Research Article Volume 7 Issue 4
College of Science, Eritrea Institute of Technology, Eritrea
Correspondence: Rama Shanker, College of Science, Eritrea Institute of Technology, Asmara, Eritrea
Received: August 13, 2018 | Published: August 29, 2018
Citation: Welday D, Shanker R. A generalized Aradhana distribution with properties and applications. Biom Biostat Int J. 2018;7(4):374-385. DOI: 10.15406/bbij.2018.07.00234
A two-parameter generalized Aradhana distribution which includes one parameter exponential and Aradhana distributions as special cases has been proposed. Its statistical properties including shapes of probability density function for varying values of parameters, coefficient of variation, skeweness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves and stress-strength reliability have been discussed. Maximum likelihood estimation has been discussed for estimating the parameters of the distribution. Applications of the distribution have been explained with two real life time data.
Keywords: aradhana distribution, statistical properties, maximum likelihood estimation, applications
In almost every fields of knowledge including engineering, biomedical science, social science, insurance, finance, etc, the statistical analysis and modeling of real life time data are crucial for researchers and policy makers. The classical one parameter life time distributions, namely exponential and Lindley, introduced by Lindley,1 are not always suitable due to theoretical or applied point of view for real lifetime data. To overcome the shortcomings of these classical one parameter distributions and have a better lifetime distribution, a number of one parameter lifetime distributions have been introduced in statistics literature and the statistics literature is flooded with a number of one parameter life time distributions. Shanker2 has introduced a one parameter lifetime distribution named Aradhana distribution having scale parameter q and defined by its probability density function (pdf) and cumulative distribution function (cdf)
f1(x;θ)=θ3θ2+2θ+2(1+x)2e−θx;x>0,θ>0f1(x;θ)=θ3θ2+2θ+2(1+x)2e−θx;x>0,θ>0 (1.1)
F1(x,θ)=1−[1+θx(θx+2θ+2)θ2+2θ+2]e−θx;x>0,θ>0F1(x,θ)=1−[1+θx(θx+2θ+2)θ2+2θ+2]e−θx;x>0,θ>0 (1.2)
The rth raw moments (moments about origin), μr′μr′ of Aradhana distribution obtained by Shanker2 is given by
μr′=r!{θ2+2(r+1)θ+(r+1)(r+2)}θr(θ2+2θ+2);r=1,2,3,...μr′=r!{θ2+2(r+1)θ+(r+1)(r+2)}θr(θ2+2θ+2);r=1,2,3,...
Thus, the first four raw moments of Aradhana distribution are obtained as
μ1′=(θ2+4θ+6)θ(θ2+2θ+2)μ1′=(θ2+4θ+6)θ(θ2+2θ+2) μ2′=2(θ2+6θ+12)θ2(θ2+2θ+2)μ2′=2(θ2+6θ+12)θ2(θ2+2θ+2)
μ3′=6(θ2+8θ+20)θ3(θ2+2θ+2)μ3′=6(θ2+8θ+20)θ3(θ2+2θ+2) μ4′=24(θ2+10θ+30)θ4(θ2+2θ+2)μ4′=24(θ2+10θ+30)θ4(θ2+2θ+2)
Using the relationship between central moments (moments about mean) and the raw moments, the central moments of Aradhana distribution are given by
μ2=θ4+8θ3+24θ2+24θ+12θ2(θ2+2θ+2)2μ2=θ4+8θ3+24θ2+24θ+12θ2(θ2+2θ+2)2
μ3=2(θ6+12θ5+54θ4+100θ3+108θ2+72θ+24)θ3(θ2+2θ+2)3μ3=2(θ6+12θ5+54θ4+100θ3+108θ2+72θ+24)θ3(θ2+2θ+2)3
μ4=3(3θ8+48θ7α+304θ6α2+944θ5α3+1816θ4α4+2304α5θ3+1920α6θ2+960α7θ+240α8)θ4(θ2+2θ+2)4
Shanker2 has discussed various statistical properties based on moments including coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviation, Bonferoni and Lorenz curves, stress-strength reliability along with estimation of parameter and applications of Aradhana distribution for modeling real lifetime data from engineering and biomedical sciences. A discrete Poisson-Aradhana distribution, a Poisson mixture of Aradhana distribution, has been obtained by Shanker and its statistical properties, estimation of parameter along with applications to model count data are available in Shanker . The Lindley distribution is defined by its pdf and cdf
f2(x;θ)=θ2θ+1(1+x)e−θx;x>0,θ>0 (1.3)
F2(x,θ)=1−(1+θxθ+1);x>0,θ>0 (1.4)
Ghitany et al.,4 have detailed study on statistical properties, estimation of parameter and application of Lindley distribution for modeling waiting time data in a bank. Shanker et al.,5 have critical and comparative study on modeling of real lifetime data from biomedical sciences and engineering and observed that there are several lifetime data where exponential distribution gives much better fit than Lindley distribution. Recently, Berhane & Shanker6 proposed a discrete Lindley distribution using infinite series approach of discretization and studied its various statistical properties, estimation of parameter and applications.
