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31
POWER WEIGHTED AKASH DISTRIBUTION WITH PROPERTIES AND APPLICATIONS
Rama Shanker1 and Kamlesh Kumar Shukla2*
department of Statistics, Assam University, Silchar, India [email protected] 2 Department of Mathematics, School of Sciences, Noida International University, Gautam Budh
Nagar, India, [email protected] Corresponding Author
Abstract
In In this paper power weighted Akash distribution (PWAD) which includes weighted Akash distribution (WAD), power Akash distribution (PAD) and Akash distribution as particular cases has been proposed and investigated. Its moments, hazard rate function and mean residual life function have been discussed. Method of maximum likelihood estimation has been discussed for estimating the parameters of the distribution. Applications of the proposed distribution to two real lifetime datasets have been presented and compared with other one parameter, two-parameter and three-parameter well-known lifetime distributions.
Keywords: Akash distribution, Weighted Akash distribution, Power Akash distribution, Hazard rate function, stochastic ordering, Maximum Likelihood estimation, Applications.
I. Introduction
Shanker and Shukla [1] proposed a two-parameter weighted Akash distribution (WAD) having parameters 9 and a and defined by its probability density function (pdf) and cumulative distribution function (cdf)
Qa+2 „«-1 (1 1)
fifl'= (62+a2+a)T^ (.l+y>)e-^y>0,9>0,a>0 '
[d2 + a(a + 1) ]T(a, 0y) + (0y)B(0y + « + 1)
Fi(r-e-a) = 1--(fl2 + g2 + g№)-
where T(a) and T(a,z) are the complete gamma function and the upper incomplete gamma function defined as
r(a) = f e-tta-1dt;a> 0 (1.3)
Jo
r(a,z)
-f
J 7.
e-yta-1dt; a > 0,z>0
(1.4)
Its structural properties including moments, hazard rate function, mean residual life function, estimation of parameters and applications for modeling survival time data has been discussed by Shanker and Shukla [1]. Shanker and Shukla [2] discussed various moments based properties including coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of weighted Akash distribution and its applications to model lifetime data from biomedical sciences and engineering.
Shanker and Shukla [2] proposed a power Akash distribution (PAD) having parameters 8 and a and defined by its pdf and cdf
pe3
■ ■ —I— \ f— r~ II if- ~ fj -j ■ 1/
(1.5)
f2(y;e.P)-
(92 + 2)
> 0,8 > 0,p > 0
F2(y;9,p)-1
l+
8yp{8yp +2)
e2 + 2
e^'-.y
> 0,8 > 0,p > 0
(1.6)
Note that the PAD is a convex combination of Weibull (a, 6) and a generalized gamma
q2
(2, a, 8) distribution with mixing proportion^^j. Shanker and Shukla [1] has discussed the properties of PAD including the shapes of the density, hazard rate functions, moments, skewness and kurtosis measures, estimation of parameters using maximum likelihood estimation and application to model a real lifetime data from engineering. Recall that WAD and PAD reduces to Akash distribution at a = 1, and P = 1 respectively. The Akash distribution proposed by Shanker [3] is defined by its pdf and cdf
Q3
fs(y; 9) -p^(1 + y2)e-ey;y>0,8>0
F3(y;8) -1-
1 +
8y(8y + 2)
82 +2
e-9y;y> 0,8 > 0
(1.7)
(1.8)
Shanker [3] has discussed its various statistical and mathematical properties including shapes of the density. Moments and moments based measures, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability, estimation of parameter using both the method of moment and the maximum likelihood estimation and application to model lifetime data from engineering and biomedical sciences.
In the present paper, a three - parameter power weighted Akash distribution which includes Akash distribution, WAD, and PAD as particular cases, has been proposed and discussed. Its raw moments have been given. The survival function and the hazard rate function of the distribution have been derived and their shapes have been discussed for varying values of the parameters. The estimation of its parameters has been discussed using maximum likelihood method. Finally, the goodness of fit and the applications of the distribution have been explained through two real lifetime datasets and the fit has been compared with other one parameter, two-parameter and three-parameter lifetime distributions.
