L-Moments, TL-Moments Estimation and Recurrence Relations for Moments of Order Statistics from Expon

Statistics Research Letters (SRL) Volume 3, 2014

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L-Moments, TL-Moments Estimation and Recurrence Relations for Moments of Order Statistics from Exponentiated Inverted Weibull Distribution Jagdish Saran*1, Devendra Kumar2, N. Pushkarna*1, Rashmi Tiwari*1 Department of Statistics, University of Delhi, Delhi-110007, India

*1

Department of Statistics, Amity Institute of Applied Sciences, Amity University, Noida-201 303, India

2

jagdish_saran52@yahoo.co.in; 2devendrastats@gmail.com

*1

Abstract In this paper we have established exact expressions and some recurrence relations for single and product moments of order statistics from exponentiated inverted Weibull distribution. Further the characterization of this distribution has been considered on using a recurrence relation for single moments. The first four moments and variances of order statistics are computed for various values of parameters α and β . We have also obtained L-moments and TL-moments of the above distribution and used them to find the Lmoments and TL-moments estimators of the parameters α and β of the distribution.

Keywords Order Statistics; Record Values; Exponentiated Inverted Weibull Distribution; Single and Product Moments; Recurrence Relations; L-Moments; TL-Moments and Characterization

Introduction

the rank order statistics in which the order of the value of observation rather than its magnitude is considered.

It plays an important role both in model building and in statistical inference. For example: extreme values are important in oceanography (waves and tides), material strength (strength of a chain depends on the weakest link) and meteorology (extremes of temperature, pressure, etc).

Let X 1 , X 2 , , X n be a random sample of size n from a continuous probability density function ( pdf )

and the distribution function (df ) F ( x) . Then the

pdf of X r:n , 1 ≤ r ≤ n , is given by

= f r:n ( x) Cr:n [ F ( x)]r −1[1 − F ( x)]n − r f ( x) , −∞ < x < ∞ ,

given by

= f r , s:n ( x, y ) Cr , s:n [ F ( x)]r −1[ F ( y ) − F ( x)]s − r −1

ascending order of magnitude such that X 1:n ≤ X 2:n ≤  ≤ X r:n ≤ ≤ X n:n , then X r:n is called the

×[1 − F ( y )]n − s f ( x) f ( y ) , −∞ < x < y < ∞ ,

(1.2)

where

X n:n = max ( X 1 , X 2 , , X n ) are called extreme order statistics or the smallest and the largest order statistics.

The subject of order statistics deals with the properties and applications of these ordered random variables and of functions involving them (David and Nagaraja, 2003). Asymptotic theory of extremes and related developments of order statistics are well described in an applausive work of Galambos (1987). Also, references may be made to Sarhan and Greenberg (1962), Balakrishnan and Cohen (1991), Arnold et al.

(1992) and the references therein. It is different from

(1.1)

and the joint pdf of X r:n and X s:n , 1 ≤ r < s ≤ n , is

If random variables X 1 , X 2 , , X n are arranged in

r − th order statistic. X 1:n = min ( X 1 , X 2 , , X n ) and

f ( x)

C = r :n

n! = [ B(r , n − r + 1)]−1 (r − 1)!(n − r )!

and

Cr= , s:n

n! = [ B(r , s − r , n − s + 1)] −1 . (r − 1)!( s − r − 1)!(n − s )!

Several recurrence relations between raw and central moments of order statistics from different distributions are available in the literature. The main utility and advantage of such recurrence relations

63

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Statistics Research Letters (SRL) Volume 3, 2014

between moments are that having obtained one of the these results are used to evaluate the first four moments, the moments of higher/lower order can be moments and variances of order statistics and are easily obtained without indulging into copious presented in Table 2.1 and Table 2.2. We shall first computational work. Various developments on order establish the exact expression for E[ X rj:n ] . Using (1.1), statistics and related topics have been studied by Joshi we have (1978), Balakrishnan and Joshi (1982), Khan et a1.

(1983a, b), Khan and Khan (1987), Balakrishnan and E[ X rj:n ] Cr:n ∫0∞ x j [ F ( x)]r −1 f ( x)[1 − F ( x)]n − r dx . (2.1) = Malik (1985) and Mohie El-Din et a1. (1991).

