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97
Statistical properties and estimation procedures for a new flexible two parameter lifetime distribution
S. K. Singh, Suraj Yadav* and Abhimanyu Singh Yadav
Department of Statistics, Banaras Hindu University, Varanasi, India. [email protected] & [email protected] Corresponding Author's Email: [email protected]
Abstract
In this article, a new transformation technique based on the cumulative distribution function is proposed, the proposed transformation technique is very useful to generate a class of lifetime distribution. The various statistical properties of the proposed transformation method are studied. Further, the proposed technique is illustrated by considering exponential distribution as a baseline distribution. Various statistical properties such as survival and hazard rate, moments, mean deviation about mean and median, order statistics, moment generating function (MGF), Bonferroni's, and Lorenz curves, entropy, stress-strength reliability have been discussed. Different classical estimation methods are used to estimate the unknown parameters. Finally, two real data sets are considered to justify the use of the proposed distribution in real scenario.
Keywords: Transformation technique, statistical properties, classical method of estimation, and application.
1. Introduction
In lifetime analysis, various transformation techniques are used to propose the new probability distribution by adding an additional parameter to the baseline probability distribution. The significance of these probability distributions is categorized according to their use and appropriateness of different hazard rates viz. increasing, decreasing, constant, bathtub, and upside-down bathtub (UBT). Modeling of the real-life data set is based on the nature of the hazard rate function. For example, the exponential distribution is the most suitable choice whenever data exhibits the pattern of constant hazard rate. However, underlying data exhibits a non-constant hazard rate then other generalized lifetime distributions such as Weibull, Gamma, Extended Exponential, Generalized Exponential, Lindley distributions, and many others are frequently used to desirable data. To know more about monotone and non-monotone hazard rates, see [17], [32], [4], [5], [14], and [8], etc.
In statistical literature, various method has been suggested by the several authors to generate a new flexible model, viz. [26], [24], [19], and [30]. The beta generated model is used by [15] who uses the beta distribution to develop the beta generated distributions. [11] propose the Kumarswamy-G family of distributions.[12] propose a new class of distribution by adding two more parameters. [2] introduce a new method for generating families of distributions called the T-X family. Recently, the quantile function is used to generate the T-X family of distributions by [1]. For another development in the family of distributions see, [18], [23], [20], and [25], etc. These methods are most popular to propose flexible and appropriate models. Here notable thing is that all the methods of transformation discussed above introduce one additional parameter. Unquestionably, the addition of an extra parameter increases the flexibility but at the same time, it also increases complexity in further statistical inference.
Motivated by the above-mentioned literature, this article aims to propose a new transformation technique to generate the class of distributions. The proposed transformation is illustrated with an exponential baseline model and named a new two-parameter lifetime model. The proposed model through this transformation will be considered as an alternative to the Gamma, Weibull, and Extension of Exponential distributions by using the transformation method.
Let G(x) is the cumulative distribution function (CDF) of any baseline distribution, then the CDF of new distribution is proposed by,
F (x) = G (x) +T+ G(x))a f0rX G R and a > 0 (1)
Clearly, F(x) is the distribution function as it satisfy the condition to be a CDF.
(¿) lim F (x) =0 (¿¿) lim F (x) = 1 (¿¿¿) F (x) = f (x)
x—^ — TO x—^ TO
where f (x) and g(x) are the probability distribution function (PDF) of proposed and baseline distribution function respectively.
(¿v) It is well known that 0 < G (x) < 1, which implies that 0 < F (x) < 1. (v) Clearly, F (x) is a continuous function.
Now, the associated probability distribution function (PDF) f(x) for (1) is,
g (x) {(1 — G (x) + aG (x))} {(1 — G (x))(a—1}
f (x) =---2-J~ (2)
) {G (x) + (1 — G (x))a}2
The survival and hazard rates are:
S (x) = (1—G(x))a (3)
S (x) = G(x) + (1 —G(x))a (3)
h (x) = g (x) {1—G (x) +aG (x)}1 (4)
(1—G(x))G(x) + (1—G(x))a+1
To illustrate the above transformation, let us assume exponential distribution as the base line distribution. The PDF of the exponential distribution is given by,
, . f Ae—Ax for x>0, A> 0 g (x) = \ (5)
0 otherwise
and the associated CDF is:
G (x) =1—e—Ax for x>0, A> 0 (6)
here, A is rate parameter of exponential distribution. Then by transformation (1), the CDF of the new flexible distribution is,
1 e—Ax
F (x) e—Ax+e—Aax for x > 0, a > 0, A > 0 (7)
The PDF, survival and hazard rate function of proposed distribution are given as:
, , Ae-Aax (a+e-Ax-ae-Ax) f (x) =-^ v A-A-^ /or x > 0, a > 0, A > 0
and
respectively.
(1-e-Ax+e-Aax)2 S (x) =
h (x) =
-Aax
(1-e-Ax) +e-Aax
A [a+e-Ax-ae-Ax] [1-e-Ax+e-Aax ]
(8) (9) (10)
The principal objective of this paper is to propose a new transformation method and derive its various statistical properties. Specifically, we substitute (6) into (1) to get the CDF of the proposed distribution and the corresponding PDF is obtained by substituting (5) and (6) into (2). Our motivation to construct a new model is: (i) it is applicable for modeling increasing, decreasing, and constant hazard shape which provides a good fit for real data sets; (ii) In our proposed model if we put a = 1 then our proposed model reduced to baseline model; (iii) The proposed model can be considered as a good alternative model for fitting the positive data with a longer tail and (iv) The proposed model provides a better fit than some well-known lifetime models to real data sets. As, the proposed model is an alternative to Weibull, Gamma, and Extended Exponential, thus the proposed model might be a good choice for the researcher. Also, we have considered different methods of estimation to estimate the unknown value of the parameter. To check the applicability, AIC and BIC's are also constructed for the parameters of the proposed model. A simulation study has been performed to appraise the performance of the proposed estimation methods. Further, we have considered two real data set to illustrate the superiority of the proposed model and study. The plots of pdf and hazard rate function of new flexible two-parameter lifetime distribution for various values of a and A are shown in Figure 1(a) and 1(b) respectively. From the Figure 1(b), the proposed distribution has an increasing, decreasing, and constant hazard rate.
(a) (b)
Figure 1: (a) PDF of proposed distribution. (b) Hazard function of proposed distribution.
e
The content of the rest of the paper is organized as follows: In section 2, we have discussed some statistical properties such as raw moments, moment generating function (MGF), mean deviations, Bonferroni and Lorenz curves, Renyi entropy, s- entropy, cumulative residual entropy, order statistics and reliability. In section 3, different method of estimation of the proposed distribution is studied. Simulation studies are carried out, in section 4, to compare the behaviour and performance of the different estimators. In section 5, the proposed model is fitted with some competing models using two real data sets and finally, the conclusions are summarised in section
6.
