Improved Estimation of the Scale Parameter for Log-Logistic Distribution Using Balanced Ranked Set Sampling

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VOLUME 18 , ISSUE 1 (March 2017) > List of articles

Improved Estimation of the Scale Parameter for Log-Logistic Distribution Using Balanced Ranked Set Sampling

Housila P. Singh * / Vishal Mehta *

Keywords : minimum mean squared error estimator, shrinkage estimator, log-logistic distribution, best linear unbiased estimator, median ranked set sample, AMS Subject Classification: 62G30; 62H12

Citation Information : Statistics in Transition New Series. Volume 18, Issue 1, Pages 53-74, DOI: https://doi.org/10.21307/stattrans-2016-057

License : (CC BY 4.0)

Published Online: 11-July-2017

ARTICLE

ABSTRACT

In this article we have suggested some improved estimators of a scale parameter of log-logistic distribution (LLD) under a situation where the units in a sample can be ordered by judgement method without any error. We have also suggested some linear shrinkage estimator of a scale parameter of LLD. Efficiency comparisons are also made in this work.

1. Introduction

Ranked set sampling (RSS) is a method of sampling that can be advantageous when quantification of all sampling units is costly but a small set of units can be easily ranked, according to the character under investigation, without actual quantification. The technique was first introduced by McIntyre (1952) for estimating means pasture and forage yields. The theory and application of ranked set sampling given by Chen et al. (2004).

A random variable X is said to have a log-logistic distribution with the scale parameter α and the shape parameter β if its cumulative distribution function (CDF) and probability density function (PDF) are respectively given as (see, Lesitha and Thomas (2012))

(1)
F(x;α,β)=xβαβ+xβ,x>0,α>0,β>1
and
(2)
f(x;α,β)=βαβxβ-1(αβ+xβ)2,x>0,α>0,β>1.

Also, the kth moment of (2) exists only when k < β and is given by

(3)
E(Xk)=αkB(1-kβ,1+kβ),
where B denotes beta function.

The applications of log-logistic distribution are well known in a survival analysis of data sets such as survival times of cancer patients in which the hazard rate increases initially and decreases later (for example, see Bennett (1983)). In economic studies of distributions of wealth or income, it is known as Fisk distribution (see Fisk (1961)) and is considered as an equivalent alternative to a lognormal distribution. For further details on the importance and applications of a log-logistic distribution one may refer to Shoukri et al. (1988), Geskus (2001), Robson and Reed (1999) and Ahmad et al. (1988). For current reference in this context the reader is referred to Singh and Mehta (2013; 2014, a, b, 2015, 2016 a, b, c), Mehta and Singh (2014) and Mehta (2015).

If X1:n, X2:n, …, Xn:n are the order statistics of a random sample of size n drawn from (1) then

(4)
Yr:n=Xr:nα,r=1,2,,n,
are distributed as order statistics of the same sample size drawn from a LLD(1, β) with PDF given by
(5)
g(y,β)=βyβ-1(1+yβ)2,y>0,β>1.

For a detailed description of various properties of order statistics arising from a LLD(1, β) one may refer to Ragab and Green (1984). Balakrishnan and Malik (1987) have given some recurrence relations on the single and product moments of order statistics arising from a LLD(1, β). Suppose

(6)
γr:n=E(Yr:n),r=1,2,,n,
(7)
σr,s:n=Cov(Yr:n,Ys:n),1r<sn
and
(8)
σr,r:n=Var(Yr:n),1rn.

By using (4) in (6)-(8) we have

(9)
E(Xr:n)=αγr:n,1rn
(10)
Cov(Xr:n,Xs:n)=α2σr,s:n,1r<sn
(11)
Var(Xr:n)=α2σr,r:n,1rn.

Lesitha and Thomas (2012) have computed the values of γr:n and σr,s:n,1r,sn independently for n = 2(1)8 and for β = 2.5(0.5)5.0 using Mathcad software so as to use those values for the computation of BLUE of α based on order statistics. If X = (Xl:n, X2:n, …, Xn:n)′ then the mean vector E(X) and dispersion matrix D(X) of X are

E(X)=γα
and
D(X)=α2G,
where γ = (γ1:n, γ2:n, …, γn:n)′ and G = ((σr,s:n)).

Thus, by Gauss-Markov theorem Lesitha and Thomas (2012) gives the BLUE α^ based on order statistics of a random sample of size n as:

t1=α^=(γG-1γ)-1γG-1X
and
(12)
Var(t1)=(γG-1γ)-1α2=α2V1,
where V1 = (γ′G–1γ)1.

Lesitha and Thomas (2012) further estimate α based on the mean of unbiased estimators of α defined from each individual observations in the balanced ranked set sampling as:

t2=α*=1nr=1n[X(r:n)rγr:n],
with
(13)
Var(t2)=1n2r=1n[σr,r:nγr:n2]α2=α2V2,
where V2=1n2r=1n[σr,r:nγr:n2].

Lesitha and Thomas (2012) also estimate α based on BLUE in the balanced ranked set sampling as:

t3=α**=(γG1-1γ)-1γG1-1Xrss
and
(14)
Var(t3)=(γG1-1γ)-1α2=α2V3.
where γ = (γ1:n, γ2:n, …, γn:n)′, G1 = diag(σ1,1:n, σ2,2:n, …, σn,n:n) and V3=(γG1-1γ)-1.

When n is small the estimators α* and α** may not be acceptable for the expected level of precision. In such situations Lesitha and Thomas (2012) makes N cycles of RSS. For details see Chen et al. (2004). Suppose αi* and αi** denote the estimators of α corresponding to α* and α** respectively, based on the ith cycle. Then, estimators of α based on N cycles are given by:

α*¯=1Ni=1Nαi*,
and
α**¯=1Ni=1Nαi**
with
Var(α*¯)=α2Nn2r=1N[σr,r:nγr:n2],
and
Var(α**¯)=(γG1-1γ)-1α2N.

Median ranked set sampling (MRSS) was first introduced by Muttlak (1997) to estimate the mean of a normal distribution. In general, MRSS is applied as a modification of RSS when one is interested in estimating a parameter associated with the central tendency of a distribution. The procedures of MRSS are given as: Select n independent samples each with n units as in the case of RSS. Then rank the units in each sample either by judgement method or by using some inexpensive means without having actual measurement on the unit. Lesitha and Thomas (2012) used MRSS method to estimate α as:

t4=α˜={1nr=1n[X(m:n)rγm:n];whenm=n+12andnisodd(γ1G2-1γ1)-1γ1G2-1Xmrss;whenm=n2andniseven
with
(15)
Var(t4)={1n[σm,m:nγm:n2]α2;whenm=n+12andnisodd(γ1G2-1γ1)-1α2;whenm=n2andniseven=a2V4,
where
Xmrss=(X(m:n)1,X(m+1:n)2,X(m:n)3,X(m+1:n)4,,X(m+1:n)n)γ1=(γm:n,γm+1:n,γm:n,γm+1:n,,γm+1:n)G2=diag(σm,m:n,σm+1,m+1:n,σm,m:n,σm+1,m+1:n,,σm+1,m+1:n)andV4={1n[σm,m:nγm:n2];whenm=n+12andnisodd(γ1G2-1γ1)-1;whenm=n2andniseven

2. Improved estimation of the scale parameter α

Let ti, i = 1,2,3,4 be an unbiased estimator of the parameter α, then we define a class of estimators for α as

Ti=Aiti,i=1,2,3,4,
where Ais,i = 1,2,3,4 are suitably chosen constants such that mean squared error of the estimators Tis,i = 1,2,3,4 is minimum.

The biases and mean squared errors (MSEs) of Ti,i = 1,2,3,4 are respectively given by

B(Ti)=α(Ai1),
and
MSE(Ti)=Ai2Var(ti)+(Ai-1)2α2=α2[Ai2(1+Vi)-2Ai+1].

The MSE(Ti), i = 1,2,3,4 is minimized for

Ai=(1+Vi)-1,i=1,2,3,4.

Thus, the resulting minimum MSE estimator of α is given by

T0i=ti(1+Vi)1,i=1,2,3,4.

The biases and MSEs of T0i,i = 1,2,3,4 are respectively given as

(16)
B(T0i)=-α(Vi1+Vi),
and
(17)
MSE(T0i)=α2(Vi1+Vi).

We have from (12) - (15) and (17) that

(18)
Var(ti)-MSE(T0i)=α2Vi2(1+Vi)>0,i=1,2,3,4.

It follows from (18) that the proposed MMSE estimators T0is,i = 1,2,3,4 are better than the corresponding usual unbiased estimators tis,i = 1,2,3,4.

3. Improved estimation of the scale parameter α with prior information

Let ti,i = 1,2,3,4 be an unbiased estimator of the parameter α, then we define a class of estimators of α using the prior point estimate α0 of α as

(19)
T1i=α0+Biti,i=1,2,3,4,
where Bis,i = 1,2,3,4 are suitably chosen constants such that mean squared error of the estimators T1is,i = 1,2,3,4 are minimum.

