Research Article Volume 10 Issue 3
1 Department of Statistics, Assam University, Silchar, Assam, India
2 Department of Community Medicine, Noida International Institute of Medical Science, India
3 Department of Mathematics, Nalanda Open University, India
4 Department of Mathematics, G.L.A. College, N.P University, India
Correspondence: Rama Shanker, Department of Statistics, Assam University, Silchar, Assam, India
Received: June 18, 2021 | Published: August 11, 2021
Citation: Shanker R, Shukla KK, Ranjan A, et al. Adya distribution with properties and application. Biom Biostat Int J. 2021;10(3):81-88. DOI: 10.15406/bbij.2021.10.00334
In the present paper, a new one parameter lifetime distribution named, “Adya distribution’ has been proposed for modeling lifetime data from engineering. Its various statistical properties including moments and moments based measures, hazard rate function, mean residual life function, stochastic ordering, deviations from the mean and the median, Bonferroni and Lorenz curves, and stress-strength reliability have been studied. Both the method of moment and the maximum likelihood estimation have been discussed for estimating the parameter of the proposed distribution. A numerical example has been presented to test the goodness of fit of the proposed distribution over other one parameter lifetime distributions available in statistical literature.
Keywords: lifetime distributions, statistical and mathematical properties, parameter estimation, goodness of fit
The classical one parameter exponential distribution and Lindley distribution proposed by Lindley1 were useful for modeling lifetime data from engineering and biomedical. It has been observed by Shanker et al.2 that exponential and Lindley distributions are not suitable for several lifetime data. In search for better one parameter lifetime distributions, Shanker has introduced several one parameter lifetime distributions including Shanker,3 Aradhana,4 Sujatha,5 Devya.6 The probability density function (pdf) and the cumulative distribution function (cdf) of these distributions are presented in table 1.
Distributions |
Probability density functions and Cumulative distribution functions |
|
Shanker |
f(x) = θ2θ2+1 (θ+x) e−θx f(x)=θ2θ2+1(θ+x)e−θx |
|
cdf |
F(x)=1−[1+θxθ2+1] e−θxF(x)=1−[1+θxθ2+1]e−θx |
|
Aradhana |
f(x)=θ3θ2+2θ+2(1+x)2e−θx f(x)=θ3θ2+2θ+2(1+x)2e−θx |
|
cdf |
F(x)=1−[1+θx(θx+2θ+2)θ2+2θ+2]e−θx F(x)=1−[1+θx(θx+2θ+2)θ2+2θ+2]e−θx |
|
Sujatha |
f(x)=θ3θ2+θ+2(1+x+x2)e−θxf(x)=θ3θ2+θ+2(1+x+x2)e−θx |
|
cdf |
F(x)=1−[1+θx(θx+θ+2)θ2+θ+2]e−θxF(x)=1−[1+θx(θx+θ+2)θ2+θ+2]e−θx |
|
Devya |
f(x)=θ5θ4+θ3+2θ2+6θ+24(1+x+x2+x3+x4)e−θxf(x)=θ5θ4+θ3+2θ2+6θ+24(1+x+x2+x3+x4)e−θx |
|
cdf |
F(x)=1−[1+{θ4(x4+x3+x2+x)+θ3(4x3+3x2+2x)+6θ2(2x2+x)+24θx}θ4+θ3+2θ2+6θ+24]e−θxF(x)=1−⎡⎢ ⎢⎣1+{θ4(x4+x3+x2+x)+θ3(4x3+3x2+2x)+6θ2(2x2+x)+24θx}θ4+θ3+2θ2+6θ+24⎤⎥ ⎥⎦e−θx |
|
Lindley |
f(x)=θ2θ+1(1+x)e−θ xf(x)=θ2θ+1(1+x)e−θx |
|
cdf |
F(x)=1−[1+θ xθ+1]e−θ xF(x)=1−[1+θxθ+1]e−θx |
Table 1 pdf and cdf of Shanker, Aradhana, Sujatha, Devya, and Lindley distributions for x>0, θ>0x>0,θ>0
The reasons for introducing such lifetime distributions with their advantages and disadvantages, statistical properties, parameter estimation and applications are available in the respective papers.
