u
3
/
β
}, whereu
1
,
u
2
,
u
3
~
U
(
0,1
).
(2) SetT
=
y
I.Algorithm 3.
(1) Generate I from the set{
1,2
,
3
}such thatP
(
I
=
j
)
=
b
j,  j
=
1,2
,
3.(a)
IfI
=
1, sety
1
=
-
ln
⁡
u
/
β, whereu
~
U
(
0,1
).
(b)
IfI
=
2, sety
2
=
-
ln
⁡
u
/
2
β, whereu
~
U
(
0,1
).
(c)
IfI
=
3, sety
3
=
-
ln
⁡
u
1
u
2
/
2
β, whereu
1
,
u
2
~
U
(
0,1
).
(2) SetT
=
y
I.


### 3.3. The Moments and the Moment Generating Function

The moment generating function (mgf) ofN
(
β
)may be written as
(7)
M
(
t
)
=
β
1
+
β
[
β
β
-
t
+
4
β
-
t
(
2
β
-
t
)
2
]
,
t
<
β
.
Differentiating the above expression k times with respect to t and setting t to zero, we get kth moments, μ
k, as
(8)
μ
k
=
E
(
T
k
)
=
β
1
+
β
[
1
+
β
2
β
(
1
+
k
+
1
2
β
)
1
2
k
]
k
!
β
k
,
k
≥
1
.Based on the first four ordinary moments, the measures of skewness(
sk
) and kurtosis (
κ
) of N
(
β
) can be obtained using
(9)
sk
=
μ
3
-
3
μ
1
μ
2
+
2
μ
1
3
[
μ
2
-
μ
1
2
]
3
/
2
,
(10)
κ
=
μ
4
-
4
μ
1
μ
3
+
6
μ
1
2
μ
2
-
3
μ
1
4
[
μ
2
-
μ
1
2
]
2
.
Plots of the skewness and kurtosis of the distribution as a function of β are plotted in Figure 1. From the plots, sk andκare unimodal functions of β. The skewness is always positive and the kurtosis is larger than 3; therefore, N
(
β
) is positively skewed and leptokurtic.Figure 1
Skewness and kurtosis ofN
(
β
) as functionsβ.


## 3.1. Quantiles

The100
qth quantile, q
∈
(
0,1
), can be derived as follows.(1)
Letq
∈
(
0,1
);
(2)
Solve the following equation numerically inu
∈
(
0,1
)
:
(6)
[
β
+
(
1
-
ln
⁡
u
)
u
]
u
-
(
1
+
β
)
(
1
-
q
)
=
0
.
(3)
The100
qth quantile is t
q
=
-
ln
⁡
u
/
β.


## 3.2. Random Sample Generation

We provide below three equivalent algorithms to generate a random variate fromN
(
β
).Algorithm 1.
(1) Generateu
~
U
(
0,1
).
(2) Solve numerically the following equation inv
∈
(
0,1
):[
β
+
(
1
-
ln
⁡
v
)
v
]
v
-
(
1
+
β
)
u
=
0
.
(3) SetT
=
-
ln
⁡
v
/
β.Algorithm 2.
(1) Generate I from the set{
1,2
}such thatP
(
I
=
j
)
=
a
j,  j
=
1,2. (a)
IfI
=
1, sety
1
=
-
ln
⁡
u
/
β, whereu
~
U
(
0,1
).
(b)
IfI
=
2, sety
2
=
min
⁡
{
-
ln
⁡
u
1
/
β
,
-
ln
⁡
u
2
u
3
/
β
}, whereu
1
,
u
2
,
u
3
~
U
(
0,1
).
(2) SetT
=
y
I.Algorithm 3.
(1) Generate I from the set{
1,2
,
3
}such thatP
(
I
=
j
)
=
b
j,  j
=
1,2
,
3.(a)
IfI
=
1, sety
1
=
-
ln
⁡
u
/
β, whereu
~
U
(
0,1
).
(b)
IfI
=
2, sety
2
=
-
ln
⁡
u
/
2
β, whereu
~
U
(
0,1
).
(c)
IfI
=
3, sety
3
=
-
ln
⁡
u
1
u
2
/
2
β, whereu
1
,
u
2
~
U
(
0,1
).
(2) SetT
=
y
I.


## 3.3. The Moments and the Moment Generating Function

The moment generating function (mgf) ofN
(
β
)may be written as
(7)
M
(
t
)
=
β
1
+
β
[
β
β
-
t
+
4
β
-
t
(
2
β
-
t
)
2
]
,
t
<
β
.
Differentiating the above expression k times with respect to t and setting t to zero, we get kth moments, μ
k, as
(8)
μ
k
=
E
(
T
k
)
=
β
1
+
β
[
1
+
β
2
β
(
1
+
k
+
1
2
β
)
1
2
k
]
k
!
β
k
,
k
≥
1
.Based on the first four ordinary moments, the measures of skewness(
sk
) and kurtosis (
κ
) of N
(
β
) can be obtained using
(9)
sk
=
μ
3
-
3
μ
1
μ
2
+
2
μ
1
3
[
μ
2
-
μ
1
2
]
3
/
2
,
(10)
κ
=
μ
4
-
4
μ
1
μ
3
+
6
μ
1
2
μ
2
-
3
μ
1
4
[
μ
2
-
μ
1
2
]
2
.
Plots of the skewness and kurtosis of the distribution as a function of β are plotted in Figure 1. From the plots, sk andκare unimodal functions of β. The skewness is always positive and the kurtosis is larger than 3; therefore, N
(
β
) is positively skewed and leptokurtic.Figure 1
Skewness and kurtosis ofN
(
β
) as functionsβ.