On expanding binomially in (2.1), we get Balakrishnan et a1. (1988), Malik et a1. (1988), and Saran and Pushkarna (1999a, 1999b, 2000) derived n−r n−r ∞ j u + r −1 E= f ( x)dx (2.2) [ X rj:n ] Cr:n ∑ (−1)u  recurrence relations and identities for single and  ∫0 x [ F ( x)] u u =0   product moments of order statistics for specific distributions. Balakrishnan and Malik (1986) Making the substitution t = − ln F ( x) in (2.2), we find established exact and explicit expressions for the that means and product moments of order statistics from n−r  n − r  ∞ − j / β − (u + r )t the linear-exponential distribution; also they E[ X rj:n ] α j / β Cr:n ∑ (−1)u  e dt =  ∫0 t u =0 established some recurrence relations for both single  u  and product moments of order statistics for the same n−r  n − r  Γ(1 − j / β ) model. .

(2.3) = α j / β Cr:n ∑ (−1)u   1− j / β u =0  u  (u + r ) In this study we obtain some recurrence relations for The recurrence relation for single moments of order single and product moments of order statistics from statistics from exponentiated inverted Weibull exponentiated inverted Weibull distribution. Further, the distribution has been characterized on using a distribution (1.3) can be obtained in the following recurrence relation for single moments. In the later theorem. part, we have also derived L-moments and TLTheorem 2.1 For 1 ≤ r ≤ n , n ≥ 2 and j = 1, 2, , moments and the estimation of parameters based on them for exponentiated inverted Weibull distribution. j +1 (2.4) E= [ X rj++1:1n ] E [ X rj:+n1 ] + E [ X rj:+n β +1 ] . αβ r A random variable X is said to have exponentiated inverted Weibull distribution (Flaih et al., 2012) if its Proof From (1.1) and (1.5), we have pdf is of the form = E [ X rj:+n β +1 ] αβ Cr:n ∫0∞ x j [ F ( x)]r [1 − F ( x)]n − r dx . − ( β +1) −α x − β , x > 0 , α, β > 0 (1.3) f ( x) = αβ x e Integrating by parts treating x j for integration and the and the corresponding df is rest of the integrand for differentiation, we get −β

F ( x) = e −α x ,

x > 0 , α, β > 0 .

(1.4)

 [ X rj:+n β +1 ] αβ Cr:n (n − r ) ∫0∞ x j +1[ F ( x)]r E= 

(1.5)

×[1 − F ( x)]n − r −1 f ( x)dx

It is easy to see that

x β +1 f ( x) = αβ F ( x) .

The inverted Weibull and inverted distributions are considered as special exponentiated inverted Weibull when α = 1 and α= β= 1 , respectively.

exponential cases of the distribution More details

on this distribution can be found in Flaih et al. (2012).

Relations for Single Moments In this section, we have derived an exact expression for the single moments of order statistics from exponentiated inverted Weibull distribution. Further

64

 − r ∫0∞ x j +1[ F ( x)]r −1[1 − F ( x)]n − r f ( x)dx   and hence the result.

Remark 2.1 Putting α = 1 , in (2.4), we get the recurrence relation for single moments of order statistics from inverted Weibull distribution.

Remark 2.2 For α = 1 and β = 1 , the relation in (2.4), gives the recurrence relation for single moments of order statistics from inverted exponential distribution.