In this paper, a new lifetime distribution which gives better fit over several one parameter lifetime distributions are introduced. The new one parameter lifetime distribution is defined by its cdf and pdf, respectively
F(x,θ)=1−[1+θx(θx+2θ2+2)θ4+2θ2+2]e−θ x ;x>0,θ>0F(x,θ)=1−[1+θx(θx+2θ2+2)θ4+2θ2+2]e−θx;x>0,θ>0 (1.1)
f(x;θ)=θ3θ4+2θ2+2(θ+x)2e−θ x ;x>0, θ>0f(x;θ)=θ3θ4+2θ2+2(θ+x)2e−θx;x>0,θ>0 (1.2)
We name this distribution, “Adya distribution”. This is a convex combination of exponential (θ)(θ) , gamma (2,θ)(2,θ) and gamma (3,θ)(3,θ) distributions. We have
f(x;θ)=p1 g1(x;θ)+p2 g2(x;2,θ)+(1−p1−p2)g3(x;3,θ)f(x;θ)=p1g1(x;θ)+p2g2(x;2,θ)+(1−p1−p2)g3(x;3,θ)Where p1=θ4θ4+2θ2+2p1=θ4θ4+2θ2+2 ,p2=2θ4θ4+2θ2+2 p2=2θ4θ4+2θ2+2 ,g1(x;θ)=θ e−θ xg1(x;θ)=θe−θx ,g2(x;2,θ)=θ2Γ(2)x2−1 e−θ xg2(x;2,θ)=θ2Γ(2)x2−1e−θx , and g3(x;3,θ)=θ3Γ(3)x3−1 e−θxg3(x;3,θ)=θ3Γ(3)x3−1e−θx ;;x>0,θ>0;x>0,θ>0 .
The pdf and the cdf of Adya distribution for values of the parameter θθ are shown in figures 1 and 2, respectively.
The rr th moment about origin μr′ of (1.2) can be obtained as
μr′=r!{θ4+2(r+1)θ2+(r+1)(r+2)}θr(θ4+2θ2+2) ;r=1,2,3,... (2.1)
Substitutingr=1,2,3, and 4 in (2.1), the first four moments about origin of (1.2) are obtained as
μ1′=θ4+4θ2+6θ(θ4+2θ2+2) ,μ2′=2(θ4+6θ2+12)θ2(θ4+2θ2+2) ,
μ3′=6(θ4+8θ2+20)θ3(θ4+2θ2+2) ,μ4′=24(θ4+10θ2+30)θ4(θ4+2θ2+2)
Thus, the central moments of (1.2) are obtained as
μ2=θ8+8θ6+24θ4+24θ2+12θ2(θ4+2θ2+2)2 μ3=2(θ12+12θ10+54θ8+100θ6+108θ4+72θ2+24)θ3(θ4+2θ2+2)3 μ4=3(3θ16+48θ14+304θ12+944θ10+1816θ8+2304θ6+1920θ4+960θ2+240)θ4(θ4+2θ2+2)4Descriptive measures including coefficient of variation (C.V) , coefficient of skweness (√β1) , coefficient of kurtosis (β2) and index of dispersion (γ) of (1.2) are thus obtained as
C.V=σμ1′=√θ8+8θ6+24θ4+24θ2+12θ4+4θ2+6 √β1=μ3μ23/2=2(θ12+12θ10+54θ8+100θ6+108θ4+72θ2+24)(θ8+8θ6+24θ4+24θ2+12)3/2 β2=μ4μ22=3(3θ16+48θ14+304θ12+944θ10+1816θ8+2304θ6+1920θ4+960θ2+240)(θ8+8θ6+24θ4+24θ2+12)2 γ=σ2μ1′=θ8+8θ6+24θ4+24θ2+12θ(θ4+2θ2+2)(θ4+4θ2+6)The natures of these descriptive measures for values of parameter are shown in figure 3.
Figure 3 Coefficients of variation, skewness, kurtosis and index of dispersion of Adya distribution .
The condition under which Adya distribution is over-dispersed, equi-dispersed, and under-dispersed along with condition under which Shanker, Aradhana, Sujatha, Devya, Lindley and exponential distributions are over-dispersed, equi-dispersed, and under-dispersed are presented in table 2.