## 4. Power Transformation of the New Distribution

To get a more flexible distribution, we consider an extension of the new distributionN
(
β
) with the pdf (3) by using the power transformation X
=
T
1
/
α, α
>
0. The pdf of X is given by
(11)
f
(
x
)
=
α
β
x
α
-
1
1
+
β
[
β
+
(
1
+
2
β
x
α
)
e
-
β
x
α
]
e
-
β
x
α
,
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
x
≥
0
,
α
,
β
>
0
.
The density of X is plotted in Figure 2 for three choices of α when β
=
1.0, which shows that the density is symmetric when α
=
3.535, left skewed when α
<
3.535, and right skewed when α
>
3.535. This implies that the power parameter α characterizes the shape of the density function. More investigation of the density will be discussed, in the next section, based on the skewness and kurtosis measures. From now on, we will use TN(
α
,
β
) to refer to the power transformation of the new distribution N
(
β
).Figure 2
The TN(
α
,
β
) density function for some values of α when β
=
1.Interpretation. There are two different interpretations of Y as follows.(1)
The pdff
(
x
) is a mixture density of two mixture components. One component follows W
(
α
,
β
) and the other is the lifetime of a two independent component series system with W
(
α
,
2
β
) and PG(
α
,
2
,
β
) and mixture weights a
1
=
β
/
(
1
+
β
) and a
2
=
1
/
(
1
+
β
), respectively.
(2)
The random variableX can be described as a mixture of three components: W
(
α
,
β
), W
(
α
,
2
β
), and a PG(
α
,
2,2
β
) with mixture weights b
1
=
β
/
(
1
+
β
)
,
b
2
=
1
/
2
(
1
+
β
), and b
3
=
1
/
2
(
1
+
β
), respectively.Straightforward calculations yield the the survival function of TN(
α
,
β
) as
(12)
S
(
x
)
=
1
1
+
β
[
β
+
(
1
+
β
x
α
)
e
-
β
x
α
]
e
-
β
x
α
,
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
x
≥
0
,
α
,
β
>
0
.
We derive some characteristics of TN(
α
,
β
) in the next section.


## 5. Characteristics ofT
N
(
α
,
β
)

### 5.1. The Hazard Function

The hazard rate function of TN(
α
,
β
) is
(13)
h
(
x
)
=
α
β
x
α
-
1
[
β
+
(
1
+
2
β
x
α
)
e
-
β
x
α
]
β
+
(
1
+
β
x
α
)
e
-
β
x
α
,
x
≥
0
,
α
,
β
>
0
.
For α
=
1, the hazard function is unimodal. Its limiting values at zero and infinity are β, and it reaches a maximum value of
(14)
(
1
+
2
W
-
1
(
e
-
1
/
β
)
)
β
1
+
W
-
1
(
e
-
1
/
β
)
,
at
(15)
1
β
(
1
+
W
-
1
(
e
-
1
β
)
)
,
where W
-
1
(
·
) denotes the Lambert W function which is the inverse of the function x
e
x.Forα
≠
1, the shape of the hazard function is difficult to ascertain analytically. The shape was determined numerically by examining the derivative of the hazard out to the 99.99th percentile of the distribution, and the results are shown in Figure 3. For α
<
1, the hazard is decreasing except for a small region with α close to 1 and β
<
0.5 where the hazard is initially decreasing, then increasing, and finally decreasing (DID). For α
>
1, the hazard is strictly increasing for large β (β
>
2.6 in the figure). For smaller β with α close to 1, the hazard can be unimodal (for very small β) or initially increasing, then decreasing, and finally increasing (IDI) (for slightly larger β). Figure 4 shows the hazard for five choices of α and β which demonstrate the five possible shapes.Figure 3
Shapes of the hazard function of TN(
α
,
β
).The hazard function of TN(
α
,
β
) for some parameter values.
(a)
α
=
0.5,
β
=
0.1
(b)
α
=
0.9,
β
=
0.1
(c)
α
=
1,
β
=
0.1
(d)
α
=
1.2,
β
=
0.5
(e)
α
=
2.5,
β
=
0.1


### 5.2. Quantiles and Random Sample Generations

The 100qth quantile of TN(
α
,
β
), x
q, can be derived from that of N
(
β
), t
q, as follows:
(16)
x
q
=
t
q
1
/
α
.
Figure 5 depicts the three quartiles Q
1, Q
2, and Q
3, which can be obtained from the qth quantile by setting q
=
0.25,0.50, and 0.75, respectively. From Figure 5, the Interquartile range (IQR = Q
3
-
Q
1) decreases dramatically when α increases.Figure 5
The quartiles of TN(
α
,
β
) as functions of α, when β
=
1.The following algorithm generates a random variate from TN(
α
,
β
).Algorithm 4.
(1) GenerateT from N
(
β
), using one of the Algorithms 1–3;
(2) SetX
=
T
1
/
α.


