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Volume 20, Issue 2, June 2021, Pages 251 - 266

Moments of Dual Generalized Order Statistics from Inverted Kumaraswamy Distribution and Related Inference

Authors
Bushra Khatoon, M. J. S. Khan*, Zubdahe Noor
Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, 202 002, India
*Corresponding author. Email: jahangirskhan@gmail.com
Corresponding Author
M. J. S. Khan
Received 12 March 2020, Accepted 24 November 2020, Available Online 3 March 2021.
DOI
10.2991/jsta.d.210219.001How to use a DOI?
Keywords
Moments; Dual generalized order statistics; Maximum likelihood estimation; Uniformly minimum variance unbiased estimation
Abstract

In this paper, the exact and explicit expressions for single, product, and conditional moments for inverted Kumaraswamy distribution using dual generalized order statistics (dgos) are derived. Further, the expression for maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) for the parameters of inverted Kumaraswamy distribution based on dgos are deduced. Also, we obtained the results for order statistics and lower record values by putting some specific values of the parameters of dgos. Finally, a simulation study is carried out for illustrative purpose.

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

Generalized order statistics (gos) was introduced by Kamps [1] as a unified model of ordered random variables arranged in increasing order of magnitude which contains almost all models related to ordered random variables such as order statistics, records, Pfeifer records, progressive type-II censored order statistics, and sequential order statistics. But those models of ordered random variables which are in decreasing order of magnitude cannot be studied in this framework. To resolve such types of problems, Pawlas and Szynal [2] proposed the concept of dual (lower) generalized order statistics (dgos) which contains several models of ordered random variables those are arranged in decreasing order of magnitude, like reversed order statistics, lower records, and lower Pfeifer records. Burkschat et al. [3] established the direct relationship between gos and dgos in such a way that if X(1,n,m˜,k),X(2,n,m˜,k),…,X(n,n,m˜,k) are n gos with cumulative distribution function (cdf) F0(.) and Xd(1,n,m˜,k),Xd(2,n,m˜,k),…,Xd(n,n,m˜,k) are n dgos with (cdf) F(.), then

F(Xd(i,n,m˜,k))=d1−F0(X(i,n,m˜,k)),1≤i≤n
where =d denotes the convergence in distribution.

Therefore, to avoid the complications in calculations, one can use dgos when F(.) is available in compact and closed form and when a closed and compact form of (1−F(.)) is available, gos can be preferred.

Let X1,X2,…,Xn be a sequence of independent and identically distributed (iid) random variables with absolutely continuous distribution function (df) F(x) and the probability density function (pdf) f(x), x∈(−∞,∞). Further, let n∈ℕ, n≥2, k≥1, 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 Xd(r,n,m˜,k), r=1,2,…,n are said to be dgos if their joint pdf is given by

k(∏j=1n−1γj)(∏i=1n−1[F(xi)]mif(xi))[F(xn)]k−1f(xn)(1.1)
for F−1(1)>x1≥x2≥…≥xn>F−1(0).

Note that Xd(r,n,m˜,k) reduces to the (n−j+1)th order statistics from a sample of size n if γj=n−j+1 (i.e. m1=m2=…=mn−1=0 and k=1) and when m→−1,k=1, Xd(r,n,m˜,k) reduces to rth lower records.

Here, we shall consider two cases:

Case I: γi≠γj, i≠j i.e. γi's are pairwise different.

The pdf of rthdgosXd(r,n,m˜,k) is given by (Kamps [1])

fXd(r,n,m˜,k)(x)=cr−1∑i=1rai(r)[F(x)]γi−1f(x), −∞<x<∞(1.2)

And the joint pdf of Xd(r,n,m˜,k) and Xd(s,n,m˜,k) is:

fXd(r,n,m˜,k),Xd(s,n,m˜,k)(x,y)=cs−1∑i=1r∑j=r+1sai(r)aj(r)(s)[F(y)F(x)]γj[F(x)]γi  × f(x)F(x)f(y)F(y), −∞<y<x<∞(1.3)

Therefore, the conditional pdf of Xd(s,n,m˜,k) given Xd(r,n,m˜,k)=x, for 1≤r<s≤n is:

fXd(s,n,m˜,k)∣Xd(r,n,m˜,k)(y∣x)=cs−1cr−1∑j=r+1saj(r)(s)[F(y)F(x)]γjf(y)F(y), −∞<y<x<∞(1.4)
similarly, the conditional pdf of Xd(r,n,m˜,k)|X(s,n,m˜,k)=y, 1≤r<s≤n, is given by
fXd(r,n,m˜,k)|Xd(s,n,m˜,k)(x|y)=∑i=1r∑j=r+1saj(r)(s)ai(r)[F(y)F(x)]γj[F(x)]γif(x)F(x)∑t=1sat(s)[F(y)]γt,  −∞<y<x<∞(1.5)
where
ai(r)=∏ j≠ij=1r1(γj−γi), γi≠γj,1≤i≤r≤n
and
ai(r)(s)=∏ j≠ij=r+1s1(γj−γi), γi≠γj,r+1≤i≤s≤n

It may be noted that for m1=…=mn−1=m≠−1, (Khan and Khan [4])

ai(r)=(−1)r−i(m+1)s−r−1(r−1)!(r−1i)(1.6)
air(s)=(−1)r−i(m+1)s−r−1(r−1)!(s−r−1i)(1.7)

Case II: When m1=m2=…,mn−1=m:

The pdf of rthdgosXd(r,n,m,k) is given by

fXd(r,n,m,k)(x)=cr−1(r−1)![F(x)]γr−1f(x) gm′ r−1(F(x)),  −∞<x<∞(1.8)
and the joint pdf of Xd(r,n,m,k) and Xd(s,n,m,k), (1≤r<s≤n) is
fXd(r,n,m,k),Xd(s,n,m,k)(x,y)=cs−1(r−1)!(s−r−1)![hm′(F(y))−hm′(F(x))]s−r−1gm′ r−1(F(x))  × [F(y)]γs−1 [F(x)]mf(x)f(y),  −∞<y<x<∞(1.9)
where cr−1=∏i=1rγi,
hm′(x)={−1m+1xm+1, m≠−1−logx, m=−1
and gm′(x)=hm′(x)−hm′(1),x∈(0,1).

