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N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RT&A, No-3 (74)
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA Volume 18, September 2023
A STUDY ON STATISTICAL PROPERTIES OF
A NEW CLASS OF Q-EXPONENTIAL-WEIBULL DISTRIBUTION WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA
N. Sundaram1 and G. Jayakodi2
12 Department of Statistics,
Presidency College, Chennai, Tamil Nadu, India ncsundar77@gmail.com1, jayakodimcc@gmail.com2
Abstract
This article introduces a new four-parameter probability distribution called the q-Exponential-Weibull distribution based on the q-Exponential-G family of distribution. The proposed new distribution has to decrease and increase failure rates which are more common in reliability scenarios and can be used instead of Weibull and the exponential distribution. It also includes some sub-models like q-Exponential-Exponential, q-Exponential-Rayleigh, Exponential-Weibull, Exponential-Exponential and Exponential-Rayleigh lifetime distributions. Various Mathematical and statistical Properties are investigated, which include Limiting behavior, Moments and Moment Generating functions, Quantile function and Order Statistics. The Maximum Likelihood estimator is used for estimating the model parameters. This new distribution is compared with other lifetime distributions using different kinds of real-life failure time data.
Keywords: q-Exponential-Weibull, Quantile, Reliability Measures, Maximum Likelihood Estimation, failure time data.
1. Introduction
Lifetime distributions are very useful statistical tool for analyzing the various characteristics of lifetime data. The developments and applications of lifetime distribution are essential in numerous fields. Hence, the major aspects of generating new families of probability distributions are they offer greater flexibility and a better fit at the expense of one or more extra parameters.
The non-extensive statistical mechanism plays a vital growth in the past few years. This new formulation is not based on the usual statistical mechanism, provided that will give a better description of the complex system developed by [26]. In the recent decade's probability distribution, which emerge from the non-extensive statistical mechanism called q-type distribution attracted several statisticians to develop new distribution [11], [20] and [23]. Studying this type of distribution is quite interesting because of its complex system and power-law behavior. The application of this type of distribution has been found in many research areas like Physics, Biology, Mathematics, Chemistry, Economics, Medicine etc.
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RL&A No3 (74)
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA_Volume 18, September 2023
The q-Exponential distribution emerged from maximizing the non-extensive statistical mechanism under appropriate constraints [26]. This theory is a generalization of the classical Boltzmann-Gibbs (BG) statistical mechanism. So, the q-Exponential distribution has found varieties of applications in the research field including in the field of complex systems. This article introduces a new four-parameter probability distribution called the q-Exponential Weibull distribution.
The well-known q-distributions are q-Exponential distributions discussed by Malacarne et al. [15], q-Gamma distribution due to Duarte et al. [7], q-Weibull distribution due to Picoli et al. [21], q-Gaussian distribution due to Adrian et al. [1]. Picoli J.R. et al. [22] discussed q-distribution in complex systems. Ana Claudia souza [3] studied the reliability data analysis of systems in the wear-out phase using q-Exponential likelihood. Fode Zhang et al. [9] discovered the information geometry on the curved q-Exponential family with application to survival analysis. Shalizi [25] express the Maximum Likelihood Estimation for q-Exponential distribution. The geometry of q-exponential distribution with dependent competing risk and accelerated life testing is given by Fode Zhang et al. [10]. Keith Briggs [12] demonstrates the modelling train delay with the q-Exponential distribution. The reliability of stress strength and its estimation of exponentiated q-Exponential distribution is given by Mohammed et al [18]. Modelling censored survival data with q-Exponential distribution discussed by Sundaram [19].
In reliability and survival analysis most commonly, used distributions are Exponential and Weibull distributions [16], q-Exponential is an alternative one. The q-Exponential distribution is a higher version of an Exponential distribution. It has two parameters: q and a, where q is the shape parameter (entropy index/control parameter) and a is the scale parameter. As compared to the Exponential distribution that has just one parameter (a), the q-Exponential distribution has more flexibility regarding the decay of the pdf [3]. Indeed, the Exponential probability distribution is a special case of the q-Exponential when q ^ 1. Another feature of this distribution is that it does not have the limitation of a constant hazard rate as the Exponential one, thus allowing the modelling of either system improvement (1 < q < 2) or degradation (q < 1). The pdf of q-Exponential distribution [26], is given by
/q(x) = (2-q) a [1- (1-q) ax] 1 a-q) for x, a > 0 ,q<2 (1)
This can also be rewritten as
/q(x) =(2-q) a eq (-ax)
1
Where eq(x) = [1 + (1-q) x]1^ Which is the ^-exponential if q * 1. When q = 1, eq(x) is just exp(x).