In this paper a two -parameter generalized Aradhana distribution (GAD) has been proposed which includes exponential and Aradhana distributions. GAD has been found to be more general in nature and wider in scope and possess tremendous capacity to fit observed real lifetime data. Its moments and moments based measures have been obtained and discussed. Statistical properties including hazard rate function, mean residual life function, stochastic ordering, mean deviation, Bonferroni and Lorenz curves and stress -strength parameter of GAD have been discussed. Estimation of parameters has been discussed using the method of maximum likelihood. Applications of the distribution have been explained with two real lifetime data and the fit has been compared with one parameter exponential, Lindley and Aradhana distributions and a generalization of Sujatha distribution (AGSD), proposed by Shanker et al.7
A generalized Aradhana distribution (GAD) having parameters and α is defined by its pdf and cdf
f3(x;θ,α)=θ3θ2+2αθ+2α2(1+αx)2e−θ x;x>0,θ>0,α≥0. (2.1)
F3(x;θ,α)=1−[1+αθx(2θ+αθx+2α)θ2+2θα+2α2]e−θx; x>0,θ>0 (2.2)
Note that exponential distribution and Aradhana distribution are special cases of GAD for and, respectively. Further, Like Aradhana distribution, GAD is also a three-component mixture of exponential, gamma and distributions. That is
f3(x;θ,α)=p1 g1(x;θ)+p2g2(x,θ)+(1−p1−p2)g3(x,θ) (2.3)
where
p1=θ2θ2+2θα+2α2
p2=2αθθ2+2θα+2α2
g1(x;θ)=θe−θx;x>0,θ>0
g2(x;2,θ)=θ2Γ(2)e−θxx2−1;x>0,θ>0
g3(x;3,θ)=θ3Γ(3)e−θxx3−1;x>0,θ>0
The nature of the pdf and the cdf of GAD for various combinations of the parametersθ and α presented in Figure 1 & Figure 2, respectively.
Moments and moments based measures
The rth raw moment, , of GAD can be obtained as
μr′=r!(θ2+2αθ(r+1)+α2(r+2)(r+1))θr(θ2+2θα+2α2);r=1,2,3,...
The first four raw moments of GAD are given by
μ1′=(θ2+4θα+6α2)θ(θ2+2θα+2α2)
μ2′=2(θ2+6θα+12α2)θ2(θ2+2θα+2α2)
μ3′=6(θ2+8θα+20α2)θ3(θ2+2θα+2α2) .
