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Volume 19, Issue 3, September 2020, Pages 352 - 367

Inferences for the Type-II Exponentiated Log-Logistic Distribution Based on Order Statistics with Application

Authors
Devendra Kumar1, *, ORCID, Maneesh Kumar1, Sanku Dey2
1Department of Statistics, Central University of Haryana, Jant, Haryana, India
2Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India
*Corresponding author. Email: devendrastats@gmail.com
Corresponding Author
Devendra Kumar
Received 11 May 2020, Accepted 12 August 2020, Available Online 11 September 2020.
DOI
10.2991/jsta.d.200825.002How to use a DOI?
Keywords
Type-II exponentiated log-logistic distribution; Moments; Order statistics; Best linear unbiased estimators
Abstract

In this paper, we first derive the exact explicit expressions for the single and product moments of order statistics from the type-II exponentiated log-logistic distribution, and then use these results to compute the means, variances, skewness and kurtosis of rth order statistics. Besides, best linear unbiased estimators (BLUEs) for the location and scale parameters for the type-II exponentiated log-logistic distribution with known shape parameters are studied. Finally, the results are illustrated with a real data set.

Copyright
© 2020 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

Rao et al. (2012) suggested a generalization of the log-logistic distribution called type-II exponentiated log-logistic (TIIELL) distribution with probability density function (pdf)

f(x)=τηxξη−1ξ1+xξητ+1,x>0,(τ,ξ)>0,η>1.(1)

The cumulative distribution function (cdf) and quantile function are, respectively given by

F(x)=1−1+xξη−τ,x>0,(τ,ξ)>0,η>1(2)
and
F−1(x)=ξ11−x1τ−11η,(3)
where ξ is the scale parameter and η and τ are the shape parameters of the distribution. If τ=1, then equation (1) becomes log-logistic distribution, and if η=1, then TIIELL distribution becomes Pareto type-II distribution. The kth moment of the TIIELL distribution in equation (1) can be easily derived as
E(Xk)=ξkτBτ−kη,1+kη,(4)
where B(.,.) is the beta function. Note that the kth moment exists if η>max{1,k∕τ}. A more compact form of equation (4) can be derived using the fact that Γ(z)Γ(1−z)=πcsc(πz) (Abramowitz and Stegun [1]) as follows:
E(Xk)=ξkτΓτ−knΓ1+knΓ(τ+1)=ξkττ−kn−1τ−kn−2…τ−kn−(τ−1)Γτ−kn−(τ−1)knΓknτΓ(τ)=ξkknΓknΓ1−knΓ(τ)∏i=1τ−1τ−kτ−i=ξkkηπcsckπηΓ(τ)∏i=1τ−1τ−kη−i.(5)

Therefore,

E(X)=ξπηcscπηΓ(τ)∏i=1τ−1τ−1η−i,
E(X2)=ξ22πηcsc2πηΓ(τ)∏i=1τ−1τ−2η−i
and
Var(X)=ξ22πηcsc2πηΓ(τ)∏i=1τ−1τ−2η−i−ξπηcscπηΓ(τ)∏i=1τ−1τ−1η−i2.

To the best of our knowledge, the work on type-II generalized log-logistic model is scanty in literature. The few works available in literature on this distribution which includes: Rao et al. [2,3] developed the reliability and an economic reliability test plans for this distribution. Kumar [4] studied exact moments of generalized order statistics from TIIELL distribution. Recently, Rao et al. [5] studied Bootstrap confidence intervals (CIs) of the process capability index, CNpk for type-II generalized log-logistic distribution.

Order statistics and functions of these statistics occupy a place of great significance in diverse field of studies involving theoretical and practical problems such as characterization of probability distributions, entropy estimation, analysis of censored samples, reliability analysis and quality control (see Arnold et al. [6] and David and Nagaraja [7]). The moments of order statistics have applicability in areas such as quality control, reliability, etc. For example, it is seen that when the duration of the failed items is high, the reliability of an item is also high, which in turn makes the product too costly, both in terms of time and money. In such a situation the one may not know enough about the item in a short period of time and hence would require few early failures data for predicting the failure of future items. Thus moments of order statistics is useful in making these kinds of prediction in such situations.

In recent past several authors have tabulated the moments of order statistics quite extensively for several distributions and also obtained maximum likelihood estimates (MLEs) and best linear unbiased estimators (BLUEs) for the scale and location parameters of the distributions based on complete and type-II censored samples. Further, they developed point prediction and goodness-of-fit tests. In this regard, readers may refer to the works of Balakrishnan and Cohan [8], Balakrishnan and Sultan [8], Sultan and Balakrishnan [9,10], Genç [11], Jabeen et al. [12], Mir Mostafaee [13], Balakrishnan et al. [14], Sultan and AL-Thubyani [15], Kumar et al. [16], Kumar and Dey [17,18], Ahsanullah and Alzaatreh [19], Kumar and Goyal [20,21], Kumar et al. [22] and many others.

In this paper, we derive the exact expressions for the single and product moments of order statistics from TIIELLD in Sections 2 and 3. In Section 4, we obtain BLUEs for θ and ξ by using these moments. These BLUEs are then used in Section 5 to obtain (1−α)100% CIs for the location and scale parameters of the BLUEs based on the pivotal quantities. Besides, lower and upper percentage points of pivotal quantities through Edgeworth approximations are obtained and compare the results with simulated percentage points. A real data application is provided in Section 6. Finally, in Section 7, we draw a conclusion for the paper.

2. SOME RELATIONS FOR THE MOMENTS OF ORDER STATISTICS

Let X1,X2,…,Xn be n independent copies of a random variable X that follows TIIELL distribution. Let X1:n≤X2:n≤…≤Xn:n be the corresponding order statistics, then the pdf of the rth order statistic is

fXr:n(x)=Cr:nFr−1(x)1−F(x)n−rf(x),x>0,(6)
where Cr:n=n!(r−1)!(n−r)!. From equations (1), (2) and (6), the pdf of rth order statistic from TIIELL distribution is given by
fXr:n(x)=τηξCr:n∑i=0r−1r−1i(−1)ixξη−1(1+xξη)τi+τ+1+τ(n−r),x≥0.(7)

The kth moment of Xr:n can be derived by the following Theorem:

Theorem 1.

For the TIIELL distribution given in equation (1) and for 1≤r≤n and k=1,2,…,

αr:n(k)=τξkCr:n∑i=0r−1r−1i(−1)iBτ(i+n−r+1)−kη,1+kη,(8)
where B(a,b) denotes the beta function defined by B(a,b)=∫0∞xa−1(1+x)a+bdx .

Proof.

From equation (7), we have

αr:n(k)=E(Xr:nk)=τηξCr:n∑i=0r−1(−1)ir−1i∫0∞xkxξη−1(1+xξη)τi+τ+1+τ(n−r)dx=τξkCr:n∑i=0r−1(−1)ir−1i∫0∞ukη(1+u)τ(n−r+i+1)+1du,
where u=xξη. The result follows from the definition of the beta function

Theorem 2.