The cumulative distribution function (cdf) of the q-Exponential-generated family is given by.
F(x) = fGW/(1-G W)(2 - q)a[1 -(1- (2)
The simplified form of (2) is.
F(X) = 1 - [1- (1-q) a ^ x, a >0, q<2 (3)
where q is the shape parameter (entropy index) and a is the scale parameter. The corresponding probability density function is given by
f(x) = (2 - q) a 9(*\ [1 - (1-q) a (4)
W V ^ [1-G(x)]A2 ^ 1-G(*)J V '
where X>0, a >0, q<2.
2-q
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RT&A, No-3 (74)
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA_Volume 18, September 2023
The rest of the paper are as follows. In Section 2, The new class q-Exponential-Weibull distribution is introduced and presented its particular cases. The mathematical and statistical properties are discussed in section 3 and in section 4, the maximum likelihood estimation method and their asymptotic behaviors have been discussed. Simulation techniques has been explained in section 5. Real life failure time data has been applied and the results are presented in section 6. In section 7, we have discussed the conclusion of the new class of q-Exponential Weibull distribution.
2. The q-Exponential-Weibull Distribution
The q-Exponential distribution combined with Weibull distribution gives the q-Exponential Weibull distribution. Here the q-exponential is the generator distribution, and the two-parameter Weibull distribution (Waloddi Weibull, 1951) is a parent distribution whose pdf and cdf are given by
g (x, A, y) = Ayxy-% '1x7 x,y, A >0 (5)
G (x, A, y) = 1 - e-XxY (6)
using (6) in (3), we get the new cdf of q-Exponential-Weibull distribution. The simplified form of q-Exponential-Weibull distribution is
2-q
F (x, Q) = 1-[1 - (1 - q) a (eXxY - 1)] 1-9 (7)
where Q = {q, a, A, Y}be the set of parameters, here q and y are the shape parameters and a, A are the scale parameters. The equation (7) is called the cdf of q-Exponential-Weibull distribution. Substituting (5) and (6) in (4), we get the new pdf. The new pdf is,
1
f (x, Q) = (2-q) oAy eXxYxy-i[1 - (1 - q) a (eXxY - 1)] 1-« (8)
Rewriting the above equation (8), we get
f (x, Q) = (2-q) aAY eXxY xY-1 eq [- a (eXxY - 1)] (9)
The equation (8) and (9) are called the pdf of q-Exponential-Weibull distribution (q-EW). The particular case of our new q-Exponential-Weibull distribution is presented in Table 1.
Table 1: The particular case of q-Exponential-Weibull distribution
Model q a A Y Cdf References
q-Exponential-Exponential q a A 1 1- [1- (1-q) a (e^- 1)] ^ l-q New
q- Exponential-Rayleigh q a A 2 2 7 — n 1- [1- (1-q) a (e—- 1)] - 2-i i-q New
Exponential - Weibull 1 a A Y 1-e-a(eXxY-1) New
Exponential - Exponential 1 a A 1 1-e-a(eXx-1) Elgarhy et al. (2017)
Exponential- Rayleigh 1 a A 2 2 A 2 1-e-a(e2* -1) Kawsar Fatima and, S.P Ahmad (2017)
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RL&A No3 (74)
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA Volume 18, September 2023
2.1 Reliability Measures: Survival function: (survivor function)
The survivor function for the new distribution S(x) is defined to be the probability that the survival time is greater than or equal to t, and it is given by
S(x) = P (X > t) = 1- F(x)
2—q
S (x, Q) =[1 - (1 - q) a (eAx7 - 1)] 1-? (10)
2.2 Hazard function:
The hazard function is used to express the risk or hazard of an event such as death occurring at some time t, and it is given by
h (x) = ^
Substituting (8) and (10) we get the hazard function of q-Exponential-Weibull distribution. which is defined below.