Statistics Research Letters (SRL) Volume 3, 2014

TABLE 2.1: FIRST FOUR MOMENTS OF ORDER STATISTICS FROM EXPONENTIATED INVERTED WEIBULL DISTRIBUTION

n

1 2

3

4

5

n

r

j =1 , β = 2

α =1

α =2

α =3

1

1.77245

4.44288

5.44140

1

1.03828

2.60258

3.18750

2

2.50663

6.28319

7.69530

1

0.86746

2.17439

2.66308

2

1.37992

3.45896

4.23634

3

3.06998

7.69530

9.42478

1

0.78506

1.96785

2.41011

2

1.11465

2.79402

3.42196

3

1.6452

4.12390

5.05072

4

3.54491

8.88577

10.88280

1

0.73458

1.84131

2.25514

2

0.98699

2.47401

3.03003

3

1.30615

3.27403

4

1.87123

4.69048

5

3.96333

4.44288

12.16734

r

1.48635

1.61190

3

1.45032

1.93958

2.10342

1

0.89788

1.20078

1.30222

2

1.03019

1.37772

1.49410

3

1.19264

1.59497

1.72970

4

1.53621

2.05445

2.22799

1

0.87610

1.17165

1.27062

2

0.98501

1.31731

1.42858

3

1.09795

1.46834

1.59237

4

1.25576

1.67939

1.82125

5

1.60632

2.14821

2.32968

j =2, β =3

4.00985 5.74464

2

1

1.10533

4.700460

6.159340

2

4.25255

18.08418

23.69698

1

0.85158

3.621400

4.745370

2

1.61282

6.858580

8.987280

3

5.57242

23.69698

31.05183

1

0.73962

3.145270

4.121470

2

1.18747

5.049780

6.617090

3

2.03816

8.667380

11.35748

4

6.75050

28.70685

37.61661

1

0.67411

2.866710

3.756450

2

1.00164

4.259530

5.581560

3

1.46622

6.235170

8.170380

4

2.41946

10.28885

13.48221

5

7.83326

33.31134

43.65021

1.00215

1.70976

1.95718

1.70608

2.91071

3.33194

1

0.89708

1.53050

1.75198

2

1.21229

2.06827

2.36758

3

1.95298

3.33194

3.81412

1

0.84236

1.43713

1.64510

2

1.06127

1.81061

2.07263

3

1.36332

2.32593

2.66253

4

2.14953

3.66727

4.19798

1

0.80740

1.37749

1.57683

2

0.98218

1.67569

1.91818

3

1.17989

2.01299

2.30430

4

1.48560

2.53456

2.90135

5

2.31551

n

r

α =1

α =2

α =3

1

1

1.22542

1.78577

1.97628

1

0.99356

1.44789

1.60235

4.52213

2

1.45727

2.12365

2.35020

1

0.91717

1.33656

1.47915

2

1.14634

1.67054

1.84875

3

1.61274

2.35020

2.60093

1

0.87598

1.27654

1.41272

2

1.04074

1.51664

1.67844

3

1.25195

1.82444

2.01907

4

1.73300

2.52546

2.79488

1

0.84915

1.23745

1.36946

2

0.98327

1.43290

1.58576

3

1.12693

1.64224

1.81744

4

1.33530

1.94590

2.1535

5

1.83243

2.67035 j =1 , β = 5

1.11141

α =3

2

5

1.35019

2

14.92816

1

4

1.24502

α =2

2.64456

3

0.93096

11.39232

2.31024

2

1.93958

1

α =1

j =1 , β = 3

3.95045 j =1 , β = 4

1.43742

1.78850

2.67894

1.35412

5

5

1.32546

1.33735

1

1

4

4

0.99111

2

1

α =3

3

3

1

r

α =2

2

2

n

α =1

1

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2.95522

n

r

α =1

α =2

α =3

1

1

1.16423

1.55698

1.68850

3

4

5

j=2, β =4

n

r

α =1

α =2

α =3

1

1

1.77245

4.44288

5.44140

1

1.03828

2.60258

3.18750

2

2.50663

6.28319

7.69530

1

0.86746

2.17439

2.66308

2

1.37992

3.45896

4.23634

3

3.06998

7.69530

9.42478

1

0.78506

1.96785

2.41011

2

1.11465

2.79402

3.42196

3

1.64520

4.12390

5.05072

4

3.54491

8.88577

10.88280

1

0.73458

1.84131

2.25514

2

0.98699

2.47401

3.03003

3

1.30615

3.27403

4.00985

4

1.87123

4.69048

5.74464

5

3.96333

9.93459

12.16734

n

r

α =1

α =2

α =3

1

1

1.48919

2.92626

3.44152

1

1.01338

1.99130

2.34193

2

1.96500

3.86123

4.54111

1

0.88357

1.73622

2.04193

2

1.27301

2.50147

2.94192

3

2.31100

4.54111

5.34070

2

3

4

5

2

3

j =2, β =5

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4

5

1

0.81791

1.60720

1.89020

2

1.08054

2.12326

2.49712

2

1.02110

8.16179

11.28908

3

1.62580

12.99522