### 5.3. The Moments and Shape Measures

LetX follow TN(
α
,
β
). After some algebra, the kth ordinary moment of X is derived as
(17)
μ
k
=
E
(
X
k
)
=
b
1
Γ
(
k
/
α
+
1
)
β
k
/
α
+
b
1
Γ
(
k
/
α
+
1
)
(
2
β
)
k
/
α
+
b
3
Γ
(
k
/
α
+
2
)
(
2
β
)
k
/
α
=
k
Γ
(
k
/
α
)
[
2
α
(
1
+
2
k
/
α
β
)
+
k
]
2
α
2
(
1
+
β
)
(
2
β
)
k
/
α
.
Therefore, the mean and variance of X are
(18)
μ
=
Γ
(
1
/
α
)
[
2
α
(
1
+
2
1
/
α
β
)
+
1
]
2
α
2
(
1
+
β
)
(
2
β
)
1
/
α
,
σ
2
=
(
8
α
2
(
1
+
β
)
Γ
(
2
α
)
[
α
(
1
+
2
2
/
α
β
)
+
1
]
-
Γ
2
(
1
α
)
f
s
d
f
×
[
2
α
(
1
+
2
1
/
α
β
)
+
1
]
2
(
2
α
)
)
×
(
4
α
4
(
1
+
β
)
2
(
2
β
)
2
/
α
)
-
1
.
Figure 6 depicts the mean and variance of TN(
α
,
β
) as functions of α when β
=
1 which shows that the mean decreases dramatically in α and takes its minimum of 0.8298 at α
=
1.61 then it increases steadily to take its maximum of 0.9687, while the variance is decreasing.Figure 6
The mean and variance of TN(
α
,
β
) as functions of α, when β
=
1.Based on the first four ordinary moments, the measures of skewness(
sk
) and kurtosis (
κ
) of TN(
α
,
β
) can be obtained by substituting (17) into (9) and (10), respectively. Plots of the skewness and kurtosis of TN(
α
,
β
) distribution as functions of α, when β
=
1.0, are given in Figure 7. From these plots, (
1
) the skewness is positive when α
<
3.535 and negative when α
>
3.535 and the kurtosis is (i) equal to 3 when either α
=
2.4977 or α
=
4.9151 which means that the distribution is mesokurtic; (ii) greater than 3 when either α
<
2.4977 or α
>
4.9151 which means that the distribution is leptokurtic; (iii) smaller than 3 when 2.4977
<
α
<
4.9151 which means that the distribution is platykurtic. This analysis shows how the power parameter α improves N
(
β
), because the power transformation model can be used for data with a wide variety of distributional shapes.Figure 7
The skewness and kurtosis of TN(
α
,
β
) as functions of α, when β
=
1.


### 5.4. Order Statistics

Considern independent and identical components whose lifetimes, say X
1
,
X
2
,
…
,
X
n, follow TN(
α
,
β
). The following theorem gives the limiting distributions of the lifetime of the series system X
1
:
n
=
min
⁡
{
X
1
,
…
,
X
n
} and of the parallel system X
n
:
n
=
max
⁡
{
X
1
,
…
,
X
n
} consisting of these n components.Theorem 5.
The limiting distributions ofX
1
:
n and X
n
:
n are
(19)
lim
⁡
n
→
∞
P
{
X
1
:
n
-
a
n
b
n
≤
x
}
=
1
-
e
-
x
α
,
x
>
0
,
(20)
lim
⁡
n
→
∞
P
{
X
n
:
n
-
c
n
d
n
≤
x
}
=
exp
⁡
{
-
exp
⁡
(
-
x
)
}
,
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
x
∈
(
-
∞
,
∞
)
,
where a
n
=
0
,
b
n
=
F
-
1
(
1
/
n
)
,
c
n
=
F
-
1
(
1
-
1
/
n
)
,
d
n
=
1
/
n
f
(
c
n
) and F
(
x
)
=
1
-
S
(
x
) is the cdf of X.Proof.
Using L’Hospital rule,(21)
lim
⁡
t
→
0
+
F
(
t
x
)
F
(
t
)
=
lim
⁡
t
→
0
+
x
f
(
t
x
)
f
(
t
)
=
lim
⁡
t
→
0
+
x
α
[
β
+
(
1
+
2
β
(
t
x
)
α
)
e
-
β
(
t
x
)
α
]
e
-
β
(
t
x
)
α
[
β
+
(
1
+
2
β
t
α
)
e
-
β
t
α
]
e
-
β
t
α
=
x
α
.
Therefore, (19) follows by Theorem 8.3
.
6(ii) of Arnold et al. [4].
For the power transformation, we have(22)
lim
⁡
x
→
∞
d
d
x
1
h
(
x
)
=
lim
⁡
x
→
∞
-
(
α
-
1
)
(
β
+
(
1
+
β
x
α
)
e
-
β
x
α
)
α
β
x
α
(
β
+
(
1
+
2
β
x
α
)
e
-
β
x
α
)
+
lim
⁡
x
→
∞
β
x
α
1
+
2
β
x
α
+
β
e
β
x
α
+
lim
⁡
x
→
∞
β
(
2
β
x
α
-
1
)
+
(
β
x
α
(
1
+
2
β
x
α
)
-
1
)
e
-
β
x
α
[
β
+
(
1
+
2
β
x
α
)
e
-
β
x
α
]
2
e
β
x
α
=
0
.
Therefore, (20) follows by Theorem 8.3
.
3 of Arnold et al. [4].The following Theorem gives the limiting distribution of theith order statistic of the n lifetimes X
1
,
X
2
,
…
,
X
n.Theorem 6.
The limiting distributions ofX
i
:
n, 1
≤
i
≤
n are
(23)
lim
⁡
n
→
∞
⁡
P
{
X
i
:
n
-
a
n
b
n
≤
x
}
=
1
-
∑
j
=
0
i
-
1
x
j
α
j
!
e
-
x
α
,
x
>
0
,
where a
n
=
0
,
b
n
=
F
-
1
(
1
/
n
).Proof.
It follows from Theorem5 and (
8.4
.
2
) of Arnold et al. [4].Theorem5 means that (
X
1
:
n
-
a
n
)
/
b
n and (
X
n
:
n
-
c
n
)
/
d
n follow asymptotically W
(
1
,
α
) and Ext(
0,1
), respectively, while Theorem 6 means that (
X
i
:
n
-
a
n
)
/
b
n follows asymptotically PG(
α
,
i
,
1
).