For the TIIELL distribution given in equation (1) and for 1≤r≤n and k=1,2,…, the kth moment of rth order statistics can be expressed as

αr:n(k)=ξkkτn!πcsckπηη(r−1)!(n−r)!∑i=0r−1r−1i(−1)i∏j=1τ(i+n−r+1)−1τ(i+n−r+1)−j−kηΓτ(i+n−r+1)−1.(9)

Proof.

The proof is straightforward and omitted for brevity.

Note that from equation (9), the first and second moments of Xr:n are, respectively, given by

αr:n(1)=ξτn!πcscπηη(r−1)!(n−r)!∑i=0r−1r−1i(−1)i∏j=1τ(i+n−r+1)−1τ(i+n−r+1)−j−1ηΓτ(i+n−r+1)−1,
and
αr:n(2)=ξ2τn!πcsc2πηη(r−1)!(n−r)!∑i=0r−1r−1i(−1)i∏j=1τ(i+n−r+1)−1τ(i+n−r+1)−j−2ηΓτ(i+n−r+1)−1,

Some special cases from equation (9) are

  1. For τ=1, in equation (9), we get the explicit expression for order statistic of log-logistic distribution

    αr:n(k)=ξkkn!πcsckπηη(r−1)!(n−r)!(−1)r∏j=1nn−r+1−j−kηΓ(n−r+1)−1.

  2. If r=n=1, we get

    α1:1(k)=ξkτBτ−kη,1+kη
    which agrees with equation (4).

  3. If k=r=1 in equation (9), we get

    α1:n(1)=ξτnπcscπηη∏j=1τn−1τn−j−1ηΓτn−1,

  4. If k=1, r=n in equation (9), we get

    αn:n(1)=ξτnπcscπηη∑i=0n−1n−1i(−1)i∏j=1τ(i+1)−1τ(i+1)−j−1ηΓτ(i+1)−1,

  5. If k=r=n=1 in equation (9), we get

    α1:1(1)=ξτπcscπηη∏j=1τ−1τ−j−1ηΓτ−1,(10)
    which agree with equation (5) for k=1.

It is interesting to note that the equation (9) can be used easily to derive several recurrence relations for the moments of order statistics. Some of these recurrence relations already exist in the literature. Below, we provide some of these recurrence relations.

I. From equation (9), we can write

αr:n(k)=ξkτkn(n−1)!πcsckπηη(r−1)(r−2)!(n−r)!∑i=0r−1r−1i(−1)r−1+i∏j=1τ(i+n−r+1)−1τ(i+n−r+1)−j−kηΓτ(i+n−r+1)−1.

Let us define Δ(τ,η)=ξkτk(n−1)!πcsckπηη(r−2)!(n−r)!, we have

αr−1:n−1(k)=Δ(τ,η)∑i=0r−2r−2i(−1)i∏j=1τ(i+n−r+1)−1τ(i+n−r+1)−j−kηΓτ(i+n−r+1)−1=Δ(τ,η)r−20(−1)0∏j=1τ(n−r+1)−1τ(n−r+1)−j−kηΓτ(i+n−r+1)−1+Δ(τ,η)r−21(−1)1∏j=1τ(n−r+2)−1τ(n−r+2)−j−kηΓτ(n−r+2)−1⋮+Δ(τ,η)r−2r−2(−1)r−2∏j=1τ(n−1)−1τ(n−1)−j−kηΓτ(n−1)−1,
which can be written in vector form as
αr−1:n−1(k)=1′iαr−1:n−1(k),
where 1′=(1,1,…,1) and iαr−1:n−1(k) denotes a vector of order (1×r−2) and ((r−2)×1), respectively, where
iαr−1:n−1(k)=Δ(τ,η)r−20(−1)0∏j=1τ(n−r+1)−1τ(n−r+1)−j−kηΓτ(n−r+1)−1         ⋮         ⋮Δ(τ,η)r−2r−2(−1)r−2∏j=1τ(n−1)−1τ(n−1)−j−kηΓτ(n−1)−1.

Therefore, we can write αr:n(k) as

αr:n(k)=nr−1Δ(τ,η)r−1r−1(−1)r−1∏j=1τn−1τn−j−kηΓτn−1+∑i=0r−2r−1r−1−ir−2i(−1)i∏j=1τ(i+n−r+1)−1τ(i+n−r+1)−j−kηΓτ(i+n−r+1)−1=nr−1Δ(τ,η)(−1)r−1∏j=1τn−1τn−j−kηΓτn−1+nΔ(τ,η)∑i=0r−21(r−1−i)r−2i(−1)i∏j=1τ(i+n−r+1)−1τ(i+n−r+1)−j−kηΓτ(i+n−r+1)−1=nr−1Δ(τ,η)(−1)r−1∏j=1τn−1τn−j−kηΓτn−1+nv′iαr−1:n−1(k),
where v′=1r−1,1r−2,1r−3,…,1 is vector of order (1×(r−2)) II If r=1 in equation (9), we get
α1:n(k)=∏u=1τnu−kηη(n−1)∏h=1τ+1τ(i+n)−hα1:n−1(k).

3. PRODUCT MOMENTS OF ORDER STATISTICS

Let X1:n≤X2:n≤⋯≤Xn:n be the order statistics from the TIIELL distribution given in (1) with its cdf in equation (2). Then, the joint pdf of the rth and sth order statistics is

fX(r:n),X(s:n)(x,y)=Cr,s:nFr−1(x)F(y)−F(x)s−1−r1−F(y)n−sf(x)f(y),(11)
for 0<x<y, r,s=1,2,…, r<s and Cr,s:n=n!(r−1)!(s−r−1)!(n−s)!. To obtain the covariance between Xr:n and Xs:n, consider the joint pdf of Xr:n and Xs:n, 1≤r<s≤n as follows:
fr,s:n(x,y)=τ2η2ξ2Cr,s:n∑i=0r−1∑j=0s−r−1r−1is−r−1j(−1)i+j×xξη−1yξη−11+xξητ(i+s−r−j)+11+yξητ(n−s+1+j)+1.(12)

Therefore, the product moments of Xr:n and Xs:n, can be obtain from the following Theorem:

Theorem 3.

For the TIIELL distribution given in (1) and for 1≤r<s≤n,

αr,s:n=ξ2τ2Cr,s:n∑i=0r−1∑j=0s−r−1r−1is−r−1j(−1)i+jBτ(i+s−r−j)−1η,1η+1×B1η+1,τ(n−s+1+j)−1η−ψ(τ,η)×∑k=0∞B1η+1,τ(i+n−r+1)−2η+k+1,(13)
where B(a,b) denotes the beta function defined by B(a,b)=∫0∞xa−1(1+x)a+bdx and
ψ(τ,η)=τ(i+s−r−j)−1ηk1ηkτ(i+s−r−j)−1η+1kτ(i+s−r−j)−1η+1k!.

