CC BY
10
MARSHALL-OLKIN EXPONENTIATED NADARAJAH HAGHIGHI DISTRIBUTION AND ITS APPLICATIONS
Nicy Sebastian1*, Jeena Joseph2, Muhsina C. S.3 and Sandra I. S.4
•
1,2,3,4 Department of Statistics, St Thomas College, Thrissur affiliated to the University of Calicut, Kerala, India- 680 001 [email protected],[email protected], [email protected] , [email protected] "Corresponding Author
Abstract
In this paper, we introduce a new generalization of exponentiated Nadarajah Haghighi distribution, namely Marshall-Olkin exponentiated Nadarajah Haghighi (MOENH) distribution and study its properties. The stress-strength parameter estimation is also taken into account. Characterizations of the new distribution are obtained. The unknown parameters of the distribution are estimated using the maximum likelihood method. It is established how important this distribution is to the research of the minification process. Simulation studies are done, and sample path properties are explored. A real data set is fitted to the new distribution to demonstrate the model's adaptability and effectiveness.
Keywords: Marshall-Olkin family, Exponentiated Nadarajah Haghighi distribution, Stress-Strength reliability, Maximum likelihood, Minification process.
1. Introduction
Marshall and Olkin introduced a new family of distributions in 1997 with one additional parameter a, known as the Marshall-Olkin Family of Distributions. The distribution function of the family is given by,
F(x)
G(x) =-.„ v .,< x < to,a > 0 (1)
w a + (1 - a)F(x)
Here, F(x) represents the distribution function of a random variable X. The Marshall-Olkin family of distributions is a class of continuous probability distributions that are used to model lifetime data. It has the advantage of being flexible and able to fit a wide range of data sets. Cordeiro and Lemonte [1] discussed some mathematical properties of the Marshall-Olkin extended Weibull distribution and estimation of the model parameters by the maximum likelihood method. Ristic and Kundu [14] introduced a third shape parameter to the two-parameter generalized exponential distribution, adopted from the Marshall-Olkin method so that the hazard function of the proposed model can have all the four major shapes, namely increasing, decreasing, bathtub or inverted bathtub types. Recent developments in the Marshall-Olkin family of distributions can be given in George and Thobias [4], Gillariose and Tomy [5], etc.
The exponential distribution is a versatile and widely used probability distribution that has many important applications in various fields, particularly in modeling time-related events. The generalized exponential distribution, sometimes referred to as the exponentiated exponential (EE) distribution, failure rate function that might be either increasing, decreasing, or constant,
see Gupta, et al. [8]. According to Gupta and Kundu [9], the failure rate function of the EE distribution has similar behaviour to that of the gamma distribution and in many cases, it can be utilized as an alternate distribution to the gamma and Weibull distributions. Nadarajah and Haghighi [13] proposed a new generalization of the exponential distribution as an alternative to the gamma, Weibull, and EE distributions. Lemonte [11] defined a new three-parameter exponential-type distribution family, exponentiated Nadarajah Haghighi, that can be used to model survival data and reliability issues. A three-parameter distribution called Exponentiated Nadarajah and Haghighi (ENH) with cdf is given in (2)
f(x)= i(1 - e1-(1+Ax)T)ft ; x > 0, y, ft, A > 0 (2)
0 ; Otherwise,
where the parameters 7, ft control the shape of the distribution and parameter A is the scale parameter. The major goal of the present study is to create a new model of the four-parameter distribution, in the expectation that, in some instances, the new distribution will "fit better" than the exponential, ENH, Nadarajah-Haghighi (NH), and Marshall-Olkin-Nadarajah-Haghighi Distribution (MONH) distributions.
The rest of the paper is organized as follows: In Section 2, we propose a new generalization of ENH distribution, namely MOENH distribution. Various structural properties of the MOENH distribution, such as moments, quantile function, order statistics and stress-strength reliability are studied in Section 3. Characterizations of MOENH distribution are obtained in Section 4. In Section 5, we study the estimation of parameters of the MOENH distribution using the method of Maximum Likelihood. In Section 6, we discuss the MOENH Minification process and the corresponding sample paths. In Section 7, we have fitted the model to real-life data to show the flexibility of the new distribution. Concluding remarks are presented in Section 8.