## 5.1. The Hazard Function

The hazard rate function of TN(
α
,
β
) is
(13)
h
(
x
)
=
α
β
x
α
-
1
[
β
+
(
1
+
2
β
x
α
)
e
-
β
x
α
]
β
+
(
1
+
β
x
α
)
e
-
β
x
α
,
x
≥
0
,
α
,
β
>
0
.
For α
=
1, the hazard function is unimodal. Its limiting values at zero and infinity are β, and it reaches a maximum value of
(14)
(
1
+
2
W
-
1
(
e
-
1
/
β
)
)
β
1
+
W
-
1
(
e
-
1
/
β
)
,
at
(15)
1
β
(
1
+
W
-
1
(
e
-
1
β
)
)
,
where W
-
1
(
·
) denotes the Lambert W function which is the inverse of the function x
e
x.Forα
≠
1, the shape of the hazard function is difficult to ascertain analytically. The shape was determined numerically by examining the derivative of the hazard out to the 99.99th percentile of the distribution, and the results are shown in Figure 3. For α
<
1, the hazard is decreasing except for a small region with α close to 1 and β
<
0.5 where the hazard is initially decreasing, then increasing, and finally decreasing (DID). For α
>
1, the hazard is strictly increasing for large β (β
>
2.6 in the figure). For smaller β with α close to 1, the hazard can be unimodal (for very small β) or initially increasing, then decreasing, and finally increasing (IDI) (for slightly larger β). Figure 4 shows the hazard for five choices of α and β which demonstrate the five possible shapes.Figure 3
Shapes of the hazard function of TN(
α
,
β
).The hazard function of TN(
α
,
β
) for some parameter values.
(a)
α
=
0.5,
β
=
0.1
(b)
α
=
0.9,
β
=
0.1
(c)
α
=
1,
β
=
0.1
(d)
α
=
1.2,
β
=
0.5
(e)
α
=
2.5,
β
=
0.1


## 5.2. Quantiles and Random Sample Generations

The 100qth quantile of TN(
α
,
β
), x
q, can be derived from that of N
(
β
), t
q, as follows:
(16)
x
q
=
t
q
1
/
α
.
Figure 5 depicts the three quartiles Q
1, Q
2, and Q
3, which can be obtained from the qth quantile by setting q
=
0.25,0.50, and 0.75, respectively. From Figure 5, the Interquartile range (IQR = Q
3
-
Q
1) decreases dramatically when α increases.Figure 5
The quartiles of TN(
α
,
β
) as functions of α, when β
=
1.The following algorithm generates a random variate from TN(
α
,
β
).Algorithm 4.
(1) GenerateT from N
(
β
), using one of the Algorithms 1–3;
(2) SetX
=
T
1
/
α.


## 5.3. The Moments and Shape Measures

LetX follow TN(
α
,
β
). After some algebra, the kth ordinary moment of X is derived as
(17)
μ
k
=
E
(
X
k
)
=
b
1
Γ
(
k
/
α
+
1
)
β
k
/
α
+
b
1
Γ
(
k
/
α
+
1
)
(
2
β
)
k
/
α
+
b
3
Γ
(
k
/
α
+
2
)
(
2
β
)
k
/
α
=
k
Γ
(
k
/
α
)
[
2
α
(
1
+
2
k
/
α
β
)
+
k
]
2
α
2
(
1
+
β
)
(
2
β
)
k
/
α
.
Therefore, the mean and variance of X are
(18)
μ
=
Γ
(
1
/
α
)
[
2
α
(
1
+
2
1
/
α
β
)
+
1
]
2
α
2
(
1
+
β
)
(
2
β
)
1
/
α
,
σ
2
=
(
8
α
2
(
1
+
β
)
Γ
(
2
α
)
[
α
(
1
+
2
2
/
α
β
)
+
1
]
-
Γ
2
(
1
α
)
f
s
d
f
×
[
2
α
(
1
+
2
1
/
α
β
)
+
1
]
2
(
2
α
)
)
×
(
4
α
4
(
1
+
β
)
2
(
2
β
)
2
/
α
)
-
1
.
Figure 6 depicts the mean and variance of TN(
α
,
β
) as functions of α when β
=
1 which shows that the mean decreases dramatically in α and takes its minimum of 0.8298 at α
=
1.61 then it increases steadily to take its maximum of 0.9687, while the variance is decreasing.Figure 6
The mean and variance of TN(
α
,
β
) as functions of α, when β
=
1.Based on the first four ordinary moments, the measures of skewness(
sk
) and kurtosis (
κ
) of TN(
α
,
β
) can be obtained by substituting (17) into (9) and (10), respectively. Plots of the skewness and kurtosis of TN(
α
,
β
) distribution as functions of α, when β
=
1.0, are given in Figure 7. From these plots, (
1
) the skewness is positive when α
<
3.535 and negative when α
>
3.535 and the kurtosis is (i) equal to 3 when either α
=
2.4977 or α
=
4.9151 which means that the distribution is mesokurtic; (ii) greater than 3 when either α
<
2.4977 or α
>
4.9151 which means that the distribution is leptokurtic; (iii) smaller than 3 when 2.4977
<
α
<
4.9151 which means that the distribution is platykurtic. This analysis shows how the power parameter α improves N
(
β
), because the power transformation model can be used for data with a wide variety of distributional shapes.Figure 7
The skewness and kurtosis of TN(
α
,
β
) as functions of α, when β
=
1.