The biases and mean squared errors (MSEs) of T1i, i = 1,2,3,4 are respectively given by

B(T1i)=α(ϕ+Bi),
and
MSE(T1i)=α2[ϕ2+Bi2(1+Vi)+2ϕBi].
where ϕ=(α0α-1)=(λ-1) with λ=α0α.

The MSE(T1i),i = 1,2,3,4 is minimized for

(20)
Bi=ϕ(1+Vi)-1,i=1,2,3,4.

The value of Bi,i = 1,2,3,4 at (20) depends on the unknown parameter α, so an estimate of Bi,i = 1,2,3,4 based on sample data is given by

Bi*=-ϕ*(1+Vi)=-(θ20-ti)ti(1+Vi),i=1,2,3,4.

Putting Bi*,i = 1,2,3,4 in (19), we get a shrinkage estimator of tis,i = 1,2,3,4 as

(21)
T1i*=α0-(1+Vi)-1(α0-ti),i=1,2,3,4.

The biases and mean squared errors (MSEs) of the estimators T1i*s,i=1,2,3,4 are respectively given by

(22)
B(T1i*)=αϕ(Vi1+Vi),
and
(23)
MSE(T1i*)=α2Vi(ϕ2Vi+1)(1+Vi)2.

Comparisons of the proposed shrinkage estimators T1i*s,i=1,2,3,4 with that of corresponding usual unbiased estimators tis,i = 1,2,3,4 are given in the following Theorem 1.

Theorem 1: The proposed shrinkage estimators T1i*s,i=1,2,3,4 are better than the corresponding usual unbiased estimators tis,i = 1,2,3,4 if

λ(0,(1+(2+Vi))),i.e.ifα0(0,α{1+(2+Vi)}),i.e.ifα(α0{1+(2+Vi)},).

Proof: From (12) - (15) and (23), we have that

(24)
MSE(T1i*)<Var(ti),i=1,2,3,4 ifα2Vi(ϕ2Vi+1)(1+Vi)2<α2Vi,i.e.if(ϕ2Vi+1)(1+Vi)2<1i.e.ifϕ2<((1+Vi)2-1)Vi,i.e.ifϕ2<Vi(2+Vi)Vi,i.e.ifϕ2<2+Vi,i.e.if(λ-1)2<2+Vi,i.e.if{1-(2+Vi)}<λ<{1+(2+Vi)}.

Since {1-(2+Vi)}<0 and λ(= α0 / α) cannot be negative therefore (24) reduces to

0<λ<{1+(2+Vi)},
or
0<α0<α{1+(2+Vi)},
or
α0{1+(2+Vi)}<α<.
Hence the theorem. ♦

Further, we have compared the proposed shrinkage estimators T1i*s,i=1,2,3,4 with that of corresponding MMSE estimators T0is,i = 1,2,3,4 and the results are presented in Theorem 2.

Theorem 2: The proposed shrinkage estimators T1i*s,i=1,2,3,4 are better than the corresponding MMSE estimators T0is,i = 1,2,3,4 if

λ(0,2),i.e.ifα0(0,2α),i.e.ifα(α02,).

Proof: From (17) and (23) we have that

MSE(T1i*)<MSE(T0i,i=1,2,,7 ifα2Vi(ϕ2Vi+1)(1+Vi)2<α2Vi1+Vi,i.e.if(ϕ2Vi+1)(1+Vi)<1,i.e.ifϕ2<1,i.e.if-1<ϕ<1,i.e.if0<λ<2,or0<α0<2α,orα02<α<.
Hence the theorem. ♦

4. Relative efficiencies

We have computed the relative efficiencies of various suggested estimators to usual estimators by using the formulae:

e1=RE(T01,t1)=1+V1;e2=RE(T02,t2)=1+V2;e3=RE(T03,t3)=1+V3;e4=RE(T04,t4)=1+V4;e5=RE(T04,T01)=V1(1+V4)V4(1+V1);e6=RE(T04,T02)=V2(1+V4)V4(1+V2);e7=RE(T04,T03)=V3(1+V4)V4(1+V3);e8=RE(T11*,t1)=(1+V1)2(ϕ2V1+1);e9=RE(T11*,T01)=(1+V1)(ϕ2V1+1);
e10=RE(T12*,t2)=(1+V2)2(ϕ2V2+1);e11=RE(T12*,T02)=(1+V2)(ϕ2V2+1);e12=RE(T13*,t3)=(1+V3)2(ϕ2V3+1);e13=RE(T13*,T03)=(1+V3)(ϕ2V3+1);e14=RE(T14*,t4)=(1+V4)2(ϕ2V4+1);
and
e15=RE(T14*,T04)=(1+V4)(ϕ2V4+1).
  • The values of ei,i = 1,2,…,7 are shown in Table 1 for n = 2(1)8 and β = 2.5(0.5)5.

    Table 1.

    The values of eis,i = 1,2,…,7.

    N

    ß

    e1

    e2

    e3

    e4

    e5

    e6

    e7

    2

    2.5

    1.3371

    1.4025

    1.2799

    1.2799

    1.1530

    1.3124

    1.0000

     

    3.0

    1.2158

    1.1978

    1.1674

    1.1674

    1.2381

    1.1516

    1.0000

     

    3.5

    1.1514

    1.1254

    1.1135

    1.1135

    1.2901

    1.0932

    1.0000

     

    4.0

    1.1126

    1.0885

    1.0827

    1.0827

    1.3242

    1.0643

    1.0000

     

    4.5

    1.0872

    1.0665

    1.0633

    1.0633

    1.3476

    1.0474

    1.0000

     

    5.0

    1.0697

    1.0521

    1.0501

    1.0501

    1.3644

    1.0368

    1.0000

    3

    2.5

    1.2011

    1.1904

    1.1180

    1.0832

    2.1798

    2.0828

    1.3740

     

    3.0

    1.1308

    1.0970

    1.0752

    1.0543

    2.2446

    1.7159

    1.3572

     

    3.5

    1.0937

    1.0626

    1.0526

    1.0385

    2.3093

    1.5893

    1.3475

     

    4.0

    1.0706

    1.0447

    1.0391

    1.0288

    2.3527

    1.5272

    1.3422

     

    4.5

    1.0552

    1.0338

    1.0303

    1.0224

    2.3815

    1.4910

    1.3382

     

    5.0

    1.0443

    1.0266

    1.0242

    1.0180

    2.4033

    1.4675

    1.3356

    4

    2.5

    1.1392

    1.1118

    1.0648

    1.0477

    2.6846

    2.2089

    1.3376

     

    3.0

    1.0939

    1.0582

    1.0424

    1.0317

    2.7929

    1.7903

    1.3246

     

    3.5

    1.0678

    1.0380

    1.0301

    1.0227

    2.8598

    1.6495

    1.3176

     

    4.0

    1.0514

    1.0273

    1.0226

    1.0171

    2.9034

    1.5803

    1.3126

     

    4.5

    1.0413

    1.0208

    1.0176

    1.0134

    3.0062

    1.5408

    1.3102

     

    5.0

    1.0324

    1.0164

    1.0141

    1.0107

    2.9570

    1.5158

    1.3085

    5

    2.5

    1.1078

    1.0740

    1.0409

    1.0278

    3.5999

    2.5482

    1.4547

     

    3.0

    1.0733

    1.0391

    1.0272

    1.0187

    3.7126

    2.0476

    1.4397

     

    3.5

    1.0531

    1.0257

    1.0195

    1.0135

    3.7819

    1.8805

    1.4317

     

    4.0

    1.0403

    1.0186

    1.0147

    1.0101

    3.8704

    1.8188

    1.4421

     

    4.5

    1.0316

    1.0141

    1.0115

    1.0080

    3.8489

    1.7503

    1.4223

     

    5.0

    1.0256

    1.0112

    1.0092

    1.0065

    3.8785

    1.7199

    1.4196

    6

    2.5

    1.0880

    1.0528

    1.0282

    1.0196

    4.2168

    2.6150

    1.4288

     

    3.0

    1.0600

    1.0282

    1.0189

    1.0133

    4.3313

    2.0988

    1.4170

     

    3.5

    1.0437

    1.0187

    1.0136

    1.0096

    4.3995

    1.9265

    1.4101

     

    4.0

    1.0332

    1.0135

    1.0103

    1.0073

    4.4473

    1.8430

    1.4065

     

    4.5

    1.0261

    1.0103

    1.0080

    1.0057

    4.4756

    1.7943

    1.4024

     

    5.0

    1.0211

    1.0082

    1.0065

    1.0046

    4.5010

    1.7638

    1.4009

    7

    2.5

    1.0735

    1.0397

    1.0206

    1.0138

    5.0286

    2.8038

    1.4800

     

    3.0

    1.0509

    1.0214

    1.0139

    1.0094

    5.1991

    2.2498

    1.4690

     

    3.5

    1.0372

    1.0147

    1.0106

    1.0068

    5.2926

    2.1298

    1.5419

     

    4.0

    1.0282

    1.0103

    1.0076

    1.0052

    5.3054

    1.9708

    1.4542

     

    4.5

    1.0222

    1.0079

    1.0060

    1.0041

    5.3447

    1.9217

    1.4556

     

    5.0

    1.0179

    1.0062

    1.0048

    1.0033

    5.3582

    1.8854

    1.4494

    8

    2.5

    1.0643

    1.0310

    1.0157

    1.0107

    5.7315

    2.8525

    1.4632

     

    3.0

    1.0444

    1.0168

    1.0106

    1.0073

    5.8758

    2.2888

    1.4526

     

    3.5

    1.0329

    1.0112

    1.0077

    1.0053

    6.0530

    2.1067

    1.4484

     

    4.0

    1.0245

    1.0081

    1.0058

    1.0040

    5.9651

    2.0116

    1.4441

     

    4.5

    1.0190

    1.0062

    1.0046

    1.0032

    5.9067

    1.9593

    1.4396

     

    5.0

    1.0156

    1.0049

    1.0037

    1.0025

    6.0905

    1.9479

    1.4568

  • The values of ei,i = 8,9,…,15 are shown in Tables 2 to 5 for n = 2(1)8; β = 2.5(0.5)5 and different values of λ=α0α.