Distribution |
Over-dispersion (μ<σ2) |
Equi-dispersion |
Under-dispersion |
Adya |
θ<1.305719841 |
θ=1.305719841 |
θ>1.305719841 |
Shanker |
θ<1.171535555 |
θ=1.171535555 |
θ>1.171535555 |
Aradhana |
θ<1.283826505 |
θ=1.283826505 |
θ>1.283826505 |
Sujatha |
θ<1.364271174 |
θ=1.364271174 |
θ>1.364271174 |
Devya |
θ<1.451669994 |
θ=1.451669994 |
θ>1.451669994 |
Lindley |
θ<1.170086487 |
θ=1.170086487 |
θ>1.170086487 |
Exponential |
θ<1 |
θ=1 |
θ>1 |
Table 2 Over-dispersion, equi-dispersion and under-dispersion of Adya, Shanker, Aradhana, Sujatha, Devya, Lindley and exponential distributions for parameter θ
The hazard rate function and the mean residual life function of a continuous random variableX having pdf and cdf f(x) and F(x) are, respectively, defined as
h(x)=limΔx→0P(X<x+Δx |X>x)Δx=f(x)1−F(x) (3.1)
and m(x)=E[X−x|X>x] = 11−F(x)∫∞x[1−F(t)] dt (3.2)
Thus,h(x) and m(x) of (1.2) are obtained as
h(x)=θ3(θ+x)2θ2x2+2θ(θ2+1)x+(θ4+2θ2+2) (3.3)
and
m(x)=1[θ2x2+2θ(θ2+1)x+(θ4+2θ2+2)]e−θx ×∞∫x[θ2t2+2θ(θ2+1)t+(θ4+2θ2+2)]e−θtdt=θ2x2+2θ(θ2+2)x+(θ4+4θ2+6)θ[θ2x2+2θ(θ2+1)x+(θ4+2θ2+2)] (3.4)
This givesh(0)=θ5θ4+2θ2+2=f(0) and m(0)=θ4+4θ2+6θ(θ4+2θ2+2)=μ1′ . The hazard rate function and mean residual life function of Adya distribution are shown in figure 4.
A random variable X is said to be smaller than a random variable in the
Shaked and Shanthikumar7 proposed following results for establishing stochastic ordering of distributions
X≤lrY⇒X≤hrY⇒X≤mrlY (4.1)
⇓X≤stYThe distribution (1.2) is ordered with respect to the strongest ‘likelihood ratio’ ordering.
Theorem: Suppose X ∼ Adya distributon (θ1) and Y ∼ Adya distribution (θ2) . If θ1>θ2 , then X≤lrY and hence X≤hrY ,X≤mrlY and X≤stY .
Proof: We have
fX(x)fY(x)=θ13(θ24+2θ22+2)θ23(θ14+2θ21+2)((θ1+x)2(θ2+x)2)e−(θ1−θ2) x ; x>0Now
lnfX(x)fY(x)=ln[θ13(θ24+2θ22+2)θ23(θ14+2θ21+2)]+2ln(θ1+xθ2+x)−(θ1−θ2)x.
This givesddx{lnfX(x)fY(x)}=−2(θ1−θ2)(θ1+x)(θ2+x)−(θ1−θ2)
Thus forθ1>θ2 ,ddx{lnfX(x)fY(x)}<0 . This means that X≤lrY and hence X≤hrY ,X≤mrlY and X≤stY .
The mean deviation about the mean and the mean deviation about the median are used to measure the amount of scatter in the population from the mean and the median and defined by
δ1(X)=∞∫0|x−μ| f(x)dx and δ2(X)=∞∫0|x−M| f(x)dx , respectively, where μ=E(X) and M=Median (X) . The computation of these measures are simplified as
δ1(X)=μ∫0(μ−x)f(x)dx+∞∫μ(x−μ)f(x)dx=2μF(μ)−2μ∫0x f(x)dx (5.1)
and
δ2(X)=M∫0(M−x)f(x)dx+∞∫M(x−M)f(x)dx=μ−2M∫0x f(x)dx (5.2)
Using pdf (1.2) and expression for the mean of Adya distribution ,we get
μ∫0x f(x;θ)dx=μ−{θ5μ+θ4(2μ2+1)+θ3(μ3+4μ)+θ2(3μ2+μ)+6θμ+6}e−θ μθ(θ4+2θ2+2) (5.3)
M∫0x f(x;θ)dx=μ−{θ5M+θ4(2M2+1)+θ3(M3+4M)+θ2(3M2+M)+6θ M+6}e−θ Mθ(θ4+2θ2+2)) (5.4)
Using expressions from (5.1), (5.2), (5.3), and (5.4), the mean deviation about mean, and the mean deviation about median, of Adya distribution are obtained as
δ1(X)=2{2θ3μ+θ2(μ2+μ)+4θμ+(θ4+6)}e−θ μθ(θ4+2θ2+2) (5.5)
δ2(X)=2{θ5M+θ4(2M2+1)+θ3(M3+4M)+θ2(3M2+M)+6 θM+6}e−θ Mθ(θ4+2θ2+2)−μ (5.6)
The Bonferroni8 and Lorenz curves are defined as
B(p)=1pμq∫0x f(x) dx=1pμ[∞∫0x f(x)dx−∞∫qx f(x) dx]=1pμ[μ−∞∫qx f(x) dx] (6.1)
and L(p)=1μq∫0x f(x) dx=1μ[∞∫0x f(x)dx−∞∫qx f(x) dx]=1μ[μ−∞∫qx f(x) dx] (6.2)
respectively or equivalently
B(p)=1pμp∫0F−1(x) dx (6.3)
and L(p)=1μp∫0F−1(x) dx (6.4)
respectively, where μ=E(X) and q=F−1(p) .
