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A TWO-PARAMETER ARADHANA DISTRIBUTION WITH APPLICATIONS TO RELIABILITY ENGINEERING
Ravi Shanker and Nitesh Kumar Soni
Department of Mathematics, G.L.A. College, Nilamber-Pitamber University, Daltonganj,
Jharkhand, India E-mail: [email protected] ; [email protected]
Rama Shanker, Mousumi Ray, and Hosenur Rahman Prodhani
Department of Statistics, Assam University, Silchar, India E-mail: [email protected]; [email protected]; [email protected]
Abstract
The search for a statistical distribution for modelling the reliability data from reliability engineering is challenging and the main cause is the stochastic nature of the data and the presence of skewness, kurtosis and over-dispersion. During recent decades several one and two-parameter statistical distributions have been proposed in statistics literature but all these distributions were unable to capture the nature of data due to the presence of skewness, kurtosis and over-dispersion in the data. In the present paper, two-parameter Aradhana distribution, which includes one parameter Aradhana distribution as a particular case, has been proposed. Using convex combination approach of deriving a new statistical distribution, a two- parameter Aradhana distribution has been proposed. Various interesting and useful statistical properties including survival function, hazard function, reverse hazard function, mean residual life function, stochastic ordering, deviation from mean and median, stress-strength reliability, Bonferroni and Lorenz curve and their indices have been discussed. The raw moments, central moments and descriptive measures based on moments of the proposed distribution have been obtained. The estimation of parameters using the maximum likelihood method has been explained. The simulation study has been presented to know the performance in terms of consistency of maximum likelihood estimators as the sample size increases and. The goodness of test of the proposed distributions has been tested using the values of Akaike Information criterion and Kolmogorov-Smirnov statistics. Finally, two examples of real lifetime datasets from reliability engineering have been presented to demonstrate its applications and the goodness of fit, and it shows a better fit over two-parameter generalized Aradhana distribution, quasi Aradhana distribution, new quasi Aradhana distribution, Power Aradhana distribution, weighted Aradhana distribution, gamma distribution and Weibull distribution. The flexibility, tractability and usefulness of the proposed distribution show that it is very much useful for modelling reliability data from reliability engineering. As this is a new distribution and it has wide applications, it will draw the attention of researchers in reliability engineering and biomedical sciences to search many more applications in the future.
Keywords: Aradhana distribution, reliability properties, maximum likelihood estimation, applications.
R. Shanker, N. K. Soni, R. Shanker, M. Ray and H. R. Prodhani RT&A, No 3 (79) A TWO-PARAMETER ARADHANA DISTRIBUTION WITH..._Volume I9, September 2024
I. INTRODUCTION
Shanker [1] proposed the Aradhana distribution, a one parameter lifetime distribution designed to characterize lifetime data originating from the fields of biomedical sciences and engineering. This distribution is characterized by its probability density function (pdf) and cumulative distribution function (cdf) as
03 9
f(x;0) = (1 + x) e°;x > 0,0 > 0
F ( x;0) = 1 -
+ 20 + 2
" 0 x(0 x + 20 + 2) 1 + ■
e0x;x > 0,0 > 0
02 + 20+2
It has been shown by Shanker [1] that in most of the real lifetime datasets it exhibited superior fit in comparison to exponential, Lindley [2], Shanker and Akash distributions introduced by Shanker [3,4]. The Aradhana distribution is a convex combination of exponential (0), gamma
02 20 2 (2,0) and gamma (3,0) distributions with their proportions —-,—-and
0L + 20 + 2 0L + 20 + 2 02 + 20 + 2 respectively.
The mean and variance of Aradhana distribution are
02 + 40 + 6 t _ 04 + 803 + 2402 + 240 +12 E(X) =—.-7 and Var(X) =-----.
00 + 20 + 2) 02 0 + 20 + 2)
Important statistical properties of Aradhana distribution including shapes for varying values of parameter, moments related measures, hazard function, mean residual life function, stochastic ordering, mean deviations, distribution of order Statistics, Bonferroni and Lorenz curves, Renyi entropy measure and stress-strength reliability have been discussed and also studied estimation of parameter and applications of Aradhana distribution for modelling lifetime data by Shanker [1].