Proof

From equation (12), we have

αr,s:n=E(Xr:nXs:n)=τ2η2Cr,s:n∑i=0r−1∑j=0s−r−1r−1is−r−1j(−1)i+j×∫0∞∫0yxξηyξη1+xξητ(i+s−r−j)+11+yξητ(n−s+1+j)+1dxdy=ξτ2η2Cr,s:n∑i=0r−1∑j=0s−r−1r−1is−r−1j(−1)i+j×∫0∞yξη1+yξητ(n−s+1+j)+11η∫0yξηu1η(1+u)τ(i+s−r−j)+1dudy=ξτ2η2Cr,s:n∑i=0r−1∑j=0s−r−1r−1is−r−1j(−1)i+j×∫0∞yξη1+yξητ(n−s+1+j)+1δ(t)dy,(14)
where xη=u and t=1u+1, it is not difficult to show that δ(t) can be simplified
δ(t)=Bτ(i+s−r−j)−1η,1η+1−11+yξητ(i+s−r−j)+1−1ητ(i+s−r−j)+1−1η×2F1τ(i+s−r−j)−1η,1η,τ(i+s−r−j)−1η+1,1(1+yξη),(15)
where pFq is the generalized hypergeometric function defined as
pFq(a1,…,b1,…,bq;x)=∑k=0∞(a1)k…(ap)k(b1)k…(bq)kxkk!,
where (f)k=f(f+1)⋯(f+k−1) denotes the ascending factorial. The result follows by using equations (14) and (15). The proof is complete.

4. ESTIMATION OF PARAMETERS

4.1. Estimation of the Location and Scale Parameters

In this section, we study parameter estimation for the TIIELL distribution based on order statistics. Let Y1≤Y2≤⋯≤Ym be a random sample of size m from TIIELL distribution, the pdf of the scale-parameter given in (1) and the pdf of the location-scale parameter TIIELL distribution is

f(x)=τηx−θξη−1ξ1+x−θξητ+1,x>θ,(τ,ξ)>0,η>1.(16)

Let Yr+1:m≤Yr+2:m≤…≤Ym−s:m denote the type-II right-censored sample from the location-scale parameter TIIELL distribution in equation (1). Let us denote Xi:m=(Yi:m−θ)∕ξ, E(Xi:m)=θi;m(1), 1≤r≤(m−s), and Cov(Xi:m,Xj:m)=ξi,j:m=θi,j:m(1,1)−θi:m(1)θj:m(1), 1≤i<j≤(m−s). Therefore

Y=(Yr+1:m,Yr+2:m,…,Ym−s:m)T,
θ=(θr+1:m,θr+2:m,…,θm−s:m)T,
1=(1,1,…,1)T︸m−s,
and
∑=((ξi,j:m)),r+1≤i,j≤m−s,
where θi:m=E(Yi:m), ξii=Var(Yi:m) and ξij=Cov(Yi:m,Yj:m), i,j=1,2,…(m−s). Then the BLUEs of θ and ξ can be computed as follows [see Balakrishnan and Cohen [23], Sultan et al. [9,10]]
θ∗=∑i=r+1m−spiYi:mandξ∗=∑i=r+1m−sqiYi:m,
where
pi=θT∑−1θ1T∑−1−θT∑−11θT∑−1(θT∑−1θ)(1T∑−11)−(θT∑−11)2,(17)
qi=1T∑−11θT∑−1−1T∑−1θ1T∑−1(θT∑−1θ)(1T∑−11)−(θT∑−11)2(18)
and the variances and covariance of these BLUEs can be computed as follows [see Balakrishnan and Cohen [23], Sultan et al. [9,10]]
Var(θ∗)=ξ2θT∑−1θ(θT∑−1θ)(1T∑−11)−(θT∑−11)2=ξ2W1,(19)
Var(ξ∗)=ξ21T∑−11(θT∑−1θ)(1T∑−11)−(θT∑−11)2=ξ2W2,(20)
and
Cov(θ∗,ξ∗)=ξ2−θT∑−11(θT∑−1θ)(1T∑−11)−(θT∑−11)2=ξ2W3,(21)
for details readers may refer to the works of David [24], Balakrishnan and Cohen [23], and Arnold et al. [6]. The values of pi and qi are displayed in Tables 3 and 4 for different values of sample sizes m=7,10 and different censoring cases s=0(1)([m∕2]−1) and for some selected values for η=2,3. The coefficient of the BLUEs pi and qi given by equations (17) and (18), respectively satisfies the conditions
∑i=r+1m−spi=1
and
∑i=r+1m−sqi=0,
which are used to check the computations accuracy.

η m s pi
2 7 0 0.013963 0.101597 0.558978 0.263992 0.051795 0.009676 −5.27E-07
1 −0.34963 0.833878 0.401217 0.108408 0.006114 1.22E-05
2 −0.11492 0.812707 0.290103 0.012084 2.69E-05
10 0 0.026562 0.067891 0.131564 0.228105 0.328165 0.132287 0.060663 0.023448 0.001311 3.53E-06
1 0.039625 0.100778 0.188777 0.284606 0.265189 0.082232 0.036252 0.002535 4.15E-06
2 0.04799 0.127099 0.250121 0.374234 0.143044 0.053643 0.003868 1.68E-06
3 0.029534 0.131456 0.511993 0.235 0.087684 0.00432 1.37E-05
4 −0.25036 0.761078 0.334894 0.145439 0.008938 9.13E-06
3 7 0 −0.46467 −0.74389 1.449078 0.545318 0.204818 0.009332 1.17E-05
1 −0.29246 −0.60828 1.444277 0.453126 0.003324 1.28E-05
2 −1.58802 1.838166 0.728422 0.021388 4.29E-05
10 0 −0.01692 −0.03763 −0.06087 −0.04818 0.118261 0.764299 0.199978 0.078079 0.002988 1.05E-06
1 0.002353 0.007739 0.019326 0.051598 0.170786 0.611624 0.131185 0.005383 5.88E-06
2 −0.01192 −0.03172 −0.03954 0.060255 0.846969 0.166772 0.009176 7.58E-06
3 −0.06968 −0.16761 −0.15093 1.136136 0.240728 0.011335 2.05E-05
4 −0.29882 −0.61856 1.540259 0.349781 0.027336 8.17E-06
Table 3

Coefficient of the best linear unbiased estimators (BLUEs) of the pi for τ=1.5

η m s qi
2 7 0 −1.42624 −2.69332 2.763219 1.092811 0.224505 0.039025 −3.63E-06
1 −5.05483 3.111726 1.515761 0.403758 0.023545 4.36E-05
2 −4.45187 3.226687 1.177263 0.047808 0.000117
10 0 −0.40703 −0.92138 −1.2938 −0.64137 2.3615 0.5141 0.274523 0.107245 0.006194 1.47E-05
1 −0.28225 −0.6877  −1.08679 −0.841  2.287784 0.409765 0.185349 0.014828 1.58E-05
2 −0.56606 −1.313  −1.47408 2.431089 0.650346 0.253884 0.017819 7.43E-06
3 −1.34466 −2.59115 2.584963 0.96291 0.369273 0.018615 5.26E-05
4 −4.68322 2.837779 1.270177 0.540726 0.034506 3.06E-05
3 7 0 −2.17816 −3.85459 4.049251 1.422057 0.537002 0.024409 3.09E-05
1 −1.65382 −3.79265 4.181823 1.255093 0.009517 3.47E-05
2 −6.66135 4.733542 1.872912 0.054788 0.000108
10 0 −0.21816 −0.51608 −0.92486 −1.24654 −0.76193 2.846231 0.570023 0.242188 0.009133 2.23E-06
1 −0.12748 −0.32472 −0.62717 −1.01443 −0.95517 2.569284 0.460068 0.019596 1.77E-05
2 −0.25649 −0.69056 −1.2704 −1.44404 3.097968 0.533783 0.029706 2.53E-05
3 −0.6069  −1.57047 −2.26163 3.697287 0.70758 0.03408 5.81E-05
4 −1.6753  −3.83228 4.469613 0.962392 0.075552 2.27E-05
Table 4

