The Prediction of Future Order Statistics from Weibull or Extreme-Value Distribution Using the Probability Paper

MEMOIRS OF SHONAN

INstITUrE OP TncHNOLOGY

Vol.30,No.1,1996

The

Prediction

Extreme-Valueof

Future

Order

Statistics

from

Weibull

or

Distribution

Using

the

Probability

Paper

TakeshiIsHIKAWA*

Thisarticle _provides a method using the probability papers forpoint and intervalpredictions of

futureorder statistics, on the basisof Type IIcensored samples from the Weibulland Extreme-Value

distribution.Firstwe _propose two-point_predictor forthe _{point prediction problem} and the _problem of

choosing plottingpositionare studied. Second we _give the method to construct the _prediction interval

of the future order statistics using the two-point _predictor.

1.

Introduction

Point and intervalpredictionare widely used

in

solving some statistical problems

in

reliabil-ityand other related _problems.

The

methods

of constructing these predictors and

intervals

have been extensively studied.

We

shall

dis-cuss the _prediction for the Weibull and

Ex-treme-Value distributionwhich are

important

distributions

in

reliability _problems.

The

Weibull

distribution

denoted

by

F(x)=1-exp[-('li-)X],

o<x,

a)

isconsidered

in

this

paper.

The

variable Z=

lnX

follows

the Extrerne-Value distribution

G(z)=i-exp[-exp(23g)1,

-oo<z<oo, -[x)<g<oo,

O<6,

{2)

where 6=::llZand

g=lno.

The

Extreme-Value

distribution

has

the advantage that

its

param-eters appear as

location

and scale _parameters.

Point

and

interval

_{prediction procedures} are

in_{general quite} complicated

for

these rnodels.

For _{point prediction} one common approach

has

been

to apply the _generalized

least

squares

method or some related method to obtain

linearestimators of the location and _scale

pa-rameters

g

and 6. Kaminsky and

Nelson5)

gave

the

best

linear unbiased _predictor

(BLUP).

*

_eswr#lsu

entyN

ilZut ₇if_1O

fi

_{13 Heetit}

Kaminsky

et at.6) _provided the best linear

in-variant _predictor

(BLIP).

Balasooriya

and

Chani)

studied a robustness of the _predictor

in

the two-parameter

Weibull

distributions.

These

approach

for

the most _part requires

knowledge

of theexpectations and covariances

of the ordered observations of the

Extreme-Value

or

Weibull

distribution,

which are avail-able up

to

sample size N=25. Engelhardt and

Bain2)

have

obtained the _prediction

intervals

for

the

Weibull _process.

Lawless8)

_provides a

technique

for

obtaining exact prediction

inter-vals

for

the

first

failure

in

a

future

sample of

size

M

from

a weibull _population. Fertiget at.4)

provided

Monte

Carlo

estimates of percentiles

of the

distribution

of a statistics

S,

that may be

used to construct _prediction

intervals

to

con-tainthe

future

observation. Furthermore they

gave an approximation

for

the

distribution

of

S.

Many

articles have been published incase of

two sample _problems,

but

_quite a few incase of

one sample _problerns.

A

_good review of the

prediction

intervals

isgiven by Pate19).

In

this

paper we consider

X(oSX(2}$

'''$Xcr)

the smallest r of

N

sample observations from a

population

having

the two-parameter

Weibull

cdf _given

by

(1).

In

other words we suppose

that testing

has

been

terminated

at

the

time

of

the r-th failure.

We

consider to_predict thes-th

order statistics

Xc,)

(r<sSM

on

the

basis

of

X{i),

Xc2),'

'',X(r).

_Most

_{of prediction methods}

_depend

on extensive computer programs, and require

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meMrNrt\SES

ee

30

8

eg

1e sample sizes of

interest

andlor are not widely

available.

Graphical

methods _{are very}

power-fultoolsfordata analysis. The compatibility of

the _proposed model with the observed

data

may be determined easily

by

_agraphical

exam-inationof the fit.When exploration of the

data

is warranted, _graphical methods are

both

useful and appropriate.

This _{paper presents} a _{simple and convenient}

point and interval _{prediction procedures}

for

the

Weibull

and

Extrme-Value

distribution

using Weibull _{probability graph paper.}

The

location

and scale _parameters of the

Extreme-Value distribution are estimated by least

square and two _point methods.