The Bonferroni and Gini indices are thus defined as
B=1−1∫0B(p) dp (6.5)
and G=1−21∫0L(p) dp (6.6)
respectively.
Using pdf (1.2), we have
∞∫qx f(x) dx={θ5q+θ4(2q2+1)+θ3(q3+4q)+θ2 (3q2+1)+6θ q+6}e−θqθ(θ4+2θ2+2) (6.7)
Now using equation (6.7) in (6.1) and (6.2), we have
B(p)=1p[1−{θ5q+ θ4(2q2+1)+θ3(q3+4q)+ θ2 (3q2+4)+6 θ q+6}e−θqθ4+4θ2+6] (6.8)
andL(p)=1−{θ5q+ θ4(2q2+1)+θ3(q3+4q)+ θ2 (3q2+4)+6 θ q+6}e−θqθ4+4θ2+6 (6.9)
Now using equations (6.8) and (6.9) in (6.5) and (6.6), the Bonferroni and Gini indices are obtained as
B=1−{θ5q+ θ4(2q2+1)+θ3(q3+4q)+ θ2 (3q2+4)+6 θ q+6}e−θqθ4+4θ2+6 (6.10)
G=2{θ5q+ θ4(2q2+1)+θ3(q3+4q)+ θ2 (3q2+4)+6 θ q+6}e−θqθ4+4θ2+6−1 (6.11)
Suppose X and Y be independent strength and stress random variables having Adya distribution with parameter θ1 and θ2 respectively. Then R=P(Y<X) is known as stress-strength parameter and is a measure of the component reliability.
Thus, R=P(Y<X)=∞∫0P(Y<X|X=x)fX(x)dx=∞∫0f(x;θ1) F(x;θ2)dx =1−θ13[24 θ22+12 θ2(θ22+θ1θ2+1)(θ1+θ2)+2(θ42+4θ1θ32+θ21θ22+2θ22+4θ1θ2+2)(θ1+θ22)+2θ1(θ24+θ1θ23+2θ2+θ1θ2+2)(θ1+θ2)3+θ21(θ42+2θ22+2)(θ1+θ2)4](θ14+2θ21+2)(θ24+2θ22+2)(θ1+θ2)5 =1−θ13[24 θ22+12 θ2(θ22+θ1θ2+1)(θ1+θ2)+2(θ42+4θ1θ32+θ21θ22+2θ22+4θ1θ2+2)(θ1+θ22)+2θ1(θ24+θ1θ23+2θ2+θ1θ2+2)(θ1+θ2)3+θ21(θ42+2θ22+2)(θ1+θ2)4](θ14+2θ21+2)(θ24+2θ22+2)(θ1+θ2)5
Since Adya distribution has one parameter, equating the population mean to the corresponding sample mean, the moment estimate ˜θ of θ is the solution of the following fifth degree polynomial equation ˉx θ5−θ4+2 ˉx θ3−4θ2+2θˉx−6=0
, where ˉx is the sample mean.
Estimation using maximum likelihood estimation
Taking (x1, x2, x3, ... ,xn) a random sample from (1.2), the natural log likelihood function of Adya distribution is
lnL=n ln(θ3θ4+2θ2+2)+2n∑i=1ln(θ+xi)−n θ ˉx .
This gives,dlnLdθ=3nθ−4 n θ(θ2+1)θ4+2θ2+2+2n∑i=11θ+xi−n ˉx ,
The MLE ˜θ of θ is the solution of dlnLdθ=0 which is given by
3nθ−4 n θ(θ2+1)θ4+2θ2+2+2n∑i=11θ+xi−n ˉx=0 .
