CC BY
101
A new method for generating distributions with an application to Weibull distribution
M. A. Lone •
Department of Statistics, University of Kashmir, Srinagar, India [email protected]
I. H. Dar •
Department of Statistics, University of Kashmir, Srinagar, India [email protected]
T. R. Jan*
Department of Statistics, University of Kashmir, Srinagar, India [email protected]
Abstract
In the literature of probability theory, it has been noticed that the classical probability distributions do not furnish an ample fit and fail to model the real-life data with a non-monotonic hazard rate behaviour. To overcome this limitation, researchers are working in the refinement of these distributions. In this paper, a new method has been presented to add an extra parameter to a family of distributions for more flexibility and potentiality. We have specialized this method to two-parametric Weibull distribution. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, reliability function, hazard rate function, the r-th moment, moment generating function, and also the order statistics. Moreover, we discussed mean residual life time, stress strength reliability and maximum likelihood estimation. The adequacy of the proposed distribution is supported by using two real lifetime data sets as well as simulated data.
Keywords: Weibull distribution, hazard rate function, survival function, mean residual life, Maximum likelihood estimation.
1. Introduction
Weibull distribution is a well known life time distribution in reliability engineering and failure analysis. The Weibull distribution is used in modelling the engineering, biological, weather forecasting and hydrogical data sets. It does not impart an admissible fit for some applications, espacially, when the hazard rates are bathtub, upside down bathtub, or bimodal shapes. To overcome these limitations, several researchers have developed various modifications and extensions of the Weibull distribution to model various types of data. Many extentions and generalizations of the Weibull distribution have accomplished the above purpose. Among these, Xie and Lai [1] introduced the additive Weibull distribution, Mudholkar et al. [2] proposed exponentiated Weibull (EW) distribution by adding an extra parameter to the Weibull distribution
which provides bathtub shaped hazard rate function. Xie et al. [3] proposed the the extended Weibull distribution. Carrasco et al. [4] presented generalized modified Weibull (GMW) distribution. Modified Weibull by Lai et al. [5]; extended flexible Weibull by Bebbington et al. [6]. The exponential-Weibull distribution by Cordeiro et al. [7]. Lee et al. [8] and Alzaatreh et al. [9] proposed methods of generalized continuous and discrete distributions.
Mahdavi and Kundu [10] proposed a method called the Alpha Power Transformation (APT) and it is useful to assimilates skewness to a family of distributions. Let F(x) be the cumulative distribution function (cdf) of a continuous random variable X, then they define the APT of F(x) for x € R as follows
Fapt<x, = {iff;a = 1 [F(x) ;a = 1
and the corresponding probability density function (pdf) as
Zapr(x) = [ ^f <x)aF<x) ; * G R+ « = 1
\ f (x) ; a = 1
They applied the proposed method to a one-parameter exponential distribution and generated a two-parameter Alpha Power Exponential distribution.
Recently, Ijaz et al. [11] proposed a new family of distributions named as New Alpha Power Trasformed family (NAPT) of distributions. They employed exponential distribution in NAPT family and derived a new distribution called New Alpha Power Trasformed exponential (NAPTE) distribution. Let F(x) be the cdf of a continuous random variable X, then they define the NAPT of F(x) for x € R as follows
Fnapt(x) = a-'0^^) ; a > 0
and the corresponding pdf as
fNAPT(x)= '«W'-y}f(x) ; « > 0
The following are the primary motivations for disposing Ratio Transformation (RT) method in practise:
• A straightforward and efficient method for adding an extra parameter to an existing distributions.
• To enhance the characteristics and flexibility of existing distributions.
• It is quite easy to use, hence it can be used quite effectively for data analysis purposes.
• To present the extended version of the baseline distribution that includes closed forms of cdf, reliability function as well as hazard rate function.
• To provide better fits than the other modified models having the same or higher number of parameters.
The remainder of the paper is organized as follows: In section 2 a new family of probability distributions called RT has been highlighted and some general properties of this family have been discussed. In section 3, RTW distribution has been considered, some special cases are presented and its structural properties including moments, moment generatin function, mean residual life and mean waiting time, order statistic and stress-strength reliability have been discussed. In section 4, Maximum likelihood estimators of unknown parameter as well as simulation study have been carried out. In secton 5, Two real life data sets have been analyzed to illustrate the potency of the proposed model. Finally, the paper is concluded in section 6.