Further, the conditional pdf of Xd(s,n,m,k) given Xd(r,n,m,k)=x, 1≤r<s≤n is

fXd(s,n,m,k)|Xd(r,n,m,k)(y|x)=cs−1cr−1(s−r−1)! [hm′(F(y))−hm′(F(x))]s−r−1  × [F(y)]γs−1 [F(x)]γr+1 f(y),  −∞<y<x<∞(1.10)

And the conditional pdf of Xd(r,n,m,k) given Xd(s,n,m,k)=x, 1≤r<s≤n is

fXd(r,n,m,k)|Xd(s,n,m,k)(x|y)=(s−1)!(r−1)!(s−r−1)! [hm′(F(y))−hm′(F(x))]s−r−1  × [gm′(F(x))]r−1 [gm′(F(y))]s−1 [F(x)]mf(x),  −∞<y<x<∞(1.11)

Here, the results are derived for case-I as it includes case-II as a particular case of it.

In the last two decades, several researchers have analyzed statistical properties of continuous distributions based on gos, dgos and hence a wide literature is available on the various developments on dgos and gos. Pawlas and Szynal [5] discussed the relation for single and product moments of gos from Pareto, generalized Pareto, and Burr distributions. Cramer et al. [6] proved the existence of moments of gos. Ahsanullah [7] characterized the uniform distribution based on dgos. Athar et al. [8] obtained the explicit expressions for ratio and inverse moments of gos from Weibull distribution. Explicit expressions for single and product moments of gos from linear exponential distribution have been derived by Ahmad [9]. Based on dgos, Khan et al. [10] established the recurrence relations for single and product moments of exponentiated Weibull distribution. Khan et al. [11] have characterized continuous distributions conditioned on a pair of nonadjacent dgos. Burkschat [12] has discussed the linear estimators and predictors from generalized Pareto distribution based on gos. Barakat and Adll [13] studied the asymptotic theory of extreme dgos. Athar and Faizan [14] deduced the explicit expression for moments of power function distribution based on dgos. Khan and Khan [4] obtained the ratio and inverse moments of gos from Burr distribution using hypergeometric functions. Domma and Hamdeni [15] discussed the truncated moments of dgos. Recurrence relations for moments of dgos from Weibull gamma distribution have been discussed by Mahmoud et al. [16]. Recurrence relation for single and product moments of dgos from a general class of distribution have been derived by Saran et al. [17]. Based on lower records, Khan and Arshad [18] have derived the uniformly minimum variance unbiased estimator(UMVUE) of reliability function and stress–strength reliability from proportional reversed hazard rate model. An exact and explicit expression of moments of Topp–Leone distribution based dgos have been deduced by Khan and Iqrar [19]. They have also derived the expressions for maximum likelihood estimator(MLE) and UMVUE for the parameter of Topp–Leone distribution based on dgos. Interval estimation for Topp–Leone generated family of distributions based on dgos have been considered by Arshad and Jamal [20]. Khatoon et al. [21] have considered the single, product, and conditional moments of gos as well as parameter estimation based on gos from Kumaraswamy power function distribution.

Kumaraswamy distribution was introduced by Kumaraswamy [21]. This distribution is applicable in many real-life situations and natural phenomenon that are restricted with lower and upper bounds. Over the last decades, there has been a great interest in studying the Kumaraswamy distribution among the researchers. Few of them are Nadarajah [22], Jones [23], Garg [24], Cordeiro and De-Castro [25], Mitnik [26], Safi and Ahmed [27], El-Deen et al. [28]. El-Deen et al. [28] studied the statistical inference for this distribution based on gos. Abd Al-Fattah et al. [29] introduced the inverted Kumaraswamy distribution as the inverted distributions have wide range of applications in real-life situations. Usman and Ahsan-ul-Haque [30], Reyad et al. [31] have studied the generalization form of inverted Kumaraswamy distribution.

The pdf of inverted Kumaraswamy distribution is given by

f(x;α,β)=αβ(1+x)−(α+1)[1−(1+x)−α]β−1,x>0,α,β>0.(1.12)

And its cdf is given by

F(x;α,β)=[1−(1+x)−α]β,x>0,α,β>0.(1.13)

This paper comprises of 5 sections. In Section 2, the results for single, product, and conditional moments of inverted Kumaraswamy distribution based on dgos have been derived. Section 3 included the theoretical setup and expressions for the MLE of both parameters of inverted Kumaraswamy distribution based on dgos. In Section 4, we have deduced the UMVUE of one shape parameter of inverted Kumaraswamy distribution by assuming that the other shape parameter is known. Finally, a simulation study is being carried out to check the validity and applicability of the derived estimators based on order statistics and lower records which is given in Section 5.

2. MOMENTS OF dgos

In this section, we have derived the exact and explicit expressions for single, product, and conditional moments of inverted Kumaraswamy distribution based on dgos. Further, by putting the specific values of the parameters m and k, we have deduced single moment, product moment, and conditional moments of order statistics and lower records from inverted Kumaraswamy distribution Table 1-6.

n r=1 r=2 r=3 r=4 r=5 r=6 r=7 r=8
1 0.1496
2 0.0493 0.25
3 0.0254 0.097 0.3265
4 0.0157 0.0546 0.1394 0.3889
5 0.0108 0.0356 0.083 0.177 0.4419
6 0.0078 0.0253 0.0563 0.1096 0.2107 0.4881
7 0.0060 0.0190 0.0411 0.0765 0.1344 0.2412 0.5292
8 0.0047 0.0148 0.0315 0.0571 0.0959 0.1576 0.2691 0.5664
Table 1

Mean of order statistics from inverted Kumaraswamy distribution for α=5,β=0.5, (m=0,k=1).

n r=1 r=2 r=3 r=4 r=5 r=6 r=7 r=8
1 0.06562
2 0.00687 0.10420
3 0.00195 0.01319 0.13210
4 0.00075 0.00422 0.01867 0.15436
5 0.00038 0.00183 0.00641 0.02317 0.17302
6 0.00021 0.00096 0.00303 0.00849 0.02721 0.18946
7 0.00013 0.00056 0.00161 0.00415 0.01034 0.03072 0.20405
8 0.00008 0.00035 0.00101 0.00234 0.00520 0.01196 0.03379 0.21729
Table 2

Variance of order statistics from inverted Kumaraswamy distribution for α=5,β=0.5, (m=0,k=1).