3

1.46548

2.87967

17.97450

3.38673

4

3.03718

24.27652

4

2.59283

33.57835

5.09492

5.99203

5

16.63675

132.97960

183.93231

1 2

0.77665

1.52612

1.79484

0.98297

1.93154

2.27164

3

1.22689

2.41084

2.83534

4

1.62454

3.19223

3.75432

5

2.83491

5.57059

6.55146

j =3 , β =4

n

r

α =1

α =2

α =3

1

1

3.62561

22.10725

29.96423

2

3

4

5

n

1 2

3

4

5

1

1.15370

7.03468

9.53483

2

6.09752

37.17981

50.39362

1

0.84886

5.17594

7.01548

2

1.76337

10.75218

14.57353

3

8.26460

50.39362

68.30367

1

0.72093

4.39590

5.95821

2

1.23264

7.51606

10.18730

3

2.29409

13.98829

18.95977

4

10.25477

62.52873

84.75163

1

0.64793

3.95075

5.35485

5

TABLE 2.2: VARIANCES OF ORDER STATISTICS FROM EXPONENTIATED INVERTED WEIBULL DISTRIBUTION

β =3

n

r

α =1

α =2

α =3

1 2

1 1 2 1 2 3 1 2 3 4 1 2 3 4 5

0.845299 0.101025 1.341841 0.046827 0.143173 1.758289 0.030050 0.061176 0.179519 2.130021 0.022215 0.036962 0.074080 0.212453 2.471673

6.055111 1.777181 9.611947 1.278970 2.580839 12.59516 1.079927 1.771471 3.257430 15.25798 0.969231 1.451593 2.183041 3.864856 17.70528 β =4

7.934462 2.328786 12.59516 1.675936 3.381845 16.50432 1.415116 2.321295 4.268414 19.99357 1.270057 1.902145 2.860582 5.064378 23.20055

3

4

5

2

1.01295

6.17650

8.37164

3

1.56218

9.52541

12.91077

4

2.78204

16.96354

22.99244

n

r

α =1

α =2

α =3

5

12.12296

100.19143

r

73.92003 j =3 , β =5

1 2

α =1

1 1 2 1 2 3 1 2 3 4 1 2 3 4 5

0.270796 0.051119 0.382994 0.026259 0.065825 0.469050 0.017719 0.031510 0.077821 0.541621 0.013524 0.020170 0.036179 0.088204 0.60553

1.253906 0.506195 1.773301 0.387997 0.668256 2.171860 0.338296 0.493823 0.795319 2.507822 0.310027 0.420808 0.577078 0.903953 2.803821 β =5

1.535717 0.619974 2.171860 0.475195 0.818463 2.659943 0.414332 0.604799 0.974076 3.071446 0.379719 0.515395 0.706762 1.107078 3.434015

α =2

α =3

1

2.21816

7.45768

9.51170

1

1.07422

3.61163

4.60636

2

3.36210

11.30372

14.41705

1

0.85628

2.87890

3.67183

2

1.51009

5.07708

6.47543

3

4.28811

14.41705

18.38785

1

0.75646

2.54331

3.24380

2

1.15573

3.88570

4.95591

3

1.86445

6.26846

7.99495

4

5.09599

17.13324

21.85216

1

0.69694

2.34319

2.98857

2

0.99454

3.34376

4.26471

3

1.39752

4.69861

5.99272

4

2.17573

7.31503

9.32977

5

5.82606

19.58780 j =4 β =5

24.98275

3

4

5

n

r

α =1

α =2

α =3

1

1

2

1 2 1

3

0.133759 0.031081 0.176495 0.016883

0.502073 0.234456 0.662498 0.186145

0.590488 0.275754 0.779139 0.218917

2 3

0.037778

0.292234

0.343698

0.207572

0.779139

0.916324

1

0.011722

0.165327

0.194423

2

0.019249

0.225148

0.264785

3

0.043090 0.232889

0.335741 0.874155

0.394868 1.028091 0.180365

n

r

α =1

α =2

α =3

1

1

4.59084

36.69518

50.75537

1

1.18856

9.50034

13.14051

2

7.99312

63.89002

88.37024

1

0.84894

6.78572

9.38575

2

1.86780

14.92958

20.65003

3

11.05578

88.37024

122.23034

4

1

0.71093

5.68257

7.85992

2

1.26298

10.09516

13.96324

1 2

0.009099

0.153356

0.012725

0.196234

0.230799

3

2.47263

19.76400

27.33681

3

0.021396

0.254818

0.299698

4

13.91684

111.23898

153.86152

4

0.047607

0.371879

0.437368

1

0.63339

5.06277

7.00263

5

0.254646

0.955784

1.124051

2

3

4

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Statistics Research Letters (SRL) Volume 3, 2014

4

5

Statistics Research Letters (SRL) Volume 3, 2014

Relations for Product Moments The recurrence relation for product moments of order statistics from exponentiated inverted Weibull distribution (1.3) has been presented in the following theorem.

