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29
TRUNCATED PRANAV DISTRIBUTION: PROPERTIES
AND APPLICATIONS
Kamlesh Kumar Shukla
Department of Community Medicine, Noida International Institute of Medical Sciences, Noida International University, Gautam Budh Nagar, India Email: [email protected]
Abstract
In this paper, truncated Pranav distribution has been proposed. The behavior of truncated Pranav distribution has been presented graphically. Moment based measures including coefficient of variation, skewness, kurtosis, and Index of dispersion have been derived and presented graphically. Nature of survival and hazard rate functions are presented graphically. Maximum likelihood method has been used to estimate the parameter of proposed model. Simulation based study of proposed distribution has also been discussed. It has been applied on two data sets and its superiority has been compare and checked using goodness of fit (AIC and K. S. test) over other truncated distributions as well as one parameter distribution, such as exponential, Lindley, Pranav, Ishita, truncated Akash, truncated Lindley, and truncated Akash distribution. It was found good fit over above-mentioned distributions. It can be considered as good lifetime distribution especially for non-skewed data.
Keywords: Akash distribution, Lindley distribution, Moments, Right Truncated, Left Truncated
1. Introduction
In the recent past decades, lifetime modeling has been becoming popular in distribution theory, where many statisticians are involved in introducing new models. Some of the life time models are very popular and applied in biological, engineering and agricultural areas, such as Lindley distribution of Lindley [1], weighted Lindley distribution introduced by [2], Akash distribution suggested by Shanker[3], Ishita distribution proposed by Shanker and Shukla [4], Pranav distribution introduced by Shukla [5], are some among others and extension of above mentioned distribution has also been becoming popular in different areas of statistics.
Shukla [5] proposed Pranav distribution convex combination of exponential and gamma distributions which is defined by its pdf and cdf
A(y;0) = (e + y3)e~ey-,y> 0,0 >0 (1)
F2(y;*) = 1 - [1 + ey(e2":++36"y+6)] e-"';y > 0,0 > 0 The rth moment about origin of Pranav distribution given as
(2)
.. ' = r!{e:+(r+i)(r+2)(r+3^ r = . „„
(3)
Shukla [5] has discussed in detailed about its mathematical and statistical properties, estimation of parameters and applications to model lifetime data from engineering and biomedical engineering. Truncated type of distribution is more effective for modeling lifetime data because its limits used as bound either upper or lower or both according to the given data. Truncated normal distribution is proposed by Johnson et al. [6]. It has wide application in economics and statistics. Many researchers have been proposed truncated type of distribution and applied in different areas of statistics, especially in censor data such as truncated Weibull distribution of Zange and Xie [7], truncated Lomax distribution of Aryuyuen and Bodhisuwan [8], truncated Pareto distribution of Janinetti and Ferraro [9], truncated Lindley distribution of Singh et al. [10]. Some researchers have proposed distribution such as Sindhu and Hussai [11] proposed a Mixture of two generalized inverted exponential distributions with censored sample model and applied on cancered data, and Shukla [12] proposed Inverse Ishita distribution and applied on using the data sets of bladder cancer patient and failure times of the air conditioning system. Recently truncated version of Akash distribution introduced by Shukla and Shanker [13] and truncated version of two parameter Pranav distribution has been proposed by Shukla [14] and its superiority has been shown in their paper over other truncated distribution. Some distributions and their introducer names have been listed in the table1.