α=2.5 α=3 α=3.5 α=4 α=4.5 α=5
r = 1 0.38730 0.29360 0.23650 0.19810 0.17050 0.14960
r = 2 0.06800 0.05510 0.04640 0.04000 0.03520 0.03140
r = 3 0.01830 0.01510 0.01280 0.01110 0.00990 0.00880
r = 4 0.00550 0.00460 0.00390 0.00340 0.00300 0.00270
r = 5 0.00180 0.00150 0.00120 0.00110 0.00097 0.00087
Table 3

Mean of lower records from inverted Kumaraswamy distribution for α=2.5:0.5:5,β=0.5, (m→−1,k=1).

α=2.5 α=3 α=3.5 α=4 α=4.5 α=5
r = 1 1.20980 0.43000 0.22167 0.13526 0.09113 0.06562
r = 2 0.02448 0.01486 0.00995 0.00720 0.00536 0.00421
r = 3 0.00267 0.00177 0.00124 0.00088 0.00071 0.00056
r = 4 0.00041 0.00027 0.00020 0.00015 0.00011 0.00009
r = 5 0.00007 0.00005 0.00003 0.00003 0.00002 0.00002
Table 4

Variance of lower records from inverted Kumaraswamy distribution for α=2.5:0.5:5,β=0.5, (m→−1,k=1).

n r=1 r=2 r=3 r=4 r=5 r=6 r=7 r=8
1 0.25
2 0.3889 0.1111
3 0.4881 0.1905 0.0714
4 0.5664 0.2531 0.1278 0.0526
5 0.6317 0.3053 0.1748 0.0965 0.0417
6 0.6879 0.3504 0.2153 0.1343 0.0776 0.0345
7 0.7376 0.3901 0.2511 0.1677 0.1093 0.0649 0.0294
8 0.7821 0.4257 0.2831 0.1976 0.1377 0.0922 0.0558 0.0256
Table 5

Mean of dgos from inverted Kumaraswamy distribution for α=5,β=0.5, (m=1,k=2).

n r=1 r=2 r=3 r=4 r=5 r=6 r=7 r=8
1 0.10420
2 0.15436 0.01546
3 0.18946 0.02511 0.00590
4 0.21729 0.03224 0.01007 0.00303
5 0.24076 0.03799 0.01334 0.00539 0.00186
6 0.26149 0.04282 0.01605 0.00736 0.00338 0.00131
7 0.27985 0.04712 0.01835 0.00898 0.00465 0.00229 0.00094
8 0.29672 0.05098 0.02045 0.01035 0.00574 0.00320 0.00169 0.00064
Table 6

Variance of dgos from inverted Kumaraswamy distribution for α=5,β=0.5, (m=1,k=2).

2.1. Single Moment

Theorem 2.1

Let X1,X2,…,Xn be n continuous iid non negative random variables and follow inverted Kumaraswamy distribution given in (1.12), then the single moment of rth dgos (1≤r≤n) is given by

μr,n,m˜,kj=E(Xdj(r,n,m˜,k)) =βcr−1∑t=1r∑i=0j(−1)i(ji)at(r)ℬ(1−j−iα,βγt), j−i<α(2.1)
and in case of lower records, the single moment is given by
μr,n,−1,kj=βr(r−1)!∑i=0j(ji)(−1)i+j+r−1∂r−1∂βr−1ℬ(1−i/α,β), i<α(2.2)
where j∈ℕ and ℬ(a,b)=∫01xa−1(1−x)b−1 is a complete beta function.

Proof.

In case of dgos, for the inverted Kumaraswamy distribution given in (1.12), the single moment of rth dgos is given by

μr,n,m˜,kj=αβcr−1∑t=1rat(r)∫0∞xj[1−(1+x)−α]βγt−1(1+x)−α−1dx

Putting (1+x)−α=t and simplifying accordingly, we get

μr,n,m˜,kj=βcr−1∑t=1rat(r)∫01(t−1/α−1)j(1−t)βγt−1dt =βcr−1∑t=1r∑i=0jat(r)(ji)(−1)i∫01t−(j−i)/α(1−t)βγt−1dt

Now, using the relation ∫01xa−1(1−x)b−1=ℬ(a,b), we obtained the result.

In case of lower records, the rth single moment from inverted Kumaraswamy distribution is given by

μr,n,−1,kj=αβ(r−1)!∫0∞xj[−ln(1−(1+x)−α)β]r−1(1+x)−α−1[1−(1+x)−α]β−1dx

Assuming (1+x)−α=u and simplifying appropriately, we get

μr,n,−1,kj=βr(r−1)!∫01(u−1/α−1)j[−ln(1−u)]r−1(1−u)β−1du=βr(r−1)!∑i=0j(ji)(−1)i+j+r−1∫01u(1−i/α)−1(1−u)β−1[ln(1−u)]r−1du

Using the relation ∫0axα−1(a−x)β−1[lnr(a−x)]dx=aα−1∂r∂βr[aβℬ(α,β)], (Prudnikov et al. [32], p. 502), we obtained the result.

It is pertinent to mention that the computation of ∂r∂βr[aβℬ(α,β)] can further be simplified by using the recurrence relation

∂rℬ(α,β)∂βr=∑k=0r−1[ψ(r−k−1)(β)−ψ(r−k−1)(α+β)]∂kℬ(α,β)∂βk
where ψ(r−k−1)(x) is the kth derivative of diagamma function given by ψ(x)=dlogΓ(x)dx=Γ′(x)Γ(x),x>0 and Γ(.) is the gamma function (Khan et al. [33]).

Corollary 2.1.

for m1=m2=…,mn−1=m≠−1, the single moment of inverted Kumaraswamy distribution is given by

μr,n,m,kj=βcr−1(m+1)r−1(r−1)!∑t=0r−1∑i=1j (−1)i+t(r−1t)(ji)ℬ(1−j−iα,βγr−t),  j−i<α(2.3)

Remark 2.1.