Coefficient of the best linear unbiased estimators (BLUEs) of the qi for τ=1.5

5. APPROXIMATE INFERENCE

Here, we derive the (1−α)100%confidence intervalCIs for the location and scale parameters of the BLUEs θ∗ and ξ∗ based on the pivotal quantities

U1=θ∗−θξW1,U2=ξ∗−ξξW2,U3=θ∗−θξ∗W1,(22)
where θ∗ and ξ∗ are the BLUEs of θ and ξ with variances ξ2W1 and ξ2W2, respectively. U1 is used to draw inference for θ when ξ is known, while U3 can be used to draw inference for θ when ξ is unknown. Similarly, U2 can be used to draw inference for θ when ξ is unknown.

To derive the CIs of the location and scale parameters based on the pivotal quantities in equation (22), the moments presented in Section 2, are used.

Hence, U1 and U2 can be rewritten as

U1=1W1∑i=r+1m−spiXi:m=U1∗W1,U2=1W2∑i=r+1m−sqiXi:m−1=U2∗−1W2,(23)
where Xi:m=(Yi:m−θ)∕ξ, i=1,2,…,m−s, is the standardized form of the available type-II right-censored sample Yi:m, i=1,2,…,m−s. Now, we consider to find the approximate distribution by using Edgeworth approximation for a statistic S (with mean 0 and variance 1) as
H(s)≈Φ(s)−ϕ(s)τ16(s2−1)+τ2−324(s3−3s)+τ172(s5−10s2+15s),(24)
where τ1 and τ2 are the coefficients of skewness and kurtosis of S, respectively and Φ(s), ϕ(s) are the cdf and pdf of the standard normal distribution, respectively.

To obtain the coefficients of skewness and kurtosis of linear functions of order statistics, single moments E(Xr:nk1) denoted by αr:n(k1), the double moments E(Xr:nk1Xs:nk2), denoted by αr,s:n(k1,k2) of the TIIELL distribution for 1≤r<s≤(m−s) are required.

Table 3 displays the values of the mean, variance, coefficients of skewness and kurtosis (τ1 and τ2) of U1∗ and U2∗. From Table 3 it is observed that the distributions of U1∗ and U2∗ and hence of U1 and U2 are positively skewed and heavier tailed than normal. Also, we can see that τ1 of U1∗ and U2∗ increases as η increases and decreases as m increases and decreases as s increases. τ2 of U1∗ decreases as m increases and increases as η increases and decreases as s increases, while τ2 of U2∗ increases as m and η increases and decreases as s increases. We also obtained the lower and upper 1%,2.5%,5%,and10% points of U1 and U2 through Edgeworth approximation (see Tables 4 and 5). From Tables 4 and 5, we can observe that the percentage points of U1 and U2 increases as η increases for m=7 and decreases for m=10 in most of the cases and increases as n increases in most of the cases for η=2, while decreases as m increases for η=3 and decreases as s increases. Similarly the percentage points of U3 increases as n increases.

η n c Var(θ*) Var(ξ*) Cov(θ*,ξ*)
2 7 0 0.074821 0.970035 0.253528
1 0.093972 1.102865 0.310651
2 0.125613 1.597531 0.426752
10 0 0.044879 0.616469 0.152306
1 0.052941 0.848929 0.185043
2 0.060478 0.878651 0.209287
3 0.073563 0.955970 0.248770
4 0.087637 1.020583 0.287920
3 7 0 0.153237 0.952101 0.377268
1 0.214149 1.459957 0.549955
2 0.248820 1.525382 0.610006
10 0 0.106367 0.810697 0.284275
1 0.132891 1.206524 0.379329
2 0.150195 1.238222 0.414675
3 0.182625 1.375398 0.488490
4 0.211345 1.424032 0.539068
Table 5

Variances and covariance of the best linear unbiased estimators (BLUEs) when τ=1.5 θ=0 and ξ=1

The performance of the developed inference can be shown from the simulated average width of CIs in Table 10. We observe that the Edgeworth approximations of the distributions of U1 and U2 both work quite satisfactory; this is also clear from the average width of the CIs based on U1 and U2 which are presented in Table 6. In addition we can see that average width decreases as η increases for most of the cases.

η n c 1% 2.50% 5% 10% 90% 95% 97.50% 99%
2 7 0 −0.7784 −0.76196 −0.73504 −0.68271 1.083601 1.925702 1.854568 4.255494
−1.03231* −0.97438* −0.90792* −0.81937* 1.083648* 1.76806* 2.560536* 3.7586*
1 −2.27628 −2.32369 −0.67909 −0.62752 0.20892 1.889575 1.822893 4.241948
−1.4765* −1.41945* −1.35722* −1.26558* 0.682143* 1.351194* 2.104918* 3.327379*
2 −2.34755 −2.41664 −2.56902 −2.86364 0.324397 0.432937 1.791339 3.749634
−2.0666* −2.01487* −1.95338* −1.8637* 0.126588* 0.818807* 1.556711* 2.749675*
10 0 −0.63703 −0.62091 −0.59465 −0.54415 1.968333 1.825218 3.499767 4.188481
−1.19869* −1.11596* −1.03013* −0.90966* 1.176293* 1.81677* 2.499808* 3.605541*
1 −2.5634 −1.08063 −1.02533 −0.92635 0.773829 1.106087 1.429446 3.896156
−1.68973* −1.60117* −1.51202* −1.38262* 0.766383* 1.428123* 2.120552* 3.137602*
2 −2.51757 −2.77431 −2.97307 −2.89647 0.704014 0.993738 1.354706 3.585678
−2.19815* −2.10884* −2.02115* −1.89652* 0.271168* 0.924273* 1.627683* 2.585719*
3 −2.53477 −3.16427 −2.9827 −2.98269 0.732411 1.042713 1.382714 1.382703
−2.72401* −2.63791* −2.55189* −2.42756* −0.25733* 0.390673* 1.068124* 2.10397*
4 −3.83866 −3.30443 −2.96218 −2.96219 0.181199 0.835414 1.163553 1.314588
−3.25567* −3.17872* −3.09859* −2.98289* −0.81876* −0.16451* 0.504467* 1.485206*
3 7 0 −0.38025 −0.37311 −0.36123 −0.33769 1.936298 1.888275 1.865233 4.491297
−0.9566* −0.90678* −0.8515* −0.77025* 1.032663* 1.716643* 2.505057* 3.743581*
1 −2.28575 −2.32319 −0.66079 −0.61545 0.161453 1.927139 1.87266 4.263209
−1.45259* −1.40138* −1.34743* −1.268* 0.586968* 1.278905* 2.080461* 3.263277*
2 −2.30645 −2.35543 −2.45118 −2.81325 0.226085 0.306988 1.837418 3.764807
−1.991* −1.94309* −1.89224* −1.81329* 0.111492* 0.792065* 1.60848* 2.764848*
10 0 −0.83021 −0.81163 −0.78146 −0.72326 1.296581 1.916373 1.828839 4.202645
−1.0873* −1.0183* −0.94733* −0.84676* 1.096046* 1.772315* 2.542991* 3.727137*
1 −2.18033 −2.21922 −2.29146 −2.39668 0.156439 1.849359 1.797111 4.137826
−1.65246* −1.57954* −1.5031* −1.39675* 0.607943* 1.262664* 2.027025* 3.137897*
2 −2.41444 −2.52619 −2.92543 −2.92543 0.480104 0.644207 1.703217 3.567014
−2.28619* −2.21093* −2.12713* −2.01619* 0.051319* 0.72511* 1.48204* 2.567054*
3 −2.44641 −2.58122 −2.94305 −2.94304 0.424358 0.740532 0.873089 1.518757
−2.94442* −2.86884* −2.78695* −2.67421* −0.57559* 0.095727* 0.809759* 1.841936*
4 −3.94473 −3.48833 −2.95145 −2.95145 −0.14498 0.510946 0.891869 1.527782
−3.48544* −3.41859* −3.34497* −3.23734* −1.14494* −0.48899* 0.243865* 1.30883*
Table 6