The

_problem

of

choosing _{plotting position}are studied.

2.

Least

Square

Predictor

We assume that the underlying

distribution

of

failure

times isa Weibull distribution

<1),

The _problem then

is

to estimate

A

and a, or

more

_generally,

to estimate the distribution

function

Prob(Failure

time$x)=F(x)

=1-exp[-(i)a].

(3)

We start the analysis by _plotting the sample

data

points on

Weibull

_{probability graph}

paper.

This

is

a special

(x,P)

coordinate _graph

paper

designed

so that the _plotof x vs.

_p=F(x)

will appear as a straight line when F(x) isa

Weibull

distribution

function.

Ifwe rewrite

(3)

as

1

lnx=lna+Tln[-ln(1-P)](4)

and then set

z=lnx, _{y=ln[-ln(1-p)],}

E=lno,

6=1/a.

(3)

is

transformed

into

z=g+by

(s)

the equation of a straight lineon rectangular

(z,y)

graph paper.

Weibull

probabilitygraph paper may

be

con-structed from rectangular _{graph paper}

by

lab-elling as x the_gridlinecorresponding toz=lnx

and labellingas

1OO

P

the _grid line

correspond-ing

toy=ln[-ln(1-P)]. Thus the piot of

the

points

(x,p),

where _p and x satisfy

(3),

on

Wei-bull _{probability graph paper}

is

equivalent to

the _plot of the _points

la,y),

where z=g+by, on

rectangular _{graph paper.}

We

note that rnost

Weibull

probability graph paper

has

its

axis

reversed; that

is,

the

horizontal

axis islabelled

x and the vertical axis islabelled

P,

Ifwe knew

_pi=F(x{i)),

for

each sample

failure

time

xc" then the

line

through the _plotted

pointswould be completely

determined

the

dis-tribution

function

F(x>.

We

must be to

deter-mine some estimate of

_Pi

_{so that} we may plot

the_points

_(x(e,pi).

Since

x(n

is

thevalue taken on

by the

Weibull

order statistics

Xc",

we may consider z(i)=lnx(i} as the value taken on

by

the

Extrerne-Value

order statistics

Z(i)

_{and yi=}

<z(i)-e/6

as

the

value taken on

by

the reduced

Extreme-Value

order statistics Yli)=(Z{i)-e/6.

Thus

we have

lnX(i)=Z(i)=g+6Yli),

lnx<i)=z{i}=g+byi.

{6>

The

sample value _yi

depends

on the unknown

parameter6 and

g,

and hence can only

be

es-timated.

In

this_paper we shall estimatePi _as

the

_mean

value of the reduced order statistics

F(X(i)).

That

is

i

Pi=E[F(X(o)]==

N+1

This

is

only one of many _{possible procedures}

for _plotting on probability graph paper

(see

Kimbal17)).

A

plot of the_points

(xo,Pi)

appear in

Fig.

1.The visually fittedline

is

an estimate of

the underlying

distribution

of

failure

times.

The

parameters of the

distribution

_{may also}

be

read from

the

_graph.

Rather

than

fit

the lineby eye, itis more

appealing touse _some analytic technique.

The

problem

is

to

fit

a

line

of the type

z-g+by

toa _given set of _points

k(i),yi),

i--

1,

2,

･ ･ ･,n

The

least

squares estirnates of

g

and

6

are

ThePnediction

of

Futare Order Statistics

_ftomWbibuU

or Extreme- VZilue p "･1"--"-e"-tt-n,o".o-o-ee"o".en"is--bO,O

1li

i

-s-,-et"l-l-eJ y ±' 1 (3,n)

Fig.1. VeibullProbabilityGraph.

(D:

90% prediction interval.

@:90%

one-sided lower _prediction interval.

where r 6Ls=

₌

_(yi-5jtae

]'=1

as=z'-6-Lsy

I=

The

estimated r

LtLl

(,ti-p)2

,

"];1

yi, z---;2Lzen.

Iine

is

then 2=gLs+6Lsy.

(7)

Hence,

immediately

we get apredictorof

statis-tics

Z(.)

such that

ZLS=gLs+6L$y,.

{8)

We call this_predictor a leastsquare _predictor

of Zcs).Therefore we _get a _predictor ef

X(,)

as

follows .]?LS=exp(2LS).

