First derived within the context of life-testing, inverse Gaussian distribution has become one of the most important and

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*Table of contents : Introduction. Inverse Gaussian Distribution and Properties. Order Statistics and Properties. Calculation of Moments and Tables. Illustrations of the Use of Tables. Applications. Bibliography. Tables. Indices.*

CRC Handbook of Tables

for Statistics from

Order Inverse Gaussian Distributions with Applications N. Balakrishnan

Department of Mathematics and Statistics McMaster University Hamilton, Ontario, Canada

William W.S. Chen Internal Revenue Service Washington, DC

CRCPress Boca Raton New York

Library

Data

~f

record is available from the Library of

book contains information obtained frmn authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references arc listed. Reasonable eff01ts have been m.ade publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. may reproduced or transmitted in any form or Neither this bcolr Inechanical. system. 1)'!1.mol_~opyiug, ruicrofilming, and recording, or by any infonnation storrtge or without prior permission writing from the publisher. creating new The conser1: of Fress Ll_,C durc;c nor extend to copying for general distrihuuon. works, or for Specific 'Jmst be obtained in writing from CRC Direct all 2000 Corporate Blvd., N.W., Boca

© 1997 by No claim to original U.S. Government works lntemalional Standard Book Number 0-8493-3! 18-8

2 3 4 5 6 7 8 9 0

Preface

Inverse Gaussian distributions have recently become one of the most commonly studied models in the theoretical literature while being commonly used in the applied literature. Though these distributions have been around for over fifty years, much of the work coneerning inferential methods for the parameters of inverse Gaussian distributions has been carried out only in the recent past. Most of these methods, particularly those based on censored samples, do involve the use of numerical methods to solve some equations. Order statistics and their moments have been studied quite extensively for numerous distributions. However, moments of order statistics from inverse Gaussian distributions have not been discussed in the statistical literature until now primarily due to the extreme eomputational eomplexity in their numerical determination. It is this challenge that motivated us to enter into this tremendous task of computing the means, variances and eovariances of order statistics for all sample sizes up to twenty five and for many choices of the shape parameter. Another major reason for taking up this study is, of eourse, to make use of the tabulated values of means, variances and covariances of order statistics in order to derive the best linear unbiased estimators of the location and scale parameters based on complete as well as Type-II censored samples. Both these goals have been achieved successfully and the extensive tables that emerged out of our efforts have been presented in this volume. We have also presented a few examples to illustrate some practical applic-ations of these tables. Our sincere thanks go to Mr. Robert Stern and Mr. Tim Pletscher (both of CRC Press, Doca Raton) for taking interest in this project and also providing great support during the course of preparation of this volume. Thanks are due to Ms. Nora Konopka and Ms. Suzanne Lassandro (both of CRC Press, Boca Raton) for helping in the production of the volume. We also thank Ms. Debbie Iscoe (Hamilton, Ontario, Canada) for the fine typesetting of the entire text. Finally, the first author acknowledges the research support received from the Natural Sciences and Engineering Research Council of Canada.

N. BALAKRISHNAN

Hamilton, Ontario, Canada

W. W. S. CHEN Washington, DC

May 1997

Contents

Prefar:e

v

List of

ix

List of

xi

1.

1

2 Inverse Gaussian Distributim"ls

Properties

3. Order Statistics and Moments

5 9

4. Best Linear Unbiased ]8stimation of [·ocation and Scale Parameters 5. KHustrative Examples

Best Linear Unbiased Prediction

7.

Pl:As and Goodness-of-I'it

17 29 37

39

41 Table:3

Plot of

UL I i and N(O,l) density functiom:

Figure

lot. uf

and N(O,l) density funnium;

Figure

of

. and N (0, 1) density functHJWi

Figure

1_,

43

45

4.

Plot of IG(0.4) and

1)

functions

5.