μ4′=24(θ2+10θα+30α2)θ4(θ2+2θα+2α2)
The central moments of GAD (2.1) are thus obtained as
μ2=θ4+8αθ3+24α2θ2+24α3θ+24α4θ2(θ2+2αθ+2α2)2
μ3=2(θ6+12θ5α+54θ4α2+100α3θ3+108α4θ2+72α5θ+24α6)θ3(θ2+2αθ+2α2)3
μ4=3(3θ8+48θ7α+304θ6α2+944θ5α3+1816θ4α4+2304α5θ3+1920α6θ2+960α7θ+240α8)θ4(θ2+2αθ+2α2)4
The expressions for various coefficients including coefficient of variation, skewness, kurtosis and index of dispersion of GAD are obtained as
C.V=σμ1′=√θ4+8θ3α+24θ2α2+24θα3+12α4θ2+4θα+6α2
√β1=μ3μ23/2=2(θ6+12θ5α+54θ4α2+100α3θ3+108α4θ2+72α5θ+24α6)(θ4+8αθ3+24α2θ2+24α3θ+24α4)3/2
β2=μ4μ22=3(3θ8+48θ7α+304θ6α2+944θ5α3+1816θ4α4+2304α5θ3+1920α6θ2+960α7θ+240α8)(θ4+8αθ3+24α2θ2+24α3θ+12α4)2
γ=σ2μ1′=θ4+8αθ3+24α2θ2+24α3θ+12α4θ(θ2+2αθ+2α2)(θ2+4θα+6α2)
The behaviors of C.V, skewness, kurtosis and index of dispersion (γ) of GAD for various combination of parameters and have been presented in Tables 1-4, respectively.
q α |
0.1 |
0.2 |
0.5 |
1 |
2 |
3 |
4 |
5 |
0.1 |
0.755 |
0.8535 |
0.9519 |
0.9843 |
0.9955 |
0.9979 |
0.9988 |
0.9992 |
0.2 |
0.6755 |
0.7552 |
0.8831 |
0.9519 |
0.9843 |
0.9924 |
0.9955 |
0.997 |
0.5 |
0.6172 |
0.6568 |
0.7551 |
0.8534 |
0.9341 |
0.9635 |
0.9771 |
0.984 |
1 |
0.597 |
0.6173 |
0.6756 |
0.7551 |
0.8535 |
0.9049 |
0.9341 |
0.952 |
2 |
0.587 |
0.5971 |
0.6273 |
0.6756 |
0.7552 |
0.8125 |
0.8535 |
0.833 |
3 |
0.583 |
0.5904 |
0.6105 |
0.6439 |
0.7048 |
0.7551 |
0.7955 |
0.8277 |
4 |
0.582 |
0.5871 |
0.6021 |
0.6273 |
0.6756 |
0.7184 |
0.7551 |
0.7863 |
5 |
0.581 |
0.5851 |
0.5971 |
0.6105 |
0.6568 |
0.6934 |
0.7262 |
0.755 |
Table 1 C.V of GAD for varying values of parameters and
q α |
0.1 |
0.2 |
0.5 |
1 |
2 |
3 |
4 |
5 |
0.1 |
1.2981 |
1.5314 |
2.0638 |
2.5955 |
3.1058 |
3.3467 |
3.4859 |
3.5763 |
0.2 |
1.1932 |
1.2981 |
1.6376 |
2.0639 |
2.5955 |
2.9057 |
3.1058 |
3.2448 |
0.5 |
1.1589 |
1.1781 |
1.2981 |
1.5313 |
1.9114 |
2.1974 |
2.4194 |
2.5955 |
1 |
1.1553 |
1.1589 |
1.1932 |
1.2981 |
1.5314 |
1.7358 |
1.9114 |
2.0639 |
2 |
1.1548 |
1.1553 |
1.1622 |
1.1932 |
1.2981 |
1.417 |
1.5314 |
1.6376 |
3 |
1.1547 |
1.1549 |
1.1573 |
1.17 |
1.224 |
1.2981 |
1.3775 |
1.456 |
4 |
1.1547 |
1.1548 |
1.1559 |
1.1622 |
1.1932 |
1.2414 |
1.2981 |
1.3576 |
5 |
1.1547 |
1.1547 |
1.1553 |
1.1589 |
1.1781 |
1.211 |
1.2523 |
1.2981 |
Table 2 of GAD for varying values of parameters and
q α |
0.1 |
0.2 |
0.5 |
1 |
2 |
3 |
4 |
5 |
0.1 |
5.3806 |
6.1353 |
7.163 |
8.4341 |
8.8155 |
8.9103 |
8.9473 |
8.9654 |
0.2 |
5.0871 |