Hence we can _get an approximate

least

square _predictor by following _graphical

tech-nique,

A

straight

line

is

fitted

tothe

data

point

by

eye. The 63.2percentileisan estirnate of

g.

The shape _parameter6

is

estimated graphically

as the reciprocal of the slope of the straight

line.The value of _predictor of statistics Z(.)is

obtained

by

entering

the

ploton

the

P

scale at

P.

=s/(?V+ 1)_going horizontally tothefittedline

and then vertically up tothe

data

scale toread

the estimate of the Z(.).The value of _predictor

of statistics X(,)isobtained by entering

verti-cally

down

tothe

data

scale.

The statistical model which leads tothe

esti-mates

(7)

assumes that the

data

points

z"}are

the values taken on

by

a set of random

varia-blesZ(i}which E(Z(i))=g+ayi,Var(ZciD=6Z and

Cov(Z(i),Zm>=O

_(itl'),

Ifsuch assumptions are

satisfied then theestirnator

gLs

and 6Lsare

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mamr*g)<#ede

za

3og

_eg

1e

Table 1. Biasesand MSEs of thePredictors

AT=:15,R=11BLIPBLUPL-STWO S=12 Biasl02 MSEt62 S==13 Biast62 MSEt62 S==14 Biast62 MSEt6Z S=15 Biasi62 MSEt62 -.O168 _{0 .0824} ,0338 .0341 .1328 =0345 0 .0996 .0761 ,0774 .2042 -,0548 _O ,1355 .1388 -.0829 _O .2465 .2539 .1126.3016 .1038.4621 .O044,0355 .O066.0829 ,O025,1516-,0291 ,2774

all

linear

estimator for

g

and 6. For our

prob-lems

Z(o

are

Extrerne-Value

order statistics.

Therefore thus assumptions do not hold. The

column

L-S

in

the

Table

1 _gives the

biast6

and

MSE/62 of the

least

square predictor

2LS.

The

values of the

MSEs

of

2LS

have

very

large

than

ones of the _predictors

BLIP

and

BLUP

of Zts).

Therefore

we need another _predictor of

Z(s}

using

Weibull

probability graph paper.

We

now _present a _predictor which

is

called two

point predictor of statistics Zcs).

3.

Two

Point

Predictor

We

consider

to

predictof statisticsZ(,)simply

in

a similar way.

We

choose any two points

which are plotted

in

the

Weibull probability

graph paper.

Next,

we

draw

a straight

line

through these two _points.

We

consider touse

this

linein _place of

the

fittedlineto the

data

points

by

the

least

square method.

The

63.2

percentileand thereciprocal of the slope of this

straight

line

are the estimates of

g

and

6,

re-spectively. But

in

this case we can _get the

mathematical formulae of the estimates of

g

and

6.

If

we _choose

the

two

_points

la{i},yi),

lae,){i)

then

the estimates of

g

and

6

are as

follows,

6(i,1')=(zo-a{i))/()ti-yi),

eri,ID=zot-6(i,j'>yi.

Therfore we _get a _predictor of statistics

Z(s)

as

follows,

2k(i,1')-g-'(iJ')+6-(i,J')y..

The

bias

B(i,isys

and mean square error

M(i,i

s)62 of

2k(i,1')

are

B(i,is)6=[(1-c)cti+ccci-as]cS,

M<i･J'･S)62=E[21s(i,j')-Z(s}]2 =[(1-c)2wti+c2w"+w.+2c(1-c)wij -2(1-c)wis-2ctvi,]62+(bias)2 where

c=(),,-y,>/()Li-y,),

_qi=E(Zm6-g)

and

,,ts,=c..(zolig , z(k3-g).

We

will choose a

_pair

of_points

izto,)tr)

and

lath,)7)

as satisfying the following condition

M<I,Ls)=

min

M<i,is).

ISi<iSr

Then

we call

a(Ll)

the two _{point predictor of}

statistics Z(,}.Calculating M(i,is> forall i,J',r,s,

N=:5(1)20,

we

find

that the second number

J

always becomes the lastnumbern Itisof

interest

to note.