Theorem 3.1

i +1 E[ X ri :+nβ +1 X sj:n ] . (3.1) αβ r

From (1.2), we have

i + β +1 E [ X= X sj:n ] r :n

Cr , s:n ∫0∞

n−s

y [1 − F ( y )] j

I ( y ) f ( y )dy , (3.2)

where

I ( y)

Theorem 4.1 Let X be a non-negative random variable having an absolutely continuous distribution function F ( x) with F (0) = 0 and 0 < F ( x) < 1 for all

x > 0 , then E= [ X rj++1:1n ] E [ X rj:+n1 ] +

For 1 ≤ r < s ≤ n , s − r ≥ 2 ,

E [= X ri ++11:n X sj:n ] E[ X ri :+n1 X sj:n ] + Proof

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j +1 E [ X rj:+n β +1 ] αβ r

(4.1)

if and only if −β

F ( x) = e −α x ,

x > 0 , α, β > 0 .

Proof The necessary part follows immediately from equation (2.4). On the other hand if the recurrence relation in equation (4.1) is satisfied, then on using equation (1.1), we have

y i + β +1 [ F ( x)]r −1[ F ( y ) − F ( x)]s − r −1 f ( x)dx ∫0 x

Cr:n ∫0∞ x j + β +1[ F ( x)]r −1[1 − F ( x)]n − r f ( x)dx

= αβ ∫0y xi [ F ( x)]r [ F ( y ) − F ( x)]s − r −1 dx , αβ (n − r ) Cr:n ∞ j +1 r n − r −1 = f ( x)dx ∫0 x [ F ( x)] [1 − F ( x)] j +1 upon using the relation in (1.5). Integrating now by parts treating xi for integration and the rest of the αβ rCr:n ∞ j +1 r −1 n−r − ∫ x [ F ( x)] [1 − F ( x)] f ( x)dx integrand for differentiation, we obtain j +1 0

αβ  y i +1 r s −r −2 αβ Cr:n ∞ j +1 r n − r −1 f ( x) ( s − r − 1) ∫0 x [ F ( x)] [ F ( y ) − F ( x)] = f ( x) ∫ x [ F ( x)] [1 − F ( x)] i +1  j +1 0  − r ∫0y xi +1[ F ( x)]r −1[ F ( y ) − F ( x)]s − r −1 f ( x)dx  .  r[1 − F ( x)]  × ( n − r ) −   dx .

F ( x)   Upon substituting the above expression for I ( y ) in Let (3.2), we have, after simplifications, the recurrence relation (3.1). −[ F ( x)]r [1 − F ( x)]n − r h( x ) = I ( y) =

Remark 3.1 Putting α = 1 , in (3.1), we deduce the recurrence relation for product moments of order statistics from inverted Weibull distribution.

Remark 3.2

For α = 1 and β = 1 , the relation in (3.1)

gives the recurrence relation for product moments of order statistics from inverted exponential distribution.

Remark 3.3

At j = 0 , Theorem 3.1 can be reduced

to Theorem 2.1.

(4.2)

(4.3)

and

 r[1 − F ( x)]  = h′( x) [ F ( x)]r f ( x)[1 − F ( x)]n − r −1 (n − r ) − .

F ( x)   Thus,

Cr:n ∫0∞ x j + β +1[ F ( x)]r −1[1 − F ( x)]n − r f ( x)dx =

αβ Cr:n j +1

∞ j +1 ∫0 x h′( x)dx .

(4.4)

Remark 3.4 The explicit expression for the single Now integrating RHS in (4.4) by parts and using the moments of order statistics allows us to evaluate the value of h( x) from (4.3), we get means and variances of all order statistics. Tables 2.1 and 2.2 present the first four moments and variances Cr:n ∫0∞ x j + β +1[ F ( x)]r −1[1 − F ( x)]n − r f ( x)dx of order statistics of exponentiated inverted Weibull distribution, respectively. = α β C ∫ ∞ x j [ F ( x)]r [1 − F ( x)]n − r dx r :n 0

Characterization In this section, we shall characterize exponentiated inverted weibull distribution based on recurrence relation for single moments of order statistics.

which reduces to

Cr:n ∫0∞ x j [ F ( x)]r −1[1 − F ( x)]n − r

0. ×{x β +1 f ( x) − αβ F ( x)} dx =

(4.5)

67

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Statistics Research Letters (SRL) Volume 3, 2014

Now applying a generalization of the Müntz-Szász Theorem (Hwang and Lin, 1984) to equation (4.5), we get

(1.3) and X 1:n ≤ X 2:n ≤  ≤ X r:n ≤ ≤ X n:n denote the corresponding order statistics. Then the rth L-moment defined by Hosking (1990) is as follows:

f ( x) = αβ x − ( β +1) F ( x) which proves that

F ( x) = e −α x ,

x > 0 , α, β > 0 .