This non-linear equation is not in compact form and its solution can be obtained analytically. We have to use non-linear optimization technique such as quasi-Newton algorithm available in R software to maximize the log-likelihood function.
A simulation study has been conducted to examine the performance of the maximum likelihood estimate (MLE) of the parameter of Adya distribution. Random number n=20,40,60 and 80 generated corresponding to the parameter θ=0.1,0.2,0.3,0.4 and 0.5 using Acceptance- Rejection method. Its Bias and Mean Square Error have been calculated and presented in the table 3.
n |
θ |
Bias (θ ) |
MSE(θ ) |
20 |
0.1 |
0.01200 |
0.00288 |
0.2 |
0.021244 |
0.009028 |
|
0.3 |
0.018753 |
0.007033 |
|
0.4 |
0.023497 |
0.0110371 |
|
40 |
0.1 |
0.004884 |
0.000954 |
0.2 |
0.007587 |
0.002302 |
|
0.3 |
0.008900 |
0.003168 |
|
0.4 |
0.010251 |
0.0042034 |
|
60 |
0.1 |
0.003484 |
0.000728 |
0.2 |
0.005428 |
0.001768 |
|
0.3 |
0.005854 |
0.002056 |
|
0.4 |
0.006242 |
0.002338 |
|
80 |
0.1 |
0.0023048 |
0.000424 |
0.2 |
0.0038444 |
0.001182 |
|
0.3 |
0.004266 |
0.001456 |
|
0.4 |
0.004735 |
0.001794 |
Table 3 Average Bias and Average Mean Square Error of simulated MLE (θ ) for fixed values θ =0.1,0.2,0.3 &0.4
The data set considered for the goodness of fit of Adya distribution is the strength data of glass of the aircraft window reported by Fuller et al.9 and are given as
18.83, 20.80, 21.657, 23.03, 23.23, 24.05, 24.321, 25.50, 25.52, 25.80, 26.69, 26.77, 26.78, 27.05, 27.67, 29.90, 31.11, 33.20, 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
The goodness of fit of the distributions are based on the values of −2lnL , AIC (Akaike Information Criterion) and K-S (Kolmogorov-Smirnov) statistic. AIC and K-S are computed using,AIC=−2lnL+2k ,K-S=Supx|Fn(x)−F0(x)| where k = the number of parameters, n = the sample size and Fn(x) is the empirical distribution function. The distribution having lower −2lnL , AIC , and K-S are said to be best distribution. The MLE (ˆθ) and standard error, S.E(ˆθ) ofθ ,−2lnL , AIC, K-S and p-value of the fitted distributions are presented in the table 4.
Distributions |
MLE(ˆθ) |
S.E(ˆθ) |
−2lnL |
AIC |
K-S |
p-value |
Adya |
0.096970 |
0.01000 |
240.63 |
242.63 |
0.298 |
0.006 |
Shanker |
0.647164 |
0.008200 |
252.35 |
254.35 |
0.358 |
0.0004 |
Aradhana |
0.094319 |
0.009780 |
242.22 |
244.22 |
0.306 |
0.0044 |
Sujatha |
0.095613 |
0.009904 |
241.50 |
243.50 |
0.303 |
0.0051 |
Devya |
0.160873 |
0.012916 |
227.68 |
229.68 |
0.422 |
0.0000 |
Lindley |
0.062992 |
0.008001 |
253.98 |
255.98 |
0.365 |
0.0003 |
Exponential |
0.032449 |
0.005822 |
274.52 |
276.53 |
0.458 |
0.0000 |
Table 4 MLE’s, S.E (ˆθ) , AIC and K-S Statistics of the fitted distributions of the given data set
Clearly Adya distribution gives better fit than Shanker, Aradhana, Sujatha, Devya, Lindley and exponential distributions.
Adya distribution, a one parameter lifetime distribution, for modeling lifetime data has been presented and studied. The statistical properties including coefficient of variation, skewness, 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. Over-dispersed, equi-dispersed, and under-dispersed of Adya distribution are presented. Method of moment and method of maximum likelihood are explained for estimating parameter. The asymptotic property of the ML estimate of the parameter has been discussed with simulation study. Finally, the goodness of fit test has been presented with a real lifetime data.
NOTE: The paper is named Adya distribution in the name of my loving niece Adya Vedanshi, the daughter of my younger brother Dr. Ravi Shanker.
None.
All authors declare that there is no conflict of interest.
©2021 Shanker, 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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