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Shukla Distribution And Its Application
Kamlesh Kumar Shukla, Rama Shanker •
Department of Statistics College of Science, Eritrea Institute of Technology, Asmara, Eritrea Email: kkshukla22@gmail.com, shankerrama2009@gmail.com
Abstract
In this paper a two-parameter lifetime distribution named, 'Shukla distribution' which includes several one parameter lifetime distributions including exponential, Shanker, Ishita, Pranav ,Rani and Ram Awadh as particular cases, has been proposed and investigated. Its moments have been obtained. The hazard rate function, mean residual life function and stochastic ordering of the distribution have been discussed. Maximum likelihood estimation has been explained for estimating the parameters of the distribution. Applications of the distribution have been explained through real life time data and its fit has been found satisfactory over well-known one parameter and two-parameter lifetime distributions.
Keywords: Lifetime distributions, Moments, Hazard rate function, Mean residual life function, Maximum likelihood estimation, Goodness of fit.
1. Introduction
In the new era of the world, it is important to study through the model for systematic approach and statistical approach. In this case approach of distribution theory is crucial to develop statistical model for knowing the occurrence of some event and their interest for some populations of individuals in almost every field of knowledge. The statistical modeling and their studies along with lifetime data has been drawn interest to researchers in engineering, biomedical science, insurance, finance, amongst others. Applications of lifetime distributions range from investigations into the endurance of manufactured items in engineering to research involving human diseases in biomedical sciences.
In the recent past years, a number of one parameter and two-parameter lifetime distributions for modeling lifetime data have been proposed by different statisticians. As we know that classical one parameter exponential distribution including other popular distribution such as Lindley, Akash, Shanker, Ishita, Pranav, Rani, Ram Awadh distributions are proposed and applied on life time data from various field. The two-parameter lifetime distributions popular in statistics are gamma, Weibull, Power Lindley, Quasi Lindley and Exponentiated exponential.
The probability density function (pdf) along with introducer (year) of exponential, Lindley, Akash, Shanker, Pranav and Ram Awadh distributions are presented in table 1.
Kamlesh Kumar Shukla, Rama Shanker RT&A, No 3 (54) SHUKLA DISTRIBUTION AND ITS APPLICATION_Volume 14, September 2019
Table 1: The pdf of exponential, Lindley, Shanker, Pranav, Rani and Ram Awadh distributions
Distributions pdf Introducer (Year)
Exponential f(x;e) = ee-9x
Lindley e2 0 f(x)=^^(1+x)e-ex Lindley (1958)
Shanker e2 a f(x)=-^l(e + x)e-ex Shanker (2015 a)
Akash e3 f(x;e) = -p^(i + x2)e-ex Shanker (2015 b)
Ishita e3 f(x;e) = -^^(e + x2)e-ex Shanker and Shukla (2017)
Pranav Q4 fix; e) = (e + x3)e-ex Shukla (2018)
Rani e5 fix; e) = Qs^A (e + x4)e-9x Shanker (2017)
Ram Awadh e6 fix; e) = 7rr—— ie + x5)e-9x J e6 + 120 Shukla(2018)
Ghitany et al (2008) have discussed various statistical properties, estimation of parameter and application of Lindley distribution to model waiting time data in a bank and showed that Lindley distribution is a suitable model over exponential distribution. Shanker et al (2015) have detailed comparative and critical study on applications of exponential and Lindley distributions for modeling real lifetime datasets from biomedical science and engineering and showed that in majority of datasets exponential distribution shows satisfactory fit over Lindley distribution.
Recently, Shanker and Shukla (2019) proposed a two-parameter lifetime distribution named Rama-Kamlesh distribution (RKD) defined by its pdf and survival function as
fix; e, a) = galna+1) i1 + xa)e-0x; x>0,e>0,a>0 (1.1)
= 9"(i+x")e-x + ar(a,9x) 0,e > 0,a > 0, (1.2)
where r(a, ex) is the lower incomplete gamma function defined as
r(a,z)=iiie-Ha-1dt (1.3)
It has been mentioned by Shanker and Shukla (2019) that RKD includes several one parameter lifetime distributions. Various interesting properties, estimation of parameters and application of the distribution have been given in Shanker and Shukla (2019).