2
II. Power weighted Akash distribution
Assuming the power transformation X = Y& in (1.1), the pdf of the random variable Xcan be obtained as
B9a+2 x13"-1
J4K ,HJ d2+a2 + a r(a)yj (2.1)
> 0,9 > 0,a> 0,fi > 0
We would call the distribution in (2.1) as the power weighted Akash distribution (PWAD). It can be easily verified that the WAD(0, a) in (1.1), PAD(d.p) in (1.5) and Akash(0)in (1.7) are the special cases of PWAD for (ft = 1),(a = 1)and (a = ft = 1), respectively.
It can be easily verified that PWAD is a convex combination of generalized gamma distribution (GGD) having parameters (9, a, ft)proposed by Stacy (1962) and GGD having parameters (9, a + 2, ft).
We have
i
where
f4 (x; 9,a,p)= pgi (x; 9,a,fi) + (1 - p)g2(x;9,a + 2, ft),
62
P='
82 + a2 + a
g1(x;9,a,p) = ^xPa-1e-6xP;x > 0,9 > 0,a >0,fi>0 g2(x;9,a,B) =£m+lxtK«+2)-ie-exP > o,g > o,a > 0,B > 0.
»2V J f(a+2) ^
Graphs of density function of PWAD for varying values of parameters 9, aandfi have been drawn and presented in figure 1. It is clear that the natures of PWAD are decreasing, positively skewed, negatively skewed, platykurtic, mesokurtic and leptokurtic for varying values of parameters and hence it can be applied to model lifetime datasets of various natures. It is observed that pdf is increasing for increased value of 9 and its pdf is increasing vastly as increased value of 9, aandftrespectively. However, role of a on the shape of the graph more as compared to other parameters.
3
Figure. 2: Graphs of the probability density function of PWAD for varying values of parameters^, aandfi
4
III. Reliability Measures
Survival function and Cumulative distribution Function
The survival function S(x; B, a, p)of PWAD can be obtained as S(x;B,a,/3) = P(X> x) = f™ f4(t;B,a,/3)dt
= , 2 , J°° (1 + t2^)e-9tPdt
(92 + a2 + a)r(a)Jx v '
= e-etPtPa-l dt + r e-et^tPa+2P-l dt\
(92 + a2 + a)r(a)LJx ->x J
1-P
Assuming u = which gives t = (u)pand dt = — (u) p du, we get
S(x;B,a,p) =
ea
(B2 + a2 + a)r(a)
L
we
e-9uua-1 du + L e-9uua+1 du P JXP
9a
(92 + a2 + a)r(a)
r(a,9xP) e-0xP (9xP + a+l)(9xP)a+a(a+1)r(a9
8a ea+2
_ (e2 + a2 + a)r(a,exP)+(exP)a(exP + a+l)e-ex^
= (02 + a2 + a)r(a) '
where T(a, 8x@) is the upper incomplete gamma function defined as r(a.dxP) = f™xp ya-1e-ydy; a > 0,8x? > 0.
It can be easily verified that at (J3 = 1),(a = l)and (a = ft = 1) the survival function of PWAD reduce to the survival function of WAD, PAD and Akash distribution.
Thus the cdf of PWAD can be given by
F4(x; 6.a.p) = 1- S(x; 8,a,/) = 1- (92+a2+a)r(a,9xl)+(9x")ai.^1)^
\ >> >rJ (e2 + a2 + a)r(a)
The natures of the cdf of PWAD for varying values of parameters 8, aand/ are shown in figure 2. From the figure 2, It is observed that distribution function is slightly increasing as increased value of 8.
1
Hazard Rate Function
The hazard rate function, h(x; 8, a,/), of PWAD can be given by
h(x-6 a B~) = f4(x-fi,a,P) _ pea+2xf>a-l(l+x2f>)e-°'P
( ; , ,3) S(x-,e,a,P) (02 + a2 + a)r(a,exP)+(exP)a(exP + a+l)e-0xp.
Graphs of h(x; 8, a,/)for varying values of parameters 8, aand/3 are shown in figure 3. The graphs of h(x; 8, a,/)shows that it takes different shapes for varying values of parameters 8, aand/ and it is observed that hazard rate is increasing as increased value of 8, aand/respectively.