Edgeworth approximate and the simulated (*) values of the distribution of U1 when τ=1.5, θ=0 and ξ=1

η n c 1% 2.50% 5% 10% 90% 95% 97.50% 99%
2 7 0 −0.68625 −0.67387 −0.65342 −0.613 1.120145 1.963351 1.917835 4.386462
−0.89515* −0.85571* −0.81026* −0.74149* 1.01628* 1.700985* 2.516495* 3.864885*
1 −2.23387 −2.26766 −0.5896 −0.54756 0.133227 1.900113 1.852216 4.326887
−1.37336* −1.33204* −1.28475* −1.2124* 0.661287* 1.334092* 2.137424* 3.36625*
2 −2.28594 −2.32889 −2.41048 −2.68876 0.190544 0.261264 1.847944 3.857855
−1.90759* −1.87523* −1.83573* −1.7718* 0.101144* 0.809872* 1.64006* 2.857924*
10 0 −0.43146 −0.42262 −0.408 −0.37914 1.928908 1.865193 1.83512 4.403044
−1.03058* −0.97355* −0.9115* −0.821* 1.084735* 1.761624* 2.52332* 3.710513*
1 −2.39482 −0.86625 −0.83388 −0.77177 0.366606 1.928228 1.824014 4.173098
−1.50745* −1.44968* −1.38733* −1.29935* 0.641782* 1.332129* 2.118734* 3.318632*
2 −2.25389 −2.31073 −2.42521 −2.8198 0.242052 0.315944 1.774996 3.702989
−2.09084* −2.0311* −1.96779* −1.87474* 0.139231* 0.812459* 1.597538* 2.70303*
3 −2.39193 −2.48068 −2.72223 −2.91628 0.413258 0.553318 0.636243 1.67989
−2.62983* −2.57057* −2.5047* −2.4097* −0.3817* 0.297366* 1.054344* 2.16912*
4 −3.95033 −3.49826 −2.94629 −2.94627 0.159258 0.734727 0.863918 0.968469
−3.14328* −3.08945* −3.02804* −2.93252* −0.8407* −0.15407* 0.56838* 1.601999*
3 7 0 −0.37383 −0.36704 −0.35582 −0.33353 1.947225 1.90334 1.882234 4.526831
−0.89584* −0.8538* −0.80822* −0.73897* 1.00372* 1.702315* 2.507245* 3.772574*
1 −2.25553 −0.60699 −0.58794 −0.55025 0.094725 1.941317 1.901147 4.364091
−1.33671* −1.30106* −1.26096* −1.19843* 0.562082* 1.280316* 2.118576* 3.364137*
2 −2.28995 −2.3318 −2.41096 −2.66945 0.186123 0.257073 1.855683 3.870927
−1.88758* −1.85427* −1.81481* −1.74995* 0.119499* 0.83215* 1.645907* 2.871001*
10 0 −0.72809 −0.71409 −0.69103 −0.64574 1.167925 1.949229 1.893921 4.330301
−0.95512* −0.90828* −0.85568* −0.77823* 1.042522* 1.747703* 2.533224* 3.81236*
1 −2.12219 −2.13917 −2.16865 −2.23302 0.037221 1.888342 1.864152 4.247133
−1.4378* −1.395* −1.34838* −1.27871* 0.511022* 1.219516* 2.01116* 3.247213*
2 −2.33214 −2.38482 −2.48992 −2.88514 0.254338 0.34656 1.841463 3.647901
−2.11302* −2.06494* −2.0116* −1.93031* −0.02184* 0.681566* 1.499523* 2.647942*
3 −2.4204 −2.50059 −3.24692 −2.94216 0.379866 0.576944 0.665016 1.735106
−2.7661* −2.71757* −2.66356* −2.5804* −0.62009* 0.089863* 0.870971* 2.015355*
4 −4.07898 −3.68032 −2.72291 −2.94198 −0.13232 0.561205 0.679897 1.720768
−3.29861* −3.25072* −3.19745* −3.115* −1.13227* −0.43873* 0.318227* 1.532725*
Table 7

Edgeworth approximate and the simulated (*) values of the distribution of U2 when τ=1.5, θ=0 and ξ=1

Tables 8 represent the simulated percentage points at τ=0.5 and sample sizes n = 7, 10. Table 9 displays the values of the mean, variance and the coefficients of skewness and kurtosis and Tables 10 represent the Average width of the Edgeworth and simulated class intervals.