The main aim of the present paper is to introduce two-parameter lifetime distribution named Shukla distribution (SD)' which includes many one parameter distributions including exponential distribution as particular case. Several other one parameter lifetime distributions can also be generated from SD. Its moments about origin and the variance have been obtained. The hazard rate function and stochastic ordering have been discussed. Maximum likelihood estimation has been discussed for estimating the parameters of the distribution. Applications of the distribution have been discussed with real lifetime dataset and the goodness of fit of the distribution has been
compared with well known one parameter and two-parameter lifetime distributions.
2. Shukla Distribution
The pdf of Shukla distribution (SD) having parameters 0 and a can be defined as
f(x; 9, a) =
ea
9a+1+r(a+1)
(9 + xa)e-0x; x > 0,9 > 0,a > 0
(2.1)
It can be easily verified that exponential, Shanker, Ishita, Pranav, Rani and Ram Awadh distributions are particular cases of SD for a = 0, a = 1 , a = 2, a = 3, a = 4 and a = 5
respectively. The pdf (2.1) can be shown as a convex combination of exponential (0) and gamma
(a,0^) distributions. We have
f(x; 9, a) = pgi(x; 9) + (1 - p)g2(x; a, 9),
where
Qa+1
P ='
— aaSx „ (v. „ a\ _ e<X+1 „-9x,,a+1-1
-, g1(x;9) = 9e-Bx, g2(x;a,d) =
r(a+1)
-e
0a +T(a +1)
Qa+i f™
S(x;9,a)=P(X>x) = j f(t; 8,a)dt = Qa+1 + f(g + ^ j (9 + t")
+ ta)e-etdt
ea
[ef e-etdt + f e-et tadt\
9a+1+r(a+1)y Jx
a+i re-6x e-6x(9x)a+ar(a,9x)
[ 1
+
9a+1+r(a+1)V 1 9a+1 J
_ 9a(9+xa)e-0x+ar(a,9x) = 9a+1+r(a+1) ,
Thus the corresponding cdf of SD can be obtained as
F(x;9,a) = 1-S(x;9,a) = 1 - ea(e+x2e~°x+ar{a'ex);x > 0,9 >0,a>0 (2.2)
v J v J ga+l+r(a+i) \ /
Behaviors of pdf and survival function of SD for varying values of parameters 0 and a have been shown in figures 1 and 2, respectively.
6=0.05
f Nv - a-
/ ^ - - a-2
---- a-3 --- a-4 a-5
I 1 . - _
L. - 1 —----
X o -
50
100
a
a. ' a. = 3 o.-4 a= 5
150
9
3. Moments
I
"r
ia+1
The r th moment about origin, pr of Shukla distribution(SD) can be obtained as
ec
Vr = , , ^ JJ хг (в + xa)e-exdx
ea+1 + Г (a + 1) ва+1Г(г + 1) + Г(а + r + 1)
er[ea+1 + ria +1)}
Thus the first four moments about origin of SD are obtained as
ea+1+Tia + 2)
Vi =
;r = 1,2,3,.
в{ва+1 + Г (a + 1)} 2ва+1 +Г(а + 3)
ß2 в2{ва+1+Г(а + 1)}
,_ 6 ва+1+Г(а + 4) ß3 = в3{ва+1 + Г(а + 1)}
- _ 24ва+1+Г(а+5)
e4{ea+1+r(a+i)}' The variance of SD can be obtained as
[29a+1 + r(a + 3)}[6a+1 + r(a + 1)} - [6a+1 + r(a + 2)}2
ß2= ß2 - (Vl)
2 _
в2{ва+1+Г(а + 1)}2
Taking r = 1,2,3 and 4, the first four moments about origin, /dr of SD can be obtained. It should be noted that the r th moment about origin, ^' of exponential, Shanker, Ishita, Pranav, Rani and Ram Awadh distribution can be obtained from the of SD by taking a = 0,1,2,3,4, and 5 .
4. Hazard Rate Function and Mean Residual Life Function
For a continuous random variable X having pdf f (x) and cdf F (x), the hazard rate function (also known as the failure rate function) , h (x) , is defined as
i—| s n P(X<x+AxlX>x) f(x)
□ (x) = l im —
Ax^0 Ax l-F(x)
Thus, hazard rate function, h (x) of Shukla distribution can be expressed as □ (x) = □ ix; e, a) =
fix;9,a) _ 9a+1i9+xa)e-Sx
1-Р(х;в,а) ва(в+ха)е-вх+аГ(а,вх)
;х > 0,в > 0,a> 0
The mean residual life function, m ( x) of Shukla distribution can be obtained as
1 Г
m(x; в, a) = —.—-—г I 1/^;в,а) dt - x S(x; в, a) Jx
ва+1 + Г(а + 1) Г°° ва+1
ва(в + ха)е-вх + аГ(а, вх) J 1'ва+г^Г(а + 1)
в
а+1
ва(в + ха)е-вх + аГ(а, вх)
J t„ " ,-~(в + 1а)е-вЫ1-х
]х в^ ' ' ^ 1
ж * ж
e-ettdt+J
х х
в\ e-bttdt + I e-btt dt
ва
ва(в + ха)е-вх + аГ(а,вх)
е-вх(вх + 1) е-вх(вх)а(вх + а + 1) + а(а + 1)Г(а,вх)
— х
9 9a+2
e-0X{da+1 + (a+1)(dx)a}+a(a+1—dx)r(afdx) e[ea(e+xa)e-0X+ar(a,ex)} .