2. General properties of RT method
Let F(x) be the cdf of a continuous random variable X, then the Ratio transformation of F(x) for x € R, is defined as follows
Frt(x) F(x) F(x); a > o (i)
1 + a — aF(x)
Clearly, FRT(x) is a proper cdf. If F(x) is an absolute continuous distribution function with the
pdf f (x), then FRT(x) is also an absolute continuous distribution function with the pdf
(l + a — aF(x)(1 — F(x)loga)) fRT(x) = f (x)±---—2-; a > 0 (2)
(1 + a — aF(x))
A useful expansion for the cdf and pdf in (1) and (2) are respectively given by
TO TO
Frt (x) = EE j (F(x))k+1 (3)
j=0 k=0
where,
= (j l°ga)k
Ujk k! (1 + a)j+1 and _ _
fRT(x) = f (x)
aF(x) to to
1 — TTa (1 — F(x)l°ga)
EE bjkFk (x) (4)
j=0 k=0
where,
= (j + 1)(j l°ga)k
bjk (1 + a)j+1 k!
The reliability function Rrt(x) is given by
D / \ 1 + a — aF(x) — F(x)
Rrt(x) =-—-w-^- ; a > 0 (5)
1 + a — aF(x)
The hazard rate function hRT(x) is given by
(1 + a — aF(x)(1 — F(x)loga))
hRT(x) = f (x) -F(x) w -W^—; a > 0 (6)
(1 + a — at(x)) (1 + a — aF(x) — F(x))
If R(x) and h(x) are the reliability and hazard rate functions of f respectively, then the hazard rate hRT(x) can be written as
(1 + a — aF(x)(1 — F(x)loga))
hRT (x) = h(x)R(x) ^--F(x) w x x ; a > 0 (7)
(1 + a — at(x)) (1 + a — aF(x) — F(x))
From (7), it is clear that
1
lim hRT(x) = - lim h(x)
x—^—to a x ?— TO
and,
lim hRT(x) = lim h(x)
x—TO^ x—TO
3. RTW distribution and its properties
Let © = (a, A, ft)T. From (2), The continuous randon variable X follows RTW distribution if its cdf, with scale parameter A > 0 and shape parameters a > 0, ft > 0, for x € R+ is given by
1- e-AxP
Frtw(x, ©) =-^ ; a > 0 (8)
1 + a — a1 e x
and the corresponding pdf is
Aftxft—1e—Axf U + a — a1—e—Aj (1 — (1 — e—AxP)'oga)) Zrtw(x,©) =-^----2-J- ; a > 0 (9)
(1 + a — al—e—Ax^
Using (3) and (4), the cdf and pdf in (8) and (9) can be respectively written as
X X k+1 ft
Frtw (x, ©) = EEE j e—1Ax'
where, and
Zrtw (x, ©) = x ft—1
where,
ajk'
j=0k=0'=0
(j'oga)k (k+ 1)(—1)' k! (1 + a)j+1
(1—e—Ax?\
1 — ^ 1 + a (1 — log* (1 — e—Axft))
XX k
EEl^b^
j=0 k=0 '=0
= Aft(j + 1)(j 'oga)k(k)(—1)' bjk' = (1 + a)j+1 k!
The reliability function Rrtw(x, ©) and the hazard rate function hRTW(x, ©) for x € R+ are, respectively, given by
J1 — a—e—Axft) + e—Axft
Rrtw(x, ©) = ^-^-; a > 0 (10)
1 + a — a1 e x
Aftxft—1 e—Axft (1 + a — a1—e—Axii (1 — (1 — e—Axft )'oga) hRTW (x, ©) ^ ; a >0
The behaviour of the hazard rate function at extremes for different values of shape parameter ft.
fx for 0 < ft < 1, (0 for 0 < ft < 1,
h(0) = f a for ft = 1, h(x) = f A for ft = 1, [0 for ft > 1, [x for ft > 1.
Remark: When a = 1, the RTW distribution becomes the Weibull distribution. In that situation
the shapes for hazard rate function are conspicuous in the literature. The seven important special cases of RTW distribution are presented in table 1
Figure 1 depicts some plots of the RTW density for selected parameter values. Plots of the hazard rate function of the RTW distribution for selected parameter values are displayed in Figure 2.