-(1+q)
h (x, Q) = (2-q) aAy eAx7xY-i[1 - (1 - q) a (eAx7 - 1)] (1-«) (11)
2.3 Reverse hazard rate function
The reverse hazard function of q-Exponential-Weibull distribution is defined by
r (x) = ™
_ (2-q) a^Y e^V-1^- (1-q) a (^ - 1)] ^ r (x, Q) 2-q (12)
1-[1- (1-q) a (eA*K - 1)]
2.4 Cumulative hazard function Cumulative hazard function is presented below,
H (x, Q) = -log (1-F(x))
2-q
H (x, Q) = -ln (s (x, Q)) = -ln [[1 - (1-q) a (eAx7 - 1)] i-i ] (13)
The above equation is known as cumulative hazard function of q-Exponential-Weibull distribution.
2.5 Graphical Study of q-Exponential Weibull distribution under various functions:
In this section, we studied the structure of the cdf, pdf, S(x) and h(x) of q-Exponential-Weibull distribution using different values of the parameters. The illustrative figures are presented below.
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RT&A, No.3 (74)
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA Vd^e 18, Sep^ta 2023
Figure 2.a: The graph of the cdf of the q-EW distribution with different values of the parameter
Figure 2.b1: Graph of the pdf of the q-EW distribution when all the parameters are changed
Figure 2.b2:The graph of the pdf of the q-EW distribution when changing first shape parameter (q) values and other parameters are fixed
Figure 2.c: The graph of the survival function of the q-EW distribution with different parameter values
Figure 2.b3: The graph of the pdf the of q-EW distribution when changing second shape parameter (y) values and other parameters are fixed
Figure 2.d: The graph of the Hazard rate of the q-EW distribution for with different parameter values
Figure 2.a cumulative density plot demonstrates the validity of the distribution as a probability distribution. The probability density function graphs (2.b1,2. b2 and 2.b3) shows that it is skewed and more adaptable for various parameter values. The graph of the hazard function (2.d) demonstrates that it can take on various shapes, including constant, increasing, and decreasing. As a result, fitting data sets of different forms may be done and which was quite well using the q-Exponential Weibull distribution.
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RT&A, N°.3 (74) WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA_Volume 18, September 2023
3.Properties
In this section we study some mathematical and statistical properties of q-Exponential-Weibull distribution.
3.1 Mixture Representation:
Several properties of the new distribution can be derived using the concept of exponentiated distribution. The mixture representation of q-exponential-Weibull distribution is derived in the following sections.
Using the generalized binomial theorem, where ß>0 is real non integer and I z I <1,
(1-z/—1 = K=o(-1)fc (V)(z)fc
f(x) =(2-q) а [l- (1-q) a since ß = ^
:r=o(-1)fc (^fc-1)(1-q) k ak™ lfc
[1— ■G(z)]"2 1
L [1— -G(z)]A2
я(ж)
(2-q) a ï^^r=o(-1)fc(^—1)(1-q) k ak [1—§-]
= (2-q) a (^—1)(1-q)k ak^
=Z?=o(-1)k 1)(1-q) k ak+i (2-q) fg^
Generalized binomial theorem
[i-c(x)]—(fc+2)=zr=o;(j) [оду
= SiT=o(-1)fc (^—1)(1-q) k ak+1 (2-q) E£o*jg(x) [G(x)]fc+
=SiT=o(-1)fc (V)(!-q) k ak+1 (2-Я) ^o^+jL) (k+j+1) g(x) [GCx)]^1"1
f (x, Q) = ££fc=o W},fc h(k+j+i) (x, Q) (14)
where Wfc= (-1)fc (^-1)(1-q) k ak+1 (2-q)-r(ki2-
ha (x, Q) = a g (x, Q) [ G (x, Q)] a-1
The q-Exponential Weibull density can be expressed as an infinite linear combination of exponentiated - G density function.
