34 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 1, 1996 MARCH

arameter-Estimation Using the Bootstrap for the 2-Parameter W eibull Distribution

Tetsurou Seki, Member IEEE

Shin-ichiro Yokoyama, Member IEEE Teikyo University of Technology, Chiba

Musashi Institute of Technology, Tokyo

Key Words - Bootstrap, Robust estimation, Outl;er, Weibull distribution, Extreme value distribution, Order statistic

Summary & Conclusions - This paper proposes bootstrap robust estimation methods for the Weibull parameters; it applies bootstrap estimators of order statistics to the parametric estima- tion procedure. Estimates of the Weibull parameters are equivalent to the estimates using the extreme value distribution. Therefore, the bootstrap estimators of order statistics for the parameters of the extreme value distribution are examined. Accuracy & robustness for outliers are examined by Monte Carlo experiments which indicate adequate efficiency of the proposed estimators for data with some outliers.

1 . INTRODUCTION

The Weibull distribution is widely used in life testing and reliability theory. Many authors, eg , [2, 81, have studied its characteristics and have proposed powerful and widely used estimators of its parameters. In general, these estimators assume that the sample size is large and/or the observations include no outliers. However, real reliability-data often contradict these assumptions; them those estimators do not always give desirable results.

One answer is to get rid of outliers, but for small sample sizes, elimination of data is difficult and not always a good idea. The alternative is to find a robust estimation method when outliers are present. Ref [9 , 101 provide an idea which is based on the bootstrap method for observations from a Gaussian distribution.

This paper treats the case where observations arise from a Weibull distribution. Section 2 states the essential ideas of [9, l o ] . Section 3 uses the bootstrap method to estimate the parameters of the extreme value distribution. These estimators are transformed to the Weibull shape & scale estimators through simple conversion formulas. Section 4 compares these Weibull estimators to other popular estimators with respect to accuracy & validity, both with & without outliers. Section 5 uses a numerical example to illustrate the calculation of our estimators.

Acronyms

BISY proposed bootstrap estimators BLUE best linear unbiased estimator

MLE maximum likelihood estimator MSE mean square error = RMSE’ RMSE root MSE.

Notation

1 index: i = 1,. . . , N unless otherwise specified order statistic i

Yil) order i bootstrap sample r?v,I,L) PdY?,) = Y ( L ) )

CBmt {BLUE, f m E [BLUE, MLEI of i+ Fw( t;6,7) Cdf ( t ] for the 2-parameter Weibull distribution

bootstrap estimator of any parameter

with shape parameter 6 and scale parameter 7 X, p, U M T ) , W d , 1/61 F,(X;~,U) Cdf{.x} of the extreme value distribution with loca-

M ( X ) median of F,(X;,K.,G) C l/ln(-ln(0.5)) = -2.7284.. . $ ( r , N ) quantile estimator of population for x ( ~ ) R( , , , , , , ) estimator of E-quantile of F,(x;,K,B) by x ( ~ ~ ) and x ( ~ ~ )

tion parameter p and scale parameter B

6(r I , r z ,q , r4 )

t‘(L,rI.rz,N,E) y .W(i -2 ,~) + (1-7) . W ( r l , ~ ) ;

estimator of B by X ( r l ) , X ( r 2 ) j X ( r 3 ) , x ( r 4 )

y = [rl + ( E - 1 ) . ( N + ~ ) ] / ( T ~ - r2) factor of order-statistic L for BlSY estimator

X ( r 4 )

6Boot(rl,rz,r3,r4) bootstrap estimator Of by X ( r , ) l X ( r z ) j X ( r 3 ) ~

OBoot BEY estimator of 6 C ( k , , k , ) estimator of by X ( k l ) , X ( k z )

Z ( . ) ( x ( . ) - P ) / @ b ( k l , k z , k 3 , k 4 )

pBoot(klrk2,k3,k4)

estimator of p by X ( k l ) ~ X ( k z ) J X ( k 3 ) j X ( k 4 )

bootstrap estimator based on b(kl ,kZrk3,k4)

p?L) 7jBoot B/SY estimator of 7

OnonB l16(q,rz ,r3,r4) t m n B eXP(f i (k l ,k2 ,k3 .k4) )

factor of order-statistic L for B/SY estimator

61,1, 7j l ,L some proper estimator of [e, 71 from the sample of Fw( t; 1 .o, 1 .O) . Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.