5
0 2 4 6 8 10 02460 10
X X
Figure 2: Graphs of the cdf of PWAD for varying values of parameters 9, aandfi
Figure 3: Graphs of hazard rate function for varying values of parametersd, aandfi.
7
Mean Residual Life Function
The mean residual life function, m(x) = m(x; 8, a, p), of PWAD can be obtained as
m(x) = m(x;8,a,p) =
ßea+2
1 -Г tf4(t;8,a,p)dt — x
S(x-ß,a,ß)Jx
(в2 + а2 + а)Г(а,вхР)+(вхР) (9xß + a+1)e-
p8a+2
-fj; tßa(l + t2ß)e-etßdt — x
jp ; л ;
e-etßtßadt+\ e-etßtßa+2ßdt
X J X
(в2 +а2 + а)Г(а, вх?) + (вх?)а(вх? + а + 1)е-вхР
1 1 1-1 Assuming и = t?, which gives t = (u)pand dt = ^ (и) p du, we get
- x
m(x) = m(x;8,a,p) =
I
J X
(82 +a2 + а)Г(а, 8xß) + (8xß)a(8xß + a + 1)e-9xß
a+i-l
1
a+2+^-1
e-8uu ß-1 du+ I e-9uua+2+ß-1 du ß Jxß
ва+2
- x
(92 + a2 + a)r(a,exß)+(exß)a(exß + a+1)e~exß
Г (a+1,8xß) Г (a + 2+1,8xß)
1
8a+ß
+
1
8a+2+ß
- x
e2r(a+1,exß)+r(a+2+1,exß)
■ — x.
92 + a2 + a)r(a,9xß)+(9xß')a (dxß + a+1)e~
The behaviors of m(x)of PWAD for varying values of its parameters 8, aandpare shown in figure 4. It is observed from the figure 4 that overall mean residual value is decreasing as increased value of 8 whereas other parameters are kept as constant; however mean residual is very much affected with value of p.
8
x
i
&(a= 0.5, p=2)
- 6=0.2 --6=0.5
V \ S N. s - - - 6=1 ■- 6=1.2 — 6=1.5
&(B=0.5, (3=3)
- 6=0.2 - - 6=0.5
-- 6=1.2
6=1.5
\ \
\ \
\ \
10
Figure 4: Graphs of mean residual life function for varying values of parametersd,aandp.
9
IV. Moments
The rth moment about origin,^ of PWAD (2.1) can be obtained as
ßßa+2 rœ
ßr' = E(Xr) = 9 ,---T I xßa+r-1 (l + x2ß)e-exßdx
(в2 +a2 + а)Г(а) J0 v 1
ß6a+2
(в2 + a2 + а)Г(а)
p œ p œ
I e-exßxßa+r-1 dx + I e-9xß xßa+2ß+r-1 00
d x
_ h! Assuming u = вх@, which gives x = (U)ßand dx = -1 (u) ß du, we get
в a
ßr =
(в2 + a2 + а)Г(а)
ва+1
(в2 + а2 + а)Г(а)
[e-uQß -1du+[ e-u(U)
1— [ e-uua+ß 1 du+--r— [
¥~ß-1Jo ea+2+ß-1->o
œ u ßa+2ß+r-1
ß du
r
_,. a+2+^-1 - u u ß d u
Vea+ß
ea+1
(в2 + а2 + а)Г(а)
ea+1
Г ( а+j) Г ( а+ 2 +
+
( в2 + а2 + а)Г( а)
r r
в^Т1 в^+Т1
в2Г + Г (а + 2+Çj
r
вa+1+ß
e2r(a+L)+r(a+2+ß _
; г = 1,2,3,.
; + a2 + a)f(a)
(4.1)
Thus the first four moments about origin of PWAD can be given by
ß1
ß2
' e2r{a+ß)+f{a+2+ß)
'- + a2 + a)r(a)
e2r(a+2)+r(a+2+
! + a2 + a)f(a)
92r(a+3)+ri a+2+3
ß3
'- + a2 + a)f(a)
ß4 =■
92r(a+ßß)+r(a+2+ßß)
' + a2 + a)T(a)
Using the relationship between moments about origin and central moments, central moments can be obtained. Since the expressions for central moments are complicated, central moments are not being given.