2-q
Then, [F (х)]й = [1 - [1 - (1-q) а (еЯжГ - 1)] ]й
Using the generalized binomial theorem, where ß>0 is real non integer and I z I <1,
(1-z/"1 = £Г=о(-1У (V)(z)'
V (2-g)'
[F(x)]fl = ^Г=о(-1)г (?) [1 - (1-q) а (еЯжГ - 1)](1-«)
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RL&A No3 (74) WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA_Volume 18, September 2023
Which can also be written as,
[F(x)]fl = ^т=о(-1)г+т (X)(1-q) m am (^ - 1)m where <9 =
-...v -.rn
[F(x)]fl = £Гш=о(-1)г+т (1-q) m am (X)
1-(1-e-A*r)
Using Generalized binomial theorem, the above equation can be written as,
№)]* = SiTm,n=o(-1)i+m(1-q) m am (^)C)(m+nn-1)[1-e-A^r]
m+n
Simply further we get, [F(x)]fl
[F(x)]fl = £Гш,п,г=о^,ш,п,г(е-Яж')г (15)
[ВД]Й = £Гт,„,г=о(-1)г+т+г (1-q) m am (^)C)(m+nn-1)(mr+n)
Where W;,m,„,r = (-1)'+-+- (1-q) m am (X)(m+nn-1)(T)
3.2 Limiting Behavior:
Lemma 1: The limit of the cdf of the q-Exponential-Weibull, F(x) as X approaches infinity, x^-œ is equal to one and limit of the cdf of the q-Exponential-Weibull, F(x) as X tends to zero, x ^ 0 is equal to zero.
lim F(x) = 1
x^ro
Proof: The cdf of the q-Exponential Weibull F(x) as X approaches infinity (x ^ œ), from 7 we get Using equation (9)
2-q
lim F (x, Q) = lim 1-[1 - (1-q) a (еЯжГ - 1)]
x^ro x^ro
1/ 2-Я
= 1-[1 - (1-q) a (еЯ(го)Г - 1)] i-ч = [1 - 0 ] = 1
Hence, the lemma is proved under limiting property.
limF(x) = 0
2-q
lim F (x, Q) = lim1-[1 - (1-q) a (еЯжГ - 1)]
= 1- [1- (1-q) a (eA(0)7 - 1)] A2-q/ a-q) = 1- [ 1- 0] = 0
Lemma 2: In probability theory, of a continuous random variable has the following property
(i) f(x) > 0; where -^<x<^
(ii) 0(x)dx = 1
Using above definition, the validity of the model f(x) is checked. In our survival model the range of x is 0<x<^.
J0°°/(x)dx = 1
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RT&A, N°.3 (74) WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA_Volume 18, September 2023
1
= J0°°(2 - q) aly eAx7x7-1[1 - (1 - q) a (eAx7 - 1)] i-«dx
1
= (2 - q) aly J" eAx7x7-1[1 - (1 - q) a (eAx7 - 1)] ^dx Now y = [1 - (1 - q) a (eAx7 - 1)]
= 0 - (1-q)alyeAx7x7-1 = dx
VV-1
oo eAx7xy-i d
(2 — q) aly f™ „ 1 *--
0 [1 — (1 — q) a (e^x7 - 1)] ^ (i-q)a^*V-1
= - i-S py^a-s) dy 1-q J0 s J
;0°°/(x)dx = -[y(2-s)/(i-s)]^ = -{ [[1— (1 — q)a(e^7- 1)](2-S)/(1-S)]x=™ -[[1— (1 — q)a(e^x7- 1)](2-S)/(1-S)]x=0l
J0°7(x)dx = - [ 0 -1] = 1
Hence q-Exponential-Weibull distribution is a valid pdf.
3.3 Quantile Function:
The quantile function of X= Q(u) = F-1(u) can be obtained by inverting equation (7) as follows,
1
]r (16)
Q(u) = [lln
1 + ■
1
(1-S)«
1-q
1 — (1 — u)2-9
Simulation of q-Exponential-Weibull random variable is straightforward. Let u be the uniform variable on the interval [0,1], then the random variable X = F-1(u) follows q-Exponential-Weibull distribution given in equation (8) with the parameters (q, a, A, y). By using equation (20), we can obtain the first, second and third quantiles by replacing u as 0.25, 0.5 and 0.75, respectively.