5.3806 |
6.4784 |
7.613 |
8.4341 |
8.7 |
8.8155 |
8.8754 |
0.5 |
5.0082 |
5.0507 |
5.3806 |
6.1353 |
7.26 |
7.8744 |
8.2225 |
8.4341 |
1 |
5.0012 |
5.0082 |
5.0871 |
5.3806 |
6.1353 |
6.779 |
7.2601 |
7.613 |
2 |
5.0002 |
5.0012 |
5.0151 |
5.087 |
5.3806 |
5.7583 |
6.1353 |
6.4784 |
3 |
5.0001 |
5.0004 |
5.005 |
5.032 |
5.1673 |
5.3806 |
5.6297 |
5.8864 |
4 |
5.00002 |
5.0002 |
5.0022 |
5.015 |
5.0871 |
5.2151 |
5.3806 |
5.5659 |
5 |
5.00001 |
5.0001 |
5.0012 |
5.0082 |
5.0507 |
5.1325 |
5.2459 |
5.3806 |
Table 3 Amplitude in mill volts of the Lead-1 of electrocardiography in sheep
*Significant (P≤0.05); NSNot significant (P>0.05)
q α |
0.1 |
0.2 |
0.5 |
1 |
2 |
3 |
4 |
5 |
0.1 |
12.5454 |
6.5556 |
2.4981 |
1.1594 |
0.5449 |
0.354 |
0.2619 |
0.2077 |
0.2 |
11.5851 |
6.2727 |
2.619 |
1.2491 |
0.5797 |
0.3717 |
0.2724 |
0.2147 |
0.5 |
10.6807 |
5.6548 |
2.509 |
1.3111 |
0.6377 |
0.4085 |
0.2969 |
0.2319 |
1 |
10.3402 |
5.3403 |
2.317 |
1.2545 |
0.6556 |
0.4336 |
0.3189 |
0.2498 |
2 |
10.1689 |
5.1701 |
2.1692 |
1.1585 |
0.6273 |
0.4325 |
0.3278 |
0.2619 |
3 |
10.1122 |
5.1129 |
2.1136 |
1.111 |
0.599 |
0.4182 |
0.3219 |
0.261 |
4 |
10.084 |
5.0844 |
2.0852 |
1.0846 |
0.5793 |
0.4049 |
0.3136 |
0.2562 |
5 |
10.067 |
5.0674 |
2.068 |
1.0681 |
0.5655 |
0.3944 |
0.306 |
0.2509 |
Table 4 of GAD for varying values of parameters and
For a fixed value of α, C.V increases as the value of increases. Again for a fixed value of , C.V decrease as the value of increases.
Clearly for any given values of parameters and , coefficient of skewness is always positive and this means that it is always positively skewed.
Since β2>3, GAD is always leptokurtic, which means that GAD is more peaked than the normal curve
As long as 0<θ<1and0<α<5 , GAD is over dispersed and for θ>2andα>0 , GAD is under-dispersed. The nature of C.V,,and of GAD for various combinations of parameters and have been shown graphically in Figures 3-6, respectively.
Hazard rate function and mean residual life function
The hazard rate function (also known as the failure rate function) and the mean residual life function
of a continuous random variable
having pdf
and cdf
are respectively defined as
h(x)=limΔx→0P(X<x+Δx|X>x)Δx=f(x)1−F(x)
and m(x)=E[X−x|X>x]=11−F(x)∫∞x[1−F(t)]dt Thus, GAD are thus obtained as
h(x)=θ3(1+αx)2θαx(2θ+αθx+2α)+(θ2+2αθ+2α2)
and
m(x)=θ2+2θα+2α2{θαx(2θ+2x+2α)+(θ2+2θα+2α2)}e−θx∞∫x({θαt(2θ+2t+2α)+(θ2+2θα+2α2)}θ2+2θα+2α2)e−θtdt.