Table

2

gives the optimul

number

I

for

sample sizes up toN=20. Itisof

interest

to note

from

Table

2

that the optimul

numbers _I

is

close _{to the} number

[ri

1

]

where

[a]

denetes

the

largest

integer

not exceed a

The column TWO in the Table 1 _gives the

bias/6

and

MSEf62

of the two point

predictor

of

Z(,)forAT=15, r=11,s=12(1)15. These values

of the MSEX62 of the two _{point predictors} have

smaller values than those of the

L-S

predictors

and show that these are very close to ones of

the

BLUP

and BLIP.

4. IntervalPrediction

We

try to construct the _prediction

interval

for

the

statistics Z(,)on a probability paper.

We

know

U==F(Z(.))

distributes

as

Beta

distribu-tionwith _parameters and

N-s+1.

Then

W=

N-s+1

U/a

-

U)

distributes

as

F-distribution

swith

degrees

of

freedom

2s

and

2(N-s+

1).

Let

1OOa

thpercentileof the

Beta

distribution

with

parameters a and

b

be

u(a;a,b) and let100a

th

71hePV'ediction

of

FutureOrderStatistics

from

Tl7laibultor Ebetreme-Value Table 2. 0ptimalnumberIand1

Ji=r

IIIIIIIV N5r4s5 1111 6455.66 11111111 7456 5,6,76,77 111 111 122 111 8456 7 5$s6,7,8788 11111 11122 12222 11111 9456 7 8 5Ss6,7,8,9 78,9899 11r1111111222212222221111122 10456789 sgs6Ss7Ss8,9,109,1010111112112222122222 111222 !145678 910 55s6gs7Sssgs91O,11lO,111111111222l1222222l222222211122222 1245678910115Ss6Ss7Ss8Ss9Ss10,11IL121211112222112222221222222211122222 I:Pi-Ci-O,5)tN

II:_Pir-(i-O.3)t(N+O.4){MedianRank) III:pi=ilaV+1)CMean Rank)

IV:_yitai. IIIIIIIV Nrs 1345Ss 111l 56$s 112 1 67$s 122 1 78Ss 1222 89Ss 2222 910Ss 2222 10IL12,13 2222 !112,13 2222 1213 2222 1445$s 1111 56STs 112 1 67Ss !221 78Ss 1222 89$s 2222 9less 2222 1011Ss 2222 1112,13 2232 14 2222 1213,14 2222 1314 2222 154sgs 11 11 56gs 112 1 67Ss 122 1 78Ss 1222 89$s 2222 910$s 2222 1011SsS14 2232 15 2222 1112$s 2232 1213,14 2232 15 2332 1314 2232 15 2322 1415 2222 1645Ss 1111 565s 112 l 67Ss 122 1 78Ss 122 2 89$s 222 2 910$s 2222 1011Ss 223 2 1112Ss 2232 1213,14 2232 l5,16 2332 1314Ss 2332 1415,16 2332 1516 2322

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湘南工科大学紀要　 第 30 巻 　第 1 号

Table　2．_（continued _｝

／＝・’r

　1：Pi＝_（i− 0．5_）μV

II：pi＝（i− 03_）1（N 十 〇．4｝（Median 　Rank _）

III：_♪尸 ゴ

1

々V十 1｝（Mean　Rank_）

IV：yi＝aゴ．

percentile　of　the　F −distribution　with 　

degrees

　of

freedom

　

k

　and 　m 　

be

　

F

（α；陀，m ）． 　

Therefore

　_we 　

know

u （α；s，

N

−s十

1

）

1

　 　 N − s十 1 1十 　 　 　 ∫ 」F（α；

2

（ハ厂一s十

1

），2s｝ 一 ₄₈ 一 1

唱

「 」 ら ． 卜 　 　 e 　 　 O 　 　 ． l 　 　 V 　 　 r 　 　 e 　 　 Syrarb ． lLO ． lnOrtOelE 一 工 工

7;hePtediction

of

FutungOrderStatistics

fivmWbibull

or Extreme-Value

Since

we can show

immediately

F-i(u)=g+

6 1n[-ln

₍₁

-u)] when u=Fla),

Then

we _get

1

-a ==

Flr

{u

(1

- -l;;a, b)SF(Zc,))

Su('3;

a, b)1

=pT

(g+6

in

in

i-u(i

-i-g,a, b)

$z(,)sg+6

1n

ln

l

ru(i , a,

b)1

where a=s, b= N-s+1.