2. Statistical properties of new flexible two parameter lifetime
distribution
In this section, we have discussed various statistical properties of our proposed new two parameter lifetime distribution like moments, moment generating function (MGF), mean deviation about mean and median, order statistics, reliability, Renyi entropy and Shannon entropy.
2.1. Raw Moments:
The rth moment about origin of the distribution with PDF (8) is obtained by,
m r Xe
f em em
¡ir=E (Xr ) =J xrf (x)dx=J x
-Xax
(a+e Xx-ae Xx)
o e-Xx +e-Xax^2
dx
After simplification the above integral, we get,
ir = E E (-1)
i=0 j=0
]
(i+1)
aT (r+1)
(1-a) r (r+1)
Xr (a+i-j+aj)r+1 Xr (1+a+i-j+aj)r+1
(11)
The respective four moments about origin can be obtained by putting r = 1,2,3, and 4. For r =1 we get mean of the distribution and is given by the following expression,
i= EE (-1);
i=0 j=0
(i+1)
(1-a)
X(a+i-j+aj)2 X(1+a+i-j+aj)2
For, r = 2,3, and 4 we can compute p2, p3, and p4 by putting these values in equation number
(11). The variance of the proposed model can be obtained using the expression,
V (X) = E (X2) - (E (X))2
Similarly, we can find other moment based, skewness, and the kurtosis of the distribution.
2.2. Moment generating function (MGF):
The moment generating function (MGF) for the proposed distribution with PDF (8) is given by;
r~ etxXe-Xxx \a+e-Xx-ae-Xx 1 Mx (t) = -l--2-Ldx
J0 _g-Xx+g-XxxJ 2
a
S. K. Singh, Suraj Yadav & Abhimanyu Singh Yadav
Statistical properties & estimation procedures for RT&A, No 4 (71)
a new flexible two parameter lifetime distribution Volume 17, December 2022
After simplification,
« E E E (—1)' (J) (J+1) ^^ tttP+V^
r=0 /=0 ¿=0 V ¿ / r! Ar (/—¿+¿a)r 1
A(1—a)eeel(—1)k (n+1)(») "-A-aA)m((v:i2>
m=0 n=0k=0 V k J m! ((n—k+ka)A)m+1
2.3. Mean Deviation (MD)
The mean deviation is the mean of the deviations. It can be calculated from the mean, median, and mode. It shows how far all the observations from the middle, on average are. The mean deviation about mean and mean deviation about median is defined as,
and,
!• TO
h (x) = J |u—x| f (x) dx
!• TO
¿2(x) = J |x-M| f(x)dx
Respectively, here v = E(X) is the mean and M = medz'an(X) is the median of the distribution. After simplification the mean deviation about mean and the mean deviation about median are
given as;
!• TO
¿1 (x) = 2^F (v) —2^+2/ xf (x) dx (13)
Now, the integral is computed as,
rto cto a e-aax Ta+e-Ax_ae-Axl
xf (x) dx=/ x --[—-aVJ
ju ju (1— e-Ax+P-Aax\2
, dx
(1—e—Ax+e—Aax)2
=E E (—1)jf j)(¿+1) x
¿=0 j=0 V J /
/•CO /»CO
Aa xe—^j+aj)Axdx+A(1—a) / xe—j+aj)Axdx By the definition of complementary incomplete gamma function and for any integer n,
r (n, x) = fxTO tn-1 e-tdt r (n, ax) = /xTO tn-1e-atdt
/»to n-± xk
r (n,x) = tn-1e-tdt= (n—1) !e-x E x- = (n-1) !e-xen-1 (x) ■!x k=0 k
here, en(x) is the exponential sum function. The above expression can be expressed as;
=E E (-DY j) (¿+1)x
i=0 j=0 V j /
ae-(a+i-j+aj)Au
-2—(1 + (a+i-j+aj)Au)
(a+i-j+aj)2 A
(1-a))e-(1+a+i-j+aj)Au / . . _ "
+—«-~—^—(1 + (a+i-j+a j)Au)
(1+a+z-j+aj) A
By using equation (13) and (14), the mean deviation S1 (x) about mean is;
¿1 (x) =2p F (p) -2p+2 £ £ (-1)j
i=0 j=0
(i+1)
ßg-(x+i-j+xj)A.p
-2— (1+ (a+i-j+aj) Ap)
(a+i-j+aj)2A
(1_a) e_(1+K+i_i+Ki)xp (1+a+i_ j+a j)2 X Similarly, the mean deviation S2(x) about median is given by,
(1 + (1+a+i-j+aj)Ap)
(15)
¿2(x) = -p+2£ £ (-1)' i=0 j=0
(i+1)
ag-(a+i-j+aj)Ap --—-(1+ (a+i-j+aj) Ap)
(a+i-j+aj)2 A
(1-a) e-(1+x+i-j+aj)ap (1+a+i-j+a j)2A
(1+ (1+a+i-j+aj) Ap)
(16)
2.4. Bonferroni and Lorenz curves
The Bonferroni [7] and Lorenz curves [21] is used to measure the inequality in the distribution of quantity in the area of economics as in term of income and wealth. The Bonferroni and Lorenz curves have various applications not only in the area of economics to study income and poverty but also in other areas like demography, medicine, insurance and reliability. Lorenz curves cannot be defined if the mean of the distribution is zero or infinite. The Bonferroni and Lorenz curves are given by,
and
1 r q
B (P) = PP Jo xf (x)dx
1 r q
L (x) = — xf (x)dx p Jo
(17)
(18)
where p = E(x) and q = F 1 (p) respectively. Now, the integral quantity in RHS is simplified as;
xf (x) dx = £ £ (-1)
i=0 j=0
(i + 1) X
ae-(a+i-j+aj)Aq
-^(1 + (a+i-j+aj)Aq)
(a+i- j+a j)2A
(1-a))e-(i+«+i-j+«j)Aq (1+a+i-j+aj)2A
(1 + (a+i-j+aj)Aq)
Hence, the Bonferroni and Lorenz curves for the new distribution are obtained as;
11 B(P)= P - PP
£ £ (-1)j v . i=0 j=0
{ae-(x+i-j+xj)Aq
-^(1 + (a+i-j+aj)Aq)
(a+i-j+a j)2A
and
L (P) = 1 — p
£ £ (-1)j i=0 j=0
(1 -a)e-(i+«+i-j+«j)Aq (1+a+i-j+a j)2A
(1 + (1+a+i-j+aj)Aq)
{ae-(a+i-j+aj)Aq
-^(1 + (a+i-j+aj)Aq)
(a+i-j+a j)2 A
CO
CO
q
Ci
C
(1 -a)e-(i+«+i-j+«j)Aq (1+a+i-j+a j)2A
(1 + (1+a+i-j+aj)Aq)
(20)
respectively.