    Table 2.

    The values of ei for i = 8 and 9

     

     

    e8=RE(T11*,t1)

    Range of λ in which T11* is efficient to t1

    n

    ß

    λ = 1.00

    λ = 1.25 and λ = 0.75

    λ = 1.50 and λ = 0.50

    λ = 1.75 and λ = 0.25

    λ = 2.00

    λ = 2.25

    2

    2.5

    1.7878

    1.7509

    1.6489

    1.5028

    1.3371

    1.1710

    (0,2.53)

     

    3.0

    1.4782

    1.4586

    1.4026

    1.3182

    1.2158

    1.1054

    (0,2.49)

     

    3.5

    1.3257

    1.3133

    1.2773

    1.2217

    1.1514

    1.0721

    (0,2.47)

     

    4.0

    1.2378

    1.2292

    1.2039

    1.1641

    1.1126

    1.0527

    (0,2.45)

     

    4.5

    1.1820

    1.1756

    1.1568

    1.1268

    1.0872

    1.0403

    (0,2.44)

     

    5.0

    1.1442

    1.1392

    1.1246

    1.1010

    1.0697

    1.0319

    (0,2.44)

    3

    2.5

    1.4426

    1.4247

    1.3735

    1.2960

    1.2011

    1.0977

    (0,2.48)

     

    3.0

    1.2786

    1.2683

    1.2382

    1.1910

    1.1308

    1.0617

    (0,2.46)

     

    3.5

    1.1961

    1.1892

    1.1688

    1.1363

    1.0937

    1.0434

    (0,2.45)

     

    4.0

    1.1461

    1.1411

    1.1263

    1.1024

    1.0706

    1.0323

    (0,2.44)

     

    4.5

    1.1133

    1.1095

    1.0982

    1.0798

    1.0552

    1.0250

    (0,2.43)

     

    5.0

    1.0906

    1.0876

    1.0787

    1.0641

    1.0443

    1.0200

    (0,2.43)

    4

    2.5

    1.2977

    1.2865

    1.2541

    1.2035

    1.1392

    1.0659

    (0,2.46)

     

    3.0

    1.1966

    1.1896

    1.1692

    1.1366

    1.0939

    1.0435

    (0,2.45)

     

    3.5

    1.1402

    1.1354

    1.1212

    1.0983

    1.0678

    1.0310

    (0,2.44)

     

    4.0

    1.1053

    1.1018

    1.0913

    1.0743

    1.0514

    1.0232

    (0,2.43)

     

    4.5

    1.0843

    1.0815

    1.0732

    1.0597

    1.0413

    1.0186

    (0,2.43)

     

    5.0

    1.0659

    1.0638

    1.0574

    1.0468

    1.0324

    1.0145

    (0,2.43)

    5

    2.5

    1.2272

    1.2190

    1.1950

    1.1570

    1.1078

    1.0503

    (0,2.45)

     

    3.0

    1.1519

    1.1466

    1.1312

    1.1063

    1.0733

    1.0336

    (0,2.44)

     

    3.5

    1.1091

    1.1054

    1.0945

    1.0769

    1.0531

    1.0241

    (0,2.43)

     

    4.0

    1.0823

    1.0796

    1.0715

    1.0583

    1.0403

    1.0181

    (0,2.43)

     

    4.5

    1.0643

    1.0622

    1.0559

    1.0457

    1.0316

    1.0141

    (0,2.43)

     

    5.0

    1.0518

    1.0501

    1.0451

    1.0369

    1.0256

    1.0114

    (0,2.42)

    6

    2.5

    1.1837

    1.1772

    1.1582

    1.1279

    1.0880

    1.0406

    (0,2.44)

     

    3.0

    1.1237

    1.1195

    1.1071

    1.0870

    1.0600

    1.0273

    (0,2.44)

     

    3.5

    1.0892

    1.0863

    1.0775

    1.0631

    1.0437

    1.0197

    (0,2.43)

     

    4.0

    1.0675

    1.0653

    1.0587

    1.0479

    1.0332

    1.0149

    (0,2.43)

     

    4.5

    1.0529

    1.0512

    1.0461

    1.0377

    1.0261

    1.0116

    (0,2.42)

     

    5.0

    1.0426

    1.0413

    1.0372

    1.0304

    1.0211

    1.0094

    (0,2.42)

    7

    2.5

    1.1524

    1.1471

    1.1316

    1.1066

    1.0735

    1.0337

    (0,2.44)

     

    3.0

    1.1043

    1.1008

    1.0905

    1.0736

    1.0509

    1.0230

    (0,2.43)

     

    3.5

    1.0759

    1.0734

    1.0659

    1.0538

    1.0372

    1.0167

    (0,2.43)

     

    4.0

    1.0572

    1.0554

    1.0498

    1.0407

    1.0282

    1.0126

    (0,2.42)

     

    4.5

    1.0449

    1.0434

    1.0391

    1.0320

    1.0222

    1.0099

    (0,2.42)

     

    5.0

    1.0362

    1.0350

    1.0316

    1.0258

    1.0179

    1.0079

    (0,2.42)

    8

    2.5

    1.1327

    1.1282

    1.1148

    1.0932

    1.0643

    1.0293

    (0,2.44)

     

    3.0

    1.0907

    1.0877

    1.0787

    1.0641

    1.0444

    1.0200

    (0,2.43)

     

    3.5

    1.0669

    1.0647

    1.0582

    1.0475

    1.0329

    1.0147

    (0,2.43)

     

    4.0

    1.0497

    1.0481

    1.0433

    1.0354

    1.0245

    1.0109

    (0,2.42)

     

    4.5

    1.0384

    1.0372

    1.0335

    1.0274

    1.0190

    1.0084

    (0,2.42)

     

    5.0

    1.0315

    1.0305

    1.0275

    1.0225

    1.0156

    1.0069

    (0,2.42)

     

     

    e9=RE(T11*,T01)

    Range of λ in which T11* is efficient to T01

    n

    ß

    λ = 1.00

    λ = 1.25 and λ = 0.75

    λ = 1.50 and λ = 0.50

    λ = 1.75 and λ = 0.25

    λ = 2.00 and λ = 0.00

    2

    2.5

    1.3371

    1.3095

    1.2332

    1.1240

    1.0000

    [0,2]

     

    3.0

    1.2158

    1.1996

    1.1536

    1.0842

    1.0000

    [0,2]

     

    3.5

    1.1514

    1.1406

    1.1094

    1.0610

    1.0000

    [0,2]

     

    4.0

    1.1126

    1.1048

    1.0821

    1.0463

    1.0000

    [0,2]

     

    4.5

    1.0872

    1.0813

    1.0640

    1.0364

    1.0000

    [0,2]

     

    5.0

    1.0697

    1.0650

    1.0514

    1.0293

    1.0000

    [0,2]

    3

    2.5

    1.2011

    1.1862

    1.1436

    1.0790

    1.0000

    [0,2]

     

    3.0

    1.1308

    1.1216

    1.0950

    1.0533

    1.0000

    [0,2]

     

    3.5

    1.0937

    1.0873

    1.0687

    1.0389

    1.0000

    [0,2]

     

    4.0

    1.0706

    1.0659

    1.0520

    1.0297

    1.0000

    [0,2]

     

    4.5

    1.0552

    1.0515

    1.0408

    1.0234

    1.0000

    [0,2]

     

    5.0

    1.0443

    1.0414

    1.0329

    1.0189

    1.0000

    [0,2]

    4

    2.5

    1.1392

    1.1294

    1.1009

    1.0565

    1.0000

    [0,2]

     

    3.0

    1.0939

    1.0875

    1.0688

    1.0390

    1.0000

    [0,2]

     

    3.5

    1.0678

    1.0633

    1.0500

    1.0286

    1.0000

    [0,2]

     

    4.0

    1.0514

    1.0480

    1.0380

    1.0218

    1.0000

    [0,2]

     

    4.5

    1.0413

    1.0386

    1.0307

    1.0177

    1.0000

    [0,2]

     

    5.0

    1.0324

    1.0304

    1.0241

    1.0139

    1.0000

    [0,2]

    5

    2.5

    1.1078

    1.1004

    1.0787

    1.0445

    1.0000

    [0,2]

     

    3.0

    1.0733

    1.0684

    1.0540

    1.0308

    1.0000

    [0,2]

     