## 5.4. Order Statistics

Considern independent and identical components whose lifetimes, say X
1
,
X
2
,
…
,
X
n, follow TN(
α
,
β
). The following theorem gives the limiting distributions of the lifetime of the series system X
1
:
n
=
min
⁡
{
X
1
,
…
,
X
n
} and of the parallel system X
n
:
n
=
max
⁡
{
X
1
,
…
,
X
n
} consisting of these n components.Theorem 5.
The limiting distributions ofX
1
:
n and X
n
:
n are
(19)
lim
⁡
n
→
∞
P
{
X
1
:
n
-
a
n
b
n
≤
x
}
=
1
-
e
-
x
α
,
x
>
0
,
(20)
lim
⁡
n
→
∞
P
{
X
n
:
n
-
c
n
d
n
≤
x
}
=
exp
⁡
{
-
exp
⁡
(
-
x
)
}
,
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
x
∈
(
-
∞
,
∞
)
,
where a
n
=
0
,
b
n
=
F
-
1
(
1
/
n
)
,
c
n
=
F
-
1
(
1
-
1
/
n
)
,
d
n
=
1
/
n
f
(
c
n
) and F
(
x
)
=
1
-
S
(
x
) is the cdf of X.Proof.
Using L’Hospital rule,(21)
lim
⁡
t
→
0
+
F
(
t
x
)
F
(
t
)
=
lim
⁡
t
→
0
+
x
f
(
t
x
)
f
(
t
)
=
lim
⁡
t
→
0
+
x
α
[
β
+
(
1
+
2
β
(
t
x
)
α
)
e
-
β
(
t
x
)
α
]
e
-
β
(
t
x
)
α
[
β
+
(
1
+
2
β
t
α
)
e
-
β
t
α
]
e
-
β
t
α
=
x
α
.
Therefore, (19) follows by Theorem 8.3
.
6(ii) of Arnold et al. [4].
For the power transformation, we have(22)
lim
⁡
x
→
∞
d
d
x
1
h
(
x
)
=
lim
⁡
x
→
∞
-
(
α
-
1
)
(
β
+
(
1
+
β
x
α
)
e
-
β
x
α
)
α
β
x
α
(
β
+
(
1
+
2
β
x
α
)
e
-
β
x
α
)
+
lim
⁡
x
→
∞
β
x
α
1
+
2
β
x
α
+
β
e
β
x
α
+
lim
⁡
x
→
∞
β
(
2
β
x
α
-
1
)
+
(
β
x
α
(
1
+
2
β
x
α
)
-
1
)
e
-
β
x
α
[
β
+
(
1
+
2
β
x
α
)
e
-
β
x
α
]
2
e
β
x
α
=
0
.
Therefore, (20) follows by Theorem 8.3
.
3 of Arnold et al. [4].The following Theorem gives the limiting distribution of theith order statistic of the n lifetimes X
1
,
X
2
,
…
,
X
n.Theorem 6.
The limiting distributions ofX
i
:
n, 1
≤
i
≤
n are
(23)
lim
⁡
n
→
∞
⁡
P
{
X
i
:
n
-
a
n
b
n
≤
x
}
=
1
-
∑
j
=
0
i
-
1
x
j
α
j
!
e
-
x
α
,
x
>
0
,
where a
n
=
0
,
b
n
=
F
-
1
(
1
/
n
).Proof.
It follows from Theorem5 and (
8.4
.
2
) of Arnold et al. [4].Theorem5 means that (
X
1
:
n
-
a
n
)
/
b
n and (
X
n
:
n
-
c
n
)
/
d
n follow asymptotically W
(
1
,
α
) and Ext(
0,1
), respectively, while Theorem 6 means that (
X
i
:
n
-
a
n
)
/
b
n follows asymptotically PG(
α
,
i
,
1
).