Shanker et al [5] have introduced a quasi Aradhana distribution (QAD) with its pdf and
cdf as
0 ?
f (x,0,a) = -(a + 0x) e~0x ; x > O,0> 0,a > 0
a + 2m + 2v 7
F ( x,0,a) = 1 -
0x (0x + 2a + 2)
a2 + 2a + 2
e~0x ; x > 0,0> 0,a > 0.
The detailed studies on various statistical properties, estimation of parameters and applications of QAD are available in Shanker et al [5]. Anthony and Elangovan [6] discussed the length-biased version of QAD and study its properties and applications. Further, Anthony and Elangovan [7] proposed a new generalization of QAD by introducing an additional parameter in the QAD and study its statistical properties and applications.
Shanker et al [8] proposed a new quasi Aradhana distribution defined by its pdf and cdf
0 ? f (x; 0, a) = —---7 (0 + ax) e~ffx ; x > 0,0 > 0, a > 0
J v ' ' ' M , -,n2„, , ">„,2 V >
F ( x;0,a) = 1 -
a3
04 + 201a + 2a Y+0 a x(0 a x + 202 + 2a) 04 + 202a + 2a2
e°;x > 0,0 > 0,a > 0.
The detailed studies on various statistical properties, estimation of parameters and applications of NQAD are available in Shanker et al [8].
In this paper an attempt has been made to suggest a two-parameter Aradhana distribution and study its statistical properties, estimation of parameters and applications. The whole paper is divided into eleven sections. Section one is introductory in nature. Section 2 deals with the derivation of pdf and cdf of a two-parameter Aradhana distribution and the behaviors of its pdf
and cdf. Descriptive measures based on moments have been discussed in section three. Reliability properties of the distribution have been studied in section four. Deviation from mean and median, Bonferroni and Lorenz curves, stress-strength reliability have been discussed in sections five, six and seven, respectively. Maximum likelihood estimation and simulation study of the proposed distribution are given in sections eight and nine respectively. Finally, the applications and concluding remarks are presented in section ten and eleven, respectively.
II.
A TWO-PARAMETER ARADHANA DISTRIBUTION
A two-parameter Aradhana distribution (ATPAD) can be defined by its pdf and cdf
f ( x;0,a)
o3
0W + 20a + 2
(a + x )2 e"0x ; x > 0, 0> 0,a> 0
F (x;0,a) = l -
1 +
0x (0x + 20a + 2)
; x > 0, 0> 0,a> 0
Aradhana distribution with parameter 0, gamma (3,0) distribution and exponential distribution are special cases of ATPAD for (a = l), (a = 0) and a ^rc. The behavior of the pdf
and the cdf of ATPAD for different values of parameters are shown in figures 1 and 2 respectively.
Figure 1: pdf of ATPAD for different values oofparameters
Figure 2: cdf of ATPAD for different values of parameters
III. DESCRIPTIVE MEASURES
The r th moment about origin of ATPAD can be obtained as
m/ = E ( Xr ) =
-Jxr (a2 + 2ax + x2)e 0xdx
02a2 + 20a + 2u r l{02a2 + 2 ( r + 1)0a + ( r +1)( r + 2)}
r = 1,2,3,...