2.5. Renyi entropy
Renyi entropy [28] is a most popular measure of average amount of uncertainty of a random variable X. If X is a random variable with probability distribution function f (x) then the Renyi entropy is defined as
Jr (y) =1_Y l0%{l fY (x) dx} (21)
where, 7> 0 and 7=1. Now, from equation (8) we get,
CC
i fY (x) dx=L
Ae
—Axx
[a+e Ax-ae Ax] (1-e-Ax +e-Aax )2
-Axn 7
dx
C C i l . , , f-2j
= AYaY+l-j ££££ (-1)j+k+l
V i J V J J \k J \l J (ja + i - k + ak + j)A
i=0 j=0 k=01=0
Y
By putting the above value in the equation (21) we get,
1
Jr(y) = T^TlogA + ( ) loga
1 - Y 1
+ 1-Y
log
1 - Y
£££££(-1)1+k+l( 2y i=0 j=0 k=0 l=0 i
k J \l J (Ya + i - k + ak + j)A
(22)
2.6. s-Entropy
Shannon entropy was proposed by [29], and is a particular case of Renyi entropy as 7 ^ 1. It can be defined as E [_logf (X)].
logf (X) = logA - aAx + loga + £
(-1)'"V „—Ax -A^n
n=1
ann
e- ae-
C
+ 2 £
m=1
Ax _e-Aaxy
m
After using the result based on series expansion of log(1 + x) and log(1 _ x) in the above expression, the expression for Shannon entropy is given by,
E [-logf (X)] = -logA + AaE (X) - loga - £ £
(-1)
n+l+1
n=1 l=0
ann
nl E e-Anx
- 2 £ £
(-1)s /m
m=1 s=0
m v s
e ^g-Amx+Asx-Aasx^ (23)
The equation number (23) can be computed with the help of following results;
Ee
- Anx
££ (-1)M;1 (i + 1)
i=0 j=0
Aa
A (1 - a)
(n + a + i - j + aj) A (n + a + 1 + i - j + aj) A
(24)
1
Ci
and
E e
,-Amx+Asx-Aasx
TO k
EE(-1)M; )(k+1) x k=0t=0 v t
1 — a
(m - s + as + k + a - t + at) (m - s + as + a + 1 + k - t + at)
(25)
respectively.
Corr: 1. The cumulative residual entropy [27] defined as, Jc = —J Pr (X > x) log (Pr (X > x) ) dx
e
—Aax
/0 (1 - e-Ax + e-Aax
y (-Aax - log (1 - e-Ax + e-Aax) ) dx
EE(-1)
i=0j=0 \J
Aa__(-1)1 (kA_1_
((a + i - j + aj) A)2 ¿0 E k U/ (a + i - j + aj + k - l + al)
2.7. Order Statistics
Suppose that X1, X2,........Xn is a random sample of size n from the proposed continuous probability distribution function (PDF), f (x). Let X1:n < X2:n < ■ ■ ■ < Xn:n denote the corresponding order statistics. We know that the probability density function of rth order statistics Xr:n , say fr(x), where the population PDF and CDF are f (X) and F(x) respectively, is given as follows,
fr (x) =
n!
- E (-1)!
(r 1)! (n-r)! E ( )
n-r i
F1
r+i—1
(x)f(x)
Consequently, using the equation (7) and (8) in (26) we get,
(26)
fr (x) =
n!
- E (-1)!
(r 1)! (n-r)! E ( )
nr
1e
Ax
1 _ e-Ax + e-Aax
and corresponding rth order statistics of CDF Fr (x) is,
r+i 1
Ae-Aax(a + e-Ax - ae-Axy
(1 - e
-Ax + e-Aax )2
(27)
n n- j
Fr (x) = EE
j=r m=0
j
n - j m
(-1)mFj+m (x)
Hence from the equation (7), equation (28) can be written as,
n n—j / w ■ \ / -1 —Ax
Fr (x) = EE (—1)m n — 1 — e
j=r m=0
mn j
n - j m
1 _ e-Ax + e-Aax
j - m
(28)
(29)
2.8. Reliability
In this section, we have discussed about reliability of a component. In the context of reliability, the stress-strength model explains the life of a component which has a random strength X that is subjected to random stress Y. The component will fail if the stress applied to it exceeds the strength and the component will work properly whenever X > Y. So, P[X > Y] is a measure of component reliability. It has various number of applications in many areas such as in science, engineering etc. In the field of stress-strength model there has been number of works as regarded
a
CO
TO /
nr
nr
X
n
estimation of reliability R when X and Y are independent random variable belonging to the same univariate family of distribution.
Here, we derive the reliability R when X and Y are independent random variables from the proposed distribution with parameter (a\, Ai) and (a2, A2) respectively. So, the reliability R is defined as bellow,
!• TO
R=P (X>Y) = J fX (x, a1, A1) Fy(x, a2, A2)dx
TO TO i k
=EEEE (-i)j+l i-
i=0 k=0 j=0 /=0
(i+1)
Ai ai
(a1+i-j+a1j) A1+ (k-/+a2/) A2
(ai+i-j+a1j) A1+ (k-/+a2/+1) A2
| + A1 (1-a1)
(1+a1+i-/+a1j) A1+ (k-/+a2/) A2
1
(1+a1+i-/+a1j) A1+ (k-/+a2/+1) A2
(30)
3. Methods of estimation
In this section, we will discuss different method of estimation namely, maximum likelihood estimation (MLE), maximum product spacing estimation (MPS), least square estimation (LSE) and weighted least square estimation (WLSE), Cramer-von-Mises estimation (CVME), Anderson Darling estimation (ADE) to estimate the unknown parameter of the considered model.
