Relationships for Moments of Generalized Order Statistics from Hjorth Distribution and Related Inference

Jagdish Saran; Kanika Verma; Narinder Pushkarna

doi:10.2991/jsta.d.210602.001

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Volume 20, Issue 2, June 2021, Pages 171 - 179

Relationships for Moments of Generalized Order Statistics from Hjorth Distribution and Related Inference

Authors

Jagdish Saran, Kanika Verma^*, Narinder Pushkarna

Department of Statistics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110007, India

^*Corresponding author. Email: kanikaverma29apr@gmail.com

Corresponding Author

Kanika Verma

Received 2 June 2018, Accepted 2 March 2019, Available Online 15 June 2021.

DOI: 10.2991/jsta.d.210602.001 How to use a DOI?
Keywords: Generalized order statistics; Hjorth distribution; single moments; product moments; recurrence relations; best linear unbiased estimators; type-II right censored samples
Abstract: In this paper some recurrence relations satisfied by single and product moments of generalized order statistics from Hjorth distribution have been obtained. Then we use these results to compute the first two moments of order statistics for some specific values of the parameters. Further, we use the results on order statistics to obtain BLUEs of location and scale parameters based on type-II right censored samples.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Let {Xn,n≥1} be a sequence of independent and identically distributed random variables with cumulative distribution function (cdf) F(x) and probability density function (pdf) f(x).

Assume that k>0, n∈N, n≥2, m˜=(m1,m2, …, mn−1)∈Rn−1, Mr=∑j=rn−1mj such that γr=k+(n−r)+Mr>0 for all r∈{1,2,…,n−1}. Then X(r,n,m˜,k), r=1,2,…,n, are called generalized order statistics if their joint pdf is given by

fX(1,n,m˜,k),…,X(n,n,m˜,k)(x1,…,xn)=k∏j=1n−1γj∏i=1n−1(1−F(xi))mif(xi)× (1−F(xn))k−1f(xn), F−1(0+)<x1≤…≤xn<F−1(1).(1)

Choosing the parameters appropriately, models such as ordinary order statistics (γi=n−i+1; i=1,2,…,n, i.e., m1=m2=…mn−1=0, k=1), k-th records values (γi=k, i.e., m1=m2=…=mn−1=−1, k∈N), sequential order statistics (γi=(n−i+1)αi; α1,α2,…,αn>0), order statistics with non-integral sample size (γi=α−i+1; α>0), Pfeifer’s record values (γi=βi; β1,β2,…,βn>0) and progressively type-II right censored order statistics (mi∈N0, k∈N) are obtained (cf. Kamps [1,2], Kamps and Cramer [3]).

The joint pdf of first r generalized order statistics is given by

fX(1,n,m˜,k),…,X(r,n,m˜,k)(x1,…,xr)=cr−1∏i=1r−1(1−F(xi))mif(xi)× (1−F(xr))k+(n−r)+Mr−1f(xr), F−1(0+)<x1≤…≤xr<F−1(1).(2)

We may consider two cases here:

Case I: m1=m2=…=mn−1=m.

Case II: γi≠γj; i≠j, i, j=1, 2,…, n−1.

For Case I, the r-th generalized order statistic will be denoted by X(r,n,m,k). The pdf of X(r,n,m,k) is given by

fX(r,n,m,k)(x)=cr−1(r−1)!(1−F(x))γr−1f(x)gmr−1(F(x)), x∈R,(3)

and the joint pdf of X(r,n,m,k) and X(s,n,m,k),1≤r<s≤n, is given by

fX(r,n,m,k),X(s,n,m,k)(x,y)=cs−1(r−1)!(s−r−1)![1−F(x)]mf(x)gmr−1(F(x))× [hm(F(y))−hm(F(x))]s−r−1[1−F(y)]γs−1f(y), x<y,(4)

where

cr−1=∏j=1rγj, γr=k+(n−r)(m+1), r=1,2,…,n,gm(x)=hm(x)−hm(0), x∈[0,1),hm(x)={−1m+1(1−x)m+1, m≠−1−log (1−x), m=−1

(cf. Kamps [1,2]).