=(θ2α2x2+2αθx(2α+θ)+(θ2+4θα+6α2))θ(θαx(2θ+2θx+2α)+(θ2+2θα+2α2))
It can be easily verified that h(0)=f(0) and m(0)=μ1′ . The behaviors of and of GAD for various combinations of parameters and have been shown graphically in Figure 7 & Figure 8, respectively. Clearly is monotonically increasing whereas h(x) is monotonically decreasing.
Stochastic orderings
Stochastic ordering of positive continuous random variables is very much useful for judging their comparative behavior. A random variable X
is said to be smaller than a random variable y in the
The following interrelationship among various stochastic orderings due to Shaker & Shanthikumar8 are well known for establishing stochastic ordering of distributions
X≤lrY⇒X≤hrY⇒X≤mrlY ⇓X≤stY
Theorem: Let x GAD(θ1,α1) andProof: We have
fX(x;θ1,α1)fY(x;θ2,α2)=θ13(θ22+2α2θ2+2α22)θ23(θ21+2α1θ1+2α21) (1+α1x1+α2x)2 e−(θ1−θ2) x ;x>0
Now lnfX(x;θ1,α1)fY(x;θ2,α2)=ln(θ13(θ22+2α2θ2+2α22)θ23(θ21+2α1θ1+2α21))+2ln(1+α1x1+α2x)−(θ1−θ2)x This gives
ddx{lnfX(x;θ1,α1)fY(x;θ2,α2)}=(α1−α2)(1+α1x)(1+α2x)−(θ1−θ2)
. Thus if α1=α2 and θ1>θ2
or ,ddxlnfX(x;θ1,α1)fY(x;θ2,α2)<0
.
This means that X≤lrY
and henceX≤hrY
, and
. Thus, GAD is ordered with respect to the strongest ‘likelihood ratio ordering’
Deviations from the Mean and the Median
The mean deviation about the mean, and the mean deviation about the median,
are defined as
δ1(X)=∞∫0|x−μ| f(x)dx
and δ2(X)=∞∫0|x−M| f(x)dx
, respectively, where μ=E(X)
and . These expressions can be further simplified as
δ1(X)=μ∫0(μ−x)f(x)dx+∞∫μ(x−μ)f(x)dx=2μF(μ)−2μ∫0x f(x)dx (3.4.1)
and
δ2(X)=M∫0(M−x)f(x)dx+∞∫M(x−M)f(x)dx=μ−2M∫0x f(x)dx (3.4.2)
Using pdf (2.1) and the mean of GAD, we get
μ∫0x f(x)dx=μ−{ θ3(μ+2αμ2+α2μ3)+θ2(1+4αμ+3α2μ2)+θ(4α+6α2μ)+6α2}e−θ μθ(θ2+2θα+2α2) (3.4.3)
M∫0x f(x)dx=μ−{θ3(M+2αM2+α2M3)+θ2(1+4αM+3α2M2)+θ(4α+6α2M)+6α2}e−θ Mθ(θ2+2θα+2α2) (3.4.4)
Using expressions from (3.4.1), (3.4.2), (3.4.3), and (3.4.4), and δ2(X) of GAD are
δ1(X)=2{ θ2(1+2αμ+α2μ2)+4αθ(1+μ)+6α2}e−θ μθ(θ2+2θα+2α2) (3.4.5)
δ2(X)=2{θ3(M+2αM2+α2M3)+θ2(1+4αM+3α2M2)+θ(4α+6α2M)+6α2}e−θ Mθ(θ2+2θα+2α2)−μ (3.4.6)
Bonferroni and lorenz curves and indices
The Bonferroni and Lorenz curves are proposed by Bonferroni9 which are used in economics to study income and poverty and other fields of knowledge including reliability, demography, insurance and medicine, some among others. The Bonferroni and Lorenz curves are defined asB(p)=1pμq∫0x f(x) dx=1pμ[∞∫0x f(x)dx−∞∫qx f(x) dx]=1pμ[μ−∞∫qx f(x) dx] (3.5.1)