When m=0,k=1, the single moment of order statistics from inverted Kumaraswamy distribution is given by

μn−r+1,n,0,1j=βn!(n−r)!(r−1)!∑t=0n−r∑i=0j(−1)i+t(n−rt)(ji)ℬ(1−j−iα,β(n−r+t+1)),  j−i<α(2.4)

2.2. Product Moments

In this section, the product moment of rth and sth dgos is computed in terms of hypergeometric function from inverted Kumaraswamy distribution. Further, by choosing the different combination of the parameters of dgos (i.e., m and k), we reduced the obtained result in terms of order statistics and lower record values. The obtained result is summarized in the form of theorem which is given below.

Theorem 2.2.

Let X1,X2,…,Xn be n continuous iid non negative random variables that follow inverted kumaraswamy distribution given in (1.12) and Xd(i,n,m˜,k),i=1,2,…,n be their corresponding dgos. For any λ,l∈ℕ and γ1:r=min(γ1,γ2,…,γr)>γr+1,…,γs, the product moment of rth and sth dgos (1≤r<s≤n) is given by

μr,s,n,m˜,kλ,l=E[Xdλ(r,n,m˜,k)Xdl(s,n,m˜,k)] =β2cs−1∑i=1r∑j=r+1s∑p=0λ∑q=0lai(r)aj(r)(s)(λp)(lq)(−1)p+q1−(λ−pα)  ×ℬ(l−λ+p−qα+2,βγj)3F2(1−λ−pα,1−β(γi−γj), l−λ+p−qα+2;2−λ−pα,l−λ+p−qα+βγj+2;1)  (2.5)
where λ,l∈ℕ and  3F2(a1,a2,a3;b1,b2;x) is generalized hypergeometric function defined as follows:
 3F2(a1,a2,a3;b1,b2;x)=∑t=0∞(a1)t(a2)t(a3)t(b1)t(b2)txtt!

(Mathai and Saxena [34]).

Proof.

In view of (1.7), for the inverted Kumaraswamy distribution given in (1.12) the product moment for sth and rth dgos is given by

μr,s,n,m˜,kλ,l=cs−1∑i=1r∑j=r+1sai(r)aj(r)(s)∫0∞∫y∞xλyl[F(y)]γj[F(x)]γi−γjf(x)F(x)f(y)F(y)dxdy0<y<x<∞

Consider

I(y)=∫y∞xλ[F(x)]γi−γjf(x)F(x)dx=αβ∫y∞xλ[1−(1+x)−α]β(γi−γj)−1(1+x)−α−1dx

Let (1+x)−α=t and simplifying accordingly, we have

I(y)=β∫0(1+y)−α(t−1/α−1)k[1−t]β(γi−γj)−1dt
=β∑p=0λ(λp)(−1)p∫0(1+y)−αt1−(λ−p)/α−1[1−t]β(γi−γj)−1dt
=β∑p=0λ(λp)(−1)pℬ(1+y)−α(1−(λ−p)α,β(γi−γj))
=β∑p=0λ(λp)(−1)p∫0(1+y)−αt1−(λ−p)/α−1[1−t]β(γi−γj)−1dt

Therefore,

μr,s,n,m˜,kλ,l=αβ2cs−1∑i=1r∑j=r+1s∑p=0λai(r)aj(r)(s)(λp)(−1)p1−(λ−pα)  ×∫0∞yl[(1+y)−α]1−(λ−pα)[1−(1+y)−α]βγj−1(1+y)−α−1  ×2F1(1−(λ−pα),1−β(γi−γj);2−(λ−pα);(1+y)−α)dy

Again putting (1+y)−α=z, the product moment of inverted Kumaraswamy distribution based on dgos is given by

μr,s,n,m˜,kλ,l=β2cs−1∑i=1r∑j=r+1s∑p=0λai(r)aj(r)(s)(λp)(−1)p1−(λ−pα)∫01(z−1/α−1)lz1−(λ−pα)  ×(1−z)βγj−1 2F1(1−(λ−pα),1−β(γi−γj);2−(λ−pα);z)dz =β2cs−1∑i=1r∑j=r+1s∑p=0λ∑q=0l(λp)(lq)ai(r)aj(r)(s)(−1)p+q1−(λ−pα)∫01z(l−q)/αz1−(λ−pα)  ×(1−z)βγj−1 2F1(1−(λ−pα),1−β(γi−γj);2−(λ−pα);z)dz
or
μr,s,n,m˜,kkλ,l=β2cs−1∑i=1r∑j=r+1s∑p=0λ∑q=0l(λp)(lq)ai(r)aj(r)(s)(−1)p+q1−(λ−pα)∫01z1+(l−q−k+p)/α  ×(1−z)βγj−1 2F1(1−(λ−pα),1−β(γi−γj);2−(λ−pα);z)dz

Now using the relation (Mathai and Saxena [34])

∫01ua−1(1−u)b−1 2F1(c,d;e;u)du=ℬ(a,b) 3F2(c,d,a;e;a+b;1)
we have,
μr,s,n,m˜,kλ,l=β2cs−1∑i=1r∑j=r+1s∑p=0λ∑q=0lai(r)aj(r)(s)(λp)(lq)(−1)p+q1−(λ−pα)  ×ℬ(l−λ+p−qα+2,βγj) 3F2(1−λ−pα,1−β(γi−γj), l−λ+p−qα+2;2−λ−pα,l−λ+p−qα+βγj+2;1)

Hence, we get the result.

However, in case of lower records, an exact expression for product moment from inverted Kumaraswamy distribution could not be obtained.

Corollary 2.2.

For m1=m2,…,mn−1=m≠−1, the product moment from inverted Kumaraswamy distribution is given as

μr,s,n,m,kλ,l=β2cs−1(m+1)s−2(r−1)!(s−r−1)!∑i=0r−1∑j=0s−r−1∑p=0λ∑q=0l(−1)p+q+i+j1−(λ−pα)  ×(λp)(lq)(r−1i)(s−r−1j)ℬ(l−λ+p−qα+2, βγs−j)  ×3F2(1−λ−pα,1−β(γr−i−γs−j),l−λ+p−qα+2; 2−λ−pα,l−λ+p−qα+β γs−j+2;1)(2.6)

Remark 2.2.