η n c 1% 2.50% 5% 10% 90% 95% 97.50% 99%
2 7 0 −8.58565 −6.00463 −4.36124 −2.92522 0.577351 0.740279 0.891971 1.127982
1 −22.0015 −14.7982 −10.3426 −7.00606 0.490966 0.754123 0.961019 1.207395
2 −64.3401 −40.636 −27.6904 −17.7657 0.128992 0.633082 1.003714 1.452917
10 0 −6.71847 −4.88497 −3.67234 −2.5574 0.671541 0.86046 1.024841 1.245431
1 −14.416 −10.4421 −7.84722 −5.55788 0.624766 0.977244 1.3142 1.778569
2 −25.1951 −18.3949 −14.0381 −10.1702 0.274881 0.778888 1.17583 1.693537
3 −43.5958 −31.043 −23.3486 −16.8049 −0.31343 0.390987 0.902069 1.484084
4 −80.3106 −55.2902 −40.0114 −27.942 −1.15339 −0.18127 0.463265 1.118295
3 7 0 −8.2837 −5.77654 −4.18322 −2.83497 0.526657 0.65697 0.744066 0.828474
1 −25.6916 −17.0079 −12.0546 −8.10941 0.431894 0.703867 0.882413 1.061739
2 −68.5725 −41.7708 −28.1831 −17.9228 0.104877 0.574845 0.890182 1.167284
10 0 −7.14072 −5.19687 −3.88199 −2.69139 0.589233 0.735267 0.842694 0.964767
1 −18.5664 −13.4642 −10.1707 −7.19092 0.482042 0.788716 1.012966 1.290519
2 −34.5303 −25.0795 −19.0127 −13.4866 0.055358 0.591351 0.969128 1.348443
3 −60.0887 −43.3305 −32.6716 −23.452 −0.75955 0.100226 0.669017 1.197819
4 −104.838 −72.0204 −52.8649 −36.8763 −1.75691 −0.58387 0.226488 0.958274
Table 8

Simulated values of the distribution of U3 when τ=1.5, θ=0 and ξ=1

U1
 U2
η n c Mean V1 τ1 τ2 Mean V1 τ1 τ2
2 7 0 0.000963 0.074656 1.90405 32.19869 1.005801 0.982068 2.114422 46.51046
1 −0.08758 0.04176 1.864872 32.9059 0.714386 0.465816 1.986304 42.78576
2 −0.15359 0.023106 1.763638 25.11762 0.498366 0.258819 1.922601 35.45901
10 0 0.000786 0.046427 1.746194 31.8938 1.001825 0.652555 2.067376 60.53579
1 −0.07138 0.026898 1.489195 12.70246 0.722236 0.351057 1.801017 24.7817
2 −0.12518 0.017567 1.485461 13.29467 0.554844 0.192955 1.758955 28.75834
3 −0.16467 0.012416 1.478552 12.92351 0.448749 0.127834 1.69991 21.34141
4 −0.1964 0.009247 1.512846 14.19956 0.37472 0.093584 1.610215 17.27234
3 7 0 0.002024 0.157938 2.234621 78.61409 1.004534 0.982242 2.312447 86.37732
1 −0.1323 0.077857 1.9914 39.44096 0.67646 0.499386 2.126234 50.97791
2 −0.21177 0.044925 1.879843 32.08649 0.503454 0.265125 1.939696 36.16978
10 0 −0.0004 0.105689 1.831395 27.65306 0.999542 0.796568 2.021134 39.482
1 −0.11958 0.052235 1.851664 37.89021 0.669847 0.405057 2.214512 75.30156
2 −0.1925 0.029811 1.632681 18.67683 0.502932 0.196356 1.870876 30.37217
3 −0.24453 0.019317 1.595066 16.9323 0.398287 0.118517 1.764992 22.59934
4 −0.28135 0.014526 1.606979 17.01038 0.331043 0.086028 1.748858 22.01017
Table 9

Mean, Variance and coefficients of skewness and kurtosis of U1∗ and U2∗ when τ=1.5, θ=0 and ξ=1

U1
 U2
 U3
n c 90% 95% 90% 95% 90% 95%
7 0 2.660745 2.67598* 2.616533 3.534921* 2.616774 2.511241* 2.59171 3.372206* 5.101516* 6.896599*
1 2.568667 2.70841* 4.14658 3.524364* 2.489716 2.618837* 4.119881 3.469468* 11.0967* 15.75922*
2 3.001957 2.772187* 4.207975 3.571577* 2.671747 2.645604* 4.176834 3.515287* 28.32346* 41.63974*
10 0 2.419866 2.846901* 4.12068 3.61577* 2.273196 2.673121* 2.257739 3.496869* 4.532798* 5.90981*
1 2.131419 2.940145* 2.510081 3.721726* 2.762113 2.719455* 2.690262 3.568417* 8.824464* 11.75634*
2 3.966812 2.945421* 4.129015 3.736519* 2.741151 2.780245* 4.085728 3.628634* 14.81699* 19.57076*
3 4.025414 2.942563* 4.546983 3.706037* 3.27555 2.802063* 3.116921 3.62491* 23.73962* 31.9451*
4 3.797594 2.934083* 4.467987 3.683186* 3.681019 2.873962* 4.36218 3.657831* 39.83013* 55.7535*
7 0 2.249506 2.568142* 2.238341 3.411832* 2.259165 2.51053* 2.249274 3.361044* 4.840185* 6.520606*
1 2.587927 2.626336* 4.195851 3.481836* 2.529257 2.541276* 2.508136 3.419633* 12.75846* 17.89027*
2 2.758169 2.684303* 4.192849 3.551572* 2.668035 2.646963* 4.187487 3.500173* 28.75793* 42.66099*
10 0 2.697834 2.719643* 2.640473 3.56129* 2.640264 2.603388* 2.608009 3.441503* 4.617257* 6.03956*
1 4.140821 2.765767* 4.016333 3.606564* 4.056995 2.567898* 4.003325 3.40616* 10.95945* 14.47721*
2 3.569634 2.852243* 4.22941 3.692967* 2.836481 2.693163* 4.22628 3.564463* 19.60404* 26.04864*
3 3.683578 2.882681* 3.454311 3.678602* 3.823864 2.753427* 3.165611 3.588544* 32.77182* 43.99951*
4 3.462393 2.855982* 4.380202 3.662456* 3.284114 2.75872* 4.360213 3.568945* 52.28107* 72.2469*
Table 10

Average width of the Edgeworth and simulated(*) C.I.’s.

6. DATA ANALYSIS

To demonstrate how the proposed methods can be used in practice, we consider the following real-life data set (see Bhaumik et al. [25]). The data set represents vinyl chloride from clean upgradient monitoring wells in mg/L. The data are:

5.1 1.2 1.3 0.6 0.5 2.4 0.5 1.1 8.0 0.8 0.4 0.6 0.9 0.4 2.0 0.5 5.3 3.2 2.7 2.9 2.5 2.3 1.0 0.2 0.1 0.1 1.8 0.9 2.0 4.0 6.8 1.2 0.4 0.2

Now a random sample of size 10 is selected from the given data set and data are 0.1, 1.1, 0.9, 2.3, 1.3, 2.5, 0.4, 2.0, 0.5, 3.2. Figure 1 shows Q-Q plot of the sample. The Kolmogorov–Smirnov (K–S) statistic is 0.17491 and the corresponding p-value is 0.8697. This shows the suitability of the TIIELL distribution for this data set.

Figure 1

Expected Cumulative Distribution Function (ECDF) and Quintile-Quintile (Q-Q) plot of the real sample based on type-II exponentiated log-logistic (TIIELL) distribution.