2. BOOTSTRAP METHOD

Ref [3, 41 introduce the bootstrap method as a resampling procedure - a simple method for calculating the approximate dispersion of the estimator.

‘Editors’ note: We have assigned this acronym BISY (bootstrap - Seki, Yokoyama) for simple, clear, unique reference to the concept.

0 0 1 8 - 9 6 2 9 / 9 6 / $ 5 . 0 0 0 1 9 9 6 IEEE

SEKUYOKOYAMA ROBUST PARAMETER-ESTIMATION FOR THE 2-PARAMETER WEIBULL DISTRIBUTION 35

2.1 Example

Assumptions

1. F is an unknown distribution. 2. Y, = yr, i = 1,2 ,..., N, are observed; Y, are i.i.d. F. 3. We wish to get the bootstrap estimator of standard error

6B of some estimator 8; it is simply estimated from bootstrap samples, Y?;,, YG), ..., YTN) i.i.d. F.

4. s-Independently , we perform a resampling procedure of some number, say B times, and by calculating,

we obtain bootstrap replications $61, 851 , . . . , QB). 5. Hence we can calculate:

r=l

CB converges to the true value as B - W. Generally, as in this example, B is a finite number in the range of 50 - 200, and the Monte Carlo experiment is used. To get a more accurate estimator, many more replications are needed.

2.2 History & Discussion

Consider the condition of infinite replication without us- ing the Monte Carlo method. Ref [9, 101 considered the mean of order statistics based on the bootstrap procedure under the condition in which B - W. Thus,

In ( l ) , the second probability is [3] :

From (1) & (2),

N

gBoot(i) = w(i ,L) ' Y ( L ) . L= 1

(3)

The estimator in (3) gives the largest weight to the order statistic i. As a result, there is a possibility of getting a robust estimator for outliers which has smaller weight around both tails of order statistics, by applying (3) to an appropriate estimator which is obtained from the order statistic.

Ref [9, 101 discussed the properties of the bootstrap estimators of the Gaussian parameters obtained by applying (3) and showed that the bootstrap estimators: 1) reduce s-bias, 2) are robust in the presence of outliers, and 3) are accurate for very small samples.

3. MAIN RESULTS

The extreme value distribution is often used to analyze Weibull data.

F,(t; 8,q) = 1 - exp[-(t/q)']; t, 8, 7 > 0. (4)

FE(x; p,a) = 1 - exp[-exp[(x-p)/all;

-w < x,p < 03, a > 0. (5 )

Therefore, estimating parameters of the extreme value distribu- tion corresponds to estimating the Weibull parameters.

The p & a can be estimated from the sample mean and sample standard deviation. However, these estimators do not always provide robust, simple estimators of 8 & q . Therefore, we consider:

M ( X ) = p + a/c (6)

for estimation of p & a.

3.1 Shape Estimator

Given and A( X) ,

6 = c . [ A ( X ) - p]. (7)

This section applies $(r,N), knowing that M(X) and p cor- respond to the 50.0% and 36.8% quantiles, respectively, to estimate a. When the necessary quantile can not be obtained directly, linear interpolation is used. A popular estimator is:

$(r,N) = Y / ( N + ~ ) . (8)

Using (8),

i(e,r , ,r*) = Y'*x(r*) + ( l - ~ ' ) . x ( r ] )

= Y*x(rz) + (1-y) * X ( r l )

7' E ($(ri,N) + €-1)/($(rI,N) - $(r*,N))

6 ( r , , N ) 5 E 5 4(rZ;N).

Apply the bootstrap estimator of the order statistic (3) to (9),

- fBoot(or,rl,r2) - Y.iBoot(r2) + ( 1 -7) .iBoot(r~)

N = 2: (L,rI,r*,N,E) ' x ( L ) . (10)

L= 1

Since a can be estimated using (9):

6(r~,r,,r,,r4) = c' ( i ( O 50,rl,rz) - i ( O 3 6 8 , r ~ 4 ) ).