3.1. Maximum likelihood estimation (MLE)
Let x1, x2,.........xn be the random samples of size n from the proposed distribution. Then the
log-likelihood function of the proposed distribution is given as,
/ogL = nlog A - aA £ xi + £ log (a + e Axi - ae Ax'
i=1
i=1
n
) - 2 £ log (1-e-Axi+e-Aax^
i=1
Now, differentiating equation (31) with respect to parameters a and A we get,
d/ogL n n
= - A £ xi + £
1e
—Axi
da
i=1
1 (a + e-Axi - ae-Axi) g (1-e-Axi+e-Aaxi)
2£
Axie-Aaxi
and
d/ogL _ n = A
dA
= A - a £ xi + £
i=1 i
(a - 1) xie
Axi
(a + e-
Axi
i ae
Axi
-2£
n xie Axi — axie Aaxi
1 (1-e-Axi+e-Aaxi
(31)
(32)
(33)
Now, putting the equation (32) and (33) equal to zero, we have two non-liner likelihood equations. After solving these equations, we get MLEs a and A of parameters a and A. These equations are not in closed form consequently it cannot be solved analytically. Therefore, Newton-Raphson method is used to get the MLE's of the parameters a and A.
k
1
/
1
1
x
n
n
3.2. Maximum Product Spacing Estimation (MPSE)
This method is one of the most popular method of estimation and it was developed by [9]. Let x(1) < x(2) < ■ ■ ■ < x(n) be the ordered sample of size n and we define spacing as,
D =
/■x(i)
/ f (x, a, A)dx ; ¿ = 1,2,3.........(n + 1)
= F (x^, a, Aj — F (x^—1), a, a)
Where, initial conditions are F (x(0)a, A^ =0, F ^x(n+1)a, A^ and sum of all the spacing will be zero.
We are taking the observation from the proposed distribution, now from the equation (7) the D's are defined as,
1- e—Ax(0 1- e—Ax(<-1)
D¿ = -7-7---7-7- ; for all ¿ = 1,2,... .n (34)
¿ 1—e—Ax(0+e—-^p) 1 —e—Ax( ¿—1) +e—Aax(,-1) J v '
For, i = 2,3,----n. The MPS estimator amps and Amps of a and A are obtained by maximising the
geometric mean of the differences,
/n+1 \ 1/(n+D G = (tiD)
after taking logarithm of G we get,
'ogG = E 'ogD (35)
substituting the value of D from the equation (34) in equation (35) we get,
1ogG =( nrO 1og
1- e—Ax0 1- e—Ax(i-1)
1 —e—Ax(0+e—Aax(0 1_ e—Ax(i-1) +e—Aax( ¿—1)
(36)
It may be noted that from the equation (36) we can get the derivatives ^daT , ^dg^ and set it equal to zero, the equation, thus obtained, cannot solved analytically, therefore, the same numerical technique may be used to obtain the solution.
3.3. Least Squares Estimation (LSE)
This method is most popular method [31]. Let x^) < x(2)... ■ ■ ■ < x(n) be ordered sample of size n from proposed distribution. LSEs 0L\S and A/s of parameters a and A are obtained by minimizing
n 2 Z (a, A) = E (F (x^, a, A) — E [F ^)])
=1
where, E [F ^)] = ^ ;i = 1,2.......(n + 1)
then,
n / 1 - e—AxO i" Z (a, A) = E -^-A---—\ (37)
v ' ¿=1 ^1— e—aAx(,) +e—Aax(.-) n + 1y
In order to minimize Z (a, A)given in (37), we differentiate equation (37) with respect to a and A
2
and equating to zero, which results of the following equations,
C(a, A) ^ / , , W tw i
= LF* (x(i),a,A [F (x(i),a,A - n+1
dZ (a, A) d (a)
= L Fa (^¿V a, A) ( F (x(i), a, A) -
n +1
= 0
= 0
(38)
(39)
The above two non-linear equations cannot be solved analytically, therefore, numerical technique is used for solution.
3.4. Weighted Least Squares Estimation (WLSE)
The estimation procedure to obtain the estimates of the parameters through WLSE is quite similar to the LSE with a slight change that it minimizes the weighted sum of squared deviation between true and expected CDF at observed ordered sample points, where weights are inversely proportional to the var [F (x^))]. Thus, WLSE is obtained by minimizing
W(* A) = L ín++!-Í++f
f (x(¿),a,A) - n+
n+1
W a A) = L in+n-^
1- e
—Ax,
(i)
1- e—Ax(i) +e—Aax(i) n + 1
(40)
To get the WLSE estimates &WLS and AWLS of parameters a and A, differentiate equation (40) with respect to a and A and equating to zero, which results of the following equations
dW(a, A) " (n + 1)2 (n + 2) i ( J ( ^ i
= L i (n—(+1) F ^x(i),a,A F ^x(i),a,A — n+1
dW(a, A) " (n + 1)2 (n + 2)r, ( J ( ^ i
~dA = L i (n — i + 1) fa (xa),a,A F(-x(i),a,A — r+
n+1
=0
=0
(41)
(42)
Again, equation (41) and (42) cannot be solved analytically, therefore, numerical technique is used to secure the solution.
3.5. Cramer-von-Mises Estimation (CVME)
This method of estimation is proposed by [22], the method is the minimum distance method based on the difference between empirical and cumulative distribution functions. See, [10, 13] for more detail about this method. The CVM estimator of the parameters are obtained by minimizing
C (a,A) = ¿+£( f (x(i),a,A)— )
(43)
To get the CVM estimates 'aCVM and ACVM of parameters a and A, differentiate equation (43) with respect to a and A and equating to zero, which results of the following equations
= L F (xd),a,A) (F (xd),a,A) — 2J— ) dCaA1 = L FA (x(i),a,A) (F (x(i),a,A) — )
(44)
2
2
2
2
2
Again, equation (44) and (45) cannot be solved analytically, therefore, same numerical technique is used to obtain the solution.
3.6. Anderson-Darling Method of Estimation (ADE)
This method is based on the minimization criteria of Anderson-Darling statistic [3] . The ADE
estimate can be obtained by minimizing the following equation,
1 n
A (a, A) = -n - - £ (2i - 1) log(f(x(i),a, a)) + log (l (x^-),a, a)) (46)
i=1
where, F (x(n+1—¿), a, A =1 — F (x@), a, A. Therefore, the AD estimates aADE and Aade of the parameters a and A can be obtained as the solutions of the partial differentiation based on equation (46) using same iterative procedure.
4. Simulation Study
In this section, the Monte Carlo simulation study has been performed to assess the performance of the different estimators obtained via different method of estimation viz., MLE, MPS, LSE, WLSE, CVME, and ADE.