3.4 Moments:
This section provides the moment and moment generating function of q-Exponential-Weibull distribution. The moments of the functions are quantitative measures related to the shape of the function. The first four moments, skewness and kurtosis of q-Exponential-Weibull distribution can be obtained as
^; = E[xr] = j-X/Mdx
Using equation (13) we have,
/4 = J-Lx^J^oW^ V+7+1) (x,n) dx = E^Wo W),fc xrh(fc+ji+i) (x, H)dx Where /¿fc( x,Q) = J_° xrft(fc+J+i) (x, H)dx
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RL&A No3 (74)
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA Volume 18, September 2023
^ = E(xr) = E^o W-,fc /7-,fc( x, Q) (17)
The mean, variance, skewness and kurtosis can be obtained from equation (14).
when r =1 gives mean = E(x)
variance = E(x2) - [E(x)]2
skewness = few- 3 M3(e)M2(e)+2
skewness ^2(6)]3/2
kurtosis = ^4(6)- 4 Mi(e)M3(e)+6M2(6)M2(e)_ 3M4w [M2(e)_ M2(e)]2
Generally, the moment generating function of q-Exponential-Weibull distribution is obtained through the following relation
Mx(t, Q) =Z»oJ0 xr /(x)dx = 77 E(xr) = SrMJ.,fc=o — W-,fc /7-,fc( x, Q) (18)
The Characteristic function of q-Exponential-Weibull distribution is obtained through the following relation
0x(t, Q) =Z?=o ^ Jor xr /(x)dx = 2r=o ^ E(xr) = 2rr-,fc=o ^ ^,/c ( x, Q) (19)
The cumulant generating function of q-Exponential-Weibull distribution is given by
Mt, Q) = ^ [[zr=o ^ JcT /Md*] = log [ Z?=o ? E(xr)]
= [Srr,-,fc=o ^f w-,fc /7-,fc( x, Q)] (20)
3.5 Order Statistics:
Let X1:n < X2:n <X3:n <.. .<Yn:n be the order statistics of a random sample of size n following q-Exponential-Weibull distribution with the parameter a, q, A, y then the probability density function of pth order statistic can be written as,
/(^M=iiSfe ^(-1)" C -p) FWrp-1 (21)
Substituting (13) and (14) in (18) and replacing R= (v + p - 1) we get
/(xp)Rp)]--B(p,n-p + 1) ^V=o(-i) ( v ) ¿i,m,n,r=o ^¿,m,n,r(e )
/(xp)[x(p)] = B(p,^-p+1) Erfc=oX"=iiEirm,n,r=offl'ft(t+i+1)(X,fl)(e-liy)r (22)
Where = (-1)-(n-P) W-,fcWi,m,„,r
4. Method of Estimation
In this section, the maximum likelihood estimates (MLE) of the unknown parameters for the q-Exponential-Weibull distribution are determined based on complete samples. Let x1, x2...xnbe a random sample from q-Exponential-Weibull distribution with unknown parameter vector Q = {q, a, A, y}. The likelihood function for the proposed distribution X is given by
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA
RT&A, No.3 (74) Volume 18, September 2023
£(x, Q) = (2 - q)nanAnyn nf=1 eAx¡rx¿7-1[1 - (1 - q) a (eAx¡7 - 1)] i-
Then the log likelihood of the equation is
I (n) = log £(t, Q) = n log (2-q) + n log a + n log A + n log y + A Zn=1 x¿7 + (Y -1)
L=1 logx¿L=1 log [1 - (1-q) a (e
1-q
Ax;7 -
1)]
(23)
The maximum likelihood estimates of the parameters (q, a, A, y) are found by taking a partial derivative of I (H) with respect to q, a, A, y, equating the derivatives to zero, and evaluating them at a, A, f.
91 (n) _ n „n 1 9q = 2-q S' = 1 1-q
a(eAx¡r-1)
1- (1-q) a (eAxir- 1)
Zn=ilog[1- (1- q)a (eAx¡7 - 1)
91 (n)_ n „n 1 9a a '=1 1-q
(1-q)(eAxir-1)
. y
1-(1-q) a (eAxi - 1)
91 (n) n
9A A
n , vn ,, ^n j + L í=1 ¿i¿=1
1
1-q
*(1-q) x¿yeAxiy
1- (1-q) a (e^i7 - 1)
^ = n + Ln=1 iogx¿ +Ln=1^x7iogx¿ - LU^
aA(1-q) x;7eAxi7*logx;
1- (1-q) a (eAxi - 1)
(24)
(25)
(26) (27)
For solving these non-linear equation's, we can use any iteration method such as Newton-Raphson technique.