For case II, X(r,n,m˜,k) denotes the r-th generalized order statistic. The pdf of X(r,n,m˜,k) is given by

fX(r,n,m˜,k)(x)=cr−1f(x) ∑i=1rai(r)(1−F(x))γi−1, x∈R,(5)

and the joint pdf of X(r,n,m˜,k) and X(s,n,m˜,k), 1 ≤r<s≤n, is given by

fX(r,n,m˜,k),X(s,n,m˜,k)(x,y)=cs−1{∑i=r+1sai(r)(s)(1−F(y)1−F(x))γi}{∑i=1rai(r)(1−F(x))γi} × f(x)1−F(x)f(y)1−F(y), x<y,(6)

where

cr−1=∏i=1rγi, γi=k+n−i+Mi, r=1,2,…,n,ai(r)=∏j(≠i)=1r1(γj−γi), 1≤i≤r≤n

and

ai(r)(s)=∏j(≠i)=r+1s1(γj−γi), r+1≤i≤s≤n

(cf. Kamps and Cramer [3]).

Further, it can be easily proved that

ai(r)=(γr+1−γi)ai(r+1),cr−1=crγr+1, ∑i=1r+1ai(r+1)=0, ∑i=r+1sai(r)(s)=0.(7)

Also, for mi=mj=m, it can be shown that

∑i=1rai(r)(1−F(x))γi=(1−F(x))γr(r−1)!gmr−1(F(x)),(8)

and

∑i=r+1sai(r)(s)(1−F(y)1−F(x))γi=1(s−r−1)!(1−F(y)1−F(x))γs × (11−F(x))(m+1)(s−r−1)[hm(F(y))−hm(F(x))]s−r−1.(9)

Several authors like Kamps and Gather [4], Keseling [5], Cramer and Kamps [6], Ahsanullah [7], Pawlas and Szynal [8], Ahmad and Fawzy [9], Athar and Islam [10], Ahmad [11], Khan et al. [12], Khan et al. [13] and Saran and Pandey [14,15] have done some work on generalized order statistics. In this paper, in Section 3, we have established recurrence relations for single and product moments of generalized order statistics from Hjorth distribution for Case II only, i.e., for γi≠γj; i≠j, i,j=1,2,…,n−1. Then we use these results to compute means and variances of order statistics for some specific values of the parameters. Further, we use the results on order statistics to obtain BLUEs of location and scale parameters based on type-II right censored samples.

2. HJORTH DISTRIBUTION

A random variable X is said to have Hjorth distribution if its pdf is of the form

f(x)=[(1+βx)δx+θ]e−δx22(1+βx)1+θβ, x≥0, β, δ, θ>0,(10)

and the cdf is of the form

F(x)=1−e−δx2/2(1+βx)θβ.(11)

Its characterizing differential equation is given by

(1+βx)f(x)=[(1+βx)δx+θ](1−F(x)).(12)

More details on this distribution can be found in Hjorth [16].

The cdf of the location-scale parameter Hjorth distribution is given by

F(x)=1−e−δ(x−μσ)2/2(1+β(x−μσ))θβ, x≥μ, μ≥0, β, δ, θ, σ>0.(13)

3. RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF GENERALIZED ORDER STATISTICS FROM HJORTH DISTRIBUTION FOR CASE II

Theorem 3.1.

For the distribution given in (10) and n∈N, k≥1,

E[Xi(r,n,m˜,k)]+βE[Xi+1(r,n,m˜,k)] =δγr(i+2)E[Xi+2(r,n,m˜,k)]−E[Xi+2(r−1,n,m˜,k)] + δβγr(i+3)E[Xi+3(r,n,m˜,k)]−E[Xi+3(r−1,n,m˜,k)] + θγr(i+1)E[Xi+1(r,n,m˜,k)]−E[Xi+1(r−1,n,m˜,k)].(14)

Proof.