V. Maximum Likelihood Estimation
Suppose (x1,x2,x3,...,xn) be a random sample of size n from PWAD (2.1). The natural log likelihood function is thus obtained as
InL - Yin=1lnf4 (xi,e,a,p) - n[lnfi + (a + 2) In6 - ln(92 + a2 + a) - lnr(a)] + (fia-
1)I^=1ln(xXi) + Y1f=1ln(l + xi2^)-9Y1]i=1xi^The maximum likelihood estimates (MLEs) of parameters (8, a.fi) of PWAD are the solution of the following nonlinear log likelihood equations
dlnL _ n(a+2) 2n6 p _ „
' - Li=1 xi - 0
дв в e2 + a2 + a
dlnL = п1пв- n(2a+1) - пф(а) + ß YH=i ln(Xi) = 0
д a
e2+a2+a
д n L n
д ß ß
= n + а YJn=1 In xi + YJn=
2xt2ß In(xi) 1 1 + Xi2ß '
вЩ^Ы^) = 0
10
r
1
2
3
4
Rama Shanker and Kamlesh Kumar Shukla
POWER WEIGHTED AKASH DISTRIBUTION WITH PROPERTIES RT&A, No 2 (68) AND APPLICATIONS_Volume 17, June 2022
where x is the sample mean and ^(a) = — InF(a) is the digamma function. These three natural
log- likelihood equations do not seem to be solved directly because they cannot be expressed in
closed forms. However, the MLE's of parameters (6,a,ß) can be obtained directly by solving the
log likelihood equation using Newton-Raphson iteration method available in R -Software till
sufficiently close estimates of parameters are obtained.
VI. Applications
In this section, the applications and goodness of fit of the PWAD have been discussed for two real lifetime datasets. The fit is compared with one parameter lifetime distributions including exponential distribution, Lindley distribution proposed by Lindley [4] and studied by Ghitany et al [5], Akash distribution; two-parameter lifetime distributions including Weibull distribution introduced by Weibull [6], Gamma distribution, Generalized exponential distribution (GED) introduced by Gupta and Kundu [7], Power Lindley distribution (PLD) proposed by Ghitany et al [8], Shukla distribution (SD) proposed by Shukla and Shanker [9],Weighted Lindley distribution (WLD) introduced by Ghitany et al [10] and PAD and WAD and three-parameter lifetime distributions including generalized gamma distribution (GGD) introduced by Stacy[11] and generalized Lindley distribution (GLD) suggested by Zakerzadeh and Dolati [12]. Note that Shanker et al [13] and Shanker[14] have detail discussion on WLD and GLD regarding some important properties and applications for various lifetime data from engineering and biomedical sciences. The first dataset is the data reported by Efron[15] represents the survival times of a group of patients suffering from Head and Neck cancer disease and treated using a combination of radiotherapy and chemotherapy (RT+CT). The second dataset is the data which represents the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm and are available in Bader and Priest [16].
Tablel The data set 1 reported by Efron [15] represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using a combination of radiotherapy and chemotherapy (RT+CT).
12.20 23.56 23.74 25.87 31.98 37 41.35 47.38 55.46 58.36
63.47 68.46 78.26 74.47 81.43 84 92 94 110 112
119 127 130 133 140 146 155 159 173 179
194 195 209 249 281 319 339 432 469 519
633 725 817 1776
Table2 The following data set 2 represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm, Bader and Priest [16]
1.312 1.314 1.479 1.552 1.700 1.803 1.861 1.865 1.944 1.958
1.966 1.997 2.006 2.021 2.027 2.055 2.063 2.098 2.140 2.179
2.224 2.240 2.253 2.270 2.272 2.274 2.301 2.301 2.359 2.382
2.382 2.426 2.434 2.435 2.478 2.490 2.511 2.514 2.535 2.554
2.566 2.570 2.586 2.629 2.633 2.642 2.648 2.684 2.697 2.726
2.770 2.773 2.800 2.809 2.818 2.821 2.848 2.880 2.954 3.012
3.067 3.084 3.090 3.096 3.128 3.233 3.433 3.585 3.585
In order to compare the goodness of fit of these distributions for the two datasets, values of —2 In L, AIC (Akaike information criterion), K-S Statistic ( Kolmogorov-Smirnov Statistic) and p-value for two datasets have been computed The formulae for AIC and K-S Statistics are as follows: AIC = —2 InL + 2k, and ^ — S = Sup|/^(x) — F0(x)|, where k being the number of parameters
X
involved in the respective distributions, n is the sample size and /^Mis the empirical distribution function. The best distribution corresponds to the lower values of—2 In L, AIC and K-S statistic.