5. Generating random samples from q-Exponential Weibull distribution
The Inverse CDF method is used for generating random numbers from a particular distribution. In this method, random numbers from a particular distribution are generated by solving the equation obtained on equating CDF of a distribution to a number u. The number u is itself being generated from u—U(0,1). In this section we made an attempt to q-Exponential-Weibull distribution to generate the random number using equation 16 at a fixed values of parameters (q, a, A, y).
X = F-1(u)
X = Tin Í1+ 1
A (1-q)a
1-q
1 - (1 -
(28)
For uniform over (0,1) then x~q — £W(1.2,2,1.1,1.7) can be generated random sample of size 50 are presented below.
0.2471, 0.8991, 0.9387, 1.5307, 0.6133, 0.8110, 0.8077, 0.7611, 0.8320, 1.5590, 1.0539, 1.7640, 1.4155, 0.8662, 1.2408, 1.9081, 1.0625, 0.6137, 0.5943, 0.7125, 0.9593, 0.3809, 0.1623, 0.2987, 0.9664, 1.31036, 0.6269, 1.3524, 0.6302, 1.0810, 2.1260, 1.4057, 1.1020, 0.6074, 1.7022, 1.1539, 1.1613, 0.5775, 0.1133, 0.9533, 1.1283, 1.2516, 1.6930, 0.9185, 1.3880, 0.8035, 0.9471, 0.1955, 2.4077, 0.7141
Here we have used one of the goodness of fit tests "Kolmogorov-Smirnov (KS)" test for the above-generated data for testing the q-Exponential-Weibull distribution. The null hypothesis is that
q
i
7
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RT&A No3 (74) WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA_Volume 18, September 2023
the samples are drawn from the q-Exponential-Weibull distribution against the alternative
hypothesis is that the samples are not drawn from the q-Exponential-Weibull distribution. The test
statistic value of the KS test for the generated samples is (D value) 0.097 at 5% level of significance
with the p-value of 0.76. Since the p-value is greater than 0.05, the null hypothesis is accepted. Hence,
the samples are drawn from the proposed distribution. Therefore, the q-Exponential-Weibull
distribution has satisfied the goodness of fit test.
6. Application to real life data
In this section, we have used different kinds of real-life failure time data to show the suitability of the q-Exponential Weibull distribution, also we have compared to some other related distributions namely Exponentiated Weibull-Exponential (EWE) and Generalized Exponential-Weibull (GEW) distributions. The pdf of the respective distributions is represented below:
• The Exponentiated Weibull-Exponential (EWE) distribution introduced by Elgarhy et.al [8], with pdf
/(x) = qayA[eAx - 1F-1 exp[-{a[eAx - 1]r - Ax}] [ 1 - exp(-a[eAx - 1]7)f-1
x,q,a,A,y >0 (29)
• The Generalized Exponential-Weibull (GEW) distribution introduced by Dikko and Faisal [6], with the pdf
/(x) = q(a + yAxA-1)e-(ax+^xA) [1 - e-(«*+y*A)]?-1 x,a, 7, A, q >0 (30)
In order to assess the flexibility of the proposed distribution, we have considered some model selection criteria like, -2loglikelihood and AIC (Akaike Information Criterion) are used and analyses performed with the aid of R software.