(9)

36 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 1, 1996 MARCH

TABLE 1 Factors for Computing the Weibull Shape Parameter

Sample-Size, n

L 3 4 5 6 7 8 9 10 11

1 -0.64085 -0.40835 -0.18918 -0.16604 -0.15199 -0,14258 -0.13586 -0.03976 -0.03849 2 0.32043 -0.11734 -0.28 146 -0.2424 1 -0.2 1678 -0.19964 -0.18754 -0.14275 -0.13584

-0.11787 -0.167 11 -0,15704 3 0.32043 0.11734 -0.13 150 -0,14354 -0.13476 -0.12526 4 0.40835 0.12227 -0.01222 -0.04595 -0.052 11 -0.05206 -0.1220 1 -0.11473 5 0.47987 0.14505 0.02907 -0.00550 -0.01477 -0.06329 -0.06 127 6 0.41915 0.'1397 1 0.03872 0.00703 -0.02114 -0,02401

8 0.35492 0.12474 0.04397 0.01117 9 0.33656 0.13665 0.04306

10 0.36895 0.13004 11 0.35120

L 12 13 14 15 16 17 18 19 20

7 0.38068 0.13146 0.03977 0.0065 1 -0.00408

1 -0.03750 -0.00798 -0.00788 -0.00857 -0.00721 -0.00126 -0.00126 -0.0001 8 -0.0001 8 -0.02024 -0.00593 -0.00588 2 -0.13047 -0.063 10 -0.06137 -0.06594

3 -0.1493 1 -0.12601 -0.12091 -0.12841 -0.10567 -0.0663 1 -0.06448 -0.03079 -0.030 15

5 -0.05934 -0.11 119 -0.1058 1 -0.11 159 -0.09135 -0.11645 -0.10463 -0.10059 6 -0.02480 -0.06658 -0.06423 -0.06847 -0.05697 -0.09620 -0.092 17 -0.11285 -0.10800

8 0.00139 -0.00995 -0.01135 -0.01275 -0.01958 -0.03648 -0.03430 -0.06705 -0.06447 -0.03 800

11 0.12488 0.04537 0.01354 0.02417 -0.0641 1 -0.03776 -0.02126 -0 .O 1463 -0.01054 -0.01046 12 0.33741 0.13417 0.04399 0.06087 -0.05023 -0.05002 -0.03734 -0.01914

13 0.36266 0.12912 0.12661 0.04147 -0,03068 -0.03082 -0.01835 14 0.349 12 0.20144 0.20780 0.05844 -0.03193 -0.03805 -0.03 149

0.30085 0.21174 0.05493 -0.01533 -0.03955 15 0.15232 -0.01 802 16 0.11595 0.29202, 0.20639 0.06950

17 0.11118 0.28875 0.21005 0.06511 18 0.11117 0.28102 0.20512 19 0.10675 0.27886 20 0.10710

-0.05480 -0.02058

4 -0.10911 -0.14109 -0.13423 -0.14156 -0.11579 -0.10705 -0.10317 -0.07 13 1 -0.06910 -0.11 164

7 -0.00726 -0.03067 -0.03074 -0.03359 -0.03063 -0.06398 -0.06142 -0.09610 -0.09196

9 0.0 1222 0.00100 -0.00236 -0.00178 -0.025 19 -0.02250 -0.01794 -0.03938 10 0.04189 0.01338 0.00311 0.00727 -0.04457 -0.02409 -0.01379 -0.021 12 -0.01949

-0.05030

Then, 3.2 Scale Estimator

Consider eBoo, as in (13). Usually if we can get 6, we can estimate < from the well known relations between 6 and 9. For MLE, 9 = [Cy= t,8/N] lis, which is a function of 6 and thus undesirable. Therefore we develop a new estimator which is

= ~ ~ ( ~ , r i , r z , r 3 , r 4 , ~ ) ' x ( L ) . (12) independent of 6. Consider the 7 bootstrap estimator with a form similar to that of eBo,,. If the estimators of U and M ( X ) in (6) are given, then estimate 7 and p using p=ln{T}.

Harter [7] proposed the estimator of the Gaussian scale parameter which usep a sub-range of the order statistics; U can be obtained in a similar way, because the range of order statistics i, j of the extreme value distribution / x ( ~ ) -xb) I do not depend on p. Using the range x(k2) - x ( ~ ~ ) ,

- 6Boot(rl,rz,r3,r4) - c ' @Boot(O 50,rl,rz) - iBoot(O 3687374))

N

L= I

Using the estimator in (12), OBOOt(rl,r2,r3,r4) is easily calculated from (5). The { r 1 ~ r 2 j r 3 ~ r 4 } can be such that eBoot(r~,rz,r3,r4) has the minimum MSE for each size. Then,

N

L= 1

Use this and k ( X ) : and is the BISY Weibull shape estimator. The are deter- mined for each sample size through Monte Carlo experiments

(15) -

and are given in table 1. b(kI,kz,k3,k4) - ' ( 0 5o,k3,k4) - ' ( k ~ , k z ) / ~ , k2 '.