Table 1: Sub-cases of the RTW Distribution
a A ft Reduced model
- 1 - RT one-parameter Weibull distribution 1 - - Two-parameter Weibull distribution
1 1 - One-parameter Weibull distribution
- - 2 RT-Rayleigh distribution 1-2 Rayleigh distribution
- - 1 RT-exponential distribution 1-1 Exponential distribution
a = 0.5, b = 0.9 a = 6, b = 09 a = 20, b = 09 a = 6, b= 1.1 a = 30, b= 11
a = 0.5, b= 2 a = 1.5, b= 2 a = 5, b= 2 a = 20, b= 2 a = 25, b= 1.8
0
2
4
6
8
10
0
2
3
4
x
x
Figure 1: Plots of the RTW density for A = 1 and various values of a and ft.
x x
Figure 2: Plots of the RTW hazard rate function for A = 1 and various values of a and ft.
3.1. Moment and moment generating function
In this subsection, the rth moment and the moment generating function of the RTW distribution are obtained by using the following series representations.
E
k=0
TO
( - loga)kx
kvk
k!
(1 - x)-2 = E (k + 1)xk ; |x| < 1,
k=0 TO
(1 - x)-1 = E xk ; |x| < 1,
k=0
The rth moment of X can be obtained as
TO
E(Xr) = J xrf (x)dx
0
TO
j xrA?x?-1 e-Ax? + a - (1 - (1 - e-Ax? )loga^
(1 + a)2
1
By substituting 1 - e Ax? = y in (14), we get
1 + a
dx
E(xr ) = |o (t+OFT U ,0«(1 - (a'y +
a(j+1)y (j + 1)loga
1 + a
y dy
Again, substituting -¡Tlog(1 - y) = x in (15), we get the final expression as
~ Aaj(-loga)k, r _
E(Xr) = E E ^ ( ^,+1 ) r( « + 1) {A + B}
jo k=o (1 + a)j+1k! ^
where,
and
A
jk
(A(k +1)) ?
i+1
a loga (j + 1)k+T 1 + a
(A(k + 1))?+T (A(k + 2))?+\
and the moment generating function can be obtained as
X
MX (t) = I etxf (x)dx
(11) (12) (13)
(14)
(15)
by using the same procedure as above, we get the final expression for moment generating function
as
TO TO TO
E E E ¡fa>(-loga)k i_ + 1) +
i=ok=0 (1 + a)j+Tk!z! 1(? + T) + D}
—x
a
2
1
1
B
where,
and
C
(A(k + 1)) ?+1
D
loga (j + 1)k+1
1 + a \(A(k + 1))i+1 (A(k + 2))ft+1
3.2. Mean residual life and mean waiting time
Suppose that X is a continuous random variable with reliability function R(x) , the mean residual life is the expected additional lifetime given that a component has survived until time t. The mean residual life function, say p.(t) , is given by
v(t)
R(t)
E(t) — J xf (x)dx I — t
(16)
where
~ ~ Aaj(—loga)k, 1 I ik
E(t)=E E Aaj r( ft+1) ((j
+1
+
a loga (j + 1)k+1 1 + a
1
1
(A(k + 1))ft+1 (A(k + 2))ft
+1
(17)
and
1
Jx = EE jji 1 1
0 j=0k=0 (1 + a)'+'k! I A ft (k +1) 1
ft+1 Y (A(k + 1)tft,1 + 1)
+
(j + 1)loga 1 + a
j(A(k + 1)tft ,1 + 1) j(A(k + 1)tft ,1 + 1
Aft (k + 1)ft
+1
Aft (k + 2)ft
+1
Substituting (10), (17) and (18) in (16), p(t) can be written as
( ) = 1 + a — a1—e—t E E aj(—loga)k F(t) = a (1 — a——tft — + e—Atft ¿0¿0 (1 + a)j+1k!