From (5) and (12), we have

E[Xi(r,n,m˜,k)]+βE[Xi+1(r,n,m˜,k)] =cr−1∫0∞xi(1+βx)f(x)∑i=1rai(r)[1−F(x)]γi−1dx =cr−1∫0∞xi[δx+βδx2+θ]∑i=1rai(r)[1−F(x)]γidx =δcr−1∫0∞xi+1∑i=1rai(r)[1−F(x)]γidx +βδcr−1∫0∞xi+2∑i=1rai(r)[1−F(x)]γidx +θcr−1∫0∞xi∑i=1rai(r)[1−F(x)]γidx =δIi+1+βδIi+2+θIi(15)

where

Ii=cr−1∫0∞xi∑i=1rai(r)[1−F(x)]γidx=−cr−1i+1∫0∞xi+1∑i=1r−1ai(r){(γr−γi)−γr}[1−F(x)]γi−1f(x)dx+cr−1i+1∫0∞xi+1ar(r)γr[1−F(x)]γr−1f(x)dx=γr(i+1)[E[Xi(r,n,m˜,k)]−E[Xi+1(r−1,n,m˜,k)]].(16)

Putting the values of Ii, Ii+1 and Ii+2 from (16) into (15), we get the required result as given in (14).

Remark 3.1.

Putting mi=mj=m in (5) and using (8), the recurrence relation established in Theorem 3.1 reduces to the recurrence relation for single moments of generalized order statistics from Hjorth distribution for Case I, i.e., when m1=m2=…=mn−1=m.

Remark 3.2.

If we take k=1 and mi=mj=m=0 in (14), we get the recurrence relation for single moments of order statistics from Hjorth distribution. Numerical computations for the means and variances of order statistics from Hjorth distribution for arbitrarily chosen values of β, δ and θ and for sample sizes n = 1(1)10 are given in Table 1.

n	r	E(X_r:n)	Var(X_r:n)	n	r	E(X_r:n)	Var(X_r:n)
1	1	0.275075	0.07296	8	1	0.032352	0.001111
2	1	0.136533	0.019918	8	2	0.070568	0.002715
2	2	0.413617	0.087615	8	3	0.116513	0.005054
3	1	0.089588	0.008731	8	4	0.173041	0.008528
3	2	0.230425	0.029067	8	5	0.244889	0.013882
3	3	0.505213	0.091719	8	6	0.340947	0.02273
4	1	0.066369	0.004787	8	7	0.481772	0.039734
4	2	0.159242	0.014093	8	8	0.740519	0.089023
4	3	0.301608	0.033907	9	1	0.02866	0.000868
4	4	0.573081	0.092565	9	2	0.061889	0.002076
5	1	0.052615	0.002992	9	3	0.100945	0.003768
5	2	0.121387	0.008186	9	4	0.147648	0.006171
5	3	0.216024	0.017581	9	5	0.204782	0.009662
5	4	0.358664	0.036652	9	6	0.276974	0.014942
5	5	0.626685	0.092176	9	7	0.372933	0.023554
6	1	0.043547	0.002036	9	8	0.512868	0.040006
6	2	0.097956	0.005302	9	9	0.768976	0.087862
6	3	0.168248	0.010659	10	1	0.025722	0.000696
6	4	0.263799	0.019939	10	2	0.055103	0.001636
6	5	0.406097	0.03826	10	3	0.089032	0.002912
6	6	0.670803	0.091282	10	4	0.128742	0.004661
7	1	0.037129	0.001471	10	5	0.176008	0.007095
7	2	0.082054	0.003696	10	6	0.233556	0.010572
7	3	0.137711	0.007106	10	7	0.305919	0.01576
7	4	0.208965	0.012496	10	8	0.401654	0.024145
7	5	0.304925	0.021574	10	9	0.540672	0.040106
7	6	0.446565	0.039202	10	10	0.794343	0.086733
7	7	0.708176	0.090184

Table 1

Means and Variances of Order statistics from Hjorth distribution with parameters β=2, δ=3, θ=4.

Theorem 3.2.

For 1≤r<s≤n and i , j≥0,

E[Xi(r,n,m˜,k)Xj(s,n,m˜,k)]+βE[Xi(r,n,m˜,k)Xj+1(s,n,m˜,k)] =θγsj+1E[Xi(r,n,m˜,k)Xj(s,n,m˜,k)]+E[Xi(r,n,m˜,k)Xj+1(s−1,n,m˜,k)]+ δβγsj+3E[Xi(r,n,m˜,k)Xj+3(s,n,m˜,k)]+E[Xi(r,n,m˜,k)Xj+3(s−1,n,m˜,k)]+ δγsj+2E[Xi(r,n,m˜,k)Xj+2(s,n,m˜,k)]+E[Xi(r,n,m˜,k)Xj+2(s−1,n,m˜,k)](17)

Proof.