SEWIYOKOYAMA: ROBUST PARAMETER-ESTIMATION FOR THE 2-PARAMETER WEIBULL DISTRIBUTION 37

TABLE 2 Factors par;, for Computing the Weibull Shape Parameter

Sample-Size, n

L 3 4 5 6 7 8 9 10 11

1 2 3 4 5 6 7 8 9

10 11

0.23648 0.20938 0.09559 0.08494 0.07441 0.04458 0.32621 0.21099 0.17061 0.13277 0.08509 0.07484 0.43732 0.22095 0.18912 0.12603 0.07910 0.09494

0.35868 0.23643 0.16321 0.14097 0.13297 0.30825 0.24245 0.21533 0.15810

0.25060 0.21937 0.14546 0.18574 0.1311 1

0.21801

0.02267 0.07998 0.09393 0.09011 0.10983 0.14247 0.14726 0.12862 0.18513

0.01 178 0.04536 0.07128 0.10008 0.13270 0.14675 0.12518 0.08688 0.08773 0.19227

0.01780 0.0631 1 0.075 83 0.06681 0.0631 1 0.07321 0.09003 0,10915 0.13913 0.17720 0.12464

L 12 13 14 15 16 17 18 19 20 ~

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.0090 1 0.03195 0.04222 0.05188 0.07481 0.10435 0.12098 0.11478 0.10073 0.10830 0.13972 0.10128

0.0 1454 0.05008 0.05797 0.0485 3 0.04544 0.05836 0.07981 0.0944 1 0.09452 0.09167 0.10949 0.14726 0.10794

0.00437 0.03399 0.06627 0.07116 0.05294 0.03500 0.03764 0.06303 0.09373 0.10834 0.10487 0.10708 0.12873 0.09284

0.00297 0.0228 1 0.04389 0.047 19 0.03 844 0.03620 0.05352 0.08582 0.11291 0.11556 0.09441 0.07350 0.08020 0.11022 0.08235

0.0037 8 0.02875 0.05527 0.05973 0.04549 0.02853 0.02429 0.03995 0.06960 0.0960 1 0.10230 0.08771 0.07278 0.08327 0.11571 0.08684

0.00256 0.01928 0.03693 0.04 10 1 0.03609 0.03424 0.04393 0.06450 0.08633 0.09663 0.08840 0.06672 0.04736 0.04805 0.07736 0.11963 0.09099

0.00062 0.00990 0.03 155 0.05059 0.05541 0.04838 0.03952 0.03726 0.04342 0.05337 0.06034 0.06145 0.06 18 1 0.07232 0.09899 0.12707 0.11371 0.03431

0.0001 1 0.00355 0.0 1845 0.04264 0.06219 0.06612 0.05521 0.03972 0.03178 0.03802 0.05534 0.07233 0.07690 0.06607 0.05054 0.04889 0.07366 0.11250 0.08598

0.0000 1 0.00098 0.00813 0.02683 0.05222 0.07075 0.07171 0.05590 0.03513 0.02430 0.03160 0.05282 0.07371 0.07995 0.06833 0.05072 0.04647 0.06792 0.10335 0.07916

Apply the bootstrap method to (15):

Then q can be estimated from the p estimator in (16), and it becomes the optimal estimator by selecting a set {kl,k2,k3,k4} in the same manner as &,,, in section 3.1. Rewrite P(L;klik2,k3,k4,N) with such ki as PTL):

Eq (17) is used as the Weibull scale estimator and the given in table 2 .

are

4. PROPERTIES OF B/SY ESTIMATORS

This section discusses the s-efficiency of eBoo, & eBoot, their RMSE & s-bias, and the enonB & fnonB and other popular estimators, MLE & BLUE,

MSE = E{(6-O)2} = Var{e} + [Bias{e}12. (18)

The following relationships are concerned with the mean & variance of q & O estimators as follows [l]:

E{6) = O.E{g,,,}, Var{e} = 02-Vat{61,1};

~ ( r j } = V-E{$;,'~}, var{$} = q2.var{q/if}. (19)

MSE{e} = 02.MSE{61,1}, Bias(e} = 0.Bias{81,1},

MSE{rj} = q2.MSE{${,'f}, Bias{$} = q-Bias{$/,'f}.