Then by using the BLUEs coefficients in Tables 1 and 2, we have

θ∗=∑r=1nprXr:n=1.012389andξ∗=∑r=1nqrXr:n=2.443834.
r n E(X) E(X2) V(X) τ1 τ2 γ1 γ2
1 1 0.166667 0.041667 0.013889 6.124244 24.00119 2.474721 21.00119
2 0.107379 0.015625 0.004095 1.480527 5.834856 1.216769 2.834856
3 0.085248 0.009615 0.002348 0.970243 4.645654 0.985009 1.645654
4 0.072834 0.006944 0.001639 0.775930 4.337018 0.880869 1.337018
5 0.064627 0.005435 0.001258 0.690985 4.060399 0.831255 1.060399
6 0.058687 0.004464 0.001020 0.653052 3.734868 0.808116 0.734868
7 0.054132 0.003788 0.000858 0.577341 4.350134 0.759829 1.350134
8 0.050495 0.003289 0.000739 0.576750 3.853975 0.759440 0.853975
9 0.047505 0.002907 0.000650 0.534243 4.254706 0.730919 1.254706
10 0.044990 0.002604 0.000580 0.479173 3.332768 0.692223 0.332768
2 2 0.225955 0.067708 0.016652 6.669762 26.89585 2.582588 23.89585
3 0.151640 0.027644 0.004649 1.311287 5.810189 1.145115 2.810189
4 0.122490 0.017628 0.002624 0.742933 4.483691 0.861936 1.483691
5 0.105661 0.012983 0.001819 0.537407 4.080492 0.733081 1.080492
6 0.094328 0.010287 0.001389 0.445519 3.949799 0.667472 0.949799
7 0.086020 0.008523 0.001124 0.391159 3.375693 0.625427 0.375693
8 0.079588 0.007277 0.000943 0.335838 3.046048 0.579516 0.046048
9 0.074417 0.006349 0.000811 0.307304 3.904693 0.554350 0.904693
10 0.070140 0.005632 0.000712 0.278758 3.848412 0.527975 0.848412
3 3 0.263112 0.087740 0.018512 7.184192 28.97034 2.680334 25.97034
4 0.180790 0.037660 0.004975 1.322886 5.987013 1.150168 2.987013
5 0.147734 0.024596 0.002771 0.707746 4.471739 0.841277 1.471739
6 0.128326 0.018375 0.001907 0.492489 3.942891 0.701776 0.942891
7 0.115099 0.014699 0.001451 0.380277 4.011283 0.616666 1.011283
8 0.105315 0.012261 0.001170 0.312036 3.453741 0.558602 0.453741
9 0.097688 0.010522 0.000979 0.302289 3.434712 0.549808 0.434712
10 0.091521 0.009218 0.000842 0.251679 2.882750 0.501676 0.399884
4 4 0.290553 0.104434 0.020013 7.578064 30.47057 2.752828 27.47057
5 0.202827 0.046370 0.005231 1.363788 6.098496 1.167813 3.098496
6 0.167141 0.030817 0.002881 0.708859 4.536456 0.841938 1.536456
7 0.145963 0.023276 0.001971 0.489885 3.966303 0.699918 0.966303
8 0.131407 0.018762 0.001494 0.363963 3.895946 0.603293 0.895946
9 0.120569 0.015738 0.001201 0.301321 3.393636 0.548927 0.393636
10 0.112075 0.013565 0.001004 0.272530 3.072031 0.522044 0.072031
5 5 0.312484 0.118950 0.021304 7.887611 31.61464 2.808489 28.61464
6 0.220670 0.054146 0.005451 1.404782 6.200086 1.185235 3.200086
7 0.183024 0.036473 0.002975 0.717814 4.548271 0.847239 1.548271
8 0.160518 0.027791 0.002025 0.477320 4.057296 0.690884 1.057296
9 0.144955 0.022541 0.001529 0.368251 3.717102 0.606837 0.717102
10 0.133308 0.018998 0.001227 0.293470 3.286510 0.541729 0.286501
6 6 0.330847 0.131910 0.022450 8.139275 32.53265 2.852941 29.53265
7 0.235729 0.061216 0.005648 1.443160 6.280985 1.201316 3.280985
8 0.196528 0.041682 0.003059 0.735319 4.529456 0.857508 1.529456
9 0.172968 0.031990 0.002072 0.488891 4.037825 0.699208 1.037825
10 0.156602 0.026085 0.001561 0.363625 3.532627 0.603013 0.532627
7 7 0.346701 0.143693 0.023492 8.346127 33.28143 2.888966 30.28143
8 0.248795 0.067727 0.005828 1.477872 6.381171 1.215677 3.381171
9 0.208308 0.046527 0.003135 0.750557 4.606563 0.866347 1.606563
10 0.183879 0.035927 0.002116 0.499695 4.026693 0.706891 1.026693
8 8 0.360687 0.154545 0.024450 8.522843 33.91907 2.919391 30.91907
9 0.260363 0.073784 0.005995 1.505565 6.457144 1.227015 3.457144
10 0.218777 0.051071 0.003207 0.757008 4.664468 0.870062 1.664468
9 9 0.373227 0.164640 0.025342 8.674506 34.46545 2.945251 31.46545
10 0.270760 0.079463 0.006152 1.531512 6.518105 1.237543 3.518105
10 10 0.384613 0.174104 0.026177 8.808048 34.94389 2.967835 31.94389
Table 1

Expected values, second moments, variances, skewness and kurtosis of the rth order statistic from type-II exponentiated log-logistic (TIIELL) distribution for n=1,2,…,10, τ=2.5, η=2 and ξ=0.25