Using (6), we have

E[Xi(r,n,m˜,k)Xj(s,n,m˜,k)]+βE[Xi(r,n,m˜,k)Xj+1(s,n,m˜,k)] =cs−1∫0∞∫x∞xiyj(1+βy){∑i=r+1sai(r)(s)(1−F(y)1−F(x))γi}{∑i=1rai(r)(1−F(x))γi}×f(x)1−F(x)f(y)1−F(y)dydx =cs−1∫0∞xi∑i=1rai(r)(1−F(x))γif(x)1−F(x)I(x)dx,(18)

where

I(x)=∫x∞yj(1+βy)∑i=r+1sai(r)(s)(1−F(y)1−F(x))γif(y)1−F(y)dy.

Using (12), we obtain

I(x)=∫x∞yj[(1+βy)δy+θ]∑i=r+1sai(r)(s)(1−F(y)1−F(x))γidy=δ∫x∞yj+1∑i=r+1sai(r)(s)(1−F(y)1−F(x))γidy+ δβ∫x∞yj+2∑i=r+1sai(r)(s)(1−F(y)1−F(x))γidy+ θ∫x∞yj∑i=r+1sai(r)(s)(1−F(y)1−F(x))γidy=δIj+1+δβIj+2+θIj,(19)

where

Ij=∫x∞yj∑i=r+1sai(r)(s)(1−F(y)1−F(x))γidy=1(j+1)∫x∞yj+1∑i=r+1sγiai(r)(s)(1−F(y)1−F(x))γif(y)1−F(y)dy=−1(j+1)∫x∞yj+1∑i=r+1s−1ai(r)(s)(−γi−γs+γs)(1−F(y)1−F(x))γif(y)1−F(y)dy+ 1j+1∫x∞yj+1as(r)(s)γs(1−F(y)1−F(x))γsf(y)1−F(y)dy=−1(j+1)∫x∞yj+1∑i=r+1s−1ai(r)(s)(−γi+γs)(1−F(y)1−F(x))γif(y)1−F(y)dy+ γs(j+1)∫x∞yj+1∑i=r+1sai(r)(s)(1−F(y)1−F(x))γif(y)1−F(y)dy=−1(j+1)∫x∞yj+1∑i=r+1s−1ai(r)(s−1)(1−F(y)1−F(x))γif(y)1−F(y)dy+ γs(j+1)∫x∞yj+1∑i=r+1sai(r)(s)(1−F(y)1−F(x))γif(y)1−F(y)dy.(20)

Putting the values of Ij,Ij+1 and Ij+2 from (20) into (19) and then putting the value of I(x), so obtained, in (18), we get the required result as in Theorem 3.2.

Remark 3.5.

Putting mi= mj= m in (6) and using (8), and (9), the recurrence relation established in Theorem 3.2 reduces to the recurrence relation for single moments of generalized order statistics from Hjorth distribution for Case I, i.e., when m1= m2=…=mn−1=m.

4. BLUEs OF μ AND σ

Let X1:n≤X2:n≤…≤Xn−c:n, c=0,1,…,n−1, denote type-II right censored sample from the location-scale parameter Hjorth distribution in (13). Let us denote Zr:n=(Xr:n−μ)/σ, E(Zr:n)=μr:n(1), 1≤r≤(n−c), and Cov(Zr:n,Zs:n)=σr,s:n=μr,s:n(1,1)−μr:n(1)μs:n(1),1≤r<s≤(n−c). We shall use the following notations:

X=(X1:n,X2:n,…,Xn−c:n)Tμ=(μ1:n,μ2:n,…,μn−c:n)T 1= (1,1,…,1)T

and∑=(σr,s:n),1≤r,s≤n−c.