where,
x A' + B'
jk
r( ft +1)
(C' + D' ) — t
(18)
A'
jk
(A(k + 1)) ft+1
a loga (j + 1)k+1
1 + a
1
(A(k + 1))ft+1 (A(k + 2))ft
C'
7(A(k + 1)tft + 1)
Aft (k + 1)ft
+1
1
+1
k
1
1
a
1
x
B
and
D, = (j + ^ogfl 1 + a
Y (A(k + 1)t?,? + l) y (A(k + 1)t?,? + l)
A? (k + 1) i^1 A? (k + 2) ?+1
where y(a, b) = / xb 1 e xdx is the lower incomplete gamma function. 0
The mean waiting time represents the waiting time elapsed since the failure of an object on condition that this failure had occurred in the interval [0, t]. The mean waiting time of X, say
fl(t), is defined by
t
f (t) = t - f!)/xf (x)dx. (19)
Substituting (8) and (18) in (19), we get
1 + a - a1-e-At? ~ ~ ajjk(-1oga)k
f (t) =t--- ^At? E E j ( g )
1 - e-Atl j=6k=0 + a)j+1k! \ A? (k + 1) ?+1
Y (A(k + 1)t?,? + 1
1 , , (j + 1)1oga
y|V + M? + l) + i + a Y (A(k + 1)t?,? + l)'
A ? (k + 1) ? +1
A? (k + 2)1 +
3.3. Order Statistics
Let X1, X2,..., Xn be a random sample of size n, and let Xr:n denote the rth order statistic, then,
the pdf of Xr:n, say fr:n (x) is given by
n' 1
fr:n(x) = (r _ ^ - r)' F(x) f (x)(1 - F(x))n-r. (20)
Substituting (8) and (9) in (20), we get
A?x1-1e-Axl + a - a1-e-Ax? (l - (1 - e-Axl )1oga
fr:n (x)
B(r,n - r + 1) (l + a - a1-e-Axl) x (i - e-Ax?)r-1 (a (i - a-e-Ax? ^ + e^1
n+1
where B(a, b) is the beta function.
3.4. Stress Strength Reliability
Suppose X1 and X2 be independent strength and stress random variables respectively, where X1 - RTW(a1, A1,?) and X2 - RTW(a2, A2,?), then the stress strength reliability P(X1 > X2), say SSR, is defined as
SSR = J f1 (x)F2(x)dx
x
nr
Table 2: Average values ofMLEs and the corresponding MSEs(n=50).
Parameter
A a ß
A
MLE
Si
A
MSE
a
1 0.5
1.5
2 0.5
1
1.5 2
1
1.5 2
1
1.5 2
1
1.5 2
1
1.5 2
1
1.5 2
1.5
1.18927 1.22622 1.18561
1.08698 1.12510 1.13204
1.04126 1.08026 1.07287
0.98987 0.98794 0.98397
2.28613 2.16929 2.23597
2.17700 2.21341 2.15140
0.75566 0.87384 0.77796
1.18120 1.27644 1.28600
1.72249 1.78405 1.81606
2.07177 2.19104 2.19145
0.68043 0.57581 0.60438
1.19918 1.27536 1.33614
0.74107 1.51407 1.98113
1.04943 1.55446 2.04414
1.05876 1.54105 2.12707
1.07668 1.60992 2.15508
1.02709 1.57420 2.09259
1.04533 1.55452 2.09166
0.27647 0.40466 0.24700
0.23511 0.24403 0.29381
0.27080 0.28238 0.27423
0.24656 0.22690 0.22988
0.30981 0.19697 0.30316
0.51708 0.49046 0.55797
0.71977 0.58467 0.61990
1.10633 1.12121 1.45360
2.20896 2.09977 1.89261
2.17639 2.61831 2.63456
0.33733 0.27864 0.31276
1.12488 1.11322 1.57897
0.04782 0.07297 0.11475
0.03877 0.09897 0.17665
0.05698 0.10365 0.24845
0.07918 0.17018 0.31296
0.03757 0.07851 0.11952
0.03843 0.09883 0.15442
1 2.05912 1.74110 1.05653 0.55367 2.22683 0.05775 1.5 2.00179 1.58862 1.61057 0.45244 1.48239 0.15847
2 1.95160 1.56072 2.18486 0.52078 2.38803 0.27118
1
1.5 2
1.99288 2.01983 1.99596
2.15485 2.26075 2.23927
1.09058 1.58730 2.14504
0.58386 0.53255 0.58591
2.18773 2.56440 3.79841
0.10266 0.15576 0.27387
1
2
1
2
The stress strength reliability SSR, is obtained by using (8), (9), (11), (12) and (13) and is given by
-A f f f f *2km(-logon)(-loga2)m f (, + a1loga1 (j + 1)l+1 \ 1 jf k k h (1 + aj1 (1 + «2)Wl!m! \ [j + 1 + «1 )
x_A2_
[(l + 1)A1 + mA2 ] [(l + 1)A1 + (m + 1)A2 ]
_A2«1 loga\( j + 1)l+1_1
(1 + «1 )[(l + 2)A1 + (m + 1)A2] [(l + 2)A1 + mA2] |
Table 3: Average va/wes ofMLEs and the corresponding MSEs(n=100).