    3.5

    1.0531

    1.0496

    1.0393

    1.0226

    1.0000

    [0,2]

     

    4.0

    1.0403

    1.0377

    1.0300

    1.0173

    1.0000

    [0,2]

     

    4.5

    1.0316

    1.0296

    1.0235

    1.0136

    1.0000

    [0,2]

     

    5.0

    1.0256

    1.0239

    1.0191

    1.0110

    1.0000

    [0,2]

    6

    2.5

    1.0880

    1.0820

    1.0646

    1.0367

    1.0000

    [0,2]

     

    3.0

    1.0600

    1.0561

    1.0444

    1.0254

    1.0000

    [0,2]

     

    3.5

    1.0437

    1.0408

    1.0324

    1.0186

    1.0000

    [0,2]

     

    4.0

    1.0332

    1.0311

    1.0247

    1.0143

    1.0000

    [0,2]

     

    4.5

    1.0261

    1.0244

    1.0195

    1.0113

    1.0000

    [0,2]

     

    5.0

    1.0211

    1.0197

    1.0157

    1.0091

    1.0000

    [0,2]

    7

    2.5

    1.0735

    1.0686

    1.0541

    1.0309

    1.0000

    [0,2]

     

    3.0

    1.0509

    1.0475

    1.0377

    1.0216

    1.0000

    [0,2]

     

    3.5

    1.0372

    1.0348

    1.0277

    1.0160

    1.0000

    [0,2]

     

    4.0

    1.0282

    1.0264

    1.0210

    1.0122

    1.0000

    [0,2]

     

    4.5

    1.0222

    1.0208

    1.0166

    1.0096

    1.0000

    [0,2]

     

    5.0

    1.0179

    1.0168

    1.0134

    1.0078

    1.0000

    [0,2]

    8

    2.5

    1.0643

    1.0600

    1.0474

    1.0271

    1.0000

    [0,2]

     

    3.0

    1.0444

    1.0415

    1.0329

    1.0189

    1.0000

    [0,2]

     

    3.5

    1.0329

    1.0308

    1.0245

    1.0141

    1.0000

    [0,2]

     

    4.0

    1.0245

    1.0230

    1.0183

    1.0106

    1.0000

    [0,2]

     

    4.5

    1.0190

    1.0178

    1.0142

    1.0082

    1.0000

    [0,2]

     

    5.0

    1.0156

    1.0146

    1.0117

    1.0068

    1.0000

    [0,2]

    Table 3.

    The values of ei for i = 10 and 11

     

     

    e10=RE(T12*,t2)

    Range of λ in which T12* is efficient to t2

    n

    ß

    λ = 1.00

    λ = 1.25 and λ = 0.75

    λ = 1.50 and λ = 0.50

    λ = 1.75 and λ = 0.25

    λ = 2.00

    λ = 2.25

    2

    2.5

    1.9669

    1.9186

    1.7871

    1.6038

    1.4025

    1.2075

    (0,2.55)

     

    3.0

    1.4347

    1.4171

    1.3671

    1.2910

    1.1978

    1.0960

    (0,2.48)

     

    3.5

    1.2665

    1.2566

    1.2280

    1.1830

    1.1254

    1.0590

    (0,2.46)

     

    4.0

    1.1849

    1.1784

    1.1592

    1.1287

    1.0885

    1.0409

    (0,2.45)

     

    4.5

    1.1374

    1.1327

    1.1188

    1.0964

    1.0665

    1.0304

    (0,2.44)

     

    5.0

    1.1069

    1.1033

    1.0926

    1.0754

    1.0521

    1.0236

    (0,2.43)

    3

    2.5

    1.4171

    1.4004

    1.3527

    1.2800

    1.1904

    1.0922

    (0,2.48)

     

    3.0

    1.2034

    1.1961

    1.1749

    1.1411

    1.0970

    1.0450

    (0,2.45)

     

    3.5

    1.1292

    1.1248

    1.1118

    1.0908

    1.0626

    1.0285

    (0,2.44)

     

    4.0

    1.0914

    1.0884

    1.0794

    1.0646

    1.0447

    1.0202

    (0,2.43)

     

    4.5

    1.0688

    1.0665

    1.0598

    1.0488

    1.0338

    1.0151

    (0,2.43)

     

    5.0

    1.0539

    1.0522

    1.0470

    1.0384

    1.0266

    1.0119

    (0,2.42)

    4

    2.5

    1.2360

    1.2274

    1.2024

    1.1629

    1.1118

    1.0523

    (0,2.45)

     

    3.0

    1.1199

    1.1158

    1.1038

    1.0843

    1.0582

    1.0265

    (0,2.43)

     

    3.5

    1.0775

    1.0749

    1.0673

    1.0549

    1.0380

    1.0171

    (0,2.43)

     

    4.0

    1.0554

    1.0536

    1.0482

    1.0394

    1.0273

    1.0122

    (0,2.42)

     

    4.5

    1.0419

    1.0406

    1.0366

    1.0299

    1.0208

    1.0092

    (0,2.42)

     

    5.0

    1.0330

    1.0320

    1.0288

    1.0236

    1.0164

    1.0072

    (0,2.42)

    5

    2.5

    1.1534

    1.1481

    1.1325

    1.1073

    1.0740

    1.0339

    (0,2.44)

     

    3.0

    1.0798

    1.0771

    1.0693

    1.0565

    1.0391

    1.0176

    (0,2.43)

     

    3.5

    1.0521

    1.0504

    1.0454

    1.0371

    1.0257

    1.0115

    (0,2.42)

     

    4.0

    1.0375

    1.0363

    1.0327

    1.0267

    1.0186

    1.0082

    (0,2.42)

     

    4.5

    1.0285

    1.0276

    1.0249

    1.0204

    1.0141

    1.0062

    (0,2.42)

     

    5.0

    1.0225

    1.0218

    1.0196

    1.0161

    1.0112

    1.0049

    (0,2.42)

    6

    2.5

    1.1084

    1.1047

    1.0939

    1.0764

    1.0528

    1.0239

    (0,2.43)

     

    3.0

    1.0572

    1.0554

    1.0498

    1.0407

    1.0282

    1.0126

    (0,2.42)

     

    3.5

    1.0377

    1.0365

    1.0328

    1.0269

    1.0187

    1.0083

    (0,2.42)

     

    4.0

    1.0272

    1.0263

    1.0237

    1.0194

    1.0135

    1.0060

    (0,2.42)

     

    4.5

    1.0207

    1.0201

    1.0181

    1.0148

    1.0103

    1.0045

    (0,2.42)

     

    5.0

    1.0164

    1.0159

    1.0143

    1.0117

    1.0082

    1.0036

    (0,2.42)

    7

    2.5

    1.0809

    1.0783

    1.0703

    1.0573

    1.0397

    1.0178

    (0,2.43)

     

    3.0

    1.0433

    1.0419

    1.0377

    1.0308

    1.0214

    1.0095

    (0,2.42)

     

    3.5

    1.0295

    1.0286

    1.0258

    1.0211

    1.0147

    1.0065

    (0,2.42)

     

    4.0

    1.0207

    1.0200

    1.0181

    1.0148

    1.0103

    1.0045

    (0,2.42)

     

    4.5

    1.0158

    1.0153

    1.0138

    1.0113

    1.0079

    1.0035

    (0,2.42)

     

    5.0

    1.0125

    1.0121

    1.0109

    1.0090

    1.0062

    1.0027

    (0,2.42)

    8

    2.5

    1.0629

    1.0609

    1.0548

    1.0447

    1.0310

    1.0138

    (0,2.43)

     

    3.0

    1.0339

    1.0328

    1.0296

    1.0242

    1.0168

    1.0074

    (0,2.42)

     

    3.5

    1.0225

    1.0218

    1.0197

    1.0161

    1.0112

    1.0049

    (0,2.42)

     

    4.0

    1.0163

    1.0158

    1.0143

    1.0117

    1.0081

    1.0036

    (0,2.42)

     

    4.5

    1.0125

    1.0121

    1.0109

    1.0090

    1.0062

    1.0027

    (0,2.42)

     

    5.0

    1.0099

    1.0096

    1.0087

    1.0071

    1.0049

    1.0022

    (0,2.42)

     

     

    e11=RE(T12*,T02)

    Range of λ in which T12* is efficient to T02

    n

    ß

    λ = 1.00

    λ = 1.25 and λ = 0.75

    λ = 1.50 and λ = 0.50

    λ = 1.75 and λ = 0.25

    λ = 2.00 and λ = 0.00

    2

    2.5

    1.4025

    1.3680

    1.2743

    1.1436

    1.0000

    [0,2]

     

    3.0

    1.1978

    1.1831

    1.1413

    1.0779

    1.0000

    [0,2]

     

    3.5

    1.1254

    1.1166

    1.0912

    1.0512

    1.0000

    [0,2]

     

    4.0

    1.0885

    1.0825

    1.0650

    1.0369

    1.0000

    [0,2]

     

    4.5

    1.0665

    1.0621

    1.0491

    1.0280

    1.0000

    [0,2]

     

    5.0

    1.0521

    1.0487

    1.0386

    1.0221

    1.0000

    [0,2]