Note that the estimates of parameters of the considered distributions are based on maximum likelihood estimates. In this paper, the initial values of the parameters for ML estimates of PWAD have been selected asd = 1.5,a = 0.5 and/ft = 1.5for both dataset. In general, it has been observed that the initial values of the parameters can be taken as any positive real numbers, preferably from 0.5 to 5, for any dataset.
The pdf of the fitted distributions are presented in table 3. The ML estimates of parameters of the considered distributions for datasets 1 and 2 are presented in tables 4 and 5. The goodness of fit by K-S statistics for datasets 1 and 2 with considered distributions are presented in tables 6 and 7. The variance-covariance matrix of the parameters (8, a, ft) of PWAD for datasets 1 and 2 are presented in tables 8 and 9. It is obvious from the goodness of fit of the proposed distribution that in tables 4 and 5 it gives better fit than all considered distributions and competes well with GGD. Therefore, PWAD can be considered an important three-parameter lifetime distribution alternative to GGD and other lifetime distributions.
Table 3: pdf of the fitted distributions
Distributions Pdf
Weibull f(x; 8, a) = 8axa-1e-exU; x> 0,8 > 0,a> 0
Gamma 8a f(x; 8, a) = -—e-9xxa-1; x> 0,8 > 0,a> 0 r(a)
PLD f(x;9,a) = + -Q xa-1(1 + xa)e-ex"; x > 0,8>0,a>0
WLD 8a+1 xa-1 f(x; 8, a) = ---- (1 + x)e-Sx; x>0,8>0,a>0 8 + (x r(a)
GED f(x; 8, a) = 8a(l - e-9x)a-1e-9x; x> 0,8 > 0,a> 0
SD Qa+1 f(x; 8, a) =---- (1 + xa)e-ex; x > 0,6 > 0,a>0 )y '' J ga + ^(a + 1)
GGD ona f(x;8,a,ß) = ^—xßa-1e-0xß;x> 0,8 > 0,a > 0,ß> 0 r(a)
GLD 8a+1 xa-1 f(x;8,a,ß) = e+ß r{a + i)(« + ß^~eX
Lindley e2 f(x; 8) =-—- (1 + x)e ; x> 0,8 > 0 8 + 1
Table 4: Summary of the ML estimates of parameters for dataset 1
Model ML Estimates
§ a ft
PWAD 11.8734 27.3026 0.1804
GLD 0.00473 0.05243 5.07505
GGD 11.25540 27.72340 0.18220
SD 0.00458 0.02380
WAD 0.0090 0.0165 ........
PAD 0.16751 0.55764 ........
WLD 0.00531 0.21191 ......
PLD 0.05301 0.68893 .......
GED 0.00482 1.09367 ......
Gamma 0.00489 1.08501 ......
Weibull 0.00710 0.92327 .......
Akash 0.01344
Lindley 0.00892
Exponential 0.00447
Table 5: Summary of the ML estimates of parameters of dataset 2
Model ML Estimates
§ a ft
PWAD 0.2918 1.7049 2.7229
GLD 9.39076 22.71981 4.77105
GGD 0.30440 3.58610 2.64830
SD 5.9922 17.1611
WAD 9.7584 22.2327 ....
PAD 0.16964 3.06033 .....
WLD 9.62655 22.89383 .....
PLD 0.0500 3.8680 ......
GED 2.03307 87.28471 ..
Gamma 9.53843 23.38184 .....
Weibull 0.00558 5.33523 ..