In case of order statistics, i.e., m=0,k=1, the product moment from inverted Kumaraswamy distribution is given by

μn−r+1,n−s+1,n,0,1λ,l=β2n!(n−s)!(r−1)!(s−r−1)!∑i=0n−r∑j=0s−r−1∑p=0λ∑q=0l(−1)i+j+p+q1−(λ−pα)(n−ri)  ×(s−r−1j)(λp)(lq)ℬ(l−λ+p−qα+2,β(n−s+j+1))   3F2(1−λ−pα,1−β(s−r−j+i),l−λ+p−qα+2;2−λ−pα,l−λ+p−qα+β(n−s+j+1)+2;1)(2.7)

2.3. Conditional Moments

In this section, we have obtained the conditional moments of sth given rth dgos (Xd(s,n,m˜,k)|Xd(r,n,m˜,k)) and rth given sth dgos (Xd(r,n,m˜,k)|Xd(s,n,m˜,k)) from inverted Kumaraswamy Distribution in terms of incomplete beta function.

Theorem 2.3.

The qth (q∈ℕ) conditional moment of sth given rth dgos (Xd(s,n,m˜,k)|Xd(r,n,m˜,k)=x), 1≤r<s≤n, from inverted Kumaraswamy distribution is given by

μXd(s,n,m˜,k)|Xd(r,n,m˜,k)q(y|x)=βcs−1cr−1∑j=r+1saj(r)(s)∑i=0q(qi)(−1)i+1[1−(1+x)−α]−βγj  × ℬ1−(1+x)−α(βγj, 1+(i−q)α),  0<y<x<∞(2.8)
where
ℬv(a,b)=∫0vxa−1(1−x)b−1dx(2.9)
is an upper incomplete beta function.

Proof.

The conditional moment of Xd(s,n,m˜,k)|Xd(r,n,m˜,k)=x is given by

μXd(s,n,m˜,k)|Xd(r,n,m˜,k)q(y|x)=∫0xyqfs|r(y|x)dy,0<y<x<∞

From (1.4), (1.12) and (1.13), we have

μXd(s,n,m˜,k)|Xd(r,n,m˜,k)q(y|x)=αβ cs−1cr−1∑j=r+1saj(r)(s)∫0xyq[1−(1+y)−α1−(1+x)−α]βγj  ×(1+y)−α−1[1−(1+y)−α]β−1[1−(1+y)−α]βdy =αβcs−1cr−1∑j=r+1saj(r)(s)[1−(1+x)−α]−βγj  ×∫0xyq[1−(1+y)−α]βγj+β−1−β(1+y)−α−1dy.

Putting (1+y)−α=t, we get

μXd(s,n,m˜,k)|Xd(r,n,m˜,k)q(y|x)=βcs−1cr−1∑j=r+1saj(r)(s)[1−(1+x)−α]−βγj  ×∫(1+x)−α1(t−1/α−1)q(1−t)βγj−1dt =βcs−1cr−1∑j=r+1s∑l=0qaj(r)(s)(ql)(−1)l[1−(1+x)−α]−βγj  ×∫(1+x)−α1(t−1/α)q−l(1−t)βγj−1dt

Setting (1−t)=u and then using the relation given in (2.9), we obtained the result.

Theorem 2.4.

The qth (q∈ℕ), conditional moment of Xd(r,n,m˜,k)|Xd(s,n,m˜,k)=y, 1≤r<s≤n, from inverted Kumaraswamy distribution is given by

μXd(r,n,m˜,k)|Xd(s,n,m˜,k)q(x|y)=β∑i=1r∑j=r+1s∑l=0qai(r) aj(r)(s)∑t=1sat(s)(ql)(−1)l  ×[1−(1+y)−α]β(γj−γt)ℬ(1+y)−α((l−q)α+1,  β(γi−γj))(2.10)
for all γ1:r=min(γ1,γ2,…,γr)>γr+1,…,γs.

Proof.

The conditional moment of Xd(r,n,m˜,k)|Xd(s,n,m˜,k)=y, is given by

μXd(r,n,m˜,k)|Xd(s,n,m˜,k)q(x|y)=∫∞yxqfr|s(x|y)dx,0<y<x<∞

In view of (1.3), (1.12) and (1.13), we have

μXd(r,n,m˜,k)|Xd(s,n,m˜,k)q(x|y)=αβ∑i=1rai(r)∑j=r+1saj(r)(s)∑t=1sat(s)[1−(1+y)−α]β(γj−γt)  ×∫y∞xq[1−(1+x)−α]β(γi−γj−1)(1+x)−α−1[1−(1+x)−α]β−1dx

Putting (1+x)−α=z and simplifying accordingly, we get

μXd(r,n,m˜,k)|Xd(s,n,m˜,k)q(x|y)=β∑i=1r∑j=r+1sai(r)aj(r)(s)∑t=1sat(s)[1−(1+y)−α]β(γj−γt) ×∫0(1+y)−α(z−1/α−1)q(1−z)β(γi−γj)−1dz=β∑i=1r∑j=r+1sai(r)aj(r)(s)∑t=1sat(s)[1−(1+y)−α]β(γj−γt) ×∫0(1+y)−αz−q/α(1−z1/α)q(1−z)β(γi−γj)−1dz=β∑i=1r∑j=r+1s∑p=0qai(r)aj(r)(s)∑t=1sat(s)(qp)(−1)p ×[1−(1+y)−α]β(γj−γt)∫0(1+y)−αz(p−q)/α(1−z)β(γi−γj)−1dz

Using (2.9), we obtained the result.

3. MLE FOR THE SHAPE PARAMETERS α AND β

In this section, we have deduced expressions for MLE of the shape parameters of the inverted Kumaraswamy distribution based on dgos.

Estimation of parameters through MLE based on gos and dgos has been discussed by many authors. Habibullah and Ahsanullah [35] have discussed the parameters of Pareto distribution based on gos. MLE and Bayes estimator of the parameters of Burr-XII distribution based on gos have been obtained by Jaheen [36] after that Malinowska et al. [37] estimated the location and scale parameter of Burr-XII distribution for gos. Bayesian and nonBayesian estimation for Weibull distribution based on gos have been given by Abo-eleneen [38]. Abo-Elfotouh and Nassar [39] have considered the MLE and Bayesion estimation of parameters of Weibull extension model for gos. Safi and Ahmed [28] have discussed the MLE for the parameter of Kumaraswamy distribution based on gos. MLE and Bayesian estimator for the parameter of Kumaraswamy distribution based on gos have been obtained by El-Deen et al. [28]. Kim [40] have discussed the parameter estimation of generalized exponential distribution under dgos. MLE, UMVUE, and Bayes estimates of stress–strength reliability based on gos for exponential distribution have been Khan and Khatoon [41].

