Journal of Statistical Theory and Applications

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
https://doi.org/10.2991/jsta.d.210602.001How 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,n1} 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,  nN,  n2,m˜=(m1,m2,  ,mn1)Rn1, Mr=j=rn1mj such that γr=k+(nr)+Mr>0 for all r{1,2,,n1}. 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)=kj=1n1γji=1n1(1F(xi))mif(xi)×(1F(xn))k1f(xn),  F1(0+)<x1xn<F1(1).(1)

Choosing the parameters appropriately, models such as ordinary order statistics (γi=ni+1;i=1,2,,n,i.e.,m1=m2=mn1=0,k=1), k-th records values (γi=k,i.e.,m1=m2==mn1=1,kN), sequential order statistics (γi=(ni+1)αi;α1,α2,,αn>0), order statistics with non-integral sample size (γi=αi+1;α>0), Pfeifer’s record values (γi=βi;β1,β2,,βn>0) and progressively type-II right censored order statistics (miN0,kN) 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)=cr1i=1r1(1F(xi))mif(xi)×(1F(xr))k+(nr)+Mr1f(xr),  F1(0+)<x1xr<F1(1).(2)

We may consider two cases here:

Case I: m1=m2==mn1=m.

Case II: γiγj;ij,i,j=1,2,,n1.

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)=cr1(r1)!(1F(x))γr1f(x)gmr1(F(x)),       xR,(3)
and the joint pdf of X(r,n,m,k) and X(s,n,m,k),1r<sn, is given by
fX(r,n,m,k),X(s,n,m,k)(x,y)=cs1(r1)!(sr1)![1F(x)]mf(x)gmr1(F(x))×[hm(F(y))hm(F(x))]sr1[1F(y)]γs1f(y),x<y,(4)
where
cr1=j=1rγj,γr=k+(nr)(m+1),r=1,2,,n,gm(x)=hm(x)hm(0),     x[0,1),hm(x)={1m+1(1x)m+1,m1log(1x),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)=cr1f(x)i=1rai(r)(1F(x))γi1,      xR,(5)
and the joint pdf of X(r,n,m˜,k) and X(s,n,m˜,k), 1 r<sn, is given by
fX(r,n,m˜,k),X(s,n,m˜,k)(x,y)=cs1{i=r+1sai(r)(s)(1F(y)1F(x))γi}{i=1rai(r)(1F(x))γi}   ×f(x)1F(x)f(y)1F(y),x<y,(6)
where
cr1=i=1rγi,γi=k+ni+Mi,r=1,2,,n,ai(r)=j(i)=1r1(γjγi),1irn
and
ai(r)(s)=j(i)=r+1s1(γjγi),r+1isn
(cf. Kamps and Cramer [3]).

Further, it can be easily proved that

ai(r)=(γr+1γi)ai(r+1),cr1=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)(1F(x))γi=(1F(x))γr(r1)!gmr1(F(x)),(8)
and
i=r+1sai(r)(s)(1F(y)1F(x))γi=1(sr1)!(1F(y)1F(x))γs×(11F(x))(m+1)(sr1)[hm(F(y))hm(F(x))]sr1.(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 γiγj;ij,i,j=1,2,,n1. 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]eδx22(1+βx)1+θβ,     x0,      β,δ,θ>0,(10)
and the cdf is of the form
F(x)=1eδx2/2(1+βx)θβ.(11)

Its characterizing differential equation is given by

(1+βx)f(x)=[(1+βx)δx+θ](1F(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)=1eδ(xμσ)2/2(1+β(xμσ))θβ,xμ,μ0,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 nN,k1,

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(r1,n,m˜,k)]+δβγr(i+3)E[Xi+3(r,n,m˜,k)]E[Xi+3(r1,n,m˜,k)]+θγr(i+1)E[Xi+1(r,n,m˜,k)]E[Xi+1(r1,n,m˜,k)].(14)

Proof.