Tablel: Pdf of some selected distribution and their introducer's name
Distribution Pdf (probability density function) Introducer'
name name
Truncated Lindley (TLD) tr m 02(x+1)e-01 ^(X; e) = (a0 + 1)e-"2 - (ft0 + 1)e-"3 + (0 + 1)(e-"2 - e-"3) Singh et al. [10]
Truncated /(x;0) Shukla &
Akash (TAD) 03(x2 + 1)e-"x Shanker
a0(a0 + 2)e-"2 - ft0(ft0 + 2)e-"3 + (02 + 2)(e-"2 - e-"3) [13]
Akash 03 /(x;e) = z^r (i + x2)e-e; ^ > 0,0 > o 02 + 2 Shanker [3]
Ishita 03 /(x;0) = (e + x 2)e-" ;y> o,e > 0 03 + 2 Shanker& Shukla [4]
Lindley 02 /(x; 0) = (1 + x)e-te;x > 0,0 > 0 Lindley [1]
Exponential /(x;0) = 0e-te;x > 0,0 > 0
Truncated version of a continuous distribution can be defined as:
Definition1. Let X be a random variable distributed according to some pdf g(x; 0) and cdf G(x; 0), where 0 is a parameter vector of X. Let X lies within the interval [a, b], where —¥ < a < x < b < ¥ , then X, conditional on a < x < b is distributed as truncated distribution. The pdf of truncated distribution as reported by Singh et al (2014) defined by:
f(x; 0) = o(x/a < x < ft; 0) = —^^--(4)
J V ' J ' ' J G(b;0)-G(a;0) V '
where /(x; 0) = g(x; 0) for all a < x < ft and /(x; 0) = 0 elsewhere. Note that /(x; 0) in fact is a pdf of X on interval [a, ft]. We have
Shukla Kamlesh Kumar RT&A, No 1 (72)
TRUNCATED PRANAV DISTRIBUTION: PROPERTIES AND APPLICATIONS Volume 18, March 2023
f(x; 9) = (b f(x; 9) dx =---(b q(x; 9) dx
Ja J y ' J G(b;9)-G(a;9)Ja &v ' J
=---G(b;9)-G(a;9) = 1 (5)
G(b;9)-G(a;9) V ' J V ' J W
The cdf of truncated distribution is given by
F{x-9)=C f(x;9)dx = G(x;9)-G(a;9) (6)
V > J Jajy'j G(b;9)-G(a;9) V '
The main objective of this paper is to propose new truncated distribution using Pranav distribution which is called as truncated Pranav distribution, and to know the behavior and properties of proposed distribution over other truncated as well as parent distributions. Present paper has been divided into eight sections. Introduction about the paper is described in the first section. In the second section, truncated Pranav distribution has been derived. Behavior of hazard rate has been presented in third section Statistical properties including its moment have been discussed in the fourth section. Estimation of parameters of the proposed distribution has been discussed in the fifth section. Simulation study of proposed distribution has been discussed in the sixth section. Its application and comparative study with one parameter lifetime distribution have been illustrated in the section seven. Finally, the conclusion of the paper has been given in the eighth section.
2. Truncated Pranav Distribution
In this section, pdf and cdf of new truncated distribution is proposed and named Truncated Pranav distribution, using (5) & (6) of definitionl and from (1) & (2) , which is defined as : Definition 2: Let X be random variable which is distributed as Truncated Pranav distribution (TPD) with scale parameter 6 and location parameters a &b and denoted by TPD(a, b, 9). The pdf and cdf of X are derived respectively as: f(x;9)= g(x/a<x<b;9) = 3(x; 9
G(b; 9)-G(a; 9)
G(x;9)-G(a;9)
F(x; 9) = ■
Where g(x; 9) = (9 + x3)e-9x
G(b;9) - G(a; 9)
(ft + v 3)o-9x
94+6 '
G(b;9) = 1-G(a;9) = 1-
9b(92b2 + 39b + 6) 1+ —
1+
94 + 6 9a(92a2 + 39a + 6)
94 + 6
e-9b
e-9a
f(x. 8) = _"4(x3+")e-&'_
' y ; ) (a393+3a292+94+6a9 + 6)e-0a-(b393 + 3b292+94+6b9+6)e-0b V '
ff .o\ = (a393+3a292+94+6a9+6)e-Sa-(x393+3x292 + 94+6x9+6)e-Sx („)
(X; ) = (a393 + 3a292+94+6a9+6)e-6a-(b393+3b292+94+6b9+6)e-eb (8)
where < a < x < b < and 9 > 0
Three cases can be considered of proposed doubly truncated distribution as:
a) When a = 0and b = it reduced to parent model (Pranav distribution).
b) When a = 0, it is known as right (upper) truncated distribution of the parental model (Right truncated Pranav distribution)
c) When b = «>, it is known as left (lower) truncated distribution of the parent model (left truncated Pranav distribution.