If

we

know

the values

4

and 6 then we _get a

1OOa _percent confidence

(prediction)

interval

of

Z(.).Since

g

and 6 are unknown _parameters, we

need toestimate these values. Then we _get a

100a _{percent prediction} intervalof Z(.}the

fol-lowing type

_{[g+by*.g+by**]}

where _y* and _y**

are suitably choosen. Now,

1-a=Pr{g+(7y*gZ{.Eg+by"}

=Pv･

{exp(g+dy')SXc,)Sexp(g+6y**)}

.

This meas as

follows,

we _plotthesample

data

points on Weibull probability graph paper and

fita lineby eye or another suitable method.

The

upper

(lower)

limitof the _predictor of the

statistics Z{.)isobtained by entering the ploton

the y scale at _y*(y**)going horizontallyto the

fittedline and then vertically up to the data

scale to read the estimate of the upper

(lower)

limitof Z(s).The limitvalues of predictor ofthe statistics

X(.)

are obtained by entering

vertical-lydown tothe

data

scale.

We adopt the fittedlineby the method used

to _get the two _{point predictor} of the statistics

Z(s).Now.

6{L

r>== Z`"rZe ,

g-(L

r) =z(.) -

6N(L

rtvr･ :Vr-YI

Then

Zg,Elfige-

s),5<･'):-Yr-.Yi+,, where T=(Zcr)-Zto)/<Z(s)-Zm)･

Since O< T< 1,

T

distributes

approximately as

Beta

distribution

with parameterP and _q where

p

and _q are determined by using the moments

of the statistics

T

Let Then E(T);ui,E(T2)=;"2.

P=

#2-u12 ' q= #2-pt12 -P'

We know

_IZ-

Tl(1 -T)

distributes

approximate-lyas

F-distribution

with

degrees

of

freedom

2P

P

Zcs}-Z(r)

and 2q. TherefOre

-a-zc.}-zen

distributes

as

F-distribution

with degrees of freedom 2q and

2p

approximateiy. That

is

[Z(sts-uf;f'r)

-yl

plq

distributes approximately as

FLdistri-y.-Ylbution

with

degrees

of freedom 2q and 2P.

Let tbbe the 100a th percentile of the

statis-tics

Z(sZE;<'r),

we

get

ta#Yr+

/S'

tvr-Yt)F(a;

2q,

2P)

･

we expand

Z(')-Zen

in a Taylar series

Z(s)-Z(n

around the expectations of ordered statistics

in

the standard

Extreme-Value

distribution

and

then approximate itby

Z(r)LZco

;A+B(Ze5-aD+C(Zl5-a.) Zcs)-ZcD +D<Ztf)-a.) where

A=

:l!:i

,

B:=:

(.a,'--.ai

'

C==

a,lai' D=-(B+C),

and

ZIS

are ordered statistics

in

the standard

Extreme-Value distribution. Therefore we _get

E(T);A,

Var(T);B2wll+C2w.+D2w,,

+2(BCwi,+BDwts+CDw.).

Now

we _put approximately

NII-Electronic Library Service

meml*sFJk#eee

eg

3o

u

m1g

andPi(1-A) op#ln[-ln(1-Pj}], wij';

N+2

(1

-pi)(1-pj)

ln

(1

-Pi)

ln

(1

-Pf) '

(i

K-1･). Then we _get

g=

rl -1k ys'yr _.

P

pl y.-yl

And

ta=Yr+lys-Yr)F(a;2q,2P)･

Hence

1-

g

=pr

la,

_s

g(I,

r)+

S(I,

r)

×

[y,

+

(ys

-yr(ij(-S-;2q･

2p)-

i

)1}

{}

= Pr{Z,S

g'(I,

r)+6"'(I,r)

×

Ps-tys-yr(i

-]F<i -

{l";

2q,

2pD]

Therefore we can _get both the _prediction

point and interval of the stati'stics Z{,)as

fol-lows,

i) the prediction point of Z(.}isobtained by

entering the _plot on the _y scale at _y.

going horizontally tothe

fitted

line

and

vertically up tothe

datascale

toread the

value.

ii)

the upper(er

lower)

limit

of the _predictor

of

Z(.)

is

obtained

by

entering the _ploton

the _y scale at _{y**=y,+(y,-yr)[F(a12;2q,}

2P)-1]

(or

y*=ys-(ys-yr)[1-F(1-at2;

2q,2P)])

going

horizontally

to the fitted

line

and then vertically up to the

data

scale to read.