2. MOENH distribution
In this section, we discuss the MOENH distribution introduced by using (1) and (2) we have the distribution function of MOENH distribution as follows:
(1_e1-(1+A*)T )ft
m \ >-;-;-^s" ; x > 0, a, 7, ft, A > 0
G(x) = <{ a+(1-a)(1-e1-(1+Ax)Y)ft ' ' ' (3)
0 ; Otherwise.
where the parameters ft, 7 are shapes of the distribution, and a, A are the scale parameter. If ft = 1, the MOENH distribution reduces to MONH distribution. We have the NH distribution when a = 1, ft = 1. For a = 1, ft = 1, 7 = 1, we obtain the exponential distribution. The probability density function of MOENH distribution is,
aft7A(1+Ax)Y-1 (/e1-(1+Ax)7)
v ; ; x > 0, a, 7, ft, A > 0
g(x) = { (1-e1-(1+Ax)Y )1-^a+(1-«)(1-e1-(1+Ax)Y )ft) (4)
; Otherwise.
The survival function and failure rate function of MOENH distribution are respectively as,
= 1 i1 - e1-(1+AX)Y= a - a(1 - eMW)ft
( ) a + (1 - a) (1 - e1-(1+Ax)7)ft a + (1 - a)(1 - e1-(1+Ax)Y)ft' ( )
h(x) =
iftjA(1 + Ax)7-1e1-(1+Ax)T
(1 - ei-(i+Ax)7) (a + (1 - a) (1 - e1-(1+Ax)Y)ft)
1
((1 - e1-(1+Ax)Y)-ft (a + (1 - a) (1 - eM^)7)ft)) - 1
It can demonstrate that the distribution exhibits increasing, decreasing, bathtub-shaped, inverse bathtub-shaped, and constant hazard functions. In Figure 1, we can see the plots of pdf and hazard function of MOENH distribution for different values of the parameters.
Figure 1: Plots ofpdf(left) and hazard function (right) for different values of parameters.
3. Statistical properties In this section, some statistical properties of MOENH distribution are discussed.
3.1. Moment Generating Function
The moment generating function of MOENH distribution is given by
to TO TO TO TO
MX (t) = EEEEE (-I)—' f(i)(imr)
h=0 k=0 m=0 n=0 j=0 h! W \ m / A aJ!
'a - 1Y r(ßn + ß)r(m + 1)
T(pn + p - j)(j + 1)m+r
3.2. Quantile Function
The quantile function has a number of applications. It can be used to obtain median, skewness, and kurtosis and can also be used to generate random variables. The quantile function of MOENH distribution is obtained as,
X
1
1 — ln f 1- eß K™^ V - 1
A
-,0 < p < 1,a,y,ß,A > 0.
(6)
3.3. Order Statistic
Order statistic makes their appearance in many areas of statistical theory. Let X1, X2,..., Xn be a random sample from the MOENH family of distributions, and let X(i), X(2),..., X(n) be the
corresponding order statistic. The pdf of ith order statistic, say Xi:n, can be written as
gi:m (x)
m!g(x) ^ [G(x)]i-1 [1 - G(x)]m
(i - 1)! (m - i)! m!g(x)
(i - 1)! (m - 7)! JC (-1)'r [G(X)I
j+i-1
(7)
by using binomial expansion [1 - G(x)]m-i = Ej=o (-1)j (m-i) (-1)j (m-i) [G(x)]j The pdf of the 1st and nth ordered statistic will be,
m-H i\j f m - 1 |
Epc1 (-i)j (
m
g1:m(x) = --(m - 1)!j 7
_gpy\(1 + Ax)7-1 e1-(1+Ax)Y_
(1 - e1-(1+Ax)Y^ + (1 - a) (1 - e1-(1+Ax)Y
(1 - e1-(1+Ax)Yy
v --L-
a + (1 - a) (1 - e1-(1+Ax)Y f
Ef- (-1)j( m -n)m!
gn:m (x) =
(m - n)!(n - 1)! aftjA(1 + Ax)Y-1e1-(1+Ax)Y
(1 - e1-(1+Ax)7)1-ft (a + (1 - a) (1 - eM^)7)ft) 2
(l - e1-(1+Ax)7y
x-^-}--ft.
a + (1 - a) (1 - e1-(1+Ax)7)ft
3.4. Stress-strength reliability
In order to estimate the stress-strength parameter, considering two random variables X and Y with MOENH(a1, ft, 7, A) and MOENH(a2, ft, 7, A) distributions, respectively, with the same baseline parameters ft, 7, A. We assume that X and Y are independent random variables. Then the stress-strength parameter is obtained in the form
œ I" f x
R = P(Y < X) = / / gy(y)
J0 _J0
p œ
= / Gy (x)gx (x)dx = -a1 Jo
gx(x)dx
a2 (ln (a1) - ln(a2) + 1) + a1 (a1 - a2)2
(8)
4. Characterization
This section deals with the characterization of the MOENH distribution based on the ratio of two truncated moments. To present the characterization of the distribution, consider the theorem presented in Glanzel [7].