0r (02a2 + 20a + 2)
Substituting r =1,2,3,4 in the above expression, the first four moments about origin (raw moments) of ATPAD can be obtained as
M =
0 a + 40a + 6 0(02a2 + 20a + 2) ,
6 (02a2 + 80a + 20)
M 03 (02a2 + 20a + 2) ,
2 (02a2 + 60a+ 12) M2 02 (02a2 + 20a + 2) , 24 (02a2 + 100a + 30)
M 4 = "
04 (02a2 + 20a + 2)
The central moments, using relationship between central moments and raw moments, can thus be obtained as
04a4 + 80 V + 2402a2 + 240a +12 M2 = —
02 (02a2 + 20a + 2)
3
0
M =
2 (06a6 + 120V + 5404a4 + 1000 V + 1080V + 720a + 24)
M =
03 (0 V + 20a + 2 )
3(308a8 + 480V7 + 3040V6 + 9440V + 18160V + 23040V + 19200V + 9600a + 240)
04 (02a2 + 20a + 2)
Thus, the coefficient of variation (C.V), coefficient of skewness («Jft), coefficient of kurtosis (P2 ),
and index of dispersion (/) of ATPAD are obtained as a -¡0AaA + 80 V + 240V + 240a +12
C.V. = ■
M
02a2 + 40a + 6
.— M 2 (06a6 + 1205a5 + 5404a4 + 10003a3 + 10802a2 + 720a + 24)
V P = _
, 32
P =
V" (0"a4 + 803a3 + 240V + 240a +12)3'
M 33(30sas + 4807a7 + 3040V5 + 94405a5 + 18160 V + 230403a3 + 192002a2 + 9600a + 240)
M2
(04a4 + 80V + 240V + 240a +12)2
r = -
a 04a4 + 803a3 + 240V + 240a +12
' 0(0V + 20a + 2)(02a2 + 40a + 6)
Mi
The graphical relationship between mean and variance of ATPAD to see the over-dispersion, equi-dispersion and under-dispersion are shown in the following figure 3.
Figure 3: Mean and variance of ATPAD
Behavior of coefficient of variation, skewness, kurtosis and index of dispersion of ATPAD shown in figure 4.
3
Figure 4: Behaviors of coefficient of variation, skewness, kurtosis and index of dispersion of ATPAD
IV. SOME RELIABILITY PROPERTIES
I. Survival Function
The survival function of ATPAD can be obtained as
|~0x (0x + 20a + 2) + (02a2 + 20a + 2)] S(x;0,a) = 1 -F(x;0,a) = ^-. .---;x > 0, 0> 0,a > 0
02a2 + 20a + 2
II. Hazard Function and Mean Residual Life Function
The hazard function of ATPAD can be obtained as
f (x;0, a)
h( x; 0, a) = :
03(a + x)2
S (x; 0, a) 0x (0x + 20a + 2) + (02a2 + 20a + 2) The mean residual life function of ATPAD can be obtained as
m(x,0,a) = 1
-fr 1 - F (i;0,a)l dx
1 -F(x;0,a)f L V ; ,
02 x2 + 2 (0a + 2)0x + [02a2 + 40a + 6) 00 (0x + 20a + 2) + (02a2 + 20a + 2)]
The graphical representation of hazard function and mean residual life function are presented in the figure 5 and 6 respectively. From the figure 5 it is cleared that all values of the parameters 0 and a hazard function is monotonically increasing. From the figure 6 it is cleared that for all values of the parameters 0 and a mean residual life function is monotonically decreasing.
Figure 5: Hazard function of ATPAD for various values of the parameters
Figure 6: Mean residual life function of ATPAD for various values of the parameters
III. Reverse Hazard Function
Reverse hazard function of ATPAD can be obtained as
( x;0,a)
6\a + x )2 e0
f (x;0,a) =_
F (x; в,a) (в2 a2 + 20a + 2)- [Ox (Ox + 20a + 2) + (02a2 + 20a + 2)] e"
IV. Stochastic Ordering
In probability theory and statistics, a stochastic order quantifies the concept of one random variable being bigger than another. A random variable X is said to be smaller than a random variable Y in the:
i. Stochastic order (X <st Y) if F^ (x) > Fy (y) for all x
ii. Hazard rate order (X Y) if Hx (x) > Hy (y) for all x
iii. Mean residual life order (X <mr/ Y) if m% (x) > my (y) for all x
f (x)
iv. Likelihood ratio order (X <} Y) if X ^ ' decrease
( lr ) fY (y)
in x
The following results due to Shaked and Shantikumar [9] are well known for establishing stochastic ordering of distributions
X <lrY * X <hrY * X <mrlY
U
X<stY
Theorem: Let X~ ATPAD (01,a1) and Y~ ATPAD(02,a2). , If a1 >a2 and0 > 02 or 01 =02 and a1 > a2 then X <r Y hence X <hrY, X <mri Y and X <stY . Proof: We have
fx (x)_ ^ («22^22 + 2«2^2 + 2) ( a1 + x) fY(x) el (a12e12 + 2a1e1 + 2) (a2 + x)2
-(H -ei) X
Now
log^ = log
fY (X)
'Y
Therefore d/
dx
log
fx (x)
fY (x)
e3 (ai2ei2+2aiei + 2) el (a12e12+2a1e1 + 2)
2 (a2 -a1)
+ 2 log
i \ a1 + x
a2 + x
-(e-ег) x
(a1 + x )(a2 + x)
-(e-e2)
Thus, If a1 > a2 and 01 > 02 or 01 = 02 and a1 >a2 , hence X <hrY, X <mrl Y and X <stY .