Parameter
A a ß
A
MLE
A
MSE
a
1 0.5
1.5
2 0.5
1
1.5 2
1
1.5 2
1
1.5 2
1
1.5 2
1
1.5 2
1
1.5 2
1.5
1.12427 1.10945 1.03860
1.09751 1.09006 1.07148
1.04769 1.05555 1.05153
0.96890 1.02776 1.02772
2.20157 2.15936 2.20967
2.14197 2.12690 2.22353
0.85980 0.64163 0.51790
1.20162 1.14954 1.14295
1.66319 1.68631 1.64619
2.04959 2.05201 2.04339
0.64279 0.60787 0.64716
1.15711 1.14278 1.21377
1.01394 1.49813 2.00760
1.00568 1.52710 2.07225
1.01968 1.54287 2.06356
1.05881 1.52135 2.02851
1.00275 1.50924 2.00121
1.03218 1.55439 1.98709
0.16520 0.08311 0.04648
0.13615 0.14385 0.17511
0.17384 0.19368 0.17319
0.19612 0.15814 0.15842
0.20282 0.18855 0.25218
0.35039 0.36045 0.51312
0.36686 0.15675 0.11094
0.44997 0.47309 0.69167
1.08369 1.20057 1.02013
1.91723 1.90158 1.92072
0.21313 0.18872 0.27663
0.70753 0.69360 1.10397
0.02039 0.04790 0.08922
0.02630 0.05824 0.12353
0.02691 0.07301 0.14828
0.04225 0.04479 0.07969
0.01958 0.03718 0.07502
0.03321 0.06954 0.09771
1 2.09963 1.74911 1.03669 0.50675 1.89731 0.03895 1.5 2.05345 1.67112 1.55151 0.33247 0.93642 0.06657
2 2.06760 1.68920 2.06563 0.37754 1.67294 0.15132
1
1.5 2
1.92617 2.01885 1.99881
2.06013 2.20664 2.13749
1.05836 1.55748 2.07929
0.42906 0.40409 0.41542
1.98756 1.79326 2.86112
0.04258 0.08481 0.14579
1
2
1
2
4. Statistical Inference
4.1. Maximum Likelihood Estimators
Let X1, X2,..., xn be a random sample from RTW distribution, then the logarithm of the likelihood function is
/ = n/oga + n/ogß + (ß - 1) E /ogxi - Aß E xi - 2 E log I 1 + a - a
i=1 i=1 i=1 \
.. -Axß 1-e '
+ Elog
i=1
1 + a - a
—AXß/ . ß
1-e ' '1 - /oga (1 - e ;)
(21)
n
n
n
n
The MLEs of a, A and ft are obtained by partially differentiating (21) with respect to the corresponding parameters and equating to zero, we have
da _ L
P
1 + (1 - e-Xxi )2a-e ' loga
i_1 1 + a - a1-e~Xx' - (1 - e-Xxß)loga^
- 2 E
1 - (1 - e-Axi )a
> ß , -lxß
1 1 —l^x -
i_1 1 + a - a1-e '
dß _ ß +(1 - A) Ex +aXß logaE
" ß-1 -Xxß -e-lxß
xI e Axi a e i_1 i
(1 - e-Xxi )loga
1 + a - a1-e-Ax' 1 + a - a1-e-Axß - (1 - e-Xxß)logaj
— _ -+ß l xi- aloga E xie-
I a-
i_1
i_1
P^
1 + a - a1
(1 - e-lxi )loga
1 + a - a1-e~Ax' - (1 - e-Xxß )loga^j
(22)
(23)
(24)
The above three equations (22),(23) and (24) are not in closed form. thus, it is difficult to calculate the values of the parameters a, ft and A. However, R software can be used to get the MLE.
4.2. Simulation study
a
X
2
fi
Figure 3: (i) The relative histogram and the fitted RTW distribution. (ii) The fitted RTW reliability function and empirical reliability function for first data set.