    3

    2.5

    1.1904

    1.1764

    1.1363

    1.0752

    1.0000

    [0,2]

     

    3.0

    1.0970

    1.0904

    1.0710

    1.0402

    1.0000

    [0,2]

     

    3.5

    1.0626

    1.0585

    1.0463

    1.0265

    1.0000

    [0,2]

     

    4.0

    1.0447

    1.0418

    1.0332

    1.0191

    1.0000

    [0,2]

     

    4.5

    1.0338

    1.0316

    1.0252

    1.0145

    1.0000

    [0,2]

     

    5.0

    1.0266

    1.0249

    1.0198

    1.0115

    1.0000

    [0,2]

    4

    2.5

    1.1118

    1.1040

    1.0815

    1.0460

    1.0000

    [0,2]

     

    3.0

    1.0582

    1.0544

    1.0430

    1.0247

    1.0000

    [0,2]

     

    3.5

    1.0380

    1.0356

    1.0282

    1.0163

    1.0000

    [0,2]

     

    4.0

    1.0273

    1.0256

    1.0203

    1.0118

    1.0000

    [0,2]

     

    4.5

    1.0208

    1.0194

    1.0155

    1.0090

    1.0000

    [0,2]

     

    5.0

    1.0164

    1.0153

    1.0122

    1.0071

    1.0000

    [0,2]

    5

    2.5

    1.0740

    1.0690

    1.0545

    1.0311

    1.0000

    [0,2]

     

    3.0

    1.0391

    1.0366

    1.0291

    1.0167

    1.0000

    [0,2]

     

    3.5

    1.0257

    1.0241

    1.0192

    1.0111

    1.0000

    [0,2]

     

    4.0

    1.0186

    1.0174

    1.0139

    1.0080

    1.0000

    [0,2]

     

    4.5

    1.0141

    1.0132

    1.0106

    1.0061

    1.0000

    [0,2]

     

    5.0

    1.0112

    1.0105

    1.0084

    1.0049

    1.0000

    [0,2]

    6

    2.5

    1.0528

    1.0493

    1.0391

    1.0224

    1.0000

    [0,2]

     

    3.0

    1.0282

    1.0264

    1.0210

    1.0122

    1.0000

    [0,2]

     

    3.5

    1.0187

    1.0175

    1.0139

    1.0081

    1.0000

    [0,2]

     

    4.0

    1.0135

    1.0126

    1.0101

    1.0059

    1.0000

    [0,2]

     

    4.5

    1.0103

    1.0097

    1.0077

    1.0045

    1.0000

    [0,2]

     

    5.0

    1.0082

    1.0076

    1.0061

    1.0036

    1.0000

    [0,2]

    7

    2.5

    1.0397

    1.0371

    1.0295

    1.0170

    1.0000

    [0,2]

     

    3.0

    1.0214

    1.0200

    1.0160

    1.0093

    1.0000

    [0,2]

     

    3.5

    1.0147

    1.0137

    1.0110

    1.0064

    1.0000

    [0,2]

     

    4.0

    1.0103

    1.0097

    1.0077

    1.0045

    1.0000

    [0,2]

     

    4.5

    1.0079

    1.0074

    1.0059

    1.0034

    1.0000

    [0,2]

     

    5.0

    1.0062

    1.0058

    1.0047

    1.0027

    1.0000

    [0,2]

    8

    2.5

    1.0310

    1.0290

    1.0231

    1.0133

    1.0000

    [0,2]

     

    3.0

    1.0168

    1.0158

    1.0126

    1.0073

    1.0000

    [0,2]

     

    3.5

    1.0112

    1.0105

    1.0084

    1.0049

    1.0000

    [0,2]

     

    4.0

    1.0081

    1.0076

    1.0061

    1.0035

    1.0000

    [0,2]

     

    4.5

    1.0062

    1.0058

    1.0047

    1.0027

    1.0000

    [0,2]

     

    5.0

    1.0049

    1.0046

    1.0037

    1.0022

    1.0000

    [0,2]

    Table 4.

    The values of ei for i = 12 and 13

     

     

    e12=RE(T13*,t3)

    Range of λ in which T13* is efficient to t3

    n

    ß

    λ = 1.00

    λ = 1.25 and λ = 0.75

    λ = 1.50 and λ = 0.50

    λ = 1.75 and λ = 0.25

    λ = 2.00

    λ = 2.25

    2

    2.5

    1.6380

    1.6099

    1.5309

    1.4152

    1.2799

    1.1397

    (0,2.51)

     

    3.0

    1.3628

    1.3487

    1.3080

    1.2455

    1.1674

    1.0803

    (0,2.47)

     

    3.5

    1.2398

    1.2311

    1.2056

    1.1654

    1.1135

    1.0531

    (0,2.45)

     

    4.0

    1.1723

    1.1663

    1.1485

    1.1202

    1.0827

    1.0381

    (0,2.44)

     

    4.5

    1.1306

    1.1262

    1.1130

    1.0917

    1.0633

    1.0288

    (0,2.44)

     

    5.0

    1.1028

    1.0993

    1.0891

    1.0725

    1.0501

    1.0227

    (0,2.43)

    3

    2.5

    1.2499

    1.2407

    1.2141

    1.1721

    1.1180

    1.0553

    (0,2.46)

     

    3.0

    1.1560

    1.1506

    1.1347

    1.1091

    1.0752

    1.0345

    (0,2.44)

     

    3.5

    1.1080

    1.1044

    1.0936

    1.0761

    1.0526

    1.0238

    (0,2.43)

     

    4.0

    1.0797

    1.0771

    1.0692

    1.0565

    1.0391

    1.0176

    (0,2.43)

     

    4.5

    1.0614

    1.0594

    1.0535

    1.0437

    1.0303

    1.0135

    (0,2.42)

     

    5.0

    1.0489

    1.0473

    1.0426

    1.0348

    1.0242

    1.0107

    (0,2.42)

    4

    2.5

    1.1338

    1.1293

    1.1158

    1.0940

    1.0648

    1.0296

    (0,2.44)

     

    3.0

    1.0867

    1.0838

    1.0753

    1.0613

    1.0424

    1.0191

    (0,2.43)

     

    3.5

    1.0612

    1.0592

    1.0533

    1.0435

    1.0301

    1.0135

    (0,2.42)

     

    4.0

    1.0457

    1.0442

    1.0398

    1.0326

    1.0226

    1.0100

    (0,2.42)

     

    4.5

    1.0355

    1.0344

    1.0310

    1.0253

    1.0176

    1.0078

    (0,2.42)

     

    5.0

    1.0284

    1.0275

    1.0248

    1.0203

    1.0141

    1.0062

    (0,2.42)

    5

    2.5

    1.0835

    1.0808

    1.0726

    1.0592

    1.0409

    1.0184

    (0,2.43)

     

    3.0

    1.0551

    1.0533

    1.0480

    1.0392

    1.0272

    1.0121

    (0,2.42)

     

    3.5

    1.0393

    1.0381

    1.0343

    1.0281

    1.0195

    1.0086

    (0,2.42)

     

    4.0

    1.0295

    1.0286

    1.0258

    1.0211

    1.0147

    1.0065

    (0,2.42)

     

    4.5

    1.0231

    1.0223

    1.0201

    1.0165

    1.0115

    1.0051

    (0,2.42)

     

    5.0

    1.0185

    1.0179

    1.0162

    1.0133

    1.0092

    1.0041

    (0,2.42)

    6

    2.5

    1.0571

    1.0553

    1.0497

    1.0406

    1.0282

    1.0126

    (0,2.42)

     

    3.0

    1.0381

    1.0369

    1.0332

    1.0272

    1.0189

    1.0084

    (0,2.42)

     

    3.5

    1.0274

    1.0265

    1.0239

    1.0196

    1.0136

    1.0060

    (0,2.42)

     

    4.0

    1.0206

    1.0200

    1.0180

    1.0148

    1.0103

    1.0045

    (0,2.42)

     

    4.5

    1.0161

    1.0156

    1.0141

    1.0116

    1.0080

    1.0035

    (0,2.42)

     

    5.0

    1.0130

    1.0126

    1.0113

    1.0093

    1.0065

    1.0028

    (0,2.42)

    7

    2.5

    1.0415

    1.0402

    1.0362

    1.0296

    1.0206

    1.0091

    (0,2.42)

     

    3.0

    1.0279

    1.0270

    1.0244

    1.0200

    1.0139

    1.0061

    (0,2.42)

     

    3.5

    1.0213

    1.0206

    1.0186

    1.0152

    1.0106

    1.0047

    (0,2.42)

     

    4.0

    1.0152

    1.0147

    1.0133

    1.0109

    1.0076

    1.0033

    (0,2.42)

     

    4.5

    1.0119

    1.0116

    1.0104

    1.0086

    1.0060

    1.0026

    (0,2.42)

     

    5.0

    1.0096

    1.0093

    1.0084

    1.0069

    1.0048

    1.0021

    (0,2.42)

    8

    2.5

    1.0316

    1.0306

    1.0275

    1.0226

    1.0157

    1.0069

    (0,2.42)

     

    3.0

    1.0213

    1.0207

    1.0186

    1.0153

    1.0106

    1.0047

    (0,2.42)