log
fx (x)
fY (x)
< 0. This means X <!r Y
V. DEVIATION FROM MEAN AND MEDIAN
The amount of scatter in a population is an evidently measured to some extent by the totality of deviations from the mean and median. These are known as the mean deviation about the mean and mean deviation about median and are defined by
¿1(x) = 2pF(p)-2Jxf(x;0,a)dx and S2(x) = -p + 2Jxf(x;0,a)dx
0 M
Thus S1 (x) and S2 (x) of ATPAD are obtained as
2 [02p2 + 202cp + 40p + (02a2 + 40a + 6)]e"p
8 ( x ) = ■
e(e2a2 + 2ea+2)
2{e3M3 +(2ea+ 3)e2M2 +(e2a2 + 4ea + 6)eM + (e2a2 + 4ea + б)} ee
8 ( x) =-:-;--U
2W e(e2a2 + 2ea+2)
VI. BONFERRONI AND LORENZ CURVES
The Bonferroni and Lorenz curves [10] and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as
1 f
B(p) =-I xf (x)dx ■
PU{
1
PU
| xf (x)dx -1 xf (x)dx
1
PU
¡и-1 xf (x)dx
and
1 q 1 L(p) = — I xf (x)dx = —
A J„ A
J xf (x)dx - J xf (x)dx
1 да Г
/u-\ xf(x)dx
A J _ q _
respectively or equivalently.
The Bonferroni and Gini indices are obtained as i i B = 1 -JB(p)dp and G = 1 -2JL(p)dprespectively. 0 0 Using the pdf of ATPAD, we get
B( p) = -
p
1 -
L(p) = 1 -
03 +(20a + 3)02 q2 + (02a2 + 40a + б)0q + (02a2 + 40a + б)} e (O'a2 + 40a + б)
[eiqi + (20a + 3) 02 q2 + (02a2 + 40a + б) 0q + (02a2 + 40a + б)} e
0q
(02a2 + 40a + 6)
Finally, after little algebraic simplification, the Bonferroni and Gini indices of ATPAD are obtained
as
B =
G =
|03q3 +(20a + 3)02q2 + (02a2 + 40a + 6)0q + (02a2 + 40a + б)} e0q
(02a2 + 40a + б) 2 |03q3 + (20a + 3) 02q2 +(02a2 + 40a + б) 0q + (02a2 + 40a + б)} e^4 (02a2 + 40a + б)
VII. STRESS-STRENGTH RELIABILITY
The stress-strength reliability of a component illustrates the life of the component which has random strength X that is subjected to a random stress Y. When the stress of the component Y applied to it exceeds the strength of the component X , the component fails instantly, and the component will function satisfactorily until X > Y. Therefore, R = P( Y < X) is a measure of the component reliability and is known as stress-strength reliability in statistical literature. It has extensive applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels, etc.