and L(p)=1μq∫0x f(x) dx=1μ[∞∫0x f(x)dx−∞∫qx f(x) dx]=1μ[μ−∞∫qx f(x) dx] (3.5.2)
The Bonferroni and Gini indices are further simplified as
B=1−1∫0B(p) dp (3.5.3)and G=1−21∫0L(p) dp (3.5.4)
Using pdf of GAD (2.1), we get
∞∫qx f(x) dx={θ3(q+2αq2+α2q3)+θ2(1+4αq+3α2q2)+θ(4α+6α2q)+6α2}e−θ qθ(θ2+2θα+2α2) (3.5.5)
Now using equation (3.5.5) in (3.5.1) and (3.5.2), we get
B(p)=1p(1−{θ3(q+2αq2+α2q3)+θ2(1+4αq+3α2q2)+θ(4α+6α2q)+6α2}e−θ q(θ2+4θα+6α2)) (3.5.6)
and
L(p)=1−{θ3(q+2αq2+α2q3)+θ2(1+4αq+3α2q2)+θ(4α+6α2q)+6α2}e−θ q(θ2+4θα+6α2) (3.5.7)
Now using equations (3.5.6) and (3.5.7) in (3.5.3) and (3.5.4), we have
B=1−{θ3(q+2αq2+α2q3)+θ2(1+4αq+3α2q2)+θ(4α+6α2q)+6α2}e−θ q(θ2+4θα+6α2) (3.5.8)
G=2{θ3(q+2αq2+α2q3)+θ2(1+4αq+3α2q2)+θ(4α+6α2q)+6α2}e−θ q(θ2+4θα+6α2)−1 (3.5.9)
Stress-strength parameter Suppose is the random strength and be the random stress of a component. , the component fails instantly and the component will function satisfactorily till . Therefore, is a measure of component reliability and in statistics it is known as stress-strength parameter. It has applications in almost all areas of knowledge including medical science, sociology, psychology, and engineering, some among others.
Let be independent strength and stress random variables having GAD (2.1) having parameters and respectively. Then the stress-strength reliability of GAD (2.1) can be obtained as
R=P(Y<X)=∞∫0P(Y<X|X=x)fX(x)dx =∞∫0f(x;θ1,α1)F(x;θ2,α2)dx
=1−θ31(θ62+2(2θ1+(2α2+α1))θ52+2(3θ21+10α2θ1+(3α22+6α2α1+α21))θ42++(4θ31+(18α2+6α1)θ21+(18α22+28α2α1+4α21)θ1+(24α22α1+4α2α21+12α22α21))θ32+(θ41+(10α2+2α1)θ31+(20α22+20α2α1+2α21)θ21+(40α22α1+20α2α21)θ1+(40α22α21))θ22+2(θ31+(5α2+2α2α1)θ21+(10α2α1+2α21)θ1+10α2α21)α2θ1θ2+2(θ21+2α1θ1+2α21)α22θ21)(θ21+2θ1α1+2α21)(θ22+2θ2α2+2α22)(θ1+θ2)5
Clearly at (α1=θ1,α2=θ2) and(α1=0,α2=0) , the above expression reduces to the corresponding expression of R for Aradhana and exponential distributions.
In this section maximum likelihood estimation of parameters of GAD has been discussed. Suppose (x1,x2,x3,...,xn) be a random sample from GAD (2.1)). The natural log likelihood function is thus obtained as
lnL=3nlnθ+nln(θ2+2θα+2α2)+2n∑i=1ln(1+αxi)−nθˉx,
where x is the sample mean.