Theorem 1. Let (Q, G, P) be a given probability space and let H = [a, b] be an interval for some a < b (a = -to, b = to might be allowed). Let X : Q ^ H be a continuous random variable with the distribution function G and let q1 and q2 be two real functions defined on H that
E[q2(x)|X > x] = E[q1 (x)|X > x]£(x),x e H
are defined with some real function £. Assume that q1 and q2 e C1 (H), £ e C2(H) and G is a twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation £q1 = q2 has no real solution in the interior of H. Then G is uniquely determined by
x
the function qi, q2 and £, particularly
£ ' (u)
tx
G(x) = C
£ (u)qi (u) - qi(u)
exp (-S(u))du,
where the function is a solution of the differential equation S' = ££ -1 and C is the normalization
constant, such that JHdG = 1 and £(x) = E[qi(x)|X>xj '
Proposition 1. Let X : Q ^ (0, to) be a continuous random variable and
qi(x) = (a + (1 - a)(1 - e1-(1+Ax)1)P)2(1 - e1-(1+Ax)1)1-^
and
qi = q1 (x)e1-(1+Ax)Y,x > 0.
Then the random variable X has pdf (4) if and only if the function £ defined in the Theorem 1 is of the form
e1-(1+Ax)Y
£ (x) = -. (9)
Proof. Suppose the random variable X has pdf (4), then
(1 - G(x))E [q1(X) | X > x] = -Ce1-(1+Ax)Y
where, C = aß. Further,
(1 - G(x))E [q2(X) | X > x] = -2e2(1-(1+Ax)T)
£(x)q1 (x) - qi(x) = -1 q1(x)e1-(1+Ax)7 = 0,x > 0. Conversely, if £ is of the above form, then
S' (x) = jrPx¥É-r) = ^ + Ax)7-1 £(x)q1(x) - qi(x)
and hence,
S(x) = (1 + Ax)7.
Now, in view of Theorem 1, X has density (4). ■
Corollary 1. Let X : Q ^ (0, to) be a continuous random variable and q1(x) be as in Proposition 1. The pdf of X in (4) if and only if there exist functions qi and £ defined in Theorem 1 satisfying the differential equation
£ (x)q1(x)
= yA(1 + Ax)7-1, x > 0.
£(x)q1(x) - qi(x)
The general solution of the differential equation given in Corollary 1 is,
£(x) = e(1+Ax)Y - j yA(1 + Ax)7-1e-(1+Ax)1 (q1 (x))-1 qi(x)dx + D
where D is a constant. Note that a set of functions satisfying the above differential equation is given in Proposition 1 with D = 0. However, it should be noted that there are other triplets
(q1, q2, £) satisfying the conditions of Theorem 1.
Proposition 2. The MOENH density function is log-convex for fixed a say, a = 1 and if 7 < 1
and ft < 1, and it is log-concave if 7 > 1 and ft > 1.
Proof. Let z = (1 + Ax)7, which implies that z > 1 for x > 0. We have x = (z1/7 - /A. The MOENH density is now rewritten as a function of z, £(z) say, we obtain
// ,, ) \ z(7-1)/7e(1-z)
£(z)= /((z1/7 - Om) = 7Aft [1-e(1-z(]1-ft, z > 1 The result follows by noting that the second derivative of log[£(z)] is
d2log[£ (z)] dz2
(7 - 1) + (ft - 1)e1-z 7z2 [1 - e1-z]2
Proposition 3. For any A > 0, a = 1, the MOENH distribution has an increasing failure rate function if 7 > 1 and ft > 1, and it has a decreasing failure rate function if 7 < 1 and ft < 1. The failure rate function is constant if 7 = ft = 1.