Akash 0.96472
Lindley 0.65450
Exponential 0.40794
Table 6: Summary of Goodness of fit by K-S Statistic for dataset 1
Model -2lnL AIC K-S p-value
PWAD 555.67 561.67 0.081 0.921
GLD 564.09 570.09 0.150 0.248
GGD 555.64 561.64 0.079 0.921
SD 564.00 568.00 0.147 0.267
WAD 580.32 584.32 0.219 0.023
PAD 559.10 563.10 0.108 0.635
WLD 565.91 569.91 0.161 0.181
PLD 560.78 564.78 0.118 0.529
GED 563.93 567.93 0.145 0.280
Gamma 564.10 568.10 0.149 0.249
Weibull 563.71 567.71 0.298 0.005
Akash 609.92 611.92 0.279 0.001
Lindley 579.16 581.16 0.219 0.025
Exponential 564.01 566.01 0.145 0.282
Table 6 represents the goodness of fit by K-S Statistic for data set-1, It is observed that AIC and P-value from K-S test were found almost minimum and maximum in comparison to all other included distributions respectively. Therefore, it may be concluded that PWAD is better fits than other included distributions except GGD (Generalized Gamma distribution). Hence, PWAD can be considered an important lifetime distribution for modeling lifetime data.
Table 7: Summary of Goodness of fit by K-S Statistic for dataset 2
Model —2 In L AIC K-S p-value
PWAD 97.93 103.93 0.037 0.999
GLD 101.96 107.96 0.056 0.979
GGD 100.58 106.58 0.044 0.999
SD 184.35 188.35 0.290 0.000
WAD 99.95 103.95 0.057 0.976
PAD 98.02 102.02 0.038 0.999
WLD 100.04 104.04 0.058 0.974
PLD 98.12 102.12 0.044 0.998
GED 109.24 113.24 0.095 0.558
Gamma 100.07 104.07 0.058 0.973
Weibull 99.31 103.31 0.060 0.964
Akash 224.27 226.27 0.362 0.000
Lindley 238.38 240.38 0.401 0.000
Exponential 261.73 263.73 0.448 0.000
Table 7 represents the goodness of fit by K-S Statistic for data set-2, It is observed that AIC and P-value from K-S test were found almost minimum and maximum in comparison to almost all other included distributions respectively expect PAD and PLD. Therefore, it may be concluded that PWAD is a better fit than other included distributions except PAD (Power Akash distribution) and
14
Rama Shanker and Kamlesh Kumar Shukla
POWER WEIGHTED AKASH DISTRIBUTION WITH PROPERTIES RT&A, No 2 (68) AND APPLICATIONS_Volume 17, June 2022
PLD (Power Lindley distribution). Further, PWAD competing well with the considered
distributions and hence can be an important distribution for lifetime data.
Table 8: Variance-covariance matrix of the parameters 9, aand ft of PWAD for dataset 1
8af3
8 [3044.5355 4357.7480 —4.9546' a 4357.7480 62462648 —7.0633 ¡31—4.9546 —7.0633 0.0081.
Table 9: Variance-covariance matrix of the parameters 9, aand ft of PWAD for dataset 2
8aj3
12.5447 -27.6412 1.7898 -27.6412 -60.2546 3.9955 1.7898 3.9955 -0.2499
VII. Conclusions
In the present paper a three-parameter power weighted Akash distribution (PWAD) ,of which two-parameter weighted Akash distribution (WAD), two-parameter power Akash distribution (PAD) and one parameter Akash distribution are particular cases, has been introduced and studied. Its moments, hazard rate function, mean residual life function and stochastic ordering have been discussed. Maximum likelihood estimation has been discussed for estimating the parameters of the distribution. The applications of the proposed distribution have been discussed through two real lifetime datasets. The goodness of fit test of the proposed distribution is a better model for lifetime data than the other well-known one parameter, two-parameter and three-parameter lifetime distributions.
ACKNOWLEDGEMENTS: Authors are grateful to the editor in chief and the anonymous reviewers for their constructive comments to improve the quality of paper.
References
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[4] Lindley, D.V. (1958): Fiducial distributions and Bayes' Theorem, Journal of the Royal Statistical Society, Series B, 20, 102 - 107.
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[9] Shukla, K. K. and Shanker, R. (2019): Shukla distribution and its application, Reliability: Theory and Applications 54(3), 46-55.
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Rama Shanker and Kamlesh Kumar Shukla
POWER WEIGHTED AKASH DISTRIBUTION WITH PROPERTIES RT&A, No 2 (68) AND APPLICATIONS_Volume 17, June 2022
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