## 6. Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is one of the most common methods for estimating the parameters of a statistical model. Assume thatnindependent and identical items, whose lifetimes follow TN(
α
,
β
), are put on a life test simultaneously. Letx
1
,
x
2
,
…
,
x
nbe the failure times of the items and letx
=
(
x
1
,
…
,
x
n
). The likelihood function for (
α
,
β
)is
(24)
L
(
α
,
β
;
x
)
=
α
n
β
n
(
1
+
β
)
n
(
∏
i
=
1
n
‍
x
i
)
α
-
1
e
-
β
∑
i
=
1
n
x
i
α
×
∏
i
=
1
n
[
β
+
(
1
+
2
β
x
i
α
)
e
-
β
x
i
α
]
.
The log-likelihood function is
(25)
L
(
α
,
β
;
x
)
=
n
ln
⁡
α
+
n
ln
⁡
β
-
n
ln
⁡
(
1
+
β
)
+
(
α
-
1
)
∑
i
=
1
n
‍
ln
⁡
x
i
-
β
∑
i
=
1
n
‍
x
i
α
+
∑
i
=
1
n
‍
ln
⁡
A
i
(
α
,
β
)
,
where
(26)
A
i
(
α
,
β
)
=
β
+
(
1
+
2
β
x
i
α
)
e
-
β
x
i
α
,
i
=
1
,
…
,
n
.
The first partial derivatives of  L, with respect toαandβ, are
(27)
L
α
=
n
α
+
∑
i
=
1
n
‍
ln
⁡
x
i
-
β
∑
i
=
1
n
‍
x
i
α
ln
⁡
x
i
+
∑
i
=
1
n
‍
A
i
,
α
(
α
,
β
)
A
i
(
α
,
β
)
,
L
β
=
n
β
(
1
+
β
)
-
∑
i
=
1
n
‍
x
i
α
+
∑
i
=
1
n
‍
A
i
,
β
(
α
,
β
)
A
i
(
α
,
β
)
,
where
(28)
A
i
,
α
(
α
,
β
)
=
∂
A
i
(
α
,
β
)
∂
α
=
β
x
i
α
(
1
-
2
β
x
i
α
)
ln
⁡
x
i
e
-
β
x
i
α
,
A
i
,
β
(
α
,
β
)
=
∂
A
i
(
α
,
β
)
∂
β
=
1
+
(
1
-
2
β
x
i
α
)
x
i
α
e
-
β
x
i
α
.
The second partial derivatives of L are
(29)
L
α
,
α
=
-
n
α
2
-
β
∑
i
=
1
n
‍
x
i
α
(
ln
⁡
x
i
)
2
+
∑
i
=
1
n
‍
A
i
(
α
,
β
)
A
i
,
α
2
(
α
,
β
)
-
[
A
i
,
α
(
α
,
β
)
]
2
[
A
i
(
α
,
β
)
]
2
,
L
α
,
β
=
-
∑
i
=
1
n
‍
x
i
α
ln
⁡
x
i
+
∑
i
=
1
n
‍
A
i
(
α
,
β
)
A
i
,
α
β
(
α
,
β
)
-
A
i
,
α
(
α
,
β
)
A
i
,
β
(
α
,
β
)
[
A
i
(
α
,
β
)
]
2
,
L
β
,
β
=
-
n
β
2
+
n
(
1
+
β
)
2
+
∑
i
=
1
n
‍
A
i
(
α
,
β
)
A
i
,
β
2
(
α
,
β
)
-
[
A
i
,
β
(
α
,
β
)
]
2
[
A
i
(
α
,
β
)
]
2
,
where
(30)
A
i
,
α
2
(
α
,
β
)
=
β
x
i
α
(
ln
⁡
x
i
)
2
e
-
β
x
i
α
(
1
-
5
β
x
i
α
+
2
β
2
x
i
2
α
)
,
A
i
,
α
β
(
α
,
β
)
=
x
i
α
(
ln
⁡
x
i
)
e
-
β
x
i
α
(
1
-
5
β
x
i
α
+
2
β
2
x
i
2
α
)
,
A
i
,
β
2
(
α
,
β
)
=
-
x
i
2
α
e
-
β
x
i
α
(
3
-
2
β
x
i
α
)
.
The information matrix is
(31)
F
(
α
,
β
)
=
-
[
L
α
,
α
L
α
,
β
L
α
,
β
L
β
,
β
]
.
The MLE ofαandβ, sayα
^andβ
^, are the solution of the system of nonlinear equations obtained by settingL
α
=
0andL
β
=
0such that the F
(
α
^
,
β
^
) is positive definite. This system has no analytic solution, so numerical methods, such as the Newton-Raphson method, Burden and Faires [5], should be used.Large-Sample Intervals. The MLE of the parametersαandβare asymptotically normally distributed with means equal to the true values ofαandβand variances given by the inverse of the information matrix. In particular,
(32)
(
α
^
β
^
)
~
N
2
(
(
α
β
)
,
F
^
-
1
)
,
whereF
^
-
1is the inverse of F
(
α
^
,
β
^
), with main diagonal elementsF
^
11andF
^
22given by
(33)
F
^
11
=
F
^
22
F
^
11
F
^
22
-
F
^
12
2
,
F
^
22
=
F
^
11
F
^
11
F
^
22
-
F
^
12
2
.
Using (32), large-sample(
1
-
ϑ
)
100
%confidence intervals forαand β are
(34)
α
^
±
Z
ϑ
/
2
F
^
11
,
β
^
±
Z
ϑ
/
2
F
^
22
,
whereZ
ϑ
/
2is the upper100
ϑ
/
2quantile of the standard normal distribution.