Proof. Using the log-convexity of the density function, the conclusion is valid. ■
5. Estimation of parameters
There are several methods in the literature for estimating unknown parameters. In this section, maximum likelihood method of estimation is used for estimating the parameters of MOENH distribution. Let us consider x1, x2,..., xn be the random variables having MOENH distribution. Then the likelihood function is given by,
r, „ n -rr aft7A(1 + Ax)7-1e1-(1+Ax)7
L (xi, ^ ^ A) = n-iW7-"-ft^2
i=1 (1 - e1-(1+Ax)7)1-ft (k + (1 - a) (1 - e1-(1+Ax)7)ft) The log-likelihood function is given by,
l =n ln(aft7A) + £ ln(1 + Ax)7-1 + £ 1 - (1 + Ax)7 i=1 i=1
n / , N ft\ n
(10)
- 2 £ ln L + (1 - a) (1 - e1-(1+Ax)7f) - £(1 - ft) ln (1 - e1-(1+Ax)7) i=1 ^ ' i=1
Partial derivatives of (10) with respect to the unknown parameters a, ft, 7, A
dL = 2 £ 1 - (1 - e1-(1+Ax)7 )ft (11)
da a £ (a + (1 - a)(1 - e1-(1+Ax)7)ft)2 ( )
da = n ft(1 - a)(1 - e1-(1+Ax)7 )(ft-1)
dft ft ¿1 a + (1 - a)(1 - e1-(1+Ax)7)ft ( )
dl n n n
= - + £ ln (1 + Ax) - £ (1 + Ax)7 ln (1 + Ax) - 2(1 - a)ft
d7 7 i=1 i=1
" e1-(1+Ax)7(1 + Ax)7 ln (1 + Ax)(1 - e1-(1+Ax)7)(ft-1) X a - (a - 1)(1 - e1-(1+Ax)7)ft
(1 - ft) ln (1 + Ax)e1-(1+Ax)7 (13)
£ e1-(1+Ax)7 (13)
i=1 c
dl n n (7 - 1)x " ^ . _ 1
dA = A + E T-^ - E Yx(1 + ^
1 = 1 1 = 1
E xyp(1 - a)e1-(1+Ax)Y(1 + Ax)7-1 (1 - e1-(1+Ax)Yp-1 a - (a - 1)(1 - e1-(1+Ax)Y)P
Yx(1 -p)(1 + Ax)Y-1e1-(1+Ax)Y
- E 1 - e1-(1+Ax)Y (14)
To obtain the maximum likelihood estimates of the unknown parameters, we equate,
da==*£=<4=0 <15)
The ML estimators are found through the solution of the nonlinear system. Hence, using R or MATLAB, a numerical approximation of the software's solution to this system of equations is possible.
6. Autoregressive Tlme Series Modelling
The autoregressive model is a stochastic process used in statistical modeling in which future values are forecasted based on a weighted sum of past values. The idea behind an autoregressive process is that the values of the past have an impact on the values of the present. A first-order autoregressive time series model with exponential stationary marginal distribution was developed by Gaver and Lewis [3]. In recent years, many authors Jayakumar and Babu [10] and Gillariose and Tomy [6] have developed various autoregressive models with minification structures. In this section, we develop various Autoregressive models of order 1 with Marshall-Olkin Exponentiated Nadarajah Haghighi as marginals, namely MIN AR (1) Model I and Model II and MAX - MIN AR (1) Model I and Model II, and explore some properties.
6.1. MIN AR(1) Model - I with MOENH Marginal Distribution
Consider an AR(1) structure,
X = i en; with probability 5 <16)
[min(Xn-1,en); with probability 1 - 5
where {en } is a sequence of iid random variables independent of {Xn } and 5 € (0,1). Then the process is Stationary Markovian with MOENH Distribution.
Theorem 2. In an AR(1) process with structure (16), {Xn } is Stationary Markovian with MOENH distribution with parameters 5, 7, p, A iff {en} is distributed as ENH(y, p, A)
Proof. Let en - ENH(y, p, A) From(16)
GXn(x) = 5Gen(x) + (1 - 5)GXn-1 (x)G-a(x).
Under stationary equilibrium,
Gx (x) = '
and hence
GX(x)
Ge (x) =
5 +(1 - 5)Gx(x)'
If e- - ENH(7, ft, A)
Ge(x) = 1 - (1 - e1-(1+Ax)7 )ft.