Plot of IG(0.5) ' ~:-:--~·

(2.8)

l). we obtain the coefficients of :·:kcwness aud kmtosis of the 15)1/2

J(h = 3 ( :.\

= 3/J¢,

=3

From (2.9), we note the following

between the coefficients of skewness and

kurtosis:

=3+

10)

As Johnson, Kotz and Balakrishnan have mentioned, the relationship in 10) t-ihows that in the Pearson ((71 the inverse GanRRian points lie between the and the lognormal lines. It io: of interest to note from CJ • oo, the coefficients of skPww :•;; "'Jd kurtosis the which are the values Lo a normal Ji :tribution. 'i~·n in (2.1), uu.::;ian distribution can be · 1; see Cheng and Amin ( 1 seen to become 15

.A) - ( .A ) ' 27r(y- 1)3

1/2

{-

reparametrizing the above model density function as

we obtain the

and (2.9), we readily obtain the

fur t J,e :2parametrized distribution

Mean, E(Y) Coe. of skewness,

'.raxiance and the

= 1 + 15 = ,u = IJ2, =

(Y) =

That 1 is the threshold parameter for the the mean, IJ is the standard deviation, and k =

12)

:~(} /8

k (say).

function in (2.12), 1 + 8 = 11 is is the shape parameter (denoting

7 the cocfficicni of , sec Chan, Cohen and Whiticn ( if we consider the standanli~ed random variable Z = - 1- li)/a, we obtain the probability [nnction of Z from (2.12) easily as

1

f(zlk) = J(27r)

c

3

+ kz)

3/2

(

e:rp

~

< z

< 1 .88) -

()

>< 42.

12().:± l >

< ~1

+ (0.022t57 >

< !Jl.84)

+

X

51.96) +

X

54.12) + (0.04130

X

55.56) + (0.04333

X

67.80)

+

X

68.64) +

X

68.64) + (0.04629

X

68.88) +

X

84.12)

+ (0.04651

>(

+

>

< 105.12) +

>< 105.84)

l

41.5:~8.

From Table 6, we compute the standard errors of the above BLUEs as

=

Next, assuming that the jl

=

-(0.23871

X

41

parameter k =

1 .88) +

+ (0.0761

= 8.905,

112

= 7.713.

we have from Tables 4 and 5

+

X

l/:2

')(' u•i

+

X

"2) X •I'1 .d

+

>

< 84.

1-

X 9:~

1'2) + (0.04516

X

X

1

+

/'

18) + (0.071D2 >

< + +

12)+

/( !Jl

XI

+

/'

+

X

+

X

1L2' >

< '12.12)

X

X

+ (0.0869() X l + (0.08921 51.96) + (().(]820() X 54.12) -r- (0.07791

X

+ 55.56) +

X

68.64) I (OJJGG08

X

68.88)--:- (0.05705 x 84.12)

+-

x S::L 12)

=

X

+

68.64) I (0.06UD8 :J(i(j

11 + (0.04640

Xi)] X

+ (0.16201

>< 1OG.l2)

67.80)

>< 100.84)

G2.750.

From Table

we compute the standard errore; of the above BLUEs I/? ~

(i2.

/2

-62.

= 13. =

14.070.

Let ns consider the life-test experiment taken in tbe last example. data on all twenty three ball :: 1ill ion rcvol u tiuu: · were '"''"'-'n'·on her• · unlcrcd form: o l co>crvcd com

The

17.88 28.!)2 41.52 42.1 l'J.(i() 48.48 51.81 ')1.96 54.12 5:3.5G \i7.80 68.64 G8.G4 G8.88 84.12 9:U 96.64 105.12 105.84 127.92 128.04 173.40

assume the three- p;uametcr invcrs(' Gaussian distribution in (4.1) for this data. t lien present hcb\\' details of the· :01 of th:· br:si. linear ul!hiascd cstinwtr•:, cf t.ll(~ mean and :-:i awlmd deviation. ·::;urning th11L i.lw parameter = 0.3, we have fro1n Tables 4 and Let ns