Properties of TPD are explained as follows:
i. TPD has three parameters, where two parameters a and b were considered lowest and largest value from the data.
ii Parameter 8 is considered as scale parameter, for the fixed value of a and b , TPD is observed decreasing and increasing as increased value of 8 for 8 < 1 and 8 > 1 respectively. Performance of pdf of TPD for varying values of parameters has been illustrated in the figure 1.
a=0,b=10
Figure 1: pdf plots of TPD for varying values of parameters
Shukla Kamlesh Kumar RT&A, No 1 (72)
TRUNCATED PRANAV DISTRIBUTION: PROPERTIES AND APPLICATIONS Volume 18, March 2023
3. Survival and Hazard function
The survival function S(x) and the hazard function h(x) of TPD are defined as
S(x) = 1 - F(x) =
(x303 + 3x202 + 04 + 6x0 + 6)е-еж - (Ь303 + 3Ь202 + 04 + 6Ь0 + 6)e (a303 + 3a202 + 04 + 6a0 + 6)e-"2 - (Ь303 + 3Ь202 + 04 + 6Ь0 + 6)e
-еь
-еь
-еж
h(x) =
/(x) _ 04(x3 + 0)e
5(x) = (x303 + 3x202 + 04 + 6x0 + 6)е-"ж - (Ь303 + 3Ь202 + 04 + 6Ь0 + 6)е-еь
It is obvious that h(x) is independent from parameter a. Behavior of Survival and hazard function of TPD for varying values of parameter are presented in figures 2&3. It was observed from the figure 3 that value of hazard rate is increasing as increased value of parameter of 0 when value of rest of the parameters (a, b fixed.
Figure 2: S(x) plots of TPD for varying values of parameter
x
Figure3: h(x) plots of TPD for varying values of parameter
4. Moments and mathematical properties
Moments of a distribution are used to study the most important characteristics of the distribution including mean, variance, skewness, kurtosis, etc. The rth moment of origin^ of TPD can be expressed in explicit expression in terms of incomplete gamma functions.
Theoreml: Suppose X9(a> 0,b~<*>) follows left (lower) truncated TPD (6,a,&>). Then the rth moment about origin of TPD is
,, ' _ 04r(r+l,0a)+r(r+4,0a) _ _ . 2 3
T,r l,Z,j,. ..
#[(a3e3+3a2e2+6ae+e4+6)e-&a-(e4+6)] '
Proof: Considering K = {(a303 + 3a282 + 6a8 + 84 + 6)e-"a - (04 + 6)} in (7), we have
"4 M
^+ =~fa x r(9 + x%) e~ dx = "4 [f~ Be &"1 xr dx + f~ e-"xxr+3 dx] Taking u = dx, x = ?
_ "4 ["#*+{foa + {C *r+3^}]
Where y(a, z) = f e-xxa-1dx, a > 0, x > 0 is the upper incomplete gamma function
_"4 |>(r+1,"a) + y(r+4,"a)l
_T[ "# e7*4 ]
> L
e4:(r+i,ea)+:(r+4,ea)i
ë# J
- _ e4:(r+i,0a)+:(r+4,0a)
= er[(a3e3+3a2e2+6ae+e4+6)e-&a-(e4+6)] Now taking r = 1,2, mean and variance can be obtained as
„ - =_e4:(2,ea)+:(5,eg)_
e[(a3e3+3a2e2+6ae+e4+6)e-S(-(e4+6)]
. - =_e4:(3,eg)+r(6,eg)_
e2[(a3e3+3a2e2+6ae+e4+6)e-ea-(e4+6)]
Variance = - (^1)2
Theorem2. Suppose (ft > 0, a~0) follows upper (right) truncated TPD(0,0, ft). Then the rth moment about origin^ of TPD is
^ 04{y(r + 1,06)} + y(r + 4,06) _
= 0+[(04 + 6) - (6303 + 36202 + 660 + 04 + 6)e-"3] ;r-1,2,3,." Proof: Considering K = {(04 + 6) - (6303 + 36202 + 660 + 04 + 6)e-"3} in (7), we have
= — (0 + x3)e-"*dx = "4 [j0" 0e-te x+dx + J0" e-"1 x++3dx] Taking u = 0x, x = ?