## 7. Applications

In this section, we analyze four data sets to illustrate the applicability of the two new distributions proposed in this paper. The first data consists of 61 observed recidivism failure times (in days) of individuals released directly from correctional institutions to parole in the District of Columbia, Columbia, USA [6]. The second data set consists of 43 active repair times (in hours) for an airborne communication transceiver [7]. The third data set consists of 57 times (in thousands of operating hours) of unscheduled maintenance actions for the number 4 diesel engine of the U.S.S. Grampus, up to 16 thousand hours of operation [8]. The forth data set consists of the tensile strength (measured in GPa) of 69 carbon fibers tested under tension at gauge lengths of 20 mm [9].We will refer to these data sets as failure times, repair times, maintenance actions, and tensile strength data, respectively. For each data set, we fit the proposed two distributions as well as Lindley and power Lindley distributions. For the sake of comparison, we apply goodness-of-fit tests to verify which distribution better fits these data sets. We consider the well-known Kolmogorov-Smirnov (K-S) statistic, the Cramér-von Mises (C-M), and Anderson-Darling (A-D) statistics [10]. Furthermore, we consider the Akaike information criterion AIC
=
-
2
L
^
+
2
q, where L
^ is the log-likelihood function at the MLE of the parameters andqis the number of model parameters. Table 1 shows the MLE of the parameters of each model, the corresponding maximum log-likelihood value, and the AIC for the four data sets. Table 2 presents the results of the goodness of fit tests for the four data sets using each model.Table 1
Parameter estimates, maximum log-likelihood, and AIC for the four data sets.
Data
Model
Parameter estimates
L
^
AIC
Failure times
L(
β
)
β
^
=
9.403
×
1
0
-
3
−386.894
775.789
PL(
α, β
)
α
^
=
0.8778, β
^
=
0.0185
−385.958
775.916
N(
β
)
β
^
=
3.61354
×
1
0
-
3
−384.872
771.743
TN(
α, β
)
α
^
=
1.2551, β
^
=
8.442
×
1
0
-
4
−382.904
769.808
Repair times
L(
β
)
β
^
=
0.449891
−104.433
210.865
PL(
α, β
)
α
^
=
0.762291, β
^
=
0.652788
−99.974
203.947
N(
β
)
β
^
=
0.226013
−100.821
203.642
TN(
α, β
)
α
^
=
0.843574, β
^
=
0.295802
−99.572
203.144
Maintenance actions
L(
β
)
β
^
=
0.217325
−170.317
342.634
PL(
α, β
)
α
^
=
1.28229, β
^
=
0.117289
−167.089
338.178
N(
β
)
β
^
=
0.0936263
−174.636
351.271
TN(
α, β
)
α
^
=
1.69539, β
^
=
0.0172624
−165.584
335.167
Tensile strength
L(
β
)
β
^
=
0.6545
−119.190
240.381
PL(
α, β
)
α
^
=
3.8678, β
^
=
0.0497
−49.060
102.119
N(
β
)
β
^
=
0.3334
−124.618
251.235
TN(
α, β
)
α
^
=
4.8853, β
^
=
0.0067
−49.976
103.952Table 2
Statistics K-S (P value), C-M, and A-D for the four data sets.
Data
Model
K-S (P value)
C-M
A-D
Failure times
L(
β
)
0.099087 (0.55362)
0.1456121
0.907295
PL(
α, β
)
0.094044 (0.61929)
0.1632937
1.013002
N(
β
)
0.103937 (0.49274)
0.1175828
0.741661
TN(
α, β
)
0.069538(0.90959)
0.0847141
0.542494
Repair times
L(
β
)
0.234632 (0.01449)
0.1810547
1.210262
PL(
α, β
)
0.120462 (0.52160)
0.1330721
0.895534
N(
β
)
0.174114 (0.13065)
0.1506954
0.993861
TN(
α, β
)
0.114174(0.58940)
0.1268951
0.840214
Maintenance actions
L(
β
)
0.122002 (0.33663)
0.187688
1.305582
PL(
α, β
)
0.119731 (0.35863)
0.178752
1.244465
N(
β
)
0.146367 (0.15728)
0.186369
1.298771
TN(
α, β
)
0.111731(0.44316)
0.160603
1.120745
Tensile strength
L(
β
)
0.386637 (1.17899
×
1
0
-
9)
0.0403019
0.3025705
PL(
α, β
)
0.044275(0.99840)
0.0181847
0.1619866
N(
β
)
0.415150 (4.57128
×
1
0
-
11)
0.0432435
0.3221342
TN(
α, β
)
0.056805 (0.96997)
0.0403920
0.3132999For every data set, we plotted(
1
) the scaled total time on test transform (TTT-transform) plot which gives qualitative information about the hazard rate shape [11]; (
2
) the hazard functions for the four fitted models; (
3
) the empirical and fitted density and distribution functions. Figures 8, 9, 10, and 11 show the four plots for the four data sets 1–4, respectively. The scaled TTT-transform plots show that the repair data set has a unimodal hazard, while the rest of data sets have increasing hazards.Figure 8
The TTT-transform, fitted hazard, pdf, and cdf of the failure times data.Figure 9
The TTT-transform, hazard, pdf, and cdf of the active repair times data.Figure 10
The TTT-transform, hazard, pdf, and cdf of the maintenance actions data.Figure 11
The TTT-transform, hazard, pdf, and cdf of the tensile strength data.The inverse of information matrix at the MLE using the four data sets are listed below.Failure times:(35)
F
^
-
1
=
[
1.717231
×
1
0
-
3
-
1.024661
×
1
0
-
5
-
1.024661
×
1
0
-
5
5.019728
×
1
0
-
7
]
.Active repair:(36)
F
^
-
1
=
[
0.00901512
-
0.00442645
-
0.00442645
0.00430030
]
.Maintenance actions:(37)
F
^
-
1
=
[
0.046064730
-
1.998013
×
1
0
-
3
-
1.998013
×
1
0
-
3
9.07815
×
1
0
-
5
]
.Tensile strength:(38)
F
^
-
1
=
[
0.521162636
-
3.846724
×
1
0
-
3
-
3.846724
×
1
0
-
3
2.892066
×
1
0
-
5
]
.For the first three data sets,TN
(
α
,
β
)model has the smallest value of the Kolmogorov-Smirnov (largest P value), the Cramér-von Mises, and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is provided by the TN model for these data sets. For the forth data set, the power Lindley model provides the best fit in the sense of having the smallest test statistics. For all data sets, TN(
α
,
β
) is a better fit than N
(
β
). For the first two data sets, N
(
β
) is a better fit than both L
(
β
)and PL(
α
,
β
)while it is the worst fit for the last two data sets. The AIC statistic is the lowest for TN(
α
,
β
)for all data sets except for tensile strength where it is slightly higher.Further, for testingN
(
β
)as a submodel of the TN(
α
,
β
), we use the likelihood ratio test statistic (LRT) to check if the fit using the TN(
α
,
β
)is statistically superior to a fit using the N
(
β
)for each data set. The LRT for testingH
0
:
α
=
1againstH
1
:
α
≠
1isΛ
=
2
(
L
^
H
1
-
L
^
H
0
), whereL
^
H
1andL
^
H
0are the maximum log-likelihood values underH
1andH
0, respectively. UnderH
0,Λ
→
d
χ
1
2. The LRT rejectsH
0ifΛ
>
χ
1
,
γ
2, whereχ
1
,
γ
2denotes the upper100
γ
%point of chi-square distribution with 1 degree of freedom. Table 3 lists the values of the LRT and the corresponding P value for the four data sets. Based on the P values, the N
(
β
)is not rejected against the TN
(
α
,
β
)to fit the repair times data set, while it is rejected, at any level of significance greater than or equal to 0.0473, to fit the other three data sets.Table 3
The LRT andP value for the four data sets.
Failure times
Repair times
Maintenance actions
Tensile strength
Λ
3.936
2.498
18.104
149.284
P value
0.0473
0.1140
2.0916
×
1
0
-
5
0