In view of (1.1) and (1.12) the likelihood function for inverted Kumaraswamy distribution is given by

L(α,β)=k αnβn(∏j=1n−1γj)(∏j=1n(1+xi)−(α+1))(∏i=1n−1[1−(1+xi)−α]β(mi+1)−1)  ×[1−(1+xn)−α]βk−1,   0<xn<xn−1<…<x2<x1<∞(3.1)

The log likelihood function of (3.1) is given by

l(α,β)=lnk+nlnα+nlnβ+∑i=1n−1lnγj−(α+1)∑i=1nln(1+xi)  +∑i=1n−1[β(mi+1)−1]ln[1−(1+xi)−α]+(βk−1)ln[1−(1+xn)−α](3.2)

Differentiating both sides of (3.2) w.r.t. α, β and equating to zero, we get

nα−∑i=1nln(1+xi)−∑i=1n−1[β(mi+1)−1](1+xi)−αln(1+xi)1−(1+xi)−α−(βk−1)(1+xn)−αln(1+xn)1−(1+xn)−α=0(3.3)
nβ+∑i=1n−1(mi+1)ln[1−(1+xi)−α]+kln[1−(1+xn)−α]=0(3.4)

In case of lower records [(XL(1),XL(2),…,XL(n))=(x1,x2,…,xn)], the likelihood equation is given by

L(α,β)=αnβn[1−(1+xn)−α]β∏i=1n(1+xi)−(α+1)1−(1+xi)−α(3.5)

And the MLE of α and β in case of lower records can be obtained by solving the normal equations

nα−∑i=1nln(1+xi)+∑i=1n(1+xi)−αln(1+xi)1−(1+xi)−α−β(1+xn)−αln(1+xn)1−(1+xn)−α=0(3.6)
and
nβ+ln[1−(1+xn)−α]=0(3.7)

From (3.3)–(3.7), the exact expression for MLE of α and β cannot be obtained directly. Therefore, an iteration method (Newton–Raphson) is applied to obtain the MLE of α and β.

4. UMVUE AND MLE OF β WHEN α IS KNOWN

In this section, we have obtained UMVUE and MLE of parameter β of the inverted Kumaraswamy distribution based on dgos when α is known say α0.

Replacing α=α0 in the equation (3.1) and re-write the joint pdf of n dgos from inverted Kumaraswamy distribution as

f(x1,x2,…,xn)= k αnβn(∏j=1n−1γj)(∏i=1n(1+xi)−(α+1[1−(1+xi)−α])e−βω(4.1)
where
ω=−∑i=1n−1(mi+1)ln[1−(1+xi)−α0]−kln[1−(1+xn)−α0]

From equation (4.1), it can be easily seen that ω is a complete sufficient statistics for β. From the normalized spacing of dgos, one can easily prove that ω follows gamma distribution with shape parameter n and scale parameter 1/β denoting G(n,1/β) (see Arshad and Jamal [20]). Therefore,

E(ω)=(n−1)/β,
and hence the UMVUE of β is
β^UMVU=n−1ω(4.2)

Further, adopting the same procedure described in the previous section (3), the MLE of β is given by

β^ML=nω(4.3)

In case of lower records, UMVUE and MLE can be obtained as

β^UMVU=n−1−ln(2XL(n)−XL(n)2)(4.4)
and
β^ML=n−ln(2XL(n)−XL(n)2)(4.5)
where XL(n),n=1,2,… denotes the lower record value of the sequence of the random variables. For more detail regarding this result, one can refer to Zghoul [42].

5. SIMULATION STUDY

In this section, the MLE for both the shape parameters, MLE and UMVUE of one shape parameter by taking another one is known of inverted Kumaraswamy distribution based on order statistics and lower records are computed. Here, R software is used for data simulation and computation purpose.

When both the shape parameters are unknown, we considered a Monte Carlo method of simulation to obtain the MLEs for the parameters by considering that (α,β)=((0.5,0.5),(0.5,1),(1,0.5),(1,1),(2.2,2.1)), sample size n=20,40,…,100 for order statistics and for lower records n=4,6,…,12. For the entire simulation, we consider 1000 replication of the process and the average estimate, bias, and mean square error (MSE) are computed for order statistics and lower records which are reported in Tables 7 and 8 respectively.

(α,β) n α
 β
MLE Bias MSE MLE Bias MSE
(0.5, 0.5) 20 0.59358 0.09358 0.06032 0.56703 0.06703 0.03495
40 0.54608 0.04608 0.02514 0.53022 0.03022 0.01283
60 0.52632 0.02632 0.01356 0.51934 0.01934 0.00714
80 0.51961 0.01961 0.00910 0.51365 0.01365 0.00526
100 0.51733 0.01733 0.00702 0.51260 0.01260 0.00437
(0.5, 1) 20 0.57640 0.07640 0.04608 1.19448 0.19448 0.26841
40 0.53440 0.03440 0.01444 1.06839 0.06839 0.06219
60 0.51871 0.01871 0.00840 1.04760 0.04760 0.03773
80 0.51417 0.01417 0.00605 1.03265 0.03265 0.02480
100 0.50881 0.00881 0.00473 1.02427 0.02427 0.01987
(1, 0.5) 20 1.16622 0.16622 0.24372 0.56160 0.06160 0.03262
40 1.08815 0.08815 0.09383 0.53022 0.03022 0.01304
60 1.04808 0.04808 0.05171 0.51787 0.01787 0.00768
80 1.04051 0.04051 0.03823 0.51435 0.01435 0.00535
100 1.02548 0.02548 0.03034 0.51053 0.01053 0.00371
(1, 1) 20 1.12171 0.12171 0.13265 1.15090 0.15090 0.17176
40 1.05221 0.05221 0.05487 1.07467 0.07466 0.06503
60 1.04386 0.04386 0.03461 1.04669 0.04669 0.03704
80 1.03591 0.03591 0.02354 1.03709 0.03709 0.02622
100 1.03215 0.03215 0.02009 1.03069 0.03069 0.02180
(2.5, 2.5) 20 2.75690 0.25690 0.54450 3.08986 0.58986 0.91808
40 2.61797 0.11797 0.21485 2.72915 0.22915 0.60108
60 2.57153 0.07153 0.14338 2.67798 0.17798 0.38361
80 2.57017 0.07017 0.10260 2.62625 0.12625 0.24075
100 2.54426 0.04426 0.07461 2.59075 0.09075 0.16908
Table 7

MLE of parameters of inverted Kumaraswamy distribution based on order statistics.