How

\'()J

1

+ (0.04:JG5 + (0.04:581 (0.04379

x 42.12) )( 01 X

X

28.92)

+ (O.Oit;-l39

X

33.00) I

X

45.60)

+

X

48.48)

01.1

+ (0.()4381

X

+ (0.04382 X + (0.0437G X

+

+ (0.04379 + (0.0438() + (() . ()43 ''') 00

X

41.52)

X

51.84)

X

I

OJ X OJ::,

25

+

I (0.04328 =

i.r:.:J4358

(()()4;3(jj X X

+ (0Jl4::l08

127.92)

96.64)

X

+ (0.04351

X

-,

i

X ]

([JJJ4248

X

+

X

105.84)

173.40)

72.069

and -(0.1363()

()

X

17.88) - (0JJ7947

)-

X ,-j X

+ + + =

+ 1:2!

1

{Ji)]7J2

I

+

127

X

33.00)- (0.04664

+ X 68.64) + (0.021 X 96.64) + (0.04081 X 128.04) +

41

X

j X

51.84)

)(

X

67.80)

X

;) X

84.12)

((].()]

>

< 51. 84) X

67.8())

+ (0.02972 X 8 1 105.12) + (0.04524 X 105.84)

68.88)

X

173.40)

1

26 =

35.658. (5.38)

From Table 6, we have the standard errors of the above BLUEs to be S.E.(p*)

= a-*(0.04347) 112 = 35.658(0.04347) 112

S.E.(u*)

= o-*(0.02387) 112 =

=

7.435,

35.658(0.02387) 112 =

5.509. (5.39)

Next, assuming that the shape parameter k p* = (0.01215

X

+ (0.04777 X + (0.04800 X + (0.04647 X + (0.04420 X + (0.04088 X =

=

1.0, we have from Tables 4 and 5

+ (0.04099 X 28.92) + (0.04506 X 33.00) + (0.04688 X 41.52) 42.12) + (0.04816 X 45.60) + (0.04827 X 48.48) + (0.04820 X 51.84) 51.96) + (0.04771 X 54.12) + (0.04735 X 55.56) + (0.04694 X 67.80) 68.64) + (0.04597 X 68.64) + (0.04543 X 68.68) + (0.04484 X 84.12) 93.12) + (0.04351 X 96.64) + (0.04275 X 105.12) + (0.04189 X 105.84) 127.92) + (0.03958 X 128.04) + (0.03700 X 173.40) 17.88)

71.980 (5.40)

and u* = -(0.28131

- (0.02110

X

+ (0.01321 X + (0.02917 X + (0.03823 X + (0.04353 X =

X

17.88)- (0.10252

42.12)- (0.00923

X

X

28.92)- (0.06129

45.60)- (0.00007

X

+ (0.01820 X 54.12) + (0.02242 X 68.64) + (0.03190 X 68.64) + (0.03428 X 93.12) + (0.03985 X 96.64) + (0.04127 X 127.92) + (0.04433 X 128.04) + (0.04430 51.96)

X

33.00)- (0.03731

X

41.52)

+ (0.00723 X 51.84) 55.56) + (0.02604 X 67.80) 68.88) + (0.03638 X 84.12) 105.12) + (0.04250 X 105.84)

48.48)

X

173.40)

36.965. (5.41)

From Table 6, we have the standard errors of the above BLUEs to be S.E.(p*) = o-*(0.04334) 112 = 36.965(0.04334) 1 12 = 7.695, S.E.(u*) = o-*(0.02635) 1 12

= 36.965(0.02635) 112

=

6.000. (5.42)

Next, assuming that the shape parameter k JL* = -(0.05157

+ (0.05706 + (0.05459

X X

X

17.88)

+ (0.04214 X

+ (0.05711 51.96) + (0.05339

42.12)

X X

=

1.5, we have from Tables 4 and 5

+ (0.05223 X 33.00) + (0.05587 X 41.52) 45.60) + (0.05655 X 48.48) + (0.05567 X 51.84) 54.12) + (0.05212 X 55.56) + (0.05082 X 67.80) 28.92)