_ "4 ["#*+ {j"' + {Jo"'
Where y(a, z) = J@z e-1xA-1dx, a > 0, x > 0 is the lower incomplete gamma function
_"4 [:(++i,"b) + :(++4,"b)]
_T[ ]
_i r"4(:(++i,"b)}+:(++4,"b^ [ ]
_ - _ "4(:(++i,"b)}+:(++4,"b)
"r[("4+6)-(b3"3 + 3b2"2+6b"+"4+6)e-«6]
Now takingr = 1,2, mean and variance can be obtained as
- _ e4r(2,eb)+r(5,eb)
Mi — ■
e[(e4+6)-(b3e3+3b2e2+6be+e4+6)e-«6]
„- =_e4r(3,eb)+r(6,eb)_
e2[(e4+6)-(b3e3+3b2e2+6be+e4+6)e-&6] Variance = - (^i)2
Theorem3: Suppose X follows dOUbly TPD C®, a, Then the rth moment about origin of TPD is
- e4{:(r+i,eb)-:(r+i,ea)}+{v(r+4,eb)-:(r+4,ea)} ■■ — — ; r = 1,2,3,...
erF(a3e3 + 3a2e2+6ae+e4+6)e-«a-V(b3e3+3b2e2+6be+e4+6)e-«6
Proof- Considerin k = f^ + 3a202 + 6ae + 04 + 6)e-ea -j
r°° . onsi ering = |(ô303 + 3ô2e2 + 6ôe + e4 + 6)e-ë3 j
in (7), we have
= C^ + x3)e-eidx
= — if3 0e-ë1 xrdx + f3 e-ë1 xr+3dxl > LJ2 J0 J
Taking u _ 9x, x _ ?
"1
>
+
[C e ?x r du-foa e ? du]
(J0"3e-uur+3du - JQ"ae-uxr+3du]
Where y (a,z) _ J@ze Xxa idx,a > 0,x > 0 is the lower incomplete gamma function
_04 r:(r+i,eb)-:(r+i,ea) + :(r+4,0b)-:(r+4,0a)] [ 0"#+4 i
_>[ J
_e4{:(r+i,0b)-:(r+i,0a)}+{:(r+4,0b)-:(r+4,0a)} V(b303+3b202+6b0+04+6)e-6,) G
Now takingr _ 1,2, mean and variance can be obtained as
> _ 04{:(2,0b)-:(2,0a)}+{:(5,0b)-:(5,0a)} 0F (a303+3a202 + 6a0+04+6)e-&a-G V(b303+3b202+6b0+04+6)e-6,) J
' _ 04{:(3,0b)-:(3,0a)}+{:(6,0b)-:(6,0a)} _ 02F(a303+3a202 + 6a0+04+6)e-Sa-G V(b303 + 3b202+6b0+04+6)e-6,) J
Variance _ - (ni)2
Similarly rest two moments of origin as well as coefficient of variation, coefficient of skewness, coefficient of kurtosis and Index of dispersion can be obtained, substituting r _ 3,4 in the equation (9), which are as follows:
^3 _
94{y(4,9b) - y(4, 9a)} + [y(7,9b) - y(7, 9a)} ](a393 + 3a292 + 6a9 + 94 + 6)e-0a -\ \(b393 + 3b292 + 6b9 + 94 + 6)e-0b )
' _ 84{y(5,9b) - y(5,9a)} + (y(8,9b) - y(8,9a)} 94\
^4 ((a393 + 3a292 + 6a9 + 94 + 6)e-0a -
{(b393 + 3b292 + 6b9 + 94 + 6)e
-0b
(I2)2) HI3 +3I2 1+ )2)
Coefficient of Variation^-.—-—, Coefficient of Skweness_—.-. , 0/, , Coefficient of
1+ (12-(Il)2)3/2 Hl4-413 h+612 (1I)2-3(1I)4)
Kurtosis_ —
(I"-(Ml)2)2
HI"-(I+)2)
Index of dispersion_ --1—- graph of above measures is presented in figures 4 to 9. From the
Ml
figure 4 &5, it was observed that mean and variance are decreasing with increased value of 9 and slightly increasing with b while parameter a is kept constant.
Coefficient of variation of TPD was found deceasing with increased value of 9 and b.