The BLUEs of μ and σ are given by

μ*={μT∑−1μ1T∑−1−μT∑−11μT∑−1(μT∑−1μ)(1T∑−11)−(μT∑−11)2}X=∑r=1n−carXr:n(21)

and

σ*={1T∑−11μT∑−1−1T∑−1μ1T∑−1(μT∑−1μ)(1T∑−11)−(μT∑−11)2}X=∑r=1n−cbrXr:n,(22)

and the variances and covariance of these BLUEs are given by

Var(μ*)=σ2{μT∑−1μ(μT∑−1μ)(1T∑−11)−(μT∑−11)2}=σ2V1,(23)

Var(σ*)=σ2{1T∑−11(μT∑−1μ)(1T∑−11)−(μT∑−11)2}=σ2V2(24)

and

Cov(μ*,σ*)=σ2{−μT∑−11(μT∑−1μ)(1T∑−11)−(μT∑−11)2}=σ2V3,(25)

respectively, (cf. Arnold et al. [17]).

Tables 2 and 3 display the coefficients of the BLUEs for type-II right censored samples of sizes n = 5(1)10 with β = 2, δ = 3, = 4 and different censoring cases c=0(1)([n/2]−1). The coefficients of the BLUEs in Tables 2 and 3 are checked by using the conditions

∑r=1n−car=1

and

∑r=1n−cbr=0.

β, δ, θ	n	c	a_i = 1, 2, ..., (n − c)
2, 3, 4	5	0	1.185103	−0.052238	−0.041786	−0.037652	−0.053427
		1	1.252633	−0.069672	−0.057286	−0.125675
	6	0	1.156770	−0.038448	−0.030920	−0.026106	−0.024929
			−0.036367
		1	1.199844	−0.047927	−0.039163	−0.033897	−0.078857
		2	1.267204	−0.062421	−0.051839	−0.152944
	7	0	1.136374	−0.029637	−0.024218	−0.020275	−0.017906
			−0.017852	−0.026486
		1	1.166529	−0.035450	−0.029273	−0.024876	−0.022455
			−0.054475
		2	1.2085355	−0.043385	−0.036193	−0.031207	−0.097750
	8	0	1.120713	−0.023427	−0.019590	−0.016555	−0.014305
			−0.013102	−0.013503	−0.020231
		1	1.143147	−0.027284	−0.022964	−0.019592	−0.01717
			−0.016030	−0.040105
		2	1.172097	−0.032172	−0.027250	−0.023461	−0.020845
			−0.068369
		3	1.214264	−0.039128	−0.033360	−0.028998	−0.112778
	9	0	1.108222	−0.018806	−0.016221	−0.013874	−0.011984
			−0.010652	−0.010054	−0.010624	−0.016007
		1	1.125638	−0.021504	−0.018611	−0.016019	−0.013967
			−0.012580	−0.012074	−0.030883
		2	1.146918	−0.024749	−0.021490	−0.018607	−0.016369
			−0.014927	−0.050776
		3	1.175679	−0.029041	−0.025302	−0.022042	−0.019571
			−0.079724
	10	0	1.098552	−0.015890	−0.013634	−0.011818	−0.010285
			0.009068	−0.008257	−0.007988	−0.008608	−0.013003
		1	1.112506	−0.017866	−0.015396	−0.013403	−0.011740
			−0.010446	−0.009629	−0.009450	−0.024576
		2	1.128879	−0.020154	−0.017436	−0.015243	−0.013431
			−0.012053	−0.011237	−0.039324
		3	1.149846	−0.023024	−0.020000	−0.017556	−0.015564
			−0.014085	−0.059616
		4	1.178702	−0.026892	−0.023457	−0.020681	−0.018449
			−0.089223

Table 2

Coefficients of the BLUE of location parameter.