The simulation study has been performed using R Software to show the behaviour of the MLEs in terms of the sample size n. Two sets of sample (n=50, n=100) each replicated 100 times with different values of parameters A — (1,2), a — (0.5,1,1.5,2) and ft — (1,1.5,2) were
(ii)
Empirical Reliability Function Fitted Reliability Function
Figure 4: (i) The relative histogram and the fitted RTW distribution. (ii) The fitted RTW reliability function and empirical reliability function for second data set.
(i)Q-Q Plot for RTW distribution
(ii)Q-Q Plot for RTW distribution
0.8 1.0 1.2 1.4 1.6 1.!
Theoretical Quantile
0 20 40 60 80 100 120
Theoretical Quantile
Figure 5: Q-Q plotfor the RTW distribution for data set first and data set second, respectively.
generated from RTW. In each setting, the average values of MLEs and the corresponding empirical mean squared errors (MSEs) were obtained. The simulation results are presented in table 2 and table 3. From tables 2 and 3, it can be seen that the estimates are stable and quite close to the true parameter values. As the sample size increases the MSE decreases in all the cases.
5. Applications
In this section, we analyse two data sets to describe the significance and flexibility of the RTW distribution. The data set first reported by Nassar et al. [12], orginally published by Smith and Naylor [13], corresponding to strengths of 1.5 cm glass fibers, measured at the National Physical Laboratory, England. The data are as follows: 0.55, 0.93,1.25,1.36, 1.49,1.52,1.58,1.61, 1.64, 1.68, 1.73, 1.81, 2, 0.74, 1.04, 1.27, 1.39, 1.49, 1.53,1.59, 1.61, 1.66, 1.68, 1.76, 1.82, 2.01, 0.77, 1.11, 1.28,
(i)
0
20
40
60
80
0
20
40
60
80
x
x
u
u
(i)P-P Plot for RTW distribution
(ii)P-P Plot for RTW distribution
0.2 0.4 0.6 0.8
Theoretical Cumuative Distribution
0.2 0.4 0.6
Theoretical Cumuative Distribution
Figure 6: P-P plotfor the RTW distribution for data set first and data set second, respectively. Table 4: MLEs (standard errors in parentheses), K-S Statistic, and p-valuesfor the first data set.
Model a Estimates ß A Statistics K-S p-value
RTW 9.49959 (6.00647) 3.261905 (0.69075) 0.72053 (0.40517) 0.08745 0.72090
APW 10.86178 (12.72527) 4.48322 (0.76269) 0.19483 (0.10826) 0.12249 0.30090
APIW 193.05946 (267.40709) 3.87688 (0.30960) 0.63654 (0.1823435) 0.21627 0.00551
MW 0.03088 (0.04349) 6.37442 (0.96544) 0.04087 (0.02476) 0.13341 0.21210
TW 0.92496 (0.21931) 5.97478 (0.74495) 1.80960 (0.07553) 0.15191 0.10920
LW 0.53504 (0.48673) 4.94433 (0.65927) 0.77920 (0.18296) 0.13673 0.18950
ZBLL 0.25140 (0.06121) 18.41002 (3.05420) 1.82436 (0.04629) 0.13053 0.23330
APE 145351 (23726.57) - 2.15458 (0.09901) 0.22099 0.00425
W - 5.77962 (0.57515) 0.05978 (0.02047) 0.15232 0.10750
1.42, 1.5, 1.54, 1.6, 1.62, 1.66, 1.69,1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.5, 1.55, 1.61, 1.62, 1.66, 1.7, 1.77,1.84, 0.84, 1.24, 1.3,1.48,1.51,1.55,1.61,1.63, 1.67, 1.7,1.78,1.89.
The second data set was reported by Elbatal et al. [14], orginally published by Aarset [15], which represents the failure times of 50 devices. The data are as follows: 0.1, 0.2,1,1,1,1, 1, 2, 3,
Table 5: -21(d), AIC, AICC, BICfor the first data set.
Model -2l(S) AIC AICC BIC
RTW 22.16977 28.16977 28.57655 34.59917
APW 26.94826 32.94826 33.35504 39.37766
APIW 75.77237 81.77237 82.17915 88.20177
MW 29.78938 35.78938 36.19616 42.21878
TW 30.28635 36.28635 36.69313 42.71576
LW 28.42141 34.42141 34.82819 40.85081
ZBLL 24.23729 30.23729 30.64407 36.66669
APE 67.56511 71.56511 71.76511 75.85138
W 30.41369 34.41369 34.61369 38.69995
Table 6: MLEs (standard errors in parentheses), K-S Statistic, and p-valuesfor the second data set.