/да
P (Y <X | X = x) fX (x)dx
0
03
= 1 -
2402 + 4810 2a +0 (0a +1)}(0 +0) +21022 a + 40 (0a +1) a+(02 a2 + 20a + 2)} (0 + 0 )2 +2|0 (0a + 1)a2 +a (0 2a2 + 20a + 2 )}(0 +0)32 +a2 (02 a2 + 20a + 2)(0 +0)4
(02a,2 + 20a + 2)(0 2a2 + 20a + 2)(0 +0)5
VIII. ESTIMATION AND INFERENCE
Let (x1, x2,..., xn) be a random sample from ATPAD (6, a) , the likelihood function L and the log-likelihood function, log L are given by
L =
n(«+x)'
2 „-nSx
da + 29a + 2 , ,=
logL = 3nlogd — nlog(d2a2 + 2da + 2) + 2^log(a + x)-ndx
i=1
The maximum likelihood estimates (MLEs) d and a of d and a are then the solutions of the following non-linear equations
d log L_ 3n 2na(da+1)
dd ~ d d2a2 + 2da + 2 d log L_ 2nd (da +1)
— nx = 0
n 1
2Z-1
= 0
da 02a2 + 26a + 2 a + xt These two natural log likelihood equations do not seem to be solved directly. However, the Fisher's scoring method can be applied to solve these equations. We have d2 logL_ 3n 2na6(6a + 2) d6 ~~6 + (62a2 + 26a + 2 )2 d2 logL 2na63 (6a + 2) da' (62a2 + 26a + 2 )2 52 logL_ -2n (62a2 + 46a + 2) _ d2 logL d6da (62a2 + 26a + 2 )2 dad6 The following equations can be solved for MLEs 6 and cx of 6 and a of ATPAD
f d log L^
fd2 log L d2 log LA
dd
dd da
d2 log L d2 log L
da dd
da
(;
d 9o
a—a.
dd d log L
da
IX. THE SIMULATION STUDY
In this section, we carried out simulation study to examine the performance of maximum likelihood estimators of the ATPAD. We examined the mean estimates, biases (B), mean square errors (MSEs) and variances of the MLEs. The mean, bias, MSE and variance are computed using the formulae
Mean =
In 1 n . , 1 n , „ ,
1 ^H,, 5 = 1X(Ht — H), MSE = 12[h, — H) ,
MSE = -\\H — H)
n i=1
Variance = MSE — B
where H = d,a and Hi =& a .
The simulation results for different parameter values of ATPAD are presented in tables 1 and 2 respectively. The steps for simulation study are as follows:
a. Data is generated using the acceptance-rejection method of simulation. The acceptance-rejection method is a commonly used approach in simulation studies to generate random samples from a target distribution when inverse transform method of simulation is not feasible or efficient. Acceptance rejection method for generating random samples from the
2
ATPAD consists of following steps.
i. Generate a random variable Y distributed as exponential (0)
ii. Generate U distributed as Uniform (0,1)
iii. If U < f(y) , then set X = Y ("accept the sample"); otherwise ("reject the sample")
Mg(y)
and if reject then repeat the process: step (i-iii) until getting the required samples. Where M is a constant.
b. The sample sizes are taken as n = 50,100,150,200
c. The parameter values are set as values 0 = 0.2, a = 1.8 and 0 = 0.2, a = 4.0
d. Each sample size is replicated 10000 times
The results obtained in Tables 1 and 2 show that as the sample size increases, biases, MSEs and variances of the MLEs of the parameters become smaller respectively. This result is in line with the first-order asymptotic theory.