38 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 1, 1996 MARCH

The estimators which are compared satisfy (19). To compare the accuracy of estimators, calculate MSE{d’,,,} & Bias(f?,,,} and MSE((ii,’!} & Bias($:,’!}. The accuracy of the estimator of 8 is independent of q , but the accuracy of the estimator of q depends on 8. We examine the effect of N on the performance of estimators for small N:

0 = 1.0, 2.0, 3.0;

N = 5, 10, 15, 20.

Section 4.1 considers no outliers; section 4.2 allows outliers.

4.1 Comparison of the Estimators

1 . 0 ,

0 . J El h

7 0.6 <s v)

2 0 . 4

8 3 0 .2

h

w 3 0.0

- 0 . 4 ! 5 1 0 1 5 2 0

Sample Size

Figure 1. MSE{d,,,} and s-Bias{d,,,)

To assess the performance of the BEY estimators, the MSE and s-bias are calculated from the Monte Carlo experiment from lo4 replications of the sample for each case. Figures 1 - 4 show the results of numerical computations; the BiSY estimators perform quite well when compared to the other estimators, especially when N is very small. There is fairly general agree- ment that MLE & BLUE do not always give good accuracy when N is small; this is shown in our results and in [5,6]. The B/SY estimators are s-biased but they minimize the MSE. The MLE & BLUE improve the s-bias as N increases, while the BiSY estimators do not have this tendency. When N is small, the BiSY estimators are quite accurate. Larger N do not im- prove the result. Consequently, the BiSY estimators are preferable when N 5 20.

$Boot * i B 0 0 ,

$nonB -a- 6nonB

-0- $MIe MSE G M I e bias

’?Blue * 6 B f u e

0.3 I I

-0 .2 5 1 0 1 5 2 0

Sample Size

Figure 2. MSE{fj:(!} and s-Bias{#,8} [For 8 = 1 .O]

I

-0.2 5 1 0 1 5 2 0

Sample Size

Figure 3. MSE{$::!} and s-Bias{fj:j,8} [For 0=2.0]

4.2 Robustness

Tables 3 - 6 show the MSE and s-bias of the 8 & q when the sample includes some outliers, and the relationships in (20) hold. We consider the next model for the Monte Carlo ex- periments in order to examine the effect of outliers on the ac- curacy of the estimators.

SEKUYOKOYAMA: ROBUST PARAMETER-ESTIMATION FOR THE 2-PARAMETER WEIBULL DISTRIBUTION 39

0.04 I 0.02

A ? -. <v 0.00 -- Y z Is" 8 0.02 h

- 7 -

v

P 7"

<v w .0.04 r"

-0.06

-0.08 5

fromFw(t;p,l.O), p = 2.0, 3.0, 4.0; for examining the r estimators, U observations are generated

Tables 3 - 6 show that eBoOt & 7jBoot are robust when outliers are present, and are more robust than other estimators in each examination.

from Fw(t ;p ,q) , q = 2.0, 3.0, 4.0. . 4

5 . EXAMPLE

Given

N = 7

Observations are from the Weibull distribution with 8 = 1 .O, 7=1.0.

1 0 1 5 20 Logarithms of x ( L ) , L = 1,2,. . . ,7 are ordered from smallest to largest as follows:

Sample Size

Figure 4. MSE{fji(!} and s-Bias{fj:(,8} [For 0=3.0] L l 2 3 4 5 6 7

x ( L ) -2.2766 -0.8351 0.2688 0.5384 0.6617 1.0201 1.0901

Model From (13) & (17):

A I

N = 5, 10, 1.5, 20; eBoot = 0.959, t B o o t = 1.530. sample includes U outliers, U = 1, 2, 3; (N- U) observations are generated from the original Weibull distribution Fw( t ; 1 .O, 1 .O); for examining the 8 estimators, U observations are generated

Tables 1 & 2 show coefficients of aiL) & piL, . For reference, the MLE & BLUE of the Weibull parameters based on the same data are estimated as:

TABLE 3 M SE { ffl, }with Out I iers

Shape Parameters of the Outliers,p

2.0 3.0 4.0

Number of Outliers Estimation

n Method 1 2 3 1 2 3 1 2 3

!Boot

5 enonB OMle

JBlW

!Boot 10 OnonB

&le

JBlue

!Boot

15 8nonB

@Blue

!Boot

20 OnonB

JMIe

JBIue

!MIMle

0.573 0.256 1.191 0