Thus
¿{1 - (1 - e1-(1+Ax)7)ft}
G x (x)
1 - (1 - S){1 - (1 - e1-(1+Ax)7)ft} S - ¿(1 - e1-(1+Ax)7)ft
S +(1 - S)(1 - e1-(1+Ax)7)ft' which is the survival function of MOENH(S). Conversely, if
S - s(1 - e1-(1+Ax)7)ft
G x (x)
S +(1 - S)(1 - e1-(1+Ax)7)ft'
then Ge- (x) is distributed as ENH(7, ft, A) and the process is stationary. In order to establish stationarity, assume that Xn-1 — MOENH(S) and en — ENH(7, ft, A) then,
S - S(1 - e1-(1+Ax)7)ft Gx(x) - v ;
S +(1 - S)(1 - e1-(1+Ax)7)ft'
This means that Xn is distributed as MOENH(S). ■
Remark 1. If X0 has an arbitrary distribution Gx0, the minification process is asymptotically
stationary with MOENH(S, 7,ft, A).
Since
n-1
Gx-(x) = SGe(x) £ (1 - S)!Ge(x) + (1 - S)nGX0(x)Gen(x) i=0
SGe (x)
1 - (1 - S)Ge (x) _ S - S(1 - e1-(1+Ax)7)ft = S +(1 - S)(1 - e1-(1+Ax)7)ft,
the survival function of MOENH(S, 7,ft, A).
6.2. MIN AR(1) Model - II with MOENH Distribution
Here we discuss a more general structure which allows probabilistic selection of process values, innovations and combinations of both. Consider the AR(1) structure given by
{Xn-1; with probability S1
en; with probability S2 (17)
min (Xn-1, en); with probability 1 - S1 - S2,
where S1, S2 > 0, S1 + S2 < 1 and {en } is a sequence of iid random variables independent of {Xn}. Then the process is stationary with Marshall-Olkin Exponentiated Nadaraja Haghighi distribution.
Theorem 3. In an AR(1) process with structure (17), {xn } is stationary Markovian with MOENH distribution with parameters t, 7, ft, and A iff {en } is distributed as ENH with parameters 7, ft and A, where t = 1-S" •
Proof. Let en — ENH(7,ft, A). From (17)
GXn (x) = S1GXn_ 1 (x) + S2Gen (x) + (1 - S1 - S2) GXn_ 1 (x)Gen (x)
Under stationary equilibrium we have,
TG e (x)
G x (x)
1 - (1 - t)Ge(x)
T 1 - (1 - gMW
1 - (1 - T) 1 - (1 - gl-(l+Ax)Y f
¿2 1-S1 1 - ( 1 - e1-(1+Ax)Y
1 - (l - 1 ¿2 A -h) 1 - (1 - e1-(1+Ax)T)fi'
where t = 1—, which is in the Marshall-Olkin form. Now let us assume that {Xn} MOENH(t, 7, fi, A). From (17) under stationarity,
(1 - ¿1) Gx(x)
G e (x)
¿2 + (1 - ¿1 - ¿2 ) G x (x)' Now by using Xn as MOENH (t, 7, fi, A), we have
(1 - ¿1)
G e (x)
¿2 1-¿1 1-( 1-g1-(1+Ax)T 'jfi
1-(1-1- ¿1) 1-(1-g1-(1+Ax)T
¿2 + (1 - ¿1 - ¿2 )
& [1-(1-e1-(1+Ax)T f
= 1 - (1 - e1-(1+Ax)y.
1-(1-^)[1-(1-e1-(1+Ax)Y )fi] fi
Which is the Survival function of Exponentiated Nadaraja Haghighi distribution with parameters 7, fi and A. ■
6.3. MAX-MIN AR(1) Model - I with MOENH Distribution
Consider the AR(1) structure given by,
Xn
max (Xn-i, en); with probability 51 min (Xn-i, en); with probability ¿2 Xn-1; with probability 1 — 51 — ¿2,
(18)
where 0 < ¿1, ¿2 > 1, ¿2 < ¿1, ¿1 + ¿2 < 1 and {en } is a sequence of iid random variables independent of {Xn}. Then the process is Stationary Markovian with Marshall-Olkin Exponentiated Nadaraja Haghighi distribution.