Table-1: The Mean values, Biases, MSEs and Variances of ATPAD for parameter 0 = 0.2, a = 1.8
Parameters Sample Size Mean Bias MSE Variance
0 20 0.20148 0.00148 0.000004212 0.000002011
40 0.20094 0.00094 0.000002244 0.000001354
50 0.20083 0.00083 0.000002477 0.000002009
100 0.20081 0.00081 0.000002422 0.000001785
150 0.20079 0.00079 0.000002254 0.000001591
200 0.20068 0.00068 0.000002251 0.000001557
a 20 1.72273 -0.07726 0.127852500 0.121882500
40 1.76513 -0.03486 0.062430860 0.061215480
50 1.77232 -0.02767 0.051223530 0.050457760
100 1.78933 -0.01066 0.026087550 0.025973800
150 1.79298 -0.00701 0.017392750 0.017343480
200 1.78781 -0.00121 0.015850450 0.025702030
Table-2: The Mean values, Biases, MSEs and Variances of ATPAD for parameter 0 = 0.2 ,a = 4.0
Parameters Sample Size Mean Bias MSE Variance
0 20 0.20138 0.001380 0.00000342 0.000001517
40 0.20110 0.001107 0.00000242 0.000001195
50 0.20107 0.001077 0.00000224 0.000001082
100 0.20107 0.001074 0.00000223 0.000001084
150 0.20096 0.000963 0.00000206 0.000001139
200 0.20074 0.000742 0.00000173 0.000001799
a 20 4.01832 0.018320 0.00659461 0.00625895
40 4.00915 0.009156 0.00329730 0.00321346
50 4.00726 0.007266 0.00263803 0.00258523
100 4.00484 0.004849 0.00175868 0.00173517
150 4.00364 0.003640 0.00141901 0.00140576
200 4.00363 0.003630 0.00131901 0.00130576
X. APPLICATIONS
The following real lifetime datasets have been considered for testing the goodness of fit of ATPAD over the other two-parameter lifetime distributions. The goodness of fit based on K-S statistic, fitted plots of considered distributions for the datasets, p-p plots of considered distributions for the datasets and total time in test (TTT) plots for the datasets and the ATPAD confirm that among all considered distributions, ATPAD provides much closure fit.
Data set-1: This censored tri-modal data contains 30 items that is tested when test is stopped after20-th failure. The following data discussed by Murthy et al [11] and the values are:
0.0014, 0.0623, 1.3826, 2.0130, 2.5274, 2.8221, 3.1544, 4.9835, 5.5462, 5.8196, 5.8714, 7.4710, 7.5080, 7.6667, 8.6122, 9.0442, 9.1153, 9.6477, 10.1547, 10.7582. Description of the data set-1
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0014 2.7484 5.8455 5.7081 8.7202 10.7582
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
i/n i/n
Figure 7: TTT- plot of the observed dataset 1 and simulated data of ATPAD respectively
Data set-2: The following skewed to right, a complete set of data, discussed by Murthy et al [11], and reports the failure time of 20 electric bulbs and the observations are: 1.32, 12.37, 6.56, 5.05, 11.58, 10.56, 21.82, 3.60, 1.33, 12.62, 5.36, 7.71, 3.53, 19.61, 36.63, 0.39, 21.35, 7.22, 12.42, 8.92.
Description of the data set-2
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.390 4.688 8.315 10.498 12.470 36.630
Figure 8: TTT- plot of the observed dataset 2 and simulated data of ATPAD respectively In order to compare lifetime distributions, values of -2 log L, Akaike Information Criterion (AIC), Kolmogorov-Smirnov Statistics (K-S) and the corresponding probability value (p-value) for the above data set has been computed. The formulae for computing AIC and K-S are as follows:
AIC = -2logL + 2k, D = Sup \Fn (x) - F0 (x) |
x
where, k = number of parameters, n = sample size, Fn (x) = empirical cdf of considered distribution and F0 (x) = cdf of considered distribution
The distribution corresponding to the lower values of -2 log L, AIC and K-S Statistics is the best fit distribution. The MLEs of parameters of the considered distributions along with their standard error, -2 log L, AIC, K-S and p-value of the considered distributions for datasets 1 and 2 are
presented in tables 3 and 4 respectively.