     

    3.5

    1.0154

    1.0149

    1.0135

    1.0111

    1.0077

    1.0034

    (0,2.42)

     

    4.0

    1.0117

    1.0113

    1.0102

    1.0084

    1.0058

    1.0026

    (0,2.42)

     

    4.5

    1.0092

    1.0089

    1.0080

    1.0066

    1.0046

    1.0020

    (0,2.42)

     

    5.0

    1.0074

    1.0072

    1.0065

    1.0053

    1.0037

    1.0016

    (0,2.42)

     

     

    e13=RE(T13*,T03)

    Range of λ in which T13* is efficient to

    T03

    n

    ß

    λ = 1.00

    λ = 1.25 and λ = 0.75

    λ = 1.50 and λ = 0.50

    λ = 1.75 and λ = 0.25

    λ = 2.00 and λ = 0.00

    2

    2.5

    1.2799

    1.2578

    1.1962

    1.1058

    1.0000

    [0,2]

     

    3.0

    1.1674

    1.1553

    1.1205

    1.0669

    1.0000

    [0,2]

     

    3.5

    1.1135

    1.1056

    1.0828

    1.0467

    1.0000

    [0,2]

     

    4.0

    1.0827

    1.0772

    1.0608

    1.0346

    1.0000

    [0,2]

     

    4.5

    1.0633

    1.0591

    1.0467

    1.0267

    1.0000

    [0,2]

     

    5.0

    1.0501

    1.0469

    1.0371

    1.0213

    1.0000

    [0,2]

    3

    2.5

    1.1180

    1.1098

    1.0859

    1.0484

    1.0000

    [0,2]

     

    3.0

    1.0752

    1.0702

    1.0553

    1.0316

    1.0000

    [0,2]

     

    3.5

    1.0526

    1.0492

    1.0389

    1.0224

    1.0000

    [0,2]

     

    4.0

    1.0391

    1.0365

    1.0290

    1.0167

    1.0000

    [0,2]

     

    4.5

    1.0303

    1.0283

    1.0225

    1.0130

    1.0000

    [0,2]

     

    5.0

    1.0242

    1.0226

    1.0180

    1.0104

    1.0000

    [0,2]

    4

    2.5

    1.0648

    1.0605

    1.0478

    1.0274

    1.0000

    [0,2]

     

    3.0

    1.0424

    1.0397

    1.0315

    1.0181

    1.0000

    [0,2]

     

    3.5

    1.0301

    1.0282

    1.0224

    1.0130

    1.0000

    [0,2]

     

    4.0

    1.0226

    1.0211

    1.0168

    1.0098

    1.0000

    [0,2]

     

    4.5

    1.0176

    1.0165

    1.0131

    1.0076

    1.0000

    [0,2]

     

    5.0

    1.0141

    1.0132

    1.0105

    1.0061

    1.0000

    [0,2]

    5

    2.5

    1.0409

    1.0383

    1.0304

    1.0175

    1.0000

    [0,2]

     

    3.0

    1.0272

    1.0254

    1.0203

    1.0117

    1.0000

    [0,2]

     

    3.5

    1.0195

    1.0182

    1.0145

    1.0084

    1.0000

    [0,2]

     

    4.0

    1.0147

    1.0137

    1.0110

    1.0064

    1.0000

    [0,2]

     

    4.5

    1.0115

    1.0107

    1.0086

    1.0050

    1.0000

    [0,2]

     

    5.0

    1.0092

    1.0086

    1.0069

    1.0040

    1.0000

    [0,2]

    6

    2.5

    1.0282

    1.0264

    1.0210

    1.0121

    1.0000

    [0,2]

     

    3.0

    1.0189

    1.0177

    1.0141

    1.0082

    1.0000

    [0,2]

     

    3.5

    1.0136

    1.0127

    1.0102

    1.0059

    1.0000

    [0,2]

     

    4.0

    1.0103

    1.0096

    1.0077

    1.0045

    1.0000

    [0,2]

     

    4.5

    1.0080

    1.0075

    1.0060

    1.0035

    1.0000

    [0,2]

     

    5.0

    1.0065

    1.0061

    1.0048

    1.0028

    1.0000

    [0,2]

    7

    2.5

    1.0206

    1.0193

    1.0153

    1.0089

    1.0000

    [0,2]

     

    3.0

    1.0139

    1.0130

    1.0104

    1.0060

    1.0000

    [0,2]

     

    3.5

    1.0106

    1.0099

    1.0079

    1.0046

    1.0000

    [0,2]

     

    4.0

    1.0076

    1.0071

    1.0057

    1.0033

    1.0000

    [0,2]

     

    4.5

    1.0060

    1.0056

    1.0045

    1.0026

    1.0000

    [0,2]

     

    5.0

    1.0048

    1.0045

    1.0036

    1.0021

    1.0000

    [0,2]

    8

    2.5

    1.0157

    1.0147

    1.0117

    1.0068

    1.0000

    [0,2]

     

    3.0

    1.0106

    1.0099

    1.0079

    1.0046

    1.0000

    [0,2]

     

    3.5

    1.0077

    1.0072

    1.0057

    1.0033

    1.0000

    [0,2]

     

    4.0

    1.0058

    1.0055

    1.0044

    1.0025

    1.0000

    [0,2]

     

    4.5

    1.0046

    1.0043

    1.0034

    1.0020

    1.0000

    [0,2]

     

    5.0

    1.0037

    1.0035

    1.0028

    1.0016

    1.0000

    [0,2]

    Table 5.

    The values of ei for i = 14 and 15

     

     

    e14=RE(T14*,t4)

    Range of λ in which T14* is efficient to t4

    n

    ß

    λ = 1.00

    λ = 1.25 and λ = 0.75

    λ = 1.50 and λ = 0.50

    λ = 1.75 and λ = 0.25

    λ = 2.00

    λ = 2.25

    2

    2.5

    1.6380

    1.6099

    1.5309

    1.4152

    1.2799

    1.1397

    (0,2.51)

     

    3.0

    1.3628

    1.3487

    1.3080

    1.2455

    1.1674

    1.0803

    (0,2.47)

     

    3.5

    1.2398

    1.2311

    1.2056

    1.1654

    1.1135

    1.0531

    (0,2.45)

     

    4.0

    1.1723

    1.1663

    1.1485

    1.1202

    1.0827

    1.0381

    (0,2.44)

     

    4.5

    1.1306

    1.1262

    1.1130

    1.0917

    1.0633

    1.0288

    (0,2.44)

     

    5.0

    1.1028

    1.0993

    1.0891

    1.0725

    1.0501

    1.0227

    (0,2.43)

    3

    2.5

    1.1733

    1.1672

    1.1494

    1.1209

    1.0832

    1.0383

    (0,2.44)

     

    3.0

    1.1116

    1.1078

    1.0967

    1.0786

    1.0543

    1.0246

    (0,2.43)

     

    3.5

    1.0785

    1.0759

    1.0682

    1.0557

    1.0385

    1.0173

    (0,2.43)

     

    4.0

    1.0585

    1.0566

    1.0509

    1.0416

    1.0288

    1.0129

    (0,2.42)

     

    4.5

    1.0454

    1.0439

    1.0396

    1.0324

    1.0224

    1.0100

    (0,2.42)

     

    5.0

    1.0363

    1.0351

    1.0316

    1.0259

    1.0180

    1.0080

    (0,2.42)

    4

    2.5

    1.0976

    1.0944

    1.0847

    1.0690

    1.0477

    1.0215

    (0,2.43)

     

    3.0

    1.0644

    1.0623

    1.0561

    1.0458

    1.0317

    1.0142

    (0,2.43)

     

    3.5

    1.0459

    1.0445

    1.0400

    1.0327

    1.0227

    1.0101

    (0,2.42)

     

    4.0

    1.0345

    1.0334

    1.0301

    1.0247

    1.0171

    1.0076

    (0,2.42)

     

    4.5

    1.0269

    1.0261

    1.0235

    1.0193

    1.0134

    1.0059

    (0,2.42)

     

    5.0

    1.0216

    1.0209

    1.0189

    1.0155

    1.0107

    1.0047

    (0,2.42)

    5

    2.5

    1.0563

    1.0545

    1.0490

    1.0401

    1.0278

    1.0124

    (0,2.42)

     

    3.0

    1.0378

    1.0366

    1.0330

    1.0270

    1.0187

    1.0083

    (0,2.42)

     

    3.5

    1.0272

    1.0264

    1.0238

    1.0195

    1.0135

    1.0060

    (0,2.42)

     

    4.0

    1.0203

    1.0197

    1.0178

    1.0146

    1.0101

    1.0045

    (0,2.42)

     

    4.5

    1.0161

    1.0156

    1.0141

    1.0116

    1.0080

    1.0035

    (0,2.42)

     

    5.0

    1.0130

    1.0126

    1.0113

    1.0093

    1.0065

    1.0028

    (0,2.42)

    6

    2.5

    1.0395

    1.0382

    1.0344

    1.0282

    1.0196

    1.0087

    (0,2.42)

     

    3.0

    1.0267

    1.0258

    1.0233

    1.0191

    1.0133

    1.0059

    (0,2.42)

     