(α,β) n α
 β
MLE Bias MSE MLE Bias MSE
(0.5, 0.5) 4 0.95799 0.45799 1.10777 0.89634 0.39634 1.97488
6 0.83490 0.33490 0.64561 0.76025 0.26025 1.10315
8 0.80013 0.30013 0.50635 0.75618 0.25618 0.65685
10 0.75162 0.25162 0.33120 0.67128 0.17128 0.14820
12 0.68433 0.18433 0.18827 0.63725 0.13725 0.12061
(0.5, 1) 4 0.88952 0.38952 0.63288 1.89668 0.89668 5.12380
6 0.78236 0.28236 0.36934 1.67542 0.67542 2.82767
8 0.71155 0.21155 0.20371 1.48849 0.48849 1.98405
10 0.65328 0.15328 0.13289 1.30778 0.30778 0.54395
12 0.61705 0.11705 0.08913 1.28827 0.28827 0.48011
(1, 0.5) 4 1.79209 0.79209 2.96595 0.91618 0.41618 2.25161
6 1.56005 0.56005 1.36735 0.71638 0.21638 0.27241
8 1.50964 0.50964 0.99735 0.67093 0.17093 0.18000
10 1.31369 0.31369 0.65002 0.61446 0.11446 0.08548
12 1.29930 0.29930 0.49603 0.60035 0.10035 0.06708
(1, 1) 4 1.58203 0.58203 1.45794 1.74879 0.74879 2.35062
6 1.40364 0.40364 0.74666 1.61671 0.61671 1.16099
8 1.32452 0.32452 0.53731 1.51269 0.51269 0.80420
10 1.22040 0.22039 0.27974 1.31340 0.31340 0.67306
12 1.19506 0.19506 0.25015 1.24222 0.24222 0.40805
(2.5, 2.5) 4 3.00554 0.50554 1.62537 3.30193 0.80193 2.54828
6 2.95440 0.45440 1.33555 3.01969 0.51969 1.72553
8 2.88874 0.38874 0.97995 2.93177 0.43177 1.20719
10 2.81722 0.31722 0.71950 2.85483 0.35483 0.88320
12 2.77335 0.27335 0.58612 2.81584 0.31584 0.69187
Table 8

MLE of parameters of inverted Kumaraswamy distribution based on lower records.

For another case, i.e., when one shape parameter (α) is known, the data is generated from G(n,1/β) and the MLE and UMVUE are obtained for β=0.5,1,2.5 for fixed value of α, n=20,40,…,100 for order statistics and n=4,6,…,12 for lower records. Based on 1000 repetition, the average MSE for MLE and UMVUE and average bias for MLE also are noted which are listed in Tables 9 and 10 for order statistics and lower records, respectively.

β n β^MLE Bias(MLE) MSE(MLE) β^UMVUE MSE(UMVUE)
(α=0.5)
0.5 20 0.52075 0.02075 0.01589 0.49471 0.01398
40 0.50993 0.00993 0.00664 0.49718 0.00623
60 0.50549 0.00549 0.00429 0.49706 0.00412
80 0.50359 0.00359 0.00318 0.49729 0.00310
100 0.50583 0.00583 0.00271 0.50077 0.00262
1 20 1.06023 0.06023 0.06709 1.00722 0.05733
40 1.03356 0.03356 0.02779 1.00772 0.02540
60 1.01423 0.01423 0.01693 0.99732 0.01619
80 1.01169 0.01169 0.01370 0.99904 0.01322
100 1.01347 0.01347 0.01038 1.00333 0.01000
2.5 20 2.59716 0.09716 0.36898 2.46730 0.32555
40 2.55618 0.05618 0.16035 2.49227 0.14949
60 2.53333 0.03333 0.11962 2.49111 0.11467
80 2.53456 0.03456 0.08193 2.50288 0.07874
100 2.53168 0.03168 0.07473 2.50636 0.07230
(α=1)
0.5 20 0.52378 0.02378 0.01557 0.49759 0.01355
40 0.51437 0.01437 0.00723 0.50151 0.00668
60 0.50897 0.00897 0.00452 0.50049 0.00429
80 0.50565 0.00565 0.00351 0.49933 0.00339
100 0.50331 0.00331 0.00243 0.49827 0.00237
1 20 1.05252 0.05252 0.06152 0.99990 0.05303
40 1.02846 0.02846 0.02938 1.00275 0.02716
60 1.01387 0.01387 0.01828 0.99697 0.01750
80 1.01335 0.01335 0.01518 1.00069 0.01351
100 1.00906 0.00906 0.00980 0.99897 0.00953
2.5 20 2.61420 0.11420 0.37656 2.48349 0.32835
40 2.56035 0.06035 0.17277 2.49634 0.16079
60 2.54435 0.04435 0.10973 2.50194 0.10420
80 2.52224 0.02224 0.09390 2.49071 0.09117
100 2.54406 0.04406 0.06967 2.51862 0.06672
(α=2.5)
0.5 20 0.53231 0.03231 0.01783 0.50570 0.01518
40 0.51169 0.01169 0.00699 0.49890 0.00652
60 0.50990 0.00990 0.00472 0.50140 0.00447
80 0.50448 0.00448 0.00336 0.49817 0.00326
100 0.50582 0.00582 0.00259 0.50076 0.00250
1 20 1.03577 0.03577 0.05644 0.98398 0.05004
40 1.01706 0.01706 0.02824 0.99164 0.02664
60 1.01700 0.01700 0.01728 1.00005 0.01643
80 1.01001 0.01001 0.01360 0.99738 0.01317
100 1.00903 0.00903 0.00981 0.99894 0.00953
2.5 20 2.63192 0.13192 0.36477 2.50033 0.31350
40 2.59603 0.09603 0.19529 2.53113 0.17785
60 2.53868 0.03868 0.12805 2.49637 0.11271
80 2.52289 0.02289 0.07952 2.49136 0.07365
100 2.52628 0.02628 0.06834 2.50102 0.06230
Table 9

MLE and UMVUE of β when α is known for inverted Kumaraswamy distribution based on order statistics.