Dataset1: The first data set is the failure times of 84 aircraft windshields. This failure time data set is available in Murthy et al's book "Weibull Models" (2004, page 297). A large aircraft's windscreen is a sophisticated piece of equipment made up of multiple layers of material, including a very touchy outer skin with a heated layer just behind it, all laminated under high temperature and pressure. These failures do not cause aircraft damage, but they do require the repair of the windscreen. The failure times of 84 aircraft windshields are given below:
0.040,1.866, 2.385, 3.443, 0.301,1.876, 2.481, 3.467, 0.309,1.899, 2.610, 3.478, 0.557,1.911, 2.625, 3.578, 0.943, 1.912, 2.632, 3.595, 1.070,1.914, 2.646, 3.699, 1.124, 1.981, 2.661, 3.779,1.248, 2.010, 2.688, 3.924, 1.281, 2.038, 2.823, 4.035,1.281, 2.085, 2.890, 4.121,1.303, 2.089, 2.902, 4.167,1.432, 2.097, 2.934, 4.240, 1.480, 2.135, 2.962, 4.255,1.505, 2.154, 2.964, 4.278,1.506, 2.190, 3.000, 4.305,1.568, 2.194, 3.103, 4.376, 1.615, 2.223, 3.114, 4.449,1.619, 2.224, 3.117, 4.485,1.652, 2.229, 3.166, 4.570,1.652, 2.300, 3.344, 4.602, 1.757, 2.324, 3.376, 4.663
Table 2: Estimates of fitted distribution for aircraft windshield failure data
Model Estimated Parameters Model Selection
q a A ? -2LL AIC
q-EW 1.729287 4.629657 0.006168 1.539852 251 259
EWE 15.46262 1.38606 4.08592 0.07846 253 261
GEW 0.04796 0.31873 0.43050 0.68102 419 427
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RL&A No3 (74)
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA Volume 18, September 2023
Dataset 2: The second data set represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal. The data is presented below:
0.1, 0.33, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 07, .08, 1.08, 1.08, 1.09, 1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24,1.3, 1.34,1.36, 1.39, 1.44, 1.46, 1.53, 1.59,1.6, 1.63, 1.63, 1.68,1.71,1.72,1.76,1.83,1.95,1.96,1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54, 2.54, 2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55.
Table 3: Estimates of fitted distribution for guinea pig failure data
Model Estimated Parameters Model Selection
<7 a A ? -2LL AIC
q-EW 1.54493 33.49094 0.01988 2.98777 184 192
EWE 1.08287 83.46078 0.01628 2.99604 187 195
GEW 0.66675 0.07983 0.27917 0.45848 242 250
We observed from the above tables 2 and 3, the -2LL and AIC values of the q-Exponential-Weibull distribution have the smallest among the other distributions. Therefore, the q-Exponential-Weibull distribution has performed well than the other distributions. So, we conclude from this section, the q-Exponential-Weibull distribution has achieved the goal of the suitability of the different kinds of real-life failure time data.
8. Conclusion
In this research article, we have introduced a new class of four-parameter distribution referred to as "q-Exponential-Weibull distribution" by taking the Weibull distribution as the base distribution and the q-Exponential distribution as the generator distribution by using the generator technique. The q-Exponential-Weibull density can be expressed as a linear combination of exponentiated - G densities. For checking the model properties, we have derived survival, hazard, cumulative hazard and reverse hazard functions from q-Exponential-Weibull distribution, and also studied graphically. In the graphical study of the q-Exponential-Weibull distribution under various functions with different parameter values, the proposed distribution has achieved the properties of the density function. The mathematical and statistical properties are applied to q-Exponential-Weibull distribution. The q-Exponential-Weibull distribution has satisfied the above said properties. The parameters of the q-Exponential-Weibull distributions are estimated using the maximum likelihood estimation method. The random samples have been generated from the q-Exponential-Weibull distribution and the goodness of fit test has been verified using Kolmogorov-Smirnov (KS) test, also we have studied the application of real-time failure time data to q-Exponential-Weibull distribution. The proposed distribution performed well than the other distribution based on the model selection criteria. Based on the above-said results, the q-Exponential-Weibull distribution is more adaptable and more flexible to fit the real-life failure time data. We hope that the proposed distribution would draw more widespread applications in different areas of research such as reliability analysis, medicine engineering and economics etc.
N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RT&A, N°3 (74)
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA Volume 18, September 2023
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N. Sundaram, G. Jayakodi
A STUDY ON STATISTICAL PROPERTIES OF A NEW CLASS OF
Q-EXPONENTIAL-WEIBULL DISTRIBUTION RL&A No3 (74)
WITH APPLICATION TO REAL-LIFE FAILURE TIME DATA Volume 18, September 2023
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