    3.5

    1.0193

    1.0187

    1.0169

    1.0138

    1.0096

    1.0042

    (0,2.42)

     

    4.0

    1.0146

    1.0142

    1.0128

    1.0105

    1.0073

    1.0032

    (0,2.42)

     

    4.5

    1.0115

    1.0111

    1.0100

    1.0082

    1.0057

    1.0025

    (0,2.42)

     

    5.0

    1.0092

    1.0090

    1.0081

    1.0066

    1.0046

    1.0020

    (0,2.42)

    7

    2.5

    1.0278

    1.0269

    1.0243

    1.0199

    1.0138

    1.0061

    (0,2.42)

     

    3.0

    1.0189

    1.0183

    1.0165

    1.0135

    1.0094

    1.0041

    (0,2.42)

     

    3.5

    1.0137

    1.0133

    1.0120

    1.0098

    1.0068

    1.0030

    (0,2.42)

     

    4.0

    1.0104

    1.0101

    1.0091

    1.0075

    1.0052

    1.0023

    (0,2.42)

     

    4.5

    1.0082

    1.0079

    1.0071

    1.0059

    1.0041

    1.0018

    (0,2.42)

     

    5.0

    1.0066

    1.0064

    1.0058

    1.0047

    1.0033

    1.0014

    (0,2.42)

    8

    2.5

    1.0214

    1.0207

    1.0187

    1.0153

    1.0107

    1.0047

    (0,2.42)

     

    3.0

    1.0146

    1.0142

    1.0128

    1.0105

    1.0073

    1.0032

    (0,2.42)

     

    3.5

    1.0106

    1.0103

    1.0093

    1.0076

    1.0053

    1.0023

    (0,2.42)

     

    4.0

    1.0081

    1.0078

    1.0071

    1.0058

    1.0040

    1.0018

    (0,2.42)

     

    4.5

    1.0064

    1.0062

    1.0056

    1.0046

    1.0032

    1.0014

    (0,2.42)

     

    5.0

    1.0051

    1.0049

    1.0044

    1.0036

    1.0025

    1.0011

    (0,2.42)

     

     

    e15=RE(T14*,T04)

    Range of λ in which T14* is efficient to T04

    n

    ß

    λ = 1.00

    λ = 1.25 and λ = 0.75

    λ = 1.50 and λ = 0.50

    λ = 1.75 and λ = 0.25

    λ = 2.00 and λ = 0.00

    2

    2.5

    1.2799

    1.2578

    1.1962

    1.1058

    1.0000

    [0,2]

     

    3.0

    1.1674

    1.1553

    1.1205

    1.0669

    1.0000

    [0,2]

     

    3.5

    1.1135

    1.1056

    1.0828

    1.0467

    1.0000

    [0,2]

     

    4.0

    1.0827

    1.0772

    1.0608

    1.0346

    1.0000

    [0,2]

     

    4.5

    1.0633

    1.0591

    1.0467

    1.0267

    1.0000

    [0,2]

     

    5.0

    1.0501

    1.0469

    1.0371

    1.0213

    1.0000

    [0,2]

    3

    2.5

    1.0832

    1.0776

    1.0611

    1.0348

    1.0000

    [0,2]

     

    3.0

    1.0543

    1.0508

    1.0402

    1.0231

    1.0000

    [0,2]

     

    3.5

    1.0385

    1.0360

    1.0286

    1.0165

    1.0000

    [0,2]

     

    4.0

    1.0288

    1.0270

    1.0215

    1.0124

    1.0000

    [0,2]

     

    4.5

    1.0224

    1.0210

    1.0167

    1.0097

    1.0000

    [0,2]

     

    5.0

    1.0180

    1.0168

    1.0134

    1.0078

    1.0000

    [0,2]

    4

    2.5

    1.0477

    1.0446

    1.0353

    1.0203

    1.0000

    [0,2]

     

    3.0

    1.0317

    1.0297

    1.0236

    1.0136

    1.0000

    [0,2]

     

    3.5

    1.0227

    1.0213

    1.0169

    1.0098

    1.0000

    [0,2]

     

    4.0

    1.0171

    1.0160

    1.0128

    1.0074

    1.0000

    [0,2]

     

    4.5

    1.0134

    1.0125

    1.0100

    1.0058

    1.0000

    [0,2]

     

    5.0

    1.0107

    1.0101

    1.0080

    1.0047

    1.0000

    [0,2]

    5

    2.5

    1.0278

    1.0260

    1.0207

    1.0120

    1.0000

    [0,2]

     

    3.0

    1.0187

    1.0175

    1.0140

    1.0081

    1.0000

    [0,2]

     

    3.5

    1.0135

    1.0127

    1.0101

    1.0059

    1.0000

    [0,2]

     

    4.0

    1.0101

    1.0095

    1.0076

    1.0044

    1.0000

    [0,2]

     

    4.5

    1.0080

    1.0075

    1.0060

    1.0035

    1.0000

    [0,2]

     

    5.0

    1.0065

    1.0061

    1.0048

    1.0028

    1.0000

    [0,2]

    6

    2.5

    1.0196

    1.0183

    1.0146

    1.0085

    1.0000

    [0,2]

     

    3.0

    1.0133

    1.0124

    1.0099

    1.0058

    1.0000

    [0,2]

     

    3.5

    1.0096

    1.0090

    1.0072

    1.0042

    1.0000

    [0,2]

     

    4.0

    1.0073

    1.0068

    1.0055

    1.0032

    1.0000

    [0,2]

     

    4.5

    1.0057

    1.0054

    1.0043

    1.0025

    1.0000

    [0,2]

     

    5.0

    1.0046

    1.0043

    1.0035

    1.0020

    1.0000

    [0,2]

    7

    2.5

    1.0138

    1.0129

    1.0103

    1.0060

    1.0000

    [0,2]

     

    3.0

    1.0094

    1.0088

    1.0070

    1.0041

    1.0000

    [0,2]

     

    3.5

    1.0068

    1.0064

    1.0051

    1.0030

    1.0000

    [0,2]

     

    4.0

    1.0052

    1.0049

    1.0039

    1.0023

    1.0000

    [0,2]

     

    4.5

    1.0041

    1.0038

    1.0031

    1.0018

    1.0000

    [0,2]

     

    5.0

    1.0033

    1.0031

    1.0025

    1.0014

    1.0000

    [0,2]

    8

    2.5

    1.0107

    1.0100

    1.0080

    1.0046

    1.0000

    [0,2]

     

    3.0

    1.0073

    1.0068

    1.0055

    1.0032

    1.0000

    [0,2]

     

    3.5

    1.0053

    1.0050

    1.0040

    1.0023

    1.0000

    [0,2]

     

    4.0

    1.0040

    1.0038

    1.0030

    1.0018

    1.0000

    [0,2]

     

    4.5

    1.0032

    1.0030

    1.0024

    1.0014

    1.0000

    [0,2]

     

    5.0

    1.0025

    1.0024

    1.0019

    1.0011

    1.0000

    [0,2]

5. Conclusion

It is observed from Table 1 that the values of the relative efficiencies eis,i = 1,2,…,7 of the proposed minimum mean squared error (MMSE) estimators T0i, i = 1,2,3,4 with respect to Lesitha and Thomas (2012) estimators tis,i = 1,2,3,4 respectively are greater than ‘unity’. Thus, the proposed estimators T0i, i = 1,2,3,4 are more efficient than the corresponding usual estimators tis,i = 1,2,3,4 respectively.

It is further observed that the values of the relative efficiency e5 of T04 with respect to T01 are the largest among eis,i = 1,2,…,7, from which follows that the proposed MMSE estimator T04 is the best estimator among Lesitha and Thomas (2012) estimators tis,i = 1,2,3,4 and MMSE estimators T0i,i = 1,2,3,4.

Tables 2 to 5 demonstrate that for fixed (n, β) the values of relative efficiencies eis,i = 8,9,…,15 increase as λ increases up to 1, while it decreases if λ goes beyond ‘unity’. When the value of λ is unity (i.e. the guessed value α0 coincides with the true value α) a higher gain in efficiency is seen. For fixed values of (n, λ) the values of eis,i = 8,9,…,15 decrease as β increases. When (β, λ) are fixed the values of eis,i = 8,9,…,15 also decrease as sample size n increases. A higher gain in efficiency is obtained when the sample size n is small. In general, the estimators T1i*,i=1,2,3,4 are more efficient than Lesitha and Thomas (2012) estimators tis,i = 1,2,3,4 and MMSE estimators T0i, i = 1,2,3,4 respectively when λ∈(0,2.42). It is further observed that T1i*,i=1,2,3,4 are respectively better than MMSE estimators T0i,i = 1,2,3,4 when λ∈[0,2].

Acknowledgement

The authors are highly grateful to the referees for their constructive comments/ suggestions that helped in the improvement of the revised version of the paper.