E ( X i:r ) =

L-moments and TL-moments It may be mentioned that Hosking (1990) introduced the L-moments as a linear combination of probability weighted moments. Similar to ordinary moments, Lmoments can also be used for parameter estimation, interval estimation and hypothesis testing. Hosking has shown that first four L-moments of a distribution measure, respectively, the average, dispersion, symmetry and tail weight (or peakedness) of the distribution. L-moments have turned out to be a popular tool in parametric estimation and distribution identification problems in different scientific ares such as hydrology in the estimation of flood frequency, etc. (see e.g. Hosking (1990), Stedinger et al. (1993), Hosking and Wallis (1997)). In comparison to the conventional moments, L-moments have lower sample variances and are more robust against outliers. For example, L1 is the same as the population mean, is

can be derived as follows:

L= 4

random

sample

(5.4)

1 E ( X 3:3 − 2 X 2:3 + X 1:3 ) 3

(5.5)

1 E ( X 4:4 − 3 X 3:4 + 3 X 2:4 − X 1:4 ).

4

(5.6)

 1 L1 = α 1/ β Γ 1 −   β

(5.7)

  1  2 L2 = α 1/ β Γ 1 −   1−(1/ β ) − 1   β   (2) 

(5.8)

  1  6 6 L3 = α 1/ β Γ 1 −   1−(1/ β ) − 1−(1/ β ) + 1   (2)  β   (3) 

(5.9)

  1   10 15 6 L= 2α 1/ β Γ 1 −   1−(1/ β ) − 1−(1/ β ) + 1−(1/ β ) − 2  .

4   (3) (2)  β   (4)  (5.10) In

particular,

L1 , L2, L3 and

L4

are

population

measures of the location, scale, skewness and kurtosis, respectively.

The L-skewness τ 3 and L-kurtosis τ 4 of exponentiated inverted Weibull distribution will be

  6 6  1−(1/ β ) − 1−(1/ β ) + 1 (2) L3  (3)  = τ= 3 L2   2  1−(1/ β ) − 1  (2) 

from

exponentiated inverted Weibull distribution defined in

1 E ( X 2:2 − X 1:2 ) 2

The L-moments of the exponentiated inverted Weibull distribution are obtained by utilizing (5.3) - (5.6) as given below:

L-Moments for Exponentiated Inverted Weibull Distribution a

(5.3)

L= 3

alternatives to the un-scaled measures of skewness and kurtosis µ3 and µ4 , respectively (see Sillitto,

X 1 , X 2 , , X n be

L1 = E ( X 1:1 ) = L2

deviation, is defined in terms of a conceptual sample of size r = 2 Similarly, the L-moments L3 and L4 are

1969). Elamir and Seheult (2003) introduced an extension of L-moments and called them TL-moments (trimmed L-moments). TL-moments are more robust than L-moments and exist even if the distribution does not have a mean, for example the TL-moments exist for Cauchy distribution (see Abdul-Moniem and Selim, 2009 and Shabri et al., 2011). Abdul-Moniem (2007) considered L-moments and TL-moments estimation for the exponential distribution. Similar work has been done by Shahzad and Asghar (2013) for Dagum distribution.

1 ∞ i −1 r −i ∫ x [ F ( x)] [1 − F ( x)] f ( x)dx (5.2) (i − 1)!(r − i )! 0

For r = 1, 2, 3 and 4 in (5.1) , the first four L-moments

defined in terms of a conceptual sample of size r = 1 , while L2 is an alternative to the population standard

68

(5.1)

where −β

Let

1 r −1 k  r − 1 ∑ (−1)   E ( X r − k :r ) , r k =0  k 

Lr =

and

(5.11)

Statistics Research Letters (SRL) Volume 3, 2014

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 10  15 6 2  1−(1/ β ) − 1−(1/ β ) + 1−(1/ β ) − 2    (3) (2) L4  (4)  . (5.12) τ= = 4 L2   2  1−(1/ β ) − 1  (2) 

Selim, 2009). TL-moments reduce to L-moments when t1 = 0 and t2 = 0 .