Estimates Statistics
Model a (6 X K-S p-value
RTW 6.28982 0.71267 0.17523 0.16014 0.15390
(2.80293) (0.12226) (0.10967)
APW 4.51340 0.83571 0.05854 0.17492 0.09379
(4.01925) (0.13558) (0.03910)
APIW 62.22037 0.59918 1.14499 0.27478 0.00105
(86.31937) (0.05672) (0.39802)
MW 0.01863 0.37305 0.04043 0.19432 0.04583
(0.00375) (0.18838) (0.03113)
TW 0.00010 0.94905 44.91508 0.1928 0.04860
(0.42067) (0.12873) (12.90900)
LW 0.91774 0.88097 0.04050 0.18488 0.06555
(0.69388) (0.12668) (0.04259)
ZBLL 20.23812 2.25295 0.00273 0.23307 0.00874
(4.33771) (0.46228) (0.00091)
APE 2.64622 - 0.02687 0.17657 0.08851
(1.90895) (0.00474)
W - 0.94770 0.02719 0.19313 0.04800
(0.11778) (0.01375)
6, 7, 11,12,18,18,18,18,18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 60, 63, 63, 67, 67, 67, 67, 72, 75, 79, 82, 82, 83, 84, 84, 84, 85, 85, 85, 85, 85, 86, 86.
We compare the fit of the proposed RTW distribution with its sub-model Weibull (W) distribution and with several other competitive models, namely Alpha Power Weibull (APW) (see [12]), Alpha Power Inverse Weibull (APIW) (see [16]), Modified Weibull (MW) (see [17]), Transmuted Weibull (TW) (see [18]), Lindley Weibull (LW) (see [19]), Zografos-Balakrishnan log-logistic
Table 7: -2/(6), AIC, A1CC, BIC/or the second data set.
Model -2/(6) AIC AICC BIC
RTW 470.2143 476.2143 476.7360 481.9504
APW 479.2431 485.2431 485.7648 490.9791
APIW 519.9063 525.9063 526.4280 531.6423
MW 478.9685 484.9685 485.4902 490.7045
TW 482.0043 488.0043 488.5261 493.7404
LW 479.5173 485.5173 486.0390 491.2534
ZBLL 517.3178 523.3178 523.8396 529.0539
APE 480.5838 484.5838 484.8391 488.4078
W 482.0038 486.0038 486.2591 489.8278
(ZBLL) (see [20]), and Alpha Power Exponential (APE) (see [10]), their correspinding density functions for x > 0 are as follows
APW /(x) = ^A"a1-e-Ax"x"-1e-fx"
APIW /(x) = ^A"x-("+a)e^x-V-^ MW /(x) = (a + A"x"-1)e-ax-Ax"
TW /(x) = x(D "-1 e-(f)' (l - a + 2ae-(A f
LW /(x) = A"x"-1 + A2"x2"-1e-a(Ax)"
a +1
ZBLL /(x) = a4)x"-1 (1 + (A)")-2 (log (1 + (A)")) -1
APE /(x) = a°-1 Ae-fxa1-e-Ax
CO
where a,",A > 0 and r(a) = f xa-1 e-xdx is the gamma function.
0
From Table 4, Table 5, Table 6 and Table 7, it is evident that RTW distribution has lowest -2/(6), AIC, AICC, BIC, K-S values and highest p-value among all the other competitive models. Hence the proposed model yeilds the better fit than the other models for both data sets.
The relative histogram and the fitted RTW distribution of the data set first and second are shown in Figures 3(i) and 4(i), respectively. The plots of the fitted RTW reliability function and empirical reliability function of the data set first and second are shown in Figures 3(ii) and 4(ii), respectively. The Q-Q plots for data set first and second are shown in Figure 5(i) and 5(ii) respectively. Also, The P-P plots for data set first and second are shown in Figure 6(i) and 6(ii) respectively that allows us to differentiate between the empirical distribution of the data with the RTW distribution. These graphical goodness of fit measures clearly support the results in
Tables 4, Table 5, Table 6 and Table 7.