In order to perform simulation, the random sample for the for the different variation of the sample size, and parameters. In particular, n = 10,20,..., 100 and a = (0.75,1,1.5,2.5), A = 0.5 are chosen. The estimators obtained via considered methods are not assumed any explicit mathematical form and not yield closed form solution, therefore N-R method is used to secure estimates of the parameters. The average estimate, MSE of the parameters using the above methods reported in Table 1-4 based on N = 5000 replication using the following formula. Average estimates:
1 N _ 1 N _
^AE = n E ^ , aAE = N E A
Mean square error:
N ^ 1 ' AE N . !=1 !=1
1 N 1 N
«mse = N E(a — , AMSE = N E — AJ =1 =1
From Table 1,2,3 and 4 it has been overserved that the MSE of the parameter decreases as the sample size increases, which ensures the consistency of the proposed estimators. It is important to mention that ML and MPS methods are based on likelihood and others are based on a distance measure. Further, if we fixed A = 0.5 and varied a = (0.5,1.0, 1.5, 2.0,4.0) the Table 1 and 2 show that, in likelihood-based methods for shape parameter a and scale parameter A, MPS and MLE perform well respectively. Furthermore, in the considered distance measure there is no trend for the shape parameter however there is a specific trend for the scale parameter viz.
CVME < ADE < WLSE < LSE
Next, if we fixed a = 0.5 and varied A = (0.5,1.0, 1.5, 2.0,4.0) the Table 3 and 4 show that in likelihood-based methods MPS is better than MLE for shape parameter a and MLE is better than MPS for scale parameter A. Further, in the considered distance measure there is the same conclusion as in the previous case. Also, we have calculated coverage probability and average length of 95% confidence interval. The MSEs of point estimates and average lengths of interval estimates decreases with increasing sample sizes.
10
20
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30
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a
MLE
MPS
LSE
WLSE
CYME
ADE
CP
0.5 1
1.5 2 4
0.5
0.7480(0.4486) 1.2587(0.9402) 1.6001(1.1336) 1.8439(1.3678) 3.7018(7.0880)
0.4206(0.1025) 0.6711(0.3066) 0.8334(0.6915) 0.9495(1.2598) 1.7501(6.3543)
0.5093(0.1580) 0.7418(0.3047) 0.9000(0.6562) 1.0172(1.2853) 1.7955(6.0933)
0.5280(0.1919) 0.7731(0.3167) 0.9409(0.6356) 1.0650(1.2287) 1.9273(5.8014)
0.8315(0.6497) 1.2324(0.8992) 1.4972(1.0294) 1.7136(1.2342) 3.1482(5.4229)
0.6123(0.2148) 0.9776(0.4341) 1.2206(0.6386) 1.3941(0.9913) 2.6206(4.9758)
0.5 1
1.5 2 4
0.5
0.6214(0.1700) 1.1372(0.5215) 1.5862(0.8163) 1.8794(0.9814) 3.7750(5.0184)
0.4564(0.0725) 0.7987(0.2494) 1.0880(0.4970) 1.2730(0.9129) 2.4191(4.3008)
0.5348(0.1216) 0.9064(0.3177) 1.1987(0.5313) 1.3964(0.8732) 2.6459(3.9490)
0.5457(0.1189) 0.9336(0.3121) 1.2552(0.5276) 1.4651(0.8341) 2.8126(3.8230)
0.6761(0.2575) 1.1634(0.6126) 1.5510(0.8429) 1.8158(1.0075) 3.5429(4.4899)
0.5729(0.1198) 1.0094(0.3567) 1.3760(0.5689) 1.6155(0.8123) 3.1448(3.8077)
0.5 1
1.5 2 4
0.5
0.5635(0.0856) 1.0698(0.3216) 1.5375(0.5993) 1.9344(0.8011) 3.7371(3.9179)
0.4568(0.0491) 0.8346(0.1949) 1.1729(0.4137) 1.4569(0.7000) 2.7059(3.5130)
0.5153(0.0726) 0.9365(0.2681) 1.2994(0.4845) 1.5892(0.7153) 2.9295(3.3280)
0.5248(0.0710) 0.9641(0.2614) 1.3492(0.4785) 1.6585(0.6753) 3.1099(3.2099)
0.5975(0.1170) 1.1028(0.4134) 1.5419(0.6836) 1.8931(0.8565) 3.5529(3.6898)
0.5386(0.0694) 0.9939(0.2584) 1.4047(0.4811) 1.7430(0.6775) 3.2870(3.2222)
0.5 1
1.5 2 4
0.5
0.5431(0.0457) 1.0231(0.1800) 1.5300(0.4326) 1.9594(0.5990) 3.7675(2.7696)
0.4764(0.0316) 0.8725(0.1339) 1.2806(0.3240) 1.6229(0.5192) 3.0364(2.5694)
0.5157(0.0420) 0.9481(0.1800) 1.3832(0.3893) 1.7557(0.6009) 3.2692(2.6738)
0.5238(0.0399) 0.9704(0.1757) 1.4262(0.3804) 1.8135(0.5593) 3.4104(2.5143)
0.5615(0.0558) 1.0426(0.2285) 1.5297(0.4835) 1.9498(0.7049) 3.6703(2.9441)
0.5288(0.0395) 0.9806(0.1635) 1.4483(0.3800) 1.8444(0.5491) 3.4859(2.9441)
AL
0.9021 0.8964 0.8746 0.8578 0.8352
2.7740 4.9517 6.4410 7.4948 16.6406
0.9226 0.9032 0.8882 0.8668 0.8588
I.5587 3.0567 4.4341 5.3555
II.8705
0.9206 0.9148 0.8964 0.8848 3.2870
1.1230 2.2968 3.4581 4.4816 9.5240
0.9352 0.9112 0.9000 0.8974 0.8862
0.8225 1.6688 2.6398 3.4954 7.3997
0.5 0.5165(0.0202) 0.4820(0.0168) 0.5029(0.0207) 0.5095(0.0197) 0.5241(0.0236) 0.5102(0.0193) 0.9428 0.5450
1 1.0021(0.0787) 0.9188(0.0689) 0.9607(0.0833) 0.9770(0.0788) 1.0061(0.0919) 0.9800(0.0760) 0.932 1.1384 100 1-5 0.5 1.5070(0.2048) 1.3649(0.1768) 1.4444(0.2228) 1.4698(0.2080) 1.5181(0.2491) 1.4720(0.2020) 0.9288 1.8126