Figure 4: Mean of TPD (doubly truncated) for varying value of parameter
Figure5: Variance of TPD for varying value of parameter
Figure 6; Coefficient of variation of TPD (Doubly truncated) for varying value of parameter
Figure 7: Coefficient of skewness of TPD (Doubly truncated) for varying value of parameter
Figure 8: Coefficient ofkurtosis of TPD (Doubly truncated) for varying value of parameter
Figure9: Index of dispersion of TPD (doubly truncated) for varying value of parameter
5. Maximum likelihood Method Estimation
Let (xi , X2, ^3,..., ^k) be a random sample of size n from (7). The likelihood function, L of TPD is given by
7=b_"1_C"nK (fl + r.3)p-n"i
V(a3"3 + 3a2"2+"4+6a"+6)e-0(-(b3"3+3b2"2+"4+6b" + 6)e-0)C lll=i^ 9 )
The log likelihood function is thus obtained as
Zn L = n Zn -tt:-:-s:"---S-flfcC + Ei— + X;3) — n0X
V(a3e3+3a2e2+e4+6ae+6)e-«a-(33e3+332e2+e4+63e+6)e-«V ;-i v 9 J
Taking a = min(xi, x2, x3,..., xn), S = max(xi, x2, x3,..., xn), the maximum likelihood estimate 0of
nog. dd
parameter 6 is the solution of the log-likelihood equation —04" = 0.
It is obvious that —= 0 will not be in closed form and hence some numerical optimization technique can be used e the equation forfl. In this paper the nonlinear method available in R software has been used to find the MLE of the parameter fl.
6. Simulation study
In this section, simulation study has been carried out using R-software. Acceptance and rejection method is used to generate random number, where sample size,n = 20,40,60,80,100, value of 8 = 0.1,0.5,1.0,1.5& (a = 10, b = 100) have been used for calculating Bias error and MSE (Mean square error) of parameter 6, which is presented in table2.
General algorithm for generating the data which was given by Robert and Casella [15] are as follows: The constraints we impose on this candidate density fY are that:
(i) Y be simulate-able (the data simulation from Y be actually possible).
(ii) (iii)
There is a constant c with S-(1) < c for all x E Sx = {x: fx(x) > 0}
fx(x) and fY (x) have compatible supports (i.e., Sx £ SY ).
In this case, X can be simulated as follows by Accept-Reject method. First, we generate y from Y~fY and, independently, we generate u from U~U(0,1).
If u < /-(y) then we set x = y. If the inequality is not satisfied, we then discard/reject y and u and start
c/y (y)
again.
Table2: Average Bias (AB) and Average MSE (AM) of the simulated MLEs of 8 at fixed value ofa = 10, b = 100
Sample q A B AM AB AM
20 0.1 0.0606620 0.07359774 0.04834145 0.04673793
0.5 0.0509281 0.05187350 0.05092813 0.00518735
1.0 0.02592813 0.01344536 0.02592813 0.01344536
1.5 0.00092813 0.000017228 0.00092813 0.00001722
40 0.1 0.02391497 0.02287705 0.02420517 0.02343561
0.5 0.02546406 0.02593675 0.02546406 0.0259367
1.0 0.01296406 0.006722681 0.01296406 0.00672268
1.5 0.00046406 0.0000086143 0.00046406 0.00008614
60 0.1 0.01535922 0.01415435 0.01619160 0.01573008
0.5 0.016976044 0.01729117 0.01697604 0.00172911
1.0 0.008642711 0.004481787 0.00864271 0.00044817
1.5 0.000309377 0.0000057428 0.00030937 0.00005742
80
100
0.1 0.010465770 0.008762596 0.01131883 0.01024928
0.5 0.012732033 0.01296837 0.01273203 0.01296837
1.0 0.006482033 0.003361341 0.00648203 0.00336134
1.5 0.00023203 0.000004.307 0.00023204 0.00000430
0.1 0.00882808 0.007793502 0.00890854 0.07936211
0.5 0.010185626 0.01037470 0.01018562 0.01037470
1.0 0.005185626 0.002689072 0.00518562 0.00268907
1.5 0.000185626 0.0000034457 0.00018562 0.00000344
From the above table, it is observed that Bias and Mean square error are decreasing as increased value of sample size.