Theorem 4. In an AR(1) MAX-MIN process with structure (18), {Xn} is a stationary Markovian AR(1) MAX-MIN process with MOENH stationary distribution with parameters t, 7, fi and A iff {en } is distributed as ENH with parameters 7, fi and A, where t =
Proof. Let en - ENH(y,fi, A). From (18) we have,
G Xn (x) = ¿1 [1 - (1 - G Xn_ 1 (x)) (1 - (Gen (x))] + ¿2 G x„_ 1 (x)(G,n (x) + (1 - ¿1 - ¿2 ) G x„_ 1 (x).
Under stationary equilibrium,
Gx„(x)
tG.
e(x)
1 - (1 - t)Ge(x)
T[1 - (l - eMl+A*)^] 1 - (1 - T)[1 - (1 - eM^)7] _ |[1 - (l - e1-(1+Ax)Y] = 1 - (1 - £)[1 - (1 - e1-(1+Ax)Y f ]'
Where t = £ and GX„ (x) is in the form of Marshall-Olkin distribution. Now Let X„
MOENH(t, y, ft, A). Then from (18), under stationarity,
82 GX„ (x)
G e (x)
¿1 + (¿2 - ¿x)GXn(x)'
Thus, after simplification it can be written as
G e (x)
¿2 " ¿1 ¿2 1-(1-eM1+A*)Y )ft|
1-(1-¿i)[1_(1_e1-(1+Ax)T
¿1 + (¿2 - ¿1 ) | [1-(1-e1-(1+Ax)T )ft
,1-(1-¿i)[1-(1-e1-(1+Ax)Y
= 1 - (1 - e1-(1+A*)Y,
which is the survival function of Exponentiated Nadaraja Haghighi distribution with parameters Y, ft and A. ■
6.4. MAX-MIN AR(1) Model - II with MOENH Distribution
Consider the more general MAX-MIN process that includes minimum, maximum innovations and the process values, The AR(1) structure is given by
Xn
max (Xn-1,e„); with probability 51
min (X„_1,e„); with probability 82
e„; with probability 83
Xn-1; with probability 1 - 81 - 82 - 83,
(19)
where 0 < ¿1, ¿2, ¿3 < 1, ¿1 + ¿2 + ¿3 < 1 and {en} is a sequence of iid random variables independent of {Xn}. Then the process is stationary Markovian with Marshall-Olkin Exponentiated Nadaraja Haghighi distribution.
Theorem 5. AR(1) MAX-MIN process {Xn} with structure (19) is a stationary Markovian AR(1) MAX-MIN process with MOENH distribution (t, 7, ft, A) if {en} is distributed as ENH with parameters 7, ft and A where t = ¿^¿3.
Proof. Let en ^ ENH(7,ft, A). From (19) we have,
G Xn (x) = ¿1[1 - (1 - G Xn-1 (x))(1 - G en (x))] + ¿2 G Xn-1 (x)Gen (X) + ¿3 G en (X) + (1 - ¿1 - ¿2 - ¿3)Gxn_ 1 (x)
Figure 2: Sample path for AR(2) Mmficatzon Mode/-/for p = 0.6,0.7, p=0.5, 7=2.2 and A=2.
Figure 3: Samp/e path of AR(2) Mmficatzon Mode/-//for different sets of (pi, p2) = (0.3,0.4), (0.2,0.5),p = 0.5, y = 1.5, and A = 1.
Under stationary equilibrium it gives,
GXn (x) =
tG e (x)
1 - (1 - t)Ge(x)
|±|[1 - (1 - e1-(1+Ax)7)p]
1 - (1 - f±§)[1 - (1 - e1-(1+Ax)Y)p]'
where t = ¿1, which is in the form of Marshall-Olkin distribution. Now let Xn ~ MOENH(t, y, p, A), Then from (19), we have
Ge(x) =
№ + ¿3) Gx (x)
(¿1 + <%) + № - ¿1) Gx(x)
By simplifying, we get
Ge(x) =
(¿2 + ¿3) - ¿2±3 1-(1-e1-(1±Ax)Y Y '
1-(1-¿3+3 )[1-(1-e1-(1+Ax)Y )P ]
(¿1 + ¿3) + (¿2 - ¿1) " ¿2+1 [1-(1-e1-(1+Ax)Y )'
LH 1-|±|)[1-(1-e1-(1+Ax)Y Y}\
= 1 - (1 - e1-(1+A*)7)p .