The maximum likelihood estimates (MLE) (ˆθ,ˆα) of parameters(θ,α) are the solutions of the following non-linear log likelihood equations
∂lnL∂θ=3nθ−2n (θ+α)θ2+2θα+2α2−nˉx=0
∂lnL∂α=−2n(θ+2α)θ2+2θα+2α2+2n∑i=1xi1+α xi=0
It is difficult to solve these two natural log likelihood equations directly because they are not in closed forms. But these equations can be solved using Fisher’s scoring method. For, we have
∂2lnL∂θ2=−3nθ2+2nθ(θ+2 α)(θ2+2θα+2α2)2
∂2lnL∂θ ∂α=2n (θ2+4θα+2α2)(θ2+2θα+2α2)2
∂2lnL∂α2=8nα(θ+α)(θ2+2θα+2α2)2−2n∑i=1xi2(1+αxi)2
The solution of following equations gives MLE’s (ˆθ,ˆα) of (θ,α) of GAD
[∂2lnL∂θ2∂2lnL∂θ ∂α∂2lnL∂θ ∂α∂2lnL∂α2]ˆθ=θ0ˆα=α0[ˆθ−θ0ˆα−α0]=[∂lnL∂θ∂lnL∂α]ˆθ=θ0ˆα=α0
where θ0 and α0 are the initial values of θ and α , respectively. These equations are solved iteratively till sufficiently close values of ˆθ and ˆα are obtained.
The goodness of fit of GAD using maximum likelihood estimates has been discussed with two real lifetime dataset and the fit has been compared with exponential, Lindley and Aradhana distributions and a generalization of Sujatha distribution (AGSD) proposed by Shanker et al.7 It has been observed that GAD is more suitable to positively skewed lifetime data. In general, majority of the real lifetime datasets in medical science and engineering are positively skewed and hence GAD is suitable for positively skewed data in these fields of knowledge. Since GAD is a new distribution, still more research is required to find several applications of the distribution in various fields of knowledge. The real lifetime datasets which are positively skewed are as follows
Data set 1: This data represents the lifetime's data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross & Clark.10
1.1 1.4 1.3 1.7 1.9 1.8 1.6 2.2 1.7 2.7 4.1 1.8
1.5 1.2 1.4 3 1.7 2.3 1.6 2
Data set 2: This data set is the strength data of glass of the aircraft window reported by Fuller et al.11 18.830 20.800 21.657 23.030 23.230 24.050 24.321 25.500 25.520 25.800 26.690 26.770 26.780 27.050 27.670 29.900 31.110 33.200 33.730 33.760 33.890 34.760 35.750 35.910 36.980 37.080 37.090 39.580 44.045 45.29 45.381
For comparing the goodness of fit of GAD, AGSD, Aradhana, Lindley and exponential distributions, values of -2ln L , AIC (Akaike Information Criterion), kolmogorove-Smirnov Statistics (K-S Statistics) and p value of these distributions for two real lifetime datasets have been computed and presented in Table 5. Since the best fit of the distribution corresponds to the lower values of -2ln L, AIC, K-S statistics, it is obvious from table 5 that GAD provides better fit than exponential, Lindley, Aradhana and AGSD.
A generalized Aradhana distribution (GAD) has been introduced which includes both Aradhana distribution proposed by Shanker2 and Lindley distribution proposed by Lindley1as particular cases. The nature of probability density function and cumulative distribution function of GAD has been studied. The raw moments and central moments of the distribution have been obtained and behavior of coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of GAD have been studied with varying values of the parameters. Hazard rate function and mean residual life function have been studied with varying values of the parameters. The stochastic ordering, mean deviations, Bonferroni and Lorenz curves, and stress-strength reliability have also been discussed. The method of maximum likelihood has been discussed for estimating parameters. Two examples of real lifetime, one from medical science and one from engineering, have been presented to show the applications and goodness of fit of GAD over exponential, Lindley and Aradhana distributions and AGSD and it has been observed that GAD gives much better fit.
None.
The author declares there is no conflict of interest.
©2018 Welday, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.
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