β, δ, θ	n	c	bi = 1, 2, ..., (n − c)
2, 3, 4	5	0	−3.208033	0.757187	0.668368	0.688593	1.093885
		1	−4.590649	1.114139	0.985722	2.490788
	6	0	−3.211420	0.644789	0.558759	0.524623	0.569260
			0.913989
		1	−4.293977	0.883014	0.765913	0.720441	1.924610
		2	−5.937974	1.236757	1.075287	3.625931
	7	0	−3.220810	0.565699	0.490602	0.445028	0.437792
			0.491353	0.790335
		1	−4.120641	0.739160	0.641423	0.582342	0.573543
			1.584173
		2	−5.342199	0.969912	0.842661	0.766453	2.763173
	8	0	−3.231744	0.505453	0.441562	0.396476	0.373008
			0.379760	0.436104	0.699380
		1	−4.007270	0.638778	0.558222	0.501464	0.472092
			0.480982	1.355732
		2	4.985911	0.804020	0.703095	0.632261	0.596270
			2.250264
		3	−6.373787	1.032960	0.904193	0.814498	3.622136
	9	0	−3.242010	0.456373	0.403847	0.361884	0.333970
			0.323848	0.338372	0.394433	0.629282
		1	−3.926664	0.562450	0.497805	0.446202	0.411943
			0.399652	0.417769	1.190842
		2	−4.747246	0.608811	0.608811	0.546026	0.504558
			0.490148	1.910109
		3	−5.829190	0.849026	0.752203	0.675246	0.624998
			2.927717
	10	0	−3.253575	0.418371	0.372557	0.334955	0.307007
			0.290254	0.288326	0.307107	0.361682	0.573316
		1	−3.868817	0.505504	0.450223	0.404843	0.371152
			0.351006	0.348794	0.371600	1.065696
		2	−4.578786	0.604700	0.538706	0.484596	0.444504
			0.420700	0.418543	1.667036
		3	−5.467608	0.726402	0.647379	0.582682	0.534916
			0.506848	2.469382
		4	−6.662882	0.886617	0.790576	0.712094	0.654412
			3.619183

Table 3

Coefficients of the BLUE of scale parameter.

The variances and covariances of the BLUEs are presented in Table 4. We see that the variances of the BLUEs increase as the censoring level increases while the variances of the BLUEs decrease as the sample size increases. In addition, we see that the covariance of the BLUEs decreases as the censoring level increases while the covariance of the BLUEs increases as the sample size increases.

β, δ, θ	n	c	Var(μ*)	Var(σ*)	Cov (μ,σ)
2, 3, 4	5	0	0.003609	0.228627	−0.011900
		1	0.003864	0.335727	−0.017130
	6	0	0.002368	0.183293	−0.007828
		1	0.002475	0.250810	−0.010515
		2	0.002641	0.349868	−0.014574
	7	0	0.001669	0.152970	−0.005532
		1	0.001722	0.199847	−0.007103
		2	0.001795	0.261616	−0.009227
	8	0	0.001239	0.131258	−0.004114
		1	0.001268	0.165900	−0.005116
		2	0.001305	0.208513	−0.006377
		3	0.001359	0.267122	−0.008158
	9	0	0.000955	0.114940	−0.003178
		1	0.000972	0.141689	−0.003859
		2	0.000993	0.173057	−0.004672
		3	0.001022	0.213272	−0.005741
	10	0	0.000758	0.102222	−0.002526
		1	0.000769	0.123554	−0.003010
		2	0.000781	0.147716	−0.003567
		3	0.000798	0.177205	−0.004263
		4	0.000820	0.215842	−0.005196

Table 4

Variances and covariance of the BLUEs when μ = 0 and σ = 1.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

AUTHORS' CONTRIBUTIONS

All the three authors have equally contributed in the preparation of this research paper.

Funding Statement

We have solely funded the research by ourselves.

ACKNOWLEDGMENTS

The authors are grateful to the learned referees for their fruitful comments.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 20 - 2
Pages: 171 - 179
Publication Date: 2021/06/15
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.210602.001 How to use a DOI?
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TY  - JOUR
AU  - Jagdish Saran
AU  - Kanika Verma
AU  - Narinder Pushkarna
PY  - 2021
DA  - 2021/06/15
TI  - Relationships for Moments of Generalized Order Statistics from Hjorth Distribution and Related Inference
JO  - Journal of Statistical Theory and Applications
SP  - 171
EP  - 179
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210602.001
DO  - 10.2991/jsta.d.210602.001
ID  - Saran2021
ER  -

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