27

+

+ (0.04401

X

+ IP.U481·1 X 68.64) + 93.12) + (0.04259 X + (CJJJ4113

-'- (0.03793

X

127.92)

~

Hl >

< X

X

+

X

28.92) -1-

X

+ 68.64) + 96.64) + ;!J

X

33.00) 1- (0.04080

X

+ (0.04341

X

X

X

X

41.52)

>< i_)1

>< + 68.88) + 5 X 1:\4.12) 105.12) + (0.04706 X 105.84)

X X2123).

(6.2)

When the above expression is cqnated to p)], 0 = G9.CHH and solved for , we obtain the BLl!P of X21:23 to be = 112.74. l\Iext, that the pBramPter k = we compnted earlier the BLGE of jt hac.;cd on the t:nwllc:it '20 order stati,•.ti:,; to be fL2o = U9.U~1D. Now, that the c:rrwlh.;t 21 order ::ntti;:tics are availalJ]c>, we have from 7 ahle i l1e BLUE of ,u tu be JL;l

=

+ +

X

17.88)

+

X

28.92)

-j-

X

33.00)

-j-

X

42.12)

+

X

45.60)

+

X

48.48)

+

+

x;;~12)+

X

55 .56)+

+-

J+

+-

+

)( .5.1

1-

!

1-

+

X

l2D::l5

(0.04110

X

41.52)

X

51.84)

)( f3.?

+ ), 12) +

,

-1 '') /:;:,

X X2123)

= i)5.279 + (0.12935

X

'lien the ahove -,,,,, ..r""~ = 69.949 obtain the BLUP of X21:23 to be Next, assuming that the parameter k = we computed earlier the BLUE of p based on the smallest 20 order statistics to be = 70.889. assuming that the smnllt>st. 21 order c,tatistif·s are availablP. we have from Table !J the BLUE off! to be =~

-(0.00709

X

1- (0.04657 >< J:2.12) t- (0.04779

+ +

(0.04916

1- (0.12236

+

X

51.96)

X

68.64) -1-

-i- (0.04 7()1 >< 03. 12) X

+ (0.04] -1 ()

+

1/

(0.04921

+ (0.04640

X X X X

+ 54.12) + 68.64) + (0.04799 +

+

X

X

+ (().(]4890 55.56) + (0.0tl896

X

68.88)

X

)( ](),j,

+

(0JJ4754

+

X

X X

67.80)

X

84.12)

31 =

56.747 + (0.12236 x

X21:23).

(6.4)

When the above expression is equated to J.L2o = 70.889 and solved for X 21 , 23 , we obtain the BLUP of X21:23 to be X2 1 , 23 115.58. Next, assuming that the shape parameter k = 1.5, we computed earlier the BLUE of fJ based on the smallest 20 order statistics to be J.L2o = 72.659. Now, assuming that the smallest 21 order statistics are available, we have from Table 4 the BLUE of JL to be f-l~l =

-(0.08121

X

17.88)

+ (0.03477 X

+ (0.05703 X 42.12) + (0.05790 X + (0.05667 X 51.96) + (0.05569 X + (0.05214 X 68.64) + (0.05085 X + (0.04675 X 93.12) + (0.04530 X + (0.11367 x x21:23) = 59.201 + (0.11367 x X21:23).