e +r(a+2) = ^1 . The behaviors of h (x) and
= f(0) and m(0) =
Note that h(0) = nn.. r ---------- = ninn.. r
v J ea+1+r(a+1) ' K J v J 8{8a+1+r(a+1)}
m (x) of SD for varying values of parameters 0 and a have been shown in figures 3 and 4
respectively.
Fig.3: Behavior of h (x) of SD for varying values of parameters 0 and
__ m(x) at alpha=0.5 - theta=0.3 --theta=0.5 • - - theta=0.8 ■ - theta=1 theta=1.2
^llllill'lllllllllllll 1-1
m(x) at alpha=3
- theta=0.3
--theta=0.5
■ - - theta=0.8
• - theta=1
theta=1.2
- - - _ _rrm
Fig. 4: Behavior of m (x) of SD for varying values of parameters 0 and
(X
( У
5. Stochastic Ordering
Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable is said to be smaller than a random variable Y in the
(i) stochastic order iX <st Y)if Fx(x) > FYix)for all x
(ii) hazard rate order iX <hr Y)if ^x(x) > ^y(x) for all x
(iii) mean residual life order iX <mrl Y)if mxix) < mYix)for all x
(iv) likelihood ratio order iX < lr Y) if fx^x) decreases in x.
v ' fvix)
The following results due to Shaked and Shanthikumar (1994) are well known for establishing stochastic ordering of distributions
X <ir Y ^ X <^r Y ^ X <mrl Y
V
X-stY
KRD is ordered with respect to the strongest 'likelihood ratio' ordering as shown in the following theorem:
Theorem: Let X ~ RKDie1,a1) and Y Y ~ RKDie2,a2)ie2,a2). If a1 < a2 and e1> e2 , then X <lr Yand henceX <hr Y, X <mrl YandX <st Y. Proof: We have
fxix-,9i,ai) 9iai+1{92a2 + 1+r(a2 + 1)) ,9i+x°i\ -(9i-92)x. Y>a fY{x-,92,a2) 92a2+i(9iai + i+nai + 1))\92+xa2je ;X>0
Now
ln fxix9i,ai) = ^ \9iai+i(92a2+i+ria2 + 1)) f Yi x; 92, a2)
2a2+1(g1»i+1+r{a1 + 1))
This gives Txln fYix^,a2) ----ie1- e2)
Thus, fora1 < a2 and e1> e2 ,— lnfx^iC'9l'ai\ < 0. This means that X <lr Yand henceX <hr Y,
dx ^Y(x'92,a2)
X <mrl YandX <st Y. This shows flexibility of SD over one parameter exponential, Shanker, Ishita, Pranav, Rani and Ram Awadh distributions.
6. Maximum Likelihood Estimation
Let (x1, x2,x3,..., xn) be a random sample from SD (2.1). The likelihood function, L of (2.1) can be expressed as
ea+1
Kea+1~ria + 1)j
The natural log likelihood function is thus obtained as
L = ' -n + rr„ + -n) U(9 + Xia)e-n0X
/ att+1 \
lnL = nln (ea+lna+1)) + m=i ln(e + xia)- n9x
= n[(a + i)ine- in(ea+1 + r(a +1))] + zn=1 in(e + xia) - nex.