6. Conclusion
A new family of distributions has been introduced called RT method. RT method has been specialized on the two-parameter Weibull distribution and a new three-parameter RTW distribution has been introduced. We have discussed various properties of RTW distribution. It has been realized that the three-parameter RTW distribution has more flexibility in terms of the hazard rate function and the density function. The effectiveness of the proposed model is compared with other existing models by using goodness of fit measures. The model has been fitted to two different real life data sets, the figures show that the proposed model provides better fit for both data sets in comparison to all other competitive models.
References
[1] Ming Xie and Chin Diew Lai. Reliability analysis using an additive weibull model with bathtub-shaped failure rate function. Reliability Engineering & System Safety, 52(1):87-93, 1996.
[2] Govind S Mudholkar, Deo Kumar Srivastava, and Marshall Freimer. The exponentiated weibull family: A reanalysis of the bus-motor-failure data. Technometrics, 37(4):436-445,1995.
[3] Min Xie, Y Tang, and Thong Ngee Goh. A modified weibull extension with bathtub-shaped failure rate function. Reliability Engineering & System Safety, 76(3):279-285, 2002.
[4] Jalmar MF Carrasco, Edwin MM Ortega, and Gauss M Cordeiro. A generalized modified weibull distribution for lifetime modeling. Computational Statistics & Data Analysis, 53(2):450-462, 2008.
[5] CD Lai, Min Xie, and DNP Murthy. A modified weibull distribution. IEEE Transactions on reliability, 52(1):33-37, 2003.
[6] Mark Bebbington, Chin-Diew Lai, and Ricardas Zitikis. A flexible weibull extension. Reliability Engineering & System Safety, 92(6):719-726, 2007.
[7] Gauss M Cordeiro, Edwin MM Ortega, and Artur J Lemonte. The exponential-weibull lifetime distribution. Journal of Statistical Computation and simulation, 84(12):2592-2606, 2014.
[8] Carl Lee, Felix Famoye, and Ayman Y Alzaatreh. Methods for generating families of univariate continuous distributions in the recent decades. Wiley Interdisciplinary Reviews: Computational Statistics, 5(3):219-238, 2013.
[9] Ayman Alzaatreh, Carl Lee, and Felix Famoye. T-normal family of distributions: A new approach to generalize the normal distribution. Journal of Statistical Distributions and Applications, 1(1):1-18, 2014.
[10] Abbas Mahdavi and Debasis Kundu. A new method for generating distributions with an application to exponential distribution. Communications in Statistics-Theory and Methods, 46(13):6543-6557, 2017.
[11] Muhammad Ijaz, Wali Khan Mashwani, Atilla Gokta§, and Yuksel Akay Unvan. A novel alpha power transformed exponential distribution with real-life applications. Journal of Applied Statistics, pages 1-16, 2021.
[12] M Nassar, A Alzaatreh, M Mead, and O Abo-Kasem. Alpha power weibull distribution: Properties and applications. Communications in Statistics-Theory and Methods, 46(20):10236-10252, 2017.
[13] Richard L Smith and JC918854 Naylor. A comparison of maximum likelihood and bayesian estimators for the three-parameter weibull distribution. Journal of the Royal Statistical Society: Series C (Applied Statistics), 36(3):358-369, 1987.
[14] I Elbatal, M Elgarhy, and BM Golam Kibria. Alpha power transformed weibull-g family of distributions: Theory and applications. Journal of Statistical Theory and Applications, 2021.
[15] Magne Vollan Aarset. How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36(1):106-108, 1987.
[16] Abdulkareem M Basheer. Alpha power inverse weibull distribution with reliability application. Journal ofTaibah University for Science, 13(1):423—432, 2019.
[17] Ammar M Sarhan and Mazen Zaindin. Modified weibull distribution. APPS. Applied Sciences, 11:123-136, 2009.
[18] Gokarna R Aryal and Chris P Tsokos. Transmuted weibull distribution: A generalization of theweibull probability distribution. European Journal of pure and applied mathematics, 4(2):89-102, 2011.
[19] Gauss M Cordeiro, Ahmed Z Afify, Haitham M Yousof, Selen Cakmakyapan, and Gamze Ozel. The lindley weibull distribution: properties and applications. Anais da Academia Brasileira de Ciencias, 90(3):2579-2598, 2018.
[20] Konstantinos Zografos and Narayanaswamy Balakrishnan. On families of beta-and generalized gamma-generated distributions and associated inference. Statistical methodology, 6(4):344-362, 2009.