28.92)

+ (0.04887 X

+ (0.05792 X 54.12) + (0.05458 X 68.64) + (0.04952 X 96.64) + (0.04379 X 45.60)

33.00)

+ (0.05458

X

41.52)

+ (0.05745 X 51.84) 55.56) + (0.05339 X 67.80) 68.88) + (0.04815 X 84.12) 105.12) + (0.04218 X 105.84) 48.48)

(6.5)

When the above expression is equated to J.L2o = 72.659 and solved for X 21 , 23 , we obtain the BLUP of X21:23 to be X2'1:2 3 = 118.40. Next, assuming that the shape parameter k = 2.0, we computed earlier the BLUE of JL based on the smallest 20 order statistics to be J.L2o = 75.750. Now, assuming that the smallest 21 order statistics are available, we have from Table 4 the BLUE of fJ to be JL2I = -(0.21339

X

17.88)

+ (0.04301

+ (0.07568 X 42.12) + (0.07479 + (0.06738 X 51.96) + (0.06449 + (0.05605 X 68.64) + (0.05339 + (0.04597 X 93.12) + (0.04366 + (0.10496 x x21:23) = 62.953 + (0.10496 x X21:23).

X X X X

X

28.92)

+ (0.06678 X

+ (0.07276 54.12) + (0.06161 68.64) + (0.05083 96.64) + (0.04140 45.60)

33.00)

+ (0.07411

X

+ (0.07019 X 55.56) + (0.05879 X

X

68.88)

X

105.12)

X

48.48)

+ (0.04836

X

X

41.52)

51.84) 67.80) 84.12)

+ (0.03917 X

105.84)

(6.6)

When the above expression is equated to V2o = 75.750 and solved for X2 1 :23, we obtain the BLUP of X21:23 to be X2'1:2 3 121.92. Finally, assuming that the shape parameter k = 2.5, we computed earlier the BLUE of fJ based on the smallest 20 order statistics to be J.L2o = 80.575. Now, assuming that the smallest 21 order statistics are available, we have from Table 4 the BLUE of fJ to be JL;l = -(0.41925

+ (0.10414 X + (0.08128 X + (0.06033 X

X

17.88)

+ (0.06345

+ (0.09940 X 51.96) + (0.07550 X 68.64) + (0.05600 X

42.12)

X

28.92)

+ (0.09847 X

+ (0.09352 X 54.12) + (0.07007 X 68.64) + (0.05200 X 45.60)

33.00)

+ (0.10568

+ (0.0873,") 55.56) + (0.06501 68.88) + (0.04832 48.48)

X

41.52)

X

51.84)

X

67.80)

X

84.12)

32

+ (0.04492

X

93.12)

+ (0.04178

+ (0.09698

x

x21:23)

= 68.335 + (0.09698 x

96.64)

X

+ (0.03887

X

105.12)

+ (0.03618

X

105.84)

X21:23).

(6.7)

When the above expression is equated to fJ·2o = 80.575 and solved for X 21 ,23 , we obtain the BLUP of X21:23 to be X2\, 23 = 126.21. Proceeding similarly, and nsing the BLUEs of p. presented in Table 4 for the case n = 23 and s = 1, we can determine the Best Linear Unbiased Predictor of X 22 ,23 , the twenty second failure. For example, let us assume that the shape parameter k = 0.3. Then, treating the BLU predicted value of 112.74 for X 21 , 23 as its observed value, we readily have tho BLUE of Jl from (6.2) as JL2 1 = 54.921 + (0.13104 x 112.74) = 69.694. Now, assuming that the smallest 22 order statistics are available (with the BLUP value as the obsl = 56.747 + (0. x 115.58) =' "/(J.6Wl. 1'\ow. that the ·:nmllest 22 onkr ctatistici-i arc m·n.ilabk (with the l3LUP \·nluc as the ohc:nTcd value of X 21 , we have from Table 'l the BLUE of p to he p~ 2 = (0.00:~33 X 17.88) I (0.03785 X 28.92)

+ (0.04no + (0.048Gl +

12)

>< >< :)

+

(0.04746 x

I (0.04547

X

+ (0.04222 X = o0.68li +

+ (0.047D7 >, + (CUW

+ (0.04322

+ '.12) + (0.0481

(0.04/0u x G8.61)

+

X

33.00)

+ +

)(

x G6.88l

+ (0.04483 X 96.64)-'- (0.04410 115.58) + (0.07786 X X22 23)

93.12)

X

+ (0.04580

+

105.

X

41

>