Which is the Survival function of Exponentiated Nadaraja Haghighi distribution with parameters Y, p and A. Figure 2-5 displays the sample path features of the four AR(1) models created in this section and how these measures change with different parameter settings. ■
Figure 4: Sample path for AR(1)MAX-MIN Model -I/or various combinations of ( p1, p2) = (0.2,0.1), (0.3,0.2), ft 0.25, y = 1.2, and A = 1.
Figure 5: Sample path for AR(1) MAX-MIN Model -Il/or ( p1, p2, p3 ) = (0.4,0.2,0.2), (0.5,0.2,0.1), ft = 0.25,7 1.2, and A = 1.
7. Numerical illustration
7.1. Simulation study
A simulation study is conducted to evaluate the effectiveness of the MLEs for estimating the parameters of MOENH distribution. For this, we take into account, a = 0.5, ft = 1.2,7 = 0.8, and A = 0.09. For different sample sizes of n = 1000, n = 2000, and n = 4000, we simulate data from the MOENH model and determine the MLEs by maximizing the likelihood function. We carry out the procedure, 10000 times, and the results show that bias and root-mean-square error (RMSE) decreases as sample size increases. The results are in the Table 1.
Table 1: Simulation
n Parameters Estimate Bias RMSE
a 0.6352 0.0135 0.1687
1000 ft 1.2098 0.0009 0.0246
7 0.9561 0.0156 0.1306
A 0.1352 0.0045 0.0512
a 0.5895 0.0089 0.1295
2000 ft 1.2074 0.0007 0.0168
7 0.9000 0.0100 0.0995
A 0.1193 0.0029 0.0373
a 0.5509 0.0051 0.0881
4000 ft 1.2027 0.0002 0.0113
7 0.8511 0.0051 0.0652
A 0.1058 0.0015 0.0242
7.2. Data illustration
We take into account the following DARWin data set for numerical illustration. DARWin is a project, within the Swedish industry organization Swedish Energy, which collects and annually presents outage data from most of the Swedish electricity system operators. The annual reports from Swedish Energy are open-accessible and can be downloaded from the Swedish Energy website. The unplanned events from 2012 divided into voltage level (12 k V) and failure causes are given by Ekstedt et al. [2]. The parameters are estimated using the maximum likelihood method. Akaike information criteria (AIC), Bayesian information criteria (BIC), Kolmogrov-Smirnov (K-S), and p — va/Me are the goodness-of-fit metrics that we take into consideration.
Table 2: Parameter estimates and goodness offit stat/st/cs/or models fitted to the data
Model MLE -Log L AIC BIC KS p-VALUE
MOENH a = 2.000 ft = 0.2777 Y = 1.2189 A = 0.00015 73.6221 155.24 156.03 0.10819 0.9994
ENH ft = 0.2777 Y = 1.2189 A = 0.00013 78.6904 163.38 163.9725 0.2306 0.6449
NH Y = 1.2189 A = 0.0003 79.124 162.249 132.6435 0.2370 0.6122
Exp A = 0.0004 78.8909 159.78 159.982 0.2240 0.6782
Exponential P-P Plot
ENH P-P Plot
0 0 0 2
MOENH P-P Plot
NH P-P Plot
Figure 6: pp-p/ots
Table 2 lists the parameter estimates and goodness of fit statistics for DARWin voltage level (12 k V) failure data. The MOENH model is more suitable for this data since the values of — log L, AIC, BIC, K-S and p-value for the MOENH distribution are lower than those of the other competing models. Figure 6 represents the pp-plot for the fitted models.
7.3. Testing of Hypothesis
In this section, we present the likelihood ratio test procedure for testing the significance of the parameters of the MOENH model. We consider LR statistics to check if the fitted MOENH
distribution for a given data set is statistically superior to the fitted exponential, NH, ENH distributions. In any case, the hypothesis test of the type H0 : d = d0 vs Hi : d = d0 using the generalized likelihood ratio test. The test statistic is,
- 2 ln A(x) = 2[ln L(©; x) - ln L(© * ; x)] (20)
where © is the maximum likelihood estimator with no restriction, and © * is the maximum likelihood estimator with restriction. The test statistic follows a Chi-square distribution with
degrees of freedom (d/ = d/a/f — d/„„//). So here we consider the following likelihood ratio tests.