The maximum likelihood estimates (MLEs) (0 ,oc^ of parameters (0, a) of SD are the solution of
the following nonlinear log likelihood equations
d In L n(a + 1) n(a + 1)e°
+
de e ea+1 + r(a + i) Z-le + xia
i=1
n
y 1 _
} ---nx =
¿-te + xia
Kamlesh Kumar Shukla, Rama Shanker RT&A, No 3 (54) SHUKLA DISTRIBUTION AND ITS APPLICATION_Volume 14, September 2019
dlnL n[ea+1lnG +xb(a + l)] ^xialn(xi)
■ = nine- „ ,,—„/ ^ + y 4-—— = 0
da ea+1 + r(a + i) L-i e + xia
i=1
where x is the sample mean and ^(a + i) = — In T (a + i) is the digamma function. These two natural log likelihood equations do not seem to be solved directly, because they cannot be expressed in closed forms. The (MLE's) (e,a) of (0,a) can be computed directly by solving the natural log likelihood equation using Newton-Raphson iteration available in R-software till sufficiently close
values of 0 and a are obtained.
7. Data Analysis
The applications of SD have been discussed with the following dataset relating to engineering from Fuller et al (1994).This data set is the strength data of glass of the aircraft window reported by Fuller et al (1994):
18.83 20.8 21.657 23.03 23.23 24.05 24.321 25.5 25.52 25.8 26.69 26.77 26.78 27.05 27.67 29.9 31.11 33.2 33.73 33.76 33.89 34.76 35.75 35.91 36.98 37.08 37.09 39.58 44.045 45.29 45.381
For the above dataset, SD has been fitted along with two- parameter distributions including Power Lindley distribution (PLD) proposed by Ghitany et al (2013), Weibull distribution suggested by Weibull distribution(1951), gamma distribution, Quasi Lindley distribution introduced by Shanker and Mishra (2013) and generalized exponential distribution proposed by Gupta and Kundu (1999), RKD and one parameter lifetime distributions including exponential, Lindley, Shanker, Akash, Ishita, Pranav, Rani and Ram Awadh . The ML estimates, value of —2 log L, Akaike Information criteria (AIC), K-S statistics and p-value of the fitted distributions are presented in tables 2 and 3. The AIC and K-S Statistics are computed using the following formulae: AIC = -2lnL + 2k and K-S = Sup|F„(x) — F0(x)|, where k = the number of parameters, n = the sample size , F„(x)is the
X
empirical (sample) cumulative distribution function, and F0(x) is the theoretical cumulative distribution function. The best distribution is the distribution corresponding to lower values of —2 log L , AIC, and K-S statistics and higher p-value
Table 2: MLE's, Standard Errors, - 2ln L, AIC, K-S and p-values of the fitted distributions for dataset 1
Distributions ML Estimates -2 log L AIC BIC K-S p-value
SD 0 = 0.6144 208.23 212.23 216.05 0.134 0.580
a = 17.9299
PLD 0 = 0.00243 220.14 224.14 226.13 0.198 0.152
a = 1.9439
RKD 0 = 0.61361 208.23 212.23 216.05 0.134 0.580
a = 17.9060
Gamma 0 = 0.61482 208.22 212.22 216.05 0.134 0.578
a = 18.9433
Weibull 0 = 0.00203 241.61 245.61 247.61 0.353 0.000
a = 1.80566
QLD 0 = 0.03416 274.45 278.45 281.32 0.458 0.000
a = 18.9393
GED 0) = 0.16531 208.27 212.27 215.13 0.135 0.581
a = 92.0017
Exponential 0) = 0.0325 274.53 276.53 277.96 0.459 0.000
Lindley 0) = 0.0629 253.99 255.99 257.42 0.333 0.000
Akash 0) = 0.0970 240.68 242.68 244.11 0.296 0.006
Shanker 0) = 0.06471 252.35 254.35 255.78 0.357 0.000
Ishita 0) = 0.09732 240.48 242.48 243.48 0.297 0.006
Pranav 0) = 0.1298 232.77 234.77 235.77 0.253 0.030
Rani 0) = 0.1623 277.25 229.25 230.24 0.220 0.080
Ram Awadh 0) = 0.19471 223.07 225.07 226.07 0.197 0.155
It is obvious from the goodness of fit given in tables 2 that SD competes well with considered one-parameter and two-parameter lifetime distributions. Therefore, SD can be considered an important two-parameter lifetime distribution as.
8. Conclusions
In this paper a two-parameter lifetime distribution named, 'Shukla distribution (SD)' which includes one parameter lifetime distributions including exponential, Shanker, Ishita, Pranav, Rani and Ram Awadh as particular cases, has been proposed and studied. Its moments have been obtained. The hazard rate function, mean residual life function and stochastic ordering have been discussed. The estimation of its parameters using maximum likelihood estimation has been discussed. Goodness of fit has been presented with a real lifetime dataset and fit found quite satisfactory over all well- known considered lifetime distributions.
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