From (5) and (12), we have

E[Xi(r,n,m˜,k)]+βE[Xi+1(r,n,m˜,k)]=cr10xi(1+βx)f(x)i=1rai(r)[1F(x)]γi1dx=cr10xi[δx+βδx2+θ]i=1rai(r)[1F(x)]γidx=δcr10xi+1i=1rai(r)[1F(x)]γidx  +βδcr10xi+2i=1rai(r)[1F(x)]γidx  +θcr10xii=1rai(r)[1F(x)]γidx=δIi+1+βδIi+2+θIi(15)
where
Ii=cr10xii=1rai(r)[1F(x)]γidx=cr1i+10xi+1i=1r1ai(r){(γrγi)γr}[1F(x)]γi1f(x)dx+cr1i+10xi+1ar(r)γr[1F(x)]γr1f(x)dx=γr(i+1)[E[Xi(r,n,m˜,k)]E[Xi+1(r1,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==mn1=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(Xr:n) Var(Xr:n) n r E(Xr:n) Var(Xr: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 1r<sn  and  i,j0,

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(s1,n,m˜,k)]+δβγsj+3E[Xi(r,n,m˜,k)Xj+3(s,n,m˜,k)]+E[Xi(r,n,m˜,k)Xj+3(s1,n,m˜,k)]+δγsj+2E[Xi(r,n,m˜,k)Xj+2(s,n,m˜,k)]+E[Xi(r,n,m˜,k)Xj+2(s1,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)]=cs10xxiyj(1+βy){i=r+1sai(r)(s)(1F(y)1F(x))γi}{i=1rai(r)(1F(x))γi}×f(x)1F(x)f(y)1F(y)dydx=cs10xii=1rai(r)(1F(x))γif(x)1F(x)I(x)dx,(18)
where
I(x)=xyj(1+βy)i=r+1sai(r)(s)(1F(y)1F(x))γif(y)1F(y)dy.

Using (12), we obtain

I(x)=xyj[(1+βy)δy+θ]i=r+1sai(r)(s)(1F(y)1F(x))γidy=δxyj+1i=r+1sai(r)(s)(1F(y)1F(x))γidy+δβxyj+2i=r+1sai(r)(s)(1F(y)1F(x))γidy+θxyji=r+1sai(r)(s)(1F(y)1F(x))γidy=δIj+1+δβIj+2+θIj,(19)
where
Ij=xyji=r+1sai(r)(s)(1F(y)1F(x))γidy=1(j+1)xyj+1i=r+1sγiai(r)(s)(1F(y)1F(x))γif(y)1F(y)dy=1(j+1)xyj+1i=r+1s1ai(r)(s)(γiγs+γs)(1F(y)1F(x))γif(y)1F(y)dy+1j+1xyj+1as(r)(s)γs(1F(y)1F(x))γsf(y)1F(y)dy=1(j+1)xyj+1i=r+1s1ai(r)(s)(γi+γs)(1F(y)1F(x))γif(y)1F(y)dy+γs(j+1)xyj+1i=r+1sai(r)(s)(1F(y)1F(x))γif(y)1F(y)dy=1(j+1)xyj+1i=r+1s1ai(r)(s1)(1F(y)1F(x))γif(y)1F(y)dy+γs(j+1)xyj+1i=r+1sai(r)(s)(1F(y)1F(x))γif(y)1F(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==mn1=m.

4. BLUEs OF μ AND σ

Let X1:nX2:nXnc:n,c=0,1,,n1, denote type-II right censored sample from the location-scale parameter Hjorth distribution in (13). Let us denote Zr:n=(Xr:nμ)/σ,E(Zr:n)=μr:n(1),1r(nc), and Cov(Zr:n,Zs:n)=σr,s:n=μr,s:n(1,1)μr:n(1)μs:n(1),1r<s(nc). We shall use the following notations:

X=(X1:n,X2:n,,Xnc:n)Tμ=(μ1:n,μ2:n,,μnc:n)T  1=  (1,1,,1)T
and=(σr,s:n),1r,snc.

The BLUEs of μ and σ are given by

μ*={μT1μ1T1μT11μT1(μT1μ)(1T11)(μT11)2}X=r=1ncarXr:n(21)
and
σ*={1T11μT11T1μ1T1(μT1μ)(1T11)(μT11)2}X=r=1ncbrXr:n,(22)
and the variances and covariance of these BLUEs are given by
Var(μ*)=σ2{μT1μ(μT1μ)(1T11)(μT11)2}=σ2V1,(23)
Var(σ*)=σ2{1T11(μT1μ)(1T11)(μT11)2}=σ2V2(24)
and
Cov(μ*,σ*)=σ2{μT11(μT1μ)(1T11)(μT11)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=1ncar=1
and
r=1ncbr=0.

β,δ,θ n c ai = 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.

REFERENCES

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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
https://doi.org/10.2991/jsta.d.210602.001How to use a DOI?
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/).

Cite this article

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  - https://doi.org/10.2991/jsta.d.210602.001
ID  - Saran2021
ER  -