1. H01 : a = fi = 7 = 1, the sample is from Exp(A)
H11 : a = fi = y = 1, the sample is from MOENH(a, fi, 7, A)
2. H02 : a = fi = 1, the sample is from NH
H12 : a = fi = 1, the sample is from MOENH(a, fi, 7, A)
3. H03 : a = 1, the sample is from ENH
H13 : a = 1, the sample is from MOENH(a, fi, 7, A)
Table 3: Likelihood ratio test
Model Hypothesis Test statistic p-value
Exp vs MOENH H01 : a, fi, y = 1 H11 : H01 is false 10.5406 0.0145
NH vs MOENH H02 : a, fi = 1 H12 : H02 is false 11.0046 0.0041
ENH vs MOENH H03 : a = 1 H13 : H03 is false 10.1365 0.0014
The test statistic —2 ln A(x) given in (20) is asymptotically distributed as x2 with three degrees of freedom for test 1, 2 degrees of freedom for test 2, and 1 degree of freedom for test 3. The computed values of the test statistic in the case of the DARWin data set are listed in Table 3. From Table 3, we can see that p — value is less than the significant level of 0.05.LR tests reject the three sub-models in favour of the MOENH distribution. Since the critical values at the significance level 0.05 and degree of freedom three, two, and one for the two-tailed tests are 9.348, 7.378, and 5.024 respectively the null hypothesis is rejected in all cases, which shows the appropriateness of the MOENH distribution to the DARWin data.
8. Conclusion
In this paper, as a generalization of the exponential distribution, the MOENH distribution is introduced. Statistical properties, characterization properties, and autoregressive time series models of MOENH distribution are obtained. It is shown that the new model is a competitor to the exponential distribution for modeling certain types of data sets. Also, the generation of random variables from the new model is simple. The new model may attract the attention of researchers as a viable competitor to the exponential distribution.
References
[1] Cordeiro, G. M. and Lemonte, A. J. (2013). On the Marshall-Olkin extended Weibull distribution. Statistical papers, 54: 333-353.
[2] Ekstedt, N., Wallnerstrom, C. J., Babu, S., Hilber, P., Westerlund, P., Jürgensen, J. H. and Lindquist, T. M. (2014). Reliability Data: A Review of Importance, Use, and Availability. in: NORDAC 2024 (Eleventh Nordic Conference on Electricity Distribution System Management and Development, Stockholm, 8-9 September 2024).
[3] Gaver, D. P. and Lewis, P. A. (1980). First-order autoregressive gamma sequences and point processes. Advances in Applied Probability, 12(3): 727-745.
[4] George, R. and Thobias, S. (2019). Kumaraswamy Marshall-Olkin exponential distribution. Communications in Statistics-Theory and methods, 48(8): 1920-1937.
[5] Gillariose, J. and Tomy, L. (2020). The Marshall-Olkin extended power Lomax distribution with applications. Pakistan Journal ofStatistics and Operation Research, 16(2): 331-341.
[6] Gillariose, J. and Tomy, L. (2023). On an extension of the two-parameter Lindley distribution. Reliability: Theory & Applications, 18(72): 385-402.
[7] Glanzel, W. (1987). A Characterization Theorem Based on Truncated Moments and its Application to Some Distribution Families. In: Bauer, P., Konecny, F., Wertz, W. (eds) Mathematical Statistics and Probability Theory. Springer, Dordrecht: 75-84.
[8] Gupta, R. C., Gupta, P. L., and Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory and methods, 27(4): 887-904.
[9] Gupta, R. D. and Kundu, D. (2001). Exponentiated exponential family: an alternative to gamma and Weibull distributions. Biomedical Journal: Journal of Mathematical Methods in Biosciences, 43(1): 117-130.
[10] Jayakumar, K. and Babu, M. G. (2015). Some Generalizations of Weibull Distribution and Related Processes. Journal of Statistical Theory and Applications, 14(4): 425-434.
[11] Lemonte, A. J. (2013). A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function. Computational Statistics & Data Analysis, 62: 149-170.
[12] Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions, with application to the exponential and Weibull families. Biometrika, 84(3): 641-652.
[13] Nadarajah, S. and Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45(6): 543-558.
[14] Ristic, M. M. and Kundu, D. (2015). Marshall-Olkin generalized exponential distribution.
Metron, 73: 317-333.