## 8. Simulation Study

We used a simulation study to investigate the performance of the accuracy of point and interval estimates of the two parameters of TN(
α
,
β
). The following steps are as follows:(1)
Specify the values of the parametersαandβ;
(2)
Specify the sample sizen;
(3)
Use Algorithm2 and the transformation to generate a random sample with sizenfrom TN(
α
,
β
).
(a)
Calculate the MLE of the two parameters and the inverse of the Fisher matrix;
(b)
Calculate the squared deviation of the MLE from the exact value of each parameter;
(c)
Calculate a 95% CI for each parameter;
(4)
Repeat steps 2-3,Ntimes;
(5)
Calculate the mean squared error (MSE), the average of the confidence interval widths, and the coverage probability for each parameter. The MSE associated with the MLE of the parameterϑ, MSE
ϑ, is
(39)
MSE
ϑ
=
1
N
∑
i
=
1
N
‍
(
ϑ
^
i
-
ϑ
)
2
,
whereϑ
^
iis the MLE ofϑusing theith sample,i
=
1,2
,
…
,
N, andϑ
=
α
,
β. Coverage probability is the proportion of theNsimulated confidence intervals which include the true parameterϑ.The simulation study is used whenN
=
10,000, the sample sizes are 25, 50, 75, and 100, and the parameter values (
α
,
β
)
=
(
0.5,1
),  (
1,1
),  (
2,1
),  (
1.3,0.5
),  (
0.5,1.2
),  (
1,1.2
),  (
2,1.2
),(
1.3,1.5
), and(
0.9,1.2
). Some of the selected values of(
α
,
β
)give decreasing, unimodal, increasing, and increasing-decreasing-increasing hazard shapes, respectively, as shown in Figure 3. Table 4 presents the MSE, coverage probability (
CP
ϑ
), and average width (AW) of 95% confidence intervals of each parameter. As it was expected, this table shows that the MSEs of the estimates decrease as the sample size increases, that the coverage probabilities are very close to the nominal level of 95%, and that the average widths decrease as the sample size increases.Table 4
MSE, coverage probability, and average width.
α
β
n
MSE
α
MSE
β
CP
α
AW
α
CP
β
AW
β
0.5
1
25
0.008889
0.0639280
0.9550
0.335546
0.9426
0.888835
50
0.003720
0.0250429
0.9553
0.229186
0.9541
0.608075
75
0.002295
0.0158072
0.9558
0.184697
0.9527
0.491875
100
0.001690
0.0120432
0.9547
0.159216
0.9528
0.425521
1.0
1.0
25
0.035209
0.0603732
0.9552
0.669487
0.9492
0.884671
50
0.015100
0.0259297
0.9530
0.458663
0.9504
0.60922
75
0.009433
0.0165418
0.9495
0.369965
0.9516
0.493743
100
0.006956
0.0121903
0.9517
0.318365
0.9498
0.425539
2.0
1.0
25
0.144086
0.0609798
0.9518
1.342590
0.9435
0.883723
50
0.061349
0.0260667
0.9472
0.916073
0.9458
0.608569
75
0.037822
0.0163597
0.9526
0.740067
0.9508
0.492714
100
0.028013
0.0119553
0.9492
0.638045
0.9510
0.424817
1.3
0.5
25
0.058995
0.0173859
0.9532
0.866467
0.9216
0.481574
50
0.024938
0.0081550
0.9535
0.594661
0.9348
0.339487
75
0.015840
0.0052273
0.9557
0.480683
0.9400
0.277333
100
0.011542
0.0037797
0.9551
0.414303
0.9451
0.239666
0.5
1.2
25
0.009273
0.0972282
0.9480
0.335704
0.9499
1.084880
50
0.003832
0.0381008
0.9485
0.229018
0.9494
0.733431
75
0.002355
0.0239632
0.9511
0.184765
0.9565
0.593873
100
0.001787
0.0173496
0.9485
0.159225
0.9531
0.510691
1.0
1.2
25
0.036640
0.0898864
0.9502
0.670473
0.9556
1.075180
50
0.015133
0.0380503
0.9525
0.457310
0.9526
0.735221
75
0.009416
0.0240831
0.9525
0.369850
0.9547
0.594754
100
0.007055
0.0174628
0.9486
0.318305
0.9523
0.510473
2.0
1.2
25
0.148128
0.0929780
0.9515
1.344300
0.9493
1.076180
50
0.061533
0.0386808
0.9522
0.916556
0.9522
0.734676
75
0.038607
0.0248240
0.9495
0.738584
0.9492
0.593036
100
0.028413
0.0172320
0.9474
0.636959
0.9541
0.509967
1.3
1.5
25
0.063330
0.1753970
0.9534
0.875085
0.9601
1.415120
50
0.025851
0.0651604
0.9519
0.596480
0.9537
0.945666
75
0.016517
0.0404539
0.9509
0.481512
0.9536
0.759421
100
0.011529
0.0292367
0.9548
0.414153
0.9552
0.652792
0.9
1.2
25
0.029932
0.0929842
0.9499
0.604276
0.9495
1.074170
50
0.012327
0.0389142
0.9501
0.411230
0.9509
0.734209
75
0.007675
0.0238699
0.9504
0.332339
0.9531
0.594194
100
0.005652
0.0174031
0.9531
0.286597
0.9505
0.510094


## 9. Conclusion

In this paper, we have proposed new one-parameter and two-parameter distributions, called theN
(
β
)and TN
(
α
,
β
), respectively. The TN
(
α
,
β
) was obtained by using a power transformation of theN
(
β
)distributed variable. The TN
(
α
,
β
) provides more flexibility than the N
(
β
) in terms of the shape of the density and hazard rate functions as well as its skewness and kurtosis. We derived the maximum likelihood estimates of the parameters and their variance-covariance matrix. We proposed different algorithms to generate samples from the two proposed distributions. Applications of the two proposed distributions to real data sets show better fits than Lindley and power Lindley distributions. Finally, we examined the accuracy of the maximum likelihood estimators of the TN
(
α
,
β
) parameters as well as the coverage probability and average width of the confidence intervals for the parameters using simulation.



---

*Source: 532024-2014-05-18.xml*