r n E(X) E(X2) V(X) τ1 τ2 γ1 γ2
1 1 0.214757 0.062501 0.016379 1.482629 5.832059 1.217633 2.832059
2 0.145668 0.027778 0.006559 0.782284 4.113691 0.884468 1.113691
3 0.117375 0.017857 0.004080 0.626158 3.790735 0.791301 0.790735
4 0.100991 0.013158 0.002959 0.560859 3.566212 0.748905 0.566212
5 0.089980 0.010417 0.002321 0.525832 3.442966 0.725143 0.442966
6 0.081930 0.008621 0.001908 0.500077 3.502186 0.707161 0.502186
7 0.075714 0.007353 0.001620 0.495238 3.237636 0.703732 0.237636
8 0.070728 0.006410 0.001408 0.478453 3.227514 0.691703 0.227514
9 0.066612 0.005682 0.001245 0.456279 3.397797 0.675484 0.397797
10 0.063141 0.005102 0.001115 0.451449 3.303742 0.671901 0.303742
2 2 0.283846 0.097222 0.016653 1.400052 6.014629 1.183238 3.014629
3 0.202256 0.047619 0.006712 0.563894 4.001324 0.750929 1.001324
4 0.166526 0.031955 0.004224 0.391693 3.627183 0.625854 0.627183
5 0.145032 0.024123 0.003089 0.322388 3.480211 0.567792 0.480211
6 0.130231 0.019397 0.002437 0.284753 3.357070 0.533623 0.357070
7 0.119225 0.016227 0.002012 0.265080 3.246232 0.514859 0.246232
8 0.110621 0.013952 0.001715 0.245458 3.320959 0.495437 0.320959
9 0.103652 0.012238 0.001494 0.230641 3.469610 0.480251 0.469610
10 0.097858 0.010901 0.001324 0.233139 2.927740 0.482845 0.218501
3 3 0.324642 0.122024 0.016632 1.466340 6.256670 1.210925 3.256670
4 0.237985 0.063283 0.006646 0.526732 4.039598 0.725763 1.039598
5 0.198768 0.043703 0.004194 0.342041 3.552107 0.584842 0.552107
6 0.174635 0.033575 0.003078 0.263913 3.429646 0.513725 0.429646
7 0.157746 0.027320 0.002436 0.222956 3.410097 0.472182 0.410097
8 0.145037 0.023054 0.002018 0.193353 3.346856 0.439719 0.346856
9 0.135010 0.019951 0.001723 0.176914 3.377494 0.420612 0.377494
10 0.126829 0.017590 0.001504 0.175914 2.934875 0.419421 0.192158
4 4 0.353527 0.141604 0.016623 1.540551 6.459855 1.241189 3.459855
5 0.264129 0.076337 0.006573 0.522017 4.094437 0.722507 1.094437
6 0.222901 0.053831 0.004146 0.322694 3.617912 0.568062 0.617912
7 0.197152 0.041916 0.003047 0.241729 3.378629 0.491659 0.378629
8 0.178928 0.034431 0.002416 0.197982 3.283768 0.444952 0.283768
9 0.165092 0.029259 0.002004 0.179578 3.085929 0.423766 0.085929
10 0.154098 0.025460 0.001714 0.161099 3.196057 0.401371 0.196057
5 5 0.375877 0.157921 0.016637 1.608648 6.618557 1.268325 3.618557
6 0.284744 0.087590 0.006511 0.529866 4.110246 0.727919 1.110246
7 0.242213 0.062767 0.004101 0.317573 3.609708 0.563536 0.609708
8 0.215376 0.049401 0.003014 0.232813 3.364292 0.482507 0.364292
9 0.196223 0.040895 0.002392 0.189503 3.326950 0.435320 0.326950
10 0.181583 0.034958 0.001986 0.166596 3.238532 0.408162 0.238532
6 6 0.394104 0.171987 0.016669 1.668102 6.757334 1.291550 3.757334
7 0.301756 0.097519 0.006462 0.540625 4.142795 0.735272 1.142795
8 0.258315 0.070787 0.004060 0.313873 3.650085 0.560244 0.650085
9 0.230698 0.056205 0.002983 0.229549 3.451351 0.479112 0.451351
10 0.210864 0.046832 0.002368 0.180792 3.353638 0.425196 0.353638
7 7 0.409495 0.184398 0.016712 1.719701 6.877223 1.311373 3.877223
8 0.316236 0.106430 0.006425 0.551682 4.167832 0.742753 1.167832
9 0.272123 0.078077 0.004026 0.318976 3.677382 0.564780 0.677382
10 0.243921 0.062454 0.002957 0.228410 3.391861 0.477923 0.391861
8 8 0.422818 0.195537 0.016762 1.764809 6.979383 1.328461 3.979383
9 0.328840 0.114530 0.006394 0.562789 4.217233 0.750193 1.217233
10 0.284209 0.084773 0.003998 0.320279 3.693316 0.565932 0.693316
9 9 0.434565 0.205663 0.016816 1.805957 7.064301 1.343859 4.064301
10 0.339998 0.121970 0.006371 0.572928 4.244680 0.756920 1.244680
10 10 0.445072 0.214962 0.016873 1.842618 7.141398 1.357431 4.141398
Table 2

Expected values, second moments, variances, skewness and kurtosis of the rth order statistic from type-II exponentiated log-logistic (TIIELL) distribution for n=1,2,…,10, τ=5, η=2 and ξ=0.5

90% CI 95% CI
Edgeworth (0.6257, 1.13836) (0.27096, 1.14393)
Simulated (0.6275, 1.23062) (0.48281, 1.24881)

By using Table 8, the CIs for the location parameter θ as

90% CI 95% CI
Edgeworth (0.99163, 3.59568) (1.00122, 3.65746)
Simulated (1.02546, 3.59603) (0.91974, 3.67228)

By using Table 10, the CIs for the scale parameter ξ as

90% CI 95% CI
Simulated (0.56691, 2.91362) (0.48181, 3.54143)

By using Table 10, the CIs for the location parameter θ when ξ is unknown as

We note that the average width of the CIs increase as the level of significant increases.

7. CONCLUSION

In this paper, the moments and product moments of the order statistics from the TIIELL distribution are derived in explicit forms. The single and double moments are used to obtain the BLUEs of the location and scale parameters of TIIELL distribution. The variances and covariances are calculated to show the performance of the BLUEs. Next, we calculate mean, variance, coefficient of skewness and kurtosis for some linear pivotal quantities. The distributions of the pivotal quantities are calculated in terms of Edgeworth approximation based on BLUEs which in turn can be used to develop CIs. Hence, the distributions of the pivotal quantities are used to construct the interval estimation for the location and scale parameters. The accuracy of the estimated CIs is investigated in terms of the average width. Finally, one real data set has been used to obtain the MLEs of the model parameters, BLUEs of θ and ξ and CIs of θ and ξ.

CONFLICTS OF INTEREST

The authors declare that there are no conflicts of interest regarding the publication of this paper.

AUTHORS' CONTRIBUTIONS

All authors contributed equally to the writing of this paper. All authors have read and agreed to the published version of the manuscript.

FUNDING STATEMENT

There is no funding of this paper.

ACKNOWLEDGMENTS

The authors would like to express thanks to the editor and anonymous referees for useful suggestions and comments which have improved the presentation of the manuscript.

REFERENCES

1.M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publication, Inc., New York, NY, Vol. 55, 1964.
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3.G.S. Rao, R.R.L. Kantam, K. Rosaiah, and S.V.S.V.S.V. Prasad, Int. J. Eng. Appl. Sci., Vol. 3, 2013, pp. 61-68.
6.B.C. Arnold, N. Balakrishnan, and H.N. Nagaraja, A first Course in Order Statistics, Wiley, New York, NY, 1992.
12.R. Jabeen, A. Ahmad, N. Feroze, and G.M. Gilani, Int. J. Adv. Sci. Technol., Vol. 55, 2013, pp. 6780.
18.D. Kumar and S. Dey, J. Stat. Res., Vol. 51, 2017b, pp. 61-78.
19.M. Ahsanullah and A. Alzaatreh, Revstat Stat. J., Vol. 16, 2018, pp. 429443.
23.N. Balakrishnan and A.C. Cohan, Order Statistics and Inference: Estimation Methods, Academic Press, San Diego, 1991.
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Journal of Statistical Theory and Applications
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352 - 367
Publication Date
2020/09/11
ISSN (Online)
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risenwbib
TY  - JOUR
AU  - Devendra Kumar
AU  - Maneesh Kumar
AU  - Sanku Dey
PY  - 2020
DA  - 2020/09/11
TI  - Inferences for the Type-II Exponentiated Log-Logistic Distribution Based on Order Statistics with Application
JO  - Journal of Statistical Theory and Applications
SP  - 352
EP  - 367
VL  - 19
IS  - 3
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200825.002
DO  - 10.2991/jsta.d.200825.002
ID  - Kumar2020
ER  -