2 1.9760(0.3487) 1.7767(0.3169) 1.8749(0.3911) 1.9127(0.3543) 1.9750(0.4278) 1.9167(0.3448) 0.9174 2.4772 4 3.8139(1.6466) 3.3752(1.6005) 3.5492(1.8477) 3.6490(1.6550) 3.7598(1.9505) 3.6633(1.6357) 0.8970 5.2742
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A
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0.5 1
1.5 2 4
0.5
0.6673(0.3265) 0.6841(0.2847) 0.7356(0.3258) 0.7680(0.3239) 0.8259(0.6516)
0.8401(0.5701) 0.9104(0.5775) 1.0049(0.7196) 1.0590(0.7175) 1.3265(8.4690)
0.7561(0.3530) 0.8337(0.3811) 0.9328(0.4947) 1.0069(0.5788) 1.1403(0.9113)
0.7459(0.3447) 0.8189(0.3685) 0.9116(0.4639) 0.9860(0.5528) 1.1075(0.8641)
0.6104(0.2124) 0.6635(0.2175) 0.7372(0.2748) 0.7879(0.3119) 0.8682(0.4780)
0.6804(0.2809) 0.7269(0.2686) 0.7988(0.3227) 0.8585(0.3748) 0.9547(0.6153)
0.5 1
1.5 2 4
0.5
0.5911(0.1410) 0.6122(0.1256) 0.6344(0.1332) 0.6649(0.1426) 0.6755(0.1574)
0.6685(0.2076) 0.7151(0.1965) 0.7579(0.2170) 0.8052(0.2433) 0.8551(0.2959)
0.6183(0.1171) 0.6716(0.1326) 0.7201(0.1628) 0.7676(0.1929) 0.8163(0.2518)
0.6086(0.1128) 0.6586(0.1255) 0.7022(0.1527) 0.7475(0.1792) 0.7882(0.2262)
0.5557(0.0896) 0.5976(0.0963) 0.6357(0.1144) 0.6739(0.1319) 0.7021(0.1670)
0.5900(0.1039) 0.6321(0.1106) 0.6691(0.1322) 0.7096(0.1506) 0.7426(0.1912)
0.5 1
1.5 2 4
0.5
0.5717(0.0816) 0.5866(0.0731) 0.6022(0.0815) 0.6132(0.0725) 0.6352(0.0870)
0.6169(0.1016) 0.6539(0.1015) 0.6845(0.1172) 0.7068(0.1125) 0.7579(0.1514)
0.5886(0.0696) 0.6241(0.0777) 0.6557(0.0961) 0.6810(0.1005) 0.7297(0.1385)
0.5813(0.0685) 0.6119(0.0731) 0.6403(0.0882) 0.6630(0.0905) 0.7029(0.1204)
0.5487(0.0573) 0.5766(0.0613) 0.6023(0.0746) 0.6227(0.0755) 0.6580(0.1005)
0.5721(0.0656) 0.6009(0.0687) 0.6254(0.0813) 0.6457(0.0823) 0.6818(0.1090)
0.5 1
1.5 2 4
0.5
0.5393(0.0373) 0.5622(0.0387) 0.5658(0.0381) 0.5742(0.0390) 0.5918(0.0438)
0.5636(0.0414) 0.6023(0.0484) 0.6163(0.0498) 0.6324(0.0543) 0.6686(0.0695)
0.5522(0.0379) 0.5852(0.0420) 0.5989(0.0469) 0.6132(0.0504) 0.6471(0.0646)
0.5457(0.0368) 0.5759(0.0390) 0.5866(0.0423) 0.5995(0.0454) 0.6280(0.0558)
0.5297(0.0338) 0.5581(0.0359) 0.5687(0.0395) 0.5802(0.0415) 0.6066(0.0510)
0.5428(0.0362) 0.5717(0.0375) 0.5816(0.0410) 0.5934(0.0433) 0.6199(0.0528)
AL
0.9219 0.9680 0.8746 0.9986 0.9970
2.0061 2.1378 6.4410 2.3944 2.7113
0.9344 0.9628 0.9812 0.9944 0.9944
1.3010 1.3512 1.3977 1.4626 1.5231
0.9416 0.9608 0.9756 0.9922 0.9916
1.0205 1.0440 1.0649 1.0844 1.1660
0.9502 0.9634 0.9702 0.9840 0.9880
0.7331 0.7580 0.7622 0.7795 0.8447
0.5 0.5203(0.0167) 0.5316(0.0174) 0.5262(0.0167) 0.5221(0.0164) 0.5154(0.0157) 0.5217(0.0164) 0.9472 0.4975
1 0.5350(0.0165) 0.5558(0.0186) 0.5471(0.0177) 0.5413(0.0168) 0.5342(0.0161) 0.5404(0.0165) 0.9632 0.5027 100 1-5 0.5 0.5340(0.0158) 0.5609(0.0189) 0.5487(0.0180) 0.5418(0.0168) 0.5343(0.0163) 0.5413(0.0166) 0.9696 0.5028
2 0.5404(0.0170) 0.5717(0.0213) 0.5594(0.0207) 0.5512(0.0188) 0.5438(0.0185) 0.5504(0.0185) 0.9716 0.5169 4 0.5577(0.0217) 0.5990(0.0299) 0.5866(0.0294) 0.5739(0.0258) 0.5674(0.0250) 0.5725(0.0255) 0.9776 0.5664
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10 0.5
0.5 1
1.5 2 4
0.7480(0.4486) 0.8849(0.7309) 0.9648(0.8586) 1.0314(0.9580) 1.2500(2.3613)
0.4206(0.1025) 0.4884(0.1380) 0.5286(0.1485) 0.5621(0.1575) 0.6568(0.3859)
0.5093(0.1580) 0.5704(0.1908) 0.6263(0.2364) 0.6512(0.2421) 0.8041(0.6648)
0.5280(0.1919) 0.5877(0.2157) 0.6443(0.2600) 0.6718(0.2728) 0.8251(0.7489)
0.8315(0.6497) 0.9441(0.8618) 1.0448(1.0950) 1.0759(1.1039) 1.3927(3.0527)
0.6123(0.2148) 0.7115(0.3390) 0.7730(0.4007) 0.8143(0.4252) 0.9708(1.0589)
20 0.5
0.5 1
1.5 2 4
0.6214(0.1700) 0.6754(0.2220) 0.7226(0.2984) 0.7451(0.3109) 0.7940(0.4595)
0.4564(0.0725) 0.4931(0.0856) 0.5248(0.1091) 0.5400(0.1096) 0.5710(0.1594)
0.5348(0.1216) 0.5840(0.1686) 0.6145(0.2036) 0.6327(0.2048) 0.6835(0.3378)
0.5457(0.1189) 0.5932(0.1650) 0.6300(0.2045) 0.6442(0.1985) 0.6970(0.3382)
0.6761(0.2575) 0.7406(0.3659) 0.7796(0.4392) 0.8047(0.4506) 0.8773(0.7492)
0.5729(0.1198) 0.6208(0.1553) 0.6630(0.2110) 0.6790(0.2101) 0.7230(0.3191)
30 0.5
0.5 1
1.5 2 4
0.5635(0.0856) 0.6090(0.1140) 0.6233(0.1264) 0.6472(0.1460) 0.6558(0.1619)
0.4568(0.0491) 0.4915(0.0579) 0.5021(0.0626) 0.5200(0.0696) 0.5262(0.0768)
0.5153(0.0726) 0.5558(0.0986) 0.5699(0.1114) 0.5951(0.1299) 0.6018(0.1511)
0.5248(0.0710) 0.5649(0.0946) 0.5811(0.1096) 0.6052(0.1288) 0.6120(0.1465)
0.5975(0.1170) 0.6459(0.1643) 0.6631(0.1879) 0.6929(0.2193) 0.7013(0.2553)
0.5386(0.0694) 0.5806(0.0926) 0.5934(0.1025) 0.6162(0.1179) 0.6237(0.1299)
50 0.5
0.5 1
1.5 2 4