Table 3: ML estimates, Standard errors, -2log L, AIC, K-S and p-value of the considered distributions for the
dataset-1
Distribution ML estimates в SE (§) a SE (a) -2 log L AIC K-S p-value
ATPAD 0.3896 (0.0806) 2.4576 (1.9436) 105.8912 109.8912 0.1837 0.5105
GAD 0.3896 (0.0806) 0.4068 (0.3217) 105.8912 109.8912 0.1949 0.4341
QAD 0.3896 (0.0806) 0.9577 (0.6199) 105.8912 109.8912 0.1971 0.3908
NQAD 0.3896 (0.0806) 0.1585 (0.1518) 105.8912 109.8912 0.1930 0.4696
PAD 0.5935 (0.1558) 0.8366 (0.1394) 106.0269 110.0269 0.1915 0.4687
WAD 0.4535 (0.1142) 0.0100 (0.5170) 107.4734 111.4734 0.1902 0.4346
GD 0.1513 (0.0552) 0.8637 (0.2373) 109.3792 113.3792 0.2530 0.1624
WD 0.1469 (0.0721) 1.0892 (0.2209) 109.5036 113.5036 0.9000 0.0000
Table 4: ML estimates, Standard errors, -2log L, AIC, K-S and p-value of the considered distributions for the
dataset-2
Distributions ML estimates 9 SE(e) a SE (a) -2 log L AIC K-S p-value
ATPAD 0.1866 (0.0565) 8.1128(9.4833) 132.9421 136.9421 0.1220 0.9276
GAD 0.1867 (0.0565) 0.1232 (0.1440) 132.9421 136.9421 0.1355 0.8570
QAD 0.1866 (0.0565) 1.5147 (1.3803) 132.9421 136.9421 0.1307 0.8972
NQAD 0.1869 (0.0558) 0.0231 (0.0329) 132.9421 136.9421 0.1459 0.8073
PAD 0.4421 (0.1225) 0.7755 (0.1097) 133.1672 137.1672 0.1343 0.8711
WAD 0.2625 (0.0710) 0.0100 (0.6242) 137.0825 141.0825 0.1425 0.8289
GD 0.1272(0.0438) 1.3361 (0.3811) 133.0916 137.0916 0.1340 0.8729
WD 0.1000 (0.0776) 1.0150 (0.2550) 134.0518 138.0518 0.9830 0.000
From Table-3 and 4 we observed that the ATPAD has the same -2 log L, AIC values but least K-S values as compared to GAD (Generalized Aradhana Distribution) of Daniel and Shanker [12] and , QAD (Quasi Aradhana Distribution), NQAD (New Quasi Aradhana distribution) and has the least -2 log L, AIC, K-S values as compared to PAD (Power Aradhana distribution) of Shanker and Shukla [13], WAD (Weighted Aradhana distribution) by Ganaie et al [14] and subsequently critical study done by Shanker et al [15], GD (gamma Distribution ) and WD (Weibull distribution) by Weibull [16].
Hence, we may conclude that ATPAD provides the better fit than GAD, QAD, NQAD, PAD, WAD, GD and WD. Further, it is also clear from the fitted plot and P-P plot of two dataset of considered distributions in figure 9, 10 and 11, that ATPAD provides a much better fit over GAD, QAD, NQAD, PAD, WAD, GD and WD.
Figure 9: Fitted plot of the considered distribution for the data set-1 and data set-2
Sample Quant les
Figure 10: P-P plot for considerd distributions of the data set-1
Figure 11: P-P plot for considerd distributions of the data set-2
XI. CONCLUSION AND FUTURE WORKS
In this paper, a two-parameter Aradhana distribution which includes Aradhana distribution, gamma distribution and exponential distribution are proposed. Its moments and statistical properties including survival function, hazard function, mean residual life function, reverse hazard function, stochastic ordering have been discussed. Deviations from mean and median, Bonferroni and Lorenz curve and their indices, stress strength reliability have also been discussed. The parameters of this distribution have been estimated using maximum likelihood estimation. To know the performance of maximum likelihood estimates of parameters, a simulation study has been presented. Finally, two examples of real lifetime datasets have been considered for
applications and compared with GAD, QAD, NQAD, PAD, WAD, GD and WD. It has been found that ATPAD provide the best fit than the GAD, QAD, NQAD, PAD, WAD, GD and WD. As this is a new two-parameter lifetime distribution, it has the possibility of extension by adding more parameter in the distribution to see its performance over other lifetime distributions of same parameter. Further, Bayesian method of estimation and ranked set sampling method of estimation can also be considered in future to see the efficiency of these two methods of estimation over the classical maximum likelihood estimation.
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