Then

we can obtain the prediction point and

the limitvalues of the statistics X<.)

by

entering

vertically

down

to

the

data

scale x.

We

need

only

100a

th _percentile of the

F-distribution

with real valued

degrees.

These values are

given

by

using numerical

integration

and

Newton's

method. We may use approximatly

100a th _percentile of the

Rdistribution

with

integer valued

degrees

of

freedom

[2q+O.5]

and

[2P

+

O.5].

1

5.

IllustrativeExample

Consider

the Type IIcensored sample from

theWeibull

distribution,

simulated with

A

=2.0

and o=

12.38,

with N=20:

1.6003,

4.5727,

5.0263,

5.4089,

6.7246,

7.2051,

8.3590,

8.8687,

9.7873,

10.3860,

10.8011,

10.8933,

1

1.5329,

13.9679,

14.4076,

15.4168,

15.5366,

17.8901,

22.1448,

22.4281

Ifwe assume that the

last

N-r=6

(r=14>

ob-servations are censored and we want to_predict

the largestobservation. By Table 2,we _{get the}

optimal numberI=3.

Then

we

draw

a straight

line

through the two _points

_(ln

5.0263,_y3)and

(ln

13.9679,

yi4)

(Fig.

1).

21

Y3=ln

ln

ls =-1.86982,

21

yi4==lnln

₇

=O.09405. We obtain

6(3,14)==O.52,

g(3,14)=2.59,

therefore A=1.92, a=13.3. Now, s=20, y2o=ln

ln

21

=

L1

1334

2(3,14)=3.18,

g(3,14)=24.0.

sincep !; 13.0754,_{qI6,7865, we get the}

percen-tilesof the

FLdistribution

by

the numerical

methods as follows, F{O.05;26.15,13.5)=2.336,

F(O.05;

13.5,

26.15)=2.083.

F(O.10;26.15,13.5)= 1.9343 Thusy**=y,,+tv2o-yi,)[F(O,05;13.5,26.15)-1] = 2.217 _{and y*=y2o-fy2o-yi4)[1-F(O.95:13.5,}

26.15)]=O.530. Therefore we _get 90 _percentile

prediction

interval

of the statistics

Z(2o)=ln

X(2o)

as

[2,89,3.75]

and hence 90 _percentile

predic-tion

interval

of

X{2o)

becomes

[18.0,42.8].

If

we

want to_get 90 _percentileone-sided _prediction

interval

of the statistics

X(2o),

since

Y"=Y2o-tv2e-yi4)[1-F(O.90;13.5,26.15)] -O.621 ' we _get

[19.0,

oo). -50-NII-Electronic

11hePrediction

of

FlettureOrderStatistics

fromWeibult

or Extreme- Vdlue

References

1) U.Balasooriyaand L.K.Chan

{1983),

"The

dictionof Future

Order

Statistics

in

the

parameter Weibull Distribution-a Robust

Stduy,"Sankya,45,B,320-329.

2) M,Engelhardtand L.

J,

Bain

(1978}.

"Prediction

Intervals for the Weibull Process," metrics, 20,167-169,

3) M.Engelhardtand L.

J.

Bain

(1982),

"On tion LimitsforSamples from a Weibull or

treme-Value Distribution," Technometrics, 24,

147-150.

4) K.W. Fertig,M.E.Meyer and N.R.Mann (1980), "On

Constructing

PredictionIntervalsfor

plesfrom a Weibull or Extreme Value

tion,"Technometrics.22,567-573,

5) K,S.Kaminsky and P,I,Nelson

(1975a),

"Best

LinearUnbiased Predictionof Order Statistics

inLocationand ScaleFamilies,"

Jeurna]

of the

American StatisticalAssociation, 70, 145-150.

6) K. S.Kaminsky, N. R. Mann and P.I.Nelson

(1975b},

"Linear InvariantPrediction

of Order

StatisticsinLocationand ScaleFamilies,"

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Aerospace Research Laboratories,

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7) B.F.Kimball

(1960),

"On theChoice of Plotting Positions on Probability Paper." Journalof the

American StatisticalAssociation,55,546-560.

8)

J.

F.Lawless

(1973),

"On

theEstimationof Safe Life When the Underlying Life Distribution is

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9) J. K. Patel

(1989),

"Prediction

Intervals-a

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