β n β^MLE Bias(MLE) MSE(MLE) β^UMVUE MSE(UMVUE)
(α=0.5)
0.5 4 0.66557 0.16557 0.26088 0.49918 0.13132
6 0.59808 0.09808 0.09449 0.49840 0.05894
8 0.57595 0.07595 0.06511 0.50396 0.04545
10 0.55709 0.05709 0.03852 0.50138 0.02856
12 0.54336 0.04336 0.03096 0.49808 0.02444
1 4 1.30753 0.30753 1.31251 0.98065 0.68546
6 1.19347 0.19347 0.35580 0.99456 0.22112
8 1.11630 0.11630 0.24408 0.97677 0.17706
10 1.09644 0.09644 0.14771 0.98679 0.11228
12 1.09222 0.09222 0.12473 1.00120 0.09766
2.5 4 3.33735 0.83735 5.22589 2.50301 2.54518
6 3.04286 0.54286 2.41545 2.53572 1.47402
8 2.78516 0.28516 1.18280 2.43701 0.84729
10 2.75911 0.25911 0.98954 2.49220 0.74293
12 2.70940 0.20940 0.68496 2.48362 0.53898
(α=1)
0.5 4 0.66773 0.16773 0.22474 0.50080 0.11059
6 0.60390 0.10390 0.10984 0.50325 0.06879
8 0.56442 0.06442 0.05583 0.49386 0.03961
10 0.55428 0.05428 0.04078 0.49885 0.03065
12 0.55223 0.05223 0.03381 0.50621 0.02616
1 4 1.33507 0.33507 0.97265 1.00130 0.48396
6 1.21079 0.21079 0.39050 1.00899 0.24041
8 1.15320 0.15320 0.23482 1.00905 0.16190
10 1.11111 0.11111 0.16608 1.00000 0.12453
12 1.09891 0.09891 0.12318 1.00733 0.09534
2.5 4 3.08184 0.58184 3.86241 2.46138 1.89275
6 2.93978 0.43978 2.26822 2.44981 1.34336
8 2.90478 0.40478 1.46366 2.54169 0.99690
10 2.78098 0.28098 1.00504 2.50288 0.75014
12 2.73585 0.23585 0.84201 2.50786 0.66085
(α=2.5)
0.5 4 0.65761 0.15761 0.20665 0.49321 0.10231
6 0.60052 0.10052 0.09825 0.50043 0.06121
8 0.58375 0.08375 0.06552 0.51078 0.04491
10 0.54310 0.04310 0.03537 0.48879 0.02727
12 0.55120 0.05120 0.03410 0.50526 0.02648
1 4 1.33677 0.33677 0.99731 1.00257 0.49720
6 0.68309 0.18309 0.41968 0.51232 0.21737
8 0.57585 0.07585 0.06433 0.50387 0.04486
10 0.56139 0.06139 0.04769 0.50525 0.03560
12 0.54230 0.04230 0.02897 0.49710 0.02285
2.5 4 3.19019 0.69019 4.28578 2.54265 2.85432
6 2.96742 0.46742 2.41758 2.47285 1.52789
8 2.86216 0.36215 1.34216 2.50439 0.92719
10 2.80393 0.30393 0.98194 2.52354 0.75142
12 2.71570 0.21570 0.78274 2.48939 0.61874
Table 10

MLE and UMVUE of β when α is known for inverted Kumaraswamy distribution based on lower records.

  • From Tables 7 and 8, one can easily see that for various configuration of parameters, biases and MSEs are decreasing as the sample size is increasing for order statistics and lower record case. Also it is noted at (α,β)=(0.5,1), MSEs of α are smaller than MSEs of β and when (α,β)=(1,0.5), MSE (β) > MSE (α) for all choices of sample sizes. Moreover, it is noted that order statistics performed better compared to lower records based on the simulated result.

  • From Tables 9 and 10, it is evident that for all the parameter values and sample sizes, MSEUMVU<MSEML, i.e., UMVUE are more adequate than MLE for both order statistics and lower records, and also UMVUE free from bias. Based on the simulated result, we can conclude that UMVUE perform better than MLE when one shape parameter is known and in this situation, use of UMVUE is recommended.

6. CONCLUSION

In this paper, we have obtained exact and explicit expression for single, product, and conditional moments of dgos from inverted Kumaraswamy distribution. The results are obtained in terms of Gauss hypergeometric and beta functions. Further, by adjusting the parameter values of dgos, we obtained the moments of order statistics and lower record values. Based on these results, mean and variances of order statistics, lower record values, and dgos are calculated. We have also obtained MLE for all shape parameters and UMVUE for one shape parameter by taking another one is known as inverted Kumaraswamy distribution based on dgos. Based on the simulation result, we conclude that UMVUE perform better than MLE when one shape parameter is known and it is also noted that MSEs of MLE and UMVUE are decreasing as the sample size increases.

ACKNOWLEDGMENT

The authors would like to thanks the reviewers for reading the manuscript carefully, for detailed comments, and helpful suggestions which greatly helped to improve the manuscript.

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Journal
Journal of Statistical Theory and Applications
Volume-Issue
20 - 2
Pages
251 - 266
Publication Date
2021/03/03
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.210219.001How to use a DOI?
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risenwbib
TY  - JOUR
AU  - Bushra Khatoon
AU  - M. J. S. Khan
AU  - Zubdahe Noor
PY  - 2021
DA  - 2021/03/03
TI  - Moments of Dual Generalized Order Statistics from Inverted Kumaraswamy Distribution and Related Inference
JO  - Journal of Statistical Theory and Applications
SP  - 251
EP  - 266
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210219.001
DO  - 10.2991/jsta.d.210219.001
ID  - Khatoon2021
ER  -