References


  1. Ahmad, M. I., Sinclair, C. D., Werritty, A., (1988). Log-logistic flood frequency analysis. J Hydrol, 98, pp. 205–224.
    [CROSSREF]
  2. Balakrishnan, N., Malik, H. J., (1987). Best linear unbiased estimation of location and scale parameter of the log-logistic distribution. Commun Stat Theory Methods, 16, pp. 3477–3495.
    [CROSSREF]
  3. Bennett, S., (1983). Log-logistic regression models for survival data. J R Stat Soc, Ser C 32, pp. 165–171.
  4. Chen, Z., Bai, Z., Sinha, B. K., (2004). Ranked set sampling, theory and applications. Lecture Notes in Statistics, Springer, New York.
    [CROSSREF]
  5. Fisk, P. R., (1961). The graduation of income distributions. Econometrica, 29, pp. 171–185.
    [CROSSREF]
  6. Geskus, R. B., (2001). Methods for estimating the AIDS incubation time distribution when data of seroconversion is censored. Stat Med, 20, 795–812.
    [CROSSREF]
  7. Lesitha, G., Thomas, P. Y., (2012). Estimation of the scale parameter of A LOG-LOGISTICS DISTRIBUTION. METRIKA, DOI 10.1007/S00184-012-0397-5.
    [CROSSREF] [URL]
  8. Mcintyre, G. A., (1952). A method for unbiased selective sampling using ranked sets. Aust J Agric Res, 3, pp. 385–390.
    [CROSSREF]
  9. Mehta, V., (2015). Estimation in Morgenstern Type Bivariate Exponential Distribution with Known Coefficient of Variation by Ranked Set Sampling. Proceeding of the “30th M. P. Young Scientist Congress” (MPYSC-2015), M. P. Council of Science and Technology, Vigyan Bhawan, Nehru Nagar, Bhopal -462 003, Madhya Pradesh, India.
    [CROSSREF] [URL]
  10. Mehta, V., Singh, H. P., (2014). Shrinkage Estimators of Parameters of Morgenstern Type Bivariate Logistic Distribution Using Ranked Set Sampling. Journal of Basic and Applied Engineering Research (JBAER), 1 (13), pp. 1–6.
  11. Muttlak, H. A., (1997). Median ranked set sampling. J Appl Stat Sci. 6, pp. 245–255.
  12. Ragab, A., Green, J., (1984). On order statistics from the log-logistic distribution and their properties. Commun Stat Theory Methods, 13, pp. 2713–2724.
    [CROSSREF]
  13. Robson, A., Reed, D., (1999). Flood estimation handbook, 3. Statistical procedures for flood frequency estimation. Institute of Hydrology, Wallingford, UK.
  14. Shoukri, M. M., Mian, I. U. M., Tracy, D., (1988). Sampling properties of estimators of log-logistic distribution with application to Canadian precipitation data. Can J Stat, 16, pp. 223–236.
    [CROSSREF]
  15. Singh, H. P., Mehta, V., (2013). An Improved Estimation of Parameters of Morgenstern Type Bivariate Logistic Distribution Using Ranked Set Sampling. STATISTICA, 73 (4), pp. 437–461.
  16. Singh, H. P., Mehta, V., (2016a). Improved Estimation of Scale Parameter of Morgenstern Type Bivariate Uniform Distribution Using Ranked Set Sampling. Communications in Statistics – Theory and Methods, 45 (5), pp. 1466–1476.
    [CROSSREF]
  17. Singh, H. P., Mehta, V., (2014a). Linear shrinkage estimator of scale parameter of Morgenstern type bivariate logistic distribution using ranked set sampling. Model Assisted Statistics and Applications (MASA), 9, pp. 295–307.
    [CROSSREF]
  18. Singh, H. P., Mehta, V., (2014b). An Alternative Estimation of the Scale Parameter for Morgenstern Type Bivariate Log-Logistic Distribution Using Ranked Set Sampling. Journal of Reliability and Statistical Studies, 7 (1), pp. 19–29.
  19. Singh, H. P., Mehta, V., (2015). Estimation of Scale Parameter of a Morgenstern Type Bivariate Uniform Distribution Using Censored Ranked Set Samples. Model Assisted Statistics and Applications (MASA), 10, pp. 139–153.
    [CROSSREF]
  20. Singh, H. P., Mehta, V., (2016b). Some Classes of Shrinkage Estimators in the Morgenstern Type Bivariate Exponential Distribution Using Ranked Set Sampling. Hacettepe Journal of Mathematics and Statistics, 45 (2), pp. 575–591.
  21. Singh, H. P., Mehta, V., (2016c). A Class of Shrinkage Estimators of Scale Parameter of Uniform Distribution Based on K- Record Values. National Academy Science Letters, 39, pp. 221–227.
    [CROSSREF]
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FIGURES & TABLES

REFERENCES

  1. Ahmad, M. I., Sinclair, C. D., Werritty, A., (1988). Log-logistic flood frequency analysis. J Hydrol, 98, pp. 205–224.
    [CROSSREF]
  2. Balakrishnan, N., Malik, H. J., (1987). Best linear unbiased estimation of location and scale parameter of the log-logistic distribution. Commun Stat Theory Methods, 16, pp. 3477–3495.
    [CROSSREF]
  3. Bennett, S., (1983). Log-logistic regression models for survival data. J R Stat Soc, Ser C 32, pp. 165–171.
  4. Chen, Z., Bai, Z., Sinha, B. K., (2004). Ranked set sampling, theory and applications. Lecture Notes in Statistics, Springer, New York.
    [CROSSREF]
  5. Fisk, P. R., (1961). The graduation of income distributions. Econometrica, 29, pp. 171–185.
    [CROSSREF]
  6. Geskus, R. B., (2001). Methods for estimating the AIDS incubation time distribution when data of seroconversion is censored. Stat Med, 20, 795–812.
    [CROSSREF]
  7. Lesitha, G., Thomas, P. Y., (2012). Estimation of the scale parameter of A LOG-LOGISTICS DISTRIBUTION. METRIKA, DOI 10.1007/S00184-012-0397-5.
    [CROSSREF] [URL]
  8. Mcintyre, G. A., (1952). A method for unbiased selective sampling using ranked sets. Aust J Agric Res, 3, pp. 385–390.
    [CROSSREF]
  9. Mehta, V., (2015). Estimation in Morgenstern Type Bivariate Exponential Distribution with Known Coefficient of Variation by Ranked Set Sampling. Proceeding of the “30th M. P. Young Scientist Congress” (MPYSC-2015), M. P. Council of Science and Technology, Vigyan Bhawan, Nehru Nagar, Bhopal -462 003, Madhya Pradesh, India.
    [CROSSREF] [URL]
  10. Mehta, V., Singh, H. P., (2014). Shrinkage Estimators of Parameters of Morgenstern Type Bivariate Logistic Distribution Using Ranked Set Sampling. Journal of Basic and Applied Engineering Research (JBAER), 1 (13), pp. 1–6.
  11. Muttlak, H. A., (1997). Median ranked set sampling. J Appl Stat Sci. 6, pp. 245–255.
  12. Ragab, A., Green, J., (1984). On order statistics from the log-logistic distribution and their properties. Commun Stat Theory Methods, 13, pp. 2713–2724.
    [CROSSREF]
  13. Robson, A., Reed, D., (1999). Flood estimation handbook, 3. Statistical procedures for flood frequency estimation. Institute of Hydrology, Wallingford, UK.
  14. Shoukri, M. M., Mian, I. U. M., Tracy, D., (1988). Sampling properties of estimators of log-logistic distribution with application to Canadian precipitation data. Can J Stat, 16, pp. 223–236.
    [CROSSREF]
  15. Singh, H. P., Mehta, V., (2013). An Improved Estimation of Parameters of Morgenstern Type Bivariate Logistic Distribution Using Ranked Set Sampling. STATISTICA, 73 (4), pp. 437–461.
  16. Singh, H. P., Mehta, V., (2016a). Improved Estimation of Scale Parameter of Morgenstern Type Bivariate Uniform Distribution Using Ranked Set Sampling. Communications in Statistics – Theory and Methods, 45 (5), pp. 1466–1476.
    [CROSSREF]
  17. Singh, H. P., Mehta, V., (2014a). Linear shrinkage estimator of scale parameter of Morgenstern type bivariate logistic distribution using ranked set sampling. Model Assisted Statistics and Applications (MASA), 9, pp. 295–307.
    [CROSSREF]
  18. Singh, H. P., Mehta, V., (2014b). An Alternative Estimation of the Scale Parameter for Morgenstern Type Bivariate Log-Logistic Distribution Using Ranked Set Sampling. Journal of Reliability and Statistical Studies, 7 (1), pp. 19–29.
  19. Singh, H. P., Mehta, V., (2015). Estimation of Scale Parameter of a Morgenstern Type Bivariate Uniform Distribution Using Censored Ranked Set Samples. Model Assisted Statistics and Applications (MASA), 10, pp. 139–153.
    [CROSSREF]
  20. Singh, H. P., Mehta, V., (2016b). Some Classes of Shrinkage Estimators in the Morgenstern Type Bivariate Exponential Distribution Using Ranked Set Sampling. Hacettepe Journal of Mathematics and Statistics, 45 (2), pp. 575–591.
  21. Singh, H. P., Mehta, V., (2016c). A Class of Shrinkage Estimators of Scale Parameter of Uniform Distribution Based on K- Record Values. National Academy Science Letters, 39, pp. 221–227.
    [CROSSREF]

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