0.5431(0.0457) 0.5649(0.0533) 0.5708(0.0528) 0.5871(0.0639) 0.5911(0.0676)
0.4764(0.0316) 0.4946(0.0344) 0.4996(0.0335) 0.5130(0.0391) 0.5163(0.0413)
0.5157(0.0420) 0.5370(0.0496) 0.5442(0.0527) 0.5587(0.0616) 0.5650(0.0671)
0.5238(0.0399) 0.5460(0.0485) 0.5530(0.0504) 0.5666(0.0582) 0.5721(0.0632)
0.5615(0.0558) 0.5851(0.0674) 0.5932(0.0724) 0.6091(0.0848) 0.6164(0.0926)
0.5288(0.0395) 0.5512(0.0475) 0.5570(0.0476) 0.5716(0.0569) 0.5770(0.0607)
AL
0.9021 0.9412 0.9656 0.9838 0.9870
2.7740 3.3487 3.6730 3.9668 5.0005
0.9226 0.9380 0.9516 0.9734 0.9782
1.5587 1.7072 1.8383 1.9020 2.0525
0.9206 0.9366 0.9462 0.9616 0.9624
1.1230 1.2204 1.2543 1.3059 1.3278
0.9352 0.9448 0.9534 0.9600 0.9542
0.8225 0.8580 0.8675 0.8971 0.9009
0.5 0.5165(0.0202) 0.4820(0.0168) 0.5029(0.0207) 0.5095(0.0197) 0.5241(0.0236) 0.5102(0.0193) 0.9428 0.5450
1 0.5237(0.0203) 0.4886(0.0165) 0.5133(0.0211) 0.5179(0.0199) 0.5349(0.0245) 0.5184(0.0194) 0.9504 0.5511 100 0.5 1-5 0.5341(0.0223) 0.4979(0.0174) 0.5226(0.0227) 0.5279(0.0217) 0.5447(0.0267) 0.5283(0.0212) 0.9526 0.5642
2 0.5384(0.0231) 0.5020(0.0179) 0.5250(0.0230) 0.5312(0.0222) 0.5472(0.0270) 0.5319(0.0215) 0.9584 0.5689 4 0.5401(0.0245) 0.5035(0.0189) 0.5267(0.0243) 0.5333(0.0235) 0.5490(0.0286) 0.5339(0.0231) 0.9542 0.5704
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10 0.5
0.5 1
1.5 2 4
0.6673(0.3265) 1.0886(0.5368) 1.4210(0.7104) 1.6901(0.8701) 3.2462(3.9636)
0.8401(0.5701) 1.3732(0.9025) 1.7915(1.0438) 2.1557(1.0643) 4.1322(4.5452)
0.7561(0.3530) 1.2920(0.7307) 1.7105(0.9571) 2.0771(1.0523) 3.9455(4.4978)
0.7459(0.3447) 1.2710(0.7011) 1.6764(0.9080) 2.0381(1.0064) 3.8684(4.2918)
0.6104(0.2124) 1.0419(0.4598) 1.3752(0.6680) 1.6672(0.8659) 3.1487(3.8866)
0.6804(0.2809) 1.1508(0.5634) 1.5178(0.7507) 1.8324(0.8789) 3.5183(3.9629)
20 0.5
0.5 1
1.5 2 4
0.5911(0.1410) 1.0851(0.3558) 1.5361(0.6136) 1.9234(0.7698) 3.7479(3.0296)
0.6685(0.2076) 1.2222(0.4673) 1.7292(0.7719) 2.1675(0.9012) 4.2350(3.4680)
0.6183(0.1171) 1.1608(0.3942) 1.6564(0.6916) 2.0903(0.8724) 4.0701(3.4265)
0.6086(0.1128) 1.1436(0.3778) 1.6260(0.6685) 2.0557(0.8501) 4.0019(3.3306)
0.5557(0.0896) 1.0436(0.3130) 1.4886(0.5637) 1.8759(0.7469) 3.6457(3.0171)
0.5900(0.1039) 1.1061(0.3461) 1.5723(0.6165) 1.9870(0.7958) 3.8748(3.1226)
30 0.5
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1.5 2 4
0.5717(0.0816) 1.0743(0.2605) 1.5641(0.4701) 1.9950(0.6302) 3.9617(2.5757)
0.6169(0.1016) 1.1607(0.3147) 1.6897(0.5472) 2.1572(0.7043) 4.2871(2.8736)
0.5886(0.0696) 1.1155(0.2511) 1.6331(0.4934) 2.0966(0.7009) 4.1582(2.7998)
0.5813(0.0685) 1.1002(0.2447) 1.6088(0.4780) 2.0667(0.6790) 4.0980(2.7071)
0.5487(0.0573) 1.0393(0.2135) 1.5219(0.4264) 1.9525(0.6221) 3.8725(2.5033)
0.5721(0.0656) 1.0819(0.2347) 1.5833(0.4543) 2.0341(0.6486) 4.0339(2.6208)
50 0.5
0.5 1
1.5 2 4
0.5393(0.0373) 1.0420(0.1413) 1.5299(0.2733) 2.0063(0.4690) 4.0068(1.8972)
0.5636(0.0414) 1.0903(0.1542) 1.6014(0.2941) 2.1016(0.4961) 4.1955(2.0007)
0.5522(0.0379) 1.0657(0.1376) 1.5685(0.2765) 2.0582(0.4837) 4.1190(1.9678)
0.5457(0.0368) 1.0532(0.1358) 1.5505(0.2730) 2.0352(0.4692) 4.0726(1.9180)
0.5297(0.0338) 1.0219(0.1250) 1.5038(0.2540) 1.9732(0.4503) 3.9486(1.8315)
0.5428(0.0362) 1.0470(0.1329) 1.5416(0.2676) 2.0240(0.4639) 4.0461(1.8783)
AL
0.9219 0.8852 0.8530 0.8380 0.7940
2.0061 3.2606 4.2231 5.0404 9.5658
0.9344 0.9110 0.8800 0.8742 0.8592
1.3010 2.3593 3.3108 4.1137 8.0157
0.9416 0.9234 0.9028 0.8992 0.8880
1.0205 1.9081 2.7595 3.4823 6.9524
0.9502 0.9324 0.9228 0.9120 0.9094
0.7331 1.4151 2.0756 2.7205 5.3994
0.5 0.5203(0.0167) 0.5316(0.0174) 0.5262(0.0167) 0.5221(0.0164) 0.5154(0.0157) 0.5217(0.0164) 0.9472 0.4975
1 1.0314(0.0626) 1.0541(0.0651) 1.0440(0.0639) 1.0368(0.0626) 1.0226(0.0605) 1.0359(0.0620) 0.9350 0.9803 100 0.5 1-5 1.5243(0.1433) 1.5584(0.1472) 1.5423(0.1442) 1.5319(0.1424) 1.5105(0.1379) 1.5308(0.1413) 0.9526 0.5642
2 2.0210(0.2438) 2.0665(0.2494) 2.0475(0.2470) 2.0310(0.2421) 2.0053(0.2368) 2.0293(0.2416) 0.9316 1.9212 4 4.0084(0.9497) 4.0983(0.9621) 4.0687(0.9915) 4.0342(0.9895) 3.9851(0.9550) 4.0317(0.9820) 0.9282 3.7997
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5. Real data application
In this section, the applicability of the proposed flexible extension has been discussed based on two survival data sets. The data set-I is taken from [6] which represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli. The second data set has been obtained from [16] which consists of survival times, in week, of 33 patients suffering from acute Myelogenous Leukaemia. The summary of both data sets is given in Table 5. Further, to show the superiority of the proposed model, the following well-known lifetime models are taken.