7. Applications on lifetime data
In this section, TPD has been applied to two datasets using maximum likelihood estimates. Parameter q estimated whereas another parameters a, and b are considered as lowest and highest values of data, i.e.
a = min{x1,x2,x%,..., xn )& b = ma.x(x1,x2,x%,...,xn). Goodness of fit has been decided using Akaike information criteria (AIC), Bayesian Information criteria (BIC) and Kolmogorov Simonov
Shukla Kamlesh Kumar RT&A, No 1 (72)
TRUNCATED PRANAV DISTRIBUTION: PROPERTIES AND APPLICATIONS Volume 18, March 2023 test (KS) values respectively, which are calculated for each distribution and compared with p-value. As we know that best goodness of fit of the distribution can be decided based on minimum value of KS, AIC and BIC.
Data Set 1: The data is given by Birnbaum and Saunders [16] on the fatigue life of 6061 - T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. Data Set 2: This data set is the strength data of glass of the aircraft window reported by Fuller et al. [17]:
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
Table 3: MLE's, Standard Errors, - 2ln L, AIC, K-S and p-values of the fitted distributions for data set-1 Distributions ML Estimates Standard -2 in! AIC BIC K-S p-
Errors value
TPD
TAD
TLD
Pranav
Ishita
Lindley
0 0.00330
= 0.055278
927.37 929.37 928.76 0.136 0.048
9 = 0.04387
9 = 0.03917 0.00303
939.13
941.13 942.05 0.153 0.017
9 = 0.02199 0.00273 958.88 960.88 962.31 0.186 0.001
0.00253
950.97
952.97 954.40 0.194 0.001
9 = 0.04390 0.002533 950.92 9952.92 954.35 0.194 0.001
9 = 0.02886 0.002038 983.10 985.10 986.54 0.252 0.000
Exponential 0 = 0.01463 0.001457 1044.87 1046.87 1048.30 0.336 0.000
Table 4: MLE's, Standard Errors, - 2ln L, AIC, K-S and p-values of the fitted distributions for data set-2
Distributions ML Estimates Standard
Errors
2 in! AIC BIC K-S p-
value
TPD
0.12067
0.02455
201.80 203.80 203.19 0.107 0.829
TAD
TLD
9 = 0.08776
0.024241 201.96
203.96
205.58 0.112 0.786
9 = 0.05392 0.023917 202.18 204.18 205.61 0.117 0.738
Pranav
Ishita
9 = 0.09706
0.01004
9 = 0.097328 0.01008
240.68
242.68
242.67 0.298 0.005
240.48 242.48 243.48 0.297 0.006
Lindley
9 = 0.06299 0.00800 253.98 255.98 256.98 0.365 0.000
Exponential 0 0.00582 274.52 276.52 277.52 0.458 0.000
= 0.032452
From the above tables 3&4 , it was observed that TPD gives good fit over other selected distributions. Fitted plots of the considered distributions are presented in figure 10 and 11 respectively.
IT)
O I I I I I I r~
0 20 40 60 80 100 120 140
x
FigurelO: Fitted plots of distributions for the dataset 1
CO CD CD
CD CD
X
CM CD CD
CD -1-1-1-1-1-r
20 25 30 35 40 45
x
Figurell: Fitted plot of distributions for dataset 2
7. Conclusions
In this paper, truncated Pranav distribution (TPD) has been proposed. Its statistical and mathematical properties have been discussed. Maximum likelihood method has been used for estimation of its parameter. Simulation study has also been conducted to know behavior of proposed distribution. Goodness of fit of TPD has been discussed with two lifetime datasets and superiority has been checked with truncated Akash, truncated Lindley, Pranav, Ishita, Lindley, and exponential distributions. It has been observed that TPD gives good fit on both the data sets. In the first data set, value of AIC 929.37 and KS value- 0.136, p-value =0.048 <0.01, and for the second data set, value of AIC-203.80 and KS value- 0.107 (p-value =0.829 >0.05) were observed which were compared over
two parameter TAD (truncated Akash Distribution), TLD (truncated Lindley Distribution) and one parameter Pranav, Ishita, Lindley and exponential distribution. Therefore, it may be considered good distribution for the lifetime data specially on fixed values (lower limit, upper limit).
Acknowledgement
Author is thankful to the Editor in chief and anonymous reviewers for the given fruitful comments to improve the quality and presentation of the paper.
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