TL-moments equation (5.16) with t1 = 1 and t2 = 1 is defined as L(1,1) ≡ L(1) r r :

Sample L-moments and L-Moment Estimators

L(1) r =

The L-moments can be estimated from the sample order statistics as follows (see Asquith, 2007):

lr =

1 n r −1 k  r − 1  i − 1   n − i  ∑ ∑ (−1)     X i:n (5.13) k   r −1− k   k   n =i 1 =k 0  r  r

For r = 1, 2, 3 and 4

= L(1) 2

1 E ( X 5:6 − 3 X 4:6 + 3 X 3:6 − X 2:6 ) .

4

(5.21)

L(1) 2=

log 2 .  l2 + x  log    x 

(1) L = 4

Elamir and Seheult (2003) introduced some robust modification of (5.1) (and called it as TL-moments) in which E ( X r − k :r ) is replaced by E ( X r + t1 − k :r + t1 + t2 ) for each

r where t1 smallest and t2 largest are trimmed from the conceptual sample. The following formula gives the rth TL-moments (cf. Elamir and Seheult, 2003): 2

)

1 r −1  r − 1 = ∑  E ( X r + t − k :r + t + t ) . r k =0  k  1

1

2

(5.16)

One can observe that TL-moments are more robust than L-moments and exist even if the distribution does not have a mean, for example the TL-moments exists for Cauchy distribution (cf. Abdul-Moneiem and

(5.22)

20α 1/ β  1 Γ 1 −  3  β

 10  5 6 1 ×  1− (1/ β ) − 1− (1/ β ) − 1− (1/ β ) + 1− (1/ β )  (5) (3) (2)  (4) 

TL-Moments for Exponentiated Inverted Weibull Distribution

1

  

 α 1/ β  1   3 2 1 Γ 1 −   1− (1/ β ) − 1− (1/ β ) − 1− (1/ β )  (5.23) 2 (4) (2)  β   (3)  L(1) = 3

L(rt , t

(5.19) (5.20)

 1  1 1 L1(1)= 6α 1/ β Γ 1 −   1−(1/ β ) − 1−(1/ β )  β (3)    (2)

and

βˆ =

1 E ( X 3:4 − X 2:4 ) 2

The TL-moments of the exponentiated inverted Weibull distribution are obtained by utilizing (5.18) (5.21) and are given below:

Solving (5.14) and (5.15), we get

   l2 + x      log    log 2 log( x ) log 2  x   , = − αˆ log Γ 1 −    log 2 l +x  l +x  log  2 log  2       x   x    

(5.18)

1 E ( X 4:5 − 2 X 3:5 + X 2:5 ) 3

L(1) 3=

L(1) 4=

n   2 2 = l2 X i:n − x x  1−(1/ β ) − 1 . (5.15) ∑ (i − 1)=  (2)  n(n − 1) i =1  

in Eq (5.17), the first four TL-

L1(1) = E ( X 2:3 )

(5.14)

and

(5.17)

moments can be derived as follow:

From equations (5.7), (5.8) and (5.13), the L-moment estimators for the parameters α and β are given by

 1 n 1 l1 = ∑ X i:n = x = α 1/ β Γ 1 −  n i =1  β

1 r −1  r − 1 ∑  E ( X r +1− k :r + 2 ) . r k =0  k 

(5.24)

15α 1/ β  1 Γ 1 −  2  β

 35  14 30 10 1 ×  1− (1/ β ) − 1− (1/ β ) − 1− (1/ β ) + 1−(1/ β ) − 1−(1/ β )  .  (5)  (6) (4) (3) (2)   (5.25) The

TL-skewness

τ 3(1)

and

TL-kurtosis

τ 4(1)

of

exponentiated inverted Weibull distribution will be

τ 3(1) =

L(1) 3 L(1) 2

 10  5 6 1  1− (1/ β ) − 1− (1/ β ) − 1− (1/ β ) + 1− (1/ β )  (5) (3) (2) 40  (4)  (5.26) = 3   3 2 1  1− (1/ β ) − 1− (1/ β ) − 1− (1/ β )  (3) (4) (2)  

69

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Statistics Research Letters (SRL) Volume 3, 2014

for carefully reading the paper and for helpful suggestions which greatly improved the paper.

and

τ 4(1) =

L(1) 4 L(1) 2

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