1. Extension of exponential distribution with pdf
f (x) = aA(1 + Ax)a-1 e(1-(1+Ax)a) ; x > 0, a > 0, A > 0
2. Weibull distribution with pdf
a 1 ( x \a
f (x) = — xa-1 e-( A) ; x > 0, a > 0, A > 0
3. Gamma distribution with pdf
Aa
f (x) = xa-1e-Ax ; x > 0, a > 0, A > 0
r (a)
where, a is shape and A is the scale parameter.
The superiority of the proposed extension with the above considered families of the distribution are shown with the help of model criterion tools. Hence, the criterion like p-value, Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), and Kolmogorov-Smirnov (KS) test are taken.
To compare the models, Table 6-7, contains the values of the parameters estimated by the maximum likelihood, AIC, BIC, and KS statistics with the p-value for fitted data sets. From the K-S test statistics or associated p-value, it may be seen that the proposed model provides better fit than Weibull, Gamma, and Extension of Exponential models. Also, a similar result is concluded on the basis of negative of Log-likelihood which is higher than other three. Also based on the relative model selection criteria it is observed that the proposed model has smaller AIC and BIC in comparison to other thee considered models. Hence, the proposed mode is more suitable for considered real phenomena.
Further, the plots of the empirical cumulative distribution function (ECDF) and the fitted CDF for the considered two data set are shown in Figure 2. From Figure 2, it is concluded that the proposed model fits better to considered real data in comparison to other competitive models. Hence it may be taken as the alternative to the several lifetime models.
Table 5: Summary of the considered data sets.
Data Min. Q1 Median Mean Q3 SD Skewness Kurtosis Max.
I 0.080 1.080 1.560 1.837 2.303 1.215 1.754 7.151 7.000
II 1.00 4.00 22.00 40.88 65.00 46.703 1.164 3.122 156.00
6. Conclusion
In this article, we have introduced a new transformation technique to generate the class of lifetime distributions. Further, the purposed transformation technique is illustrated via exponential distribution as baseline distribution and named a new flexible two-parameter lifetime distribution. The
Table 6: MLE, AIC, BIC and KS statistics with the p-valuefor the data set-I.
ML Estimates KS-Test
Distributions a  -LogL Statistics p-value AIC BIC
Proposed Weibull Gamma EE 8.2801 1.6172 2.4379 5.0262 0.1257 2.0558 1.3273 0.0731 102.4141 104.0168 102.9648 109.6485 0.09658 0.11346 0.10372 0.19617 0.5127 0.3121 0.4207 0.0078 208.8283 212.0336 209.9296 223.2970 213.3816 216.5869 214.4829 227.8504
Table 7: MLE, AIC, BIC and KS statistics with the p- value for the data set-II.
ML Estimates KS-Test
Distributions a  -LogL Statistics p-value AIC BIC
Proposed Weibull Gamma EE 0.2659 0.7764 0.6877 0.4897 0.0763 35.3613 0.0168 0.0998 151.7045 153.5868 153.6737 153.7430 0.10004 0.13668 0.13900 0.13920 0.8959 0.5684 0.5466 0.5440 307.4091 311.1737 311.3473 311.4860 310.4021 314.1667 314.3403 314.4790
ECDF plot for dataset 1 ECDF plot for dataset 2
(a) (b)
Figure 2: (a) ECDF and fitted CDF plot of proposed distribution for data 1. (b) ECDF and fitted CDF plot of proposed distribution for data 2.
proposed distribution has an increasing, decreasing, and constant hazard nature see Figure 1(b). Next, the different distributional properties are derived viz. mean, moments, moment generating function (MGF), mean deviation about mean and median, Bonferroni and Lorenz curves, Renyi entropy, s-entropy, cumulative residual entropy, rth order statistics, and reliability. The unknown parameter of the proposed model is estimated by different methods of estimation namely MLE, MPS, LSE, WLSE, CVME, and ADE. To compare the performances of different estimators obtained via different estimation methods Monte Carlo simulation study has been performed in Table 1,2,3 and 4. The superiority of the present study and model has been illustrated by constructing two real data sets. From the Table 6,7 and Figure 2, It is observed that the proposed new flexible two-parameter lifetime distribution provides better fits to the considered data sets among the
most popular distributions viz., Weibull, Gamma, and extension of exponential distribution.
Therefore, we may conclude that the proposed model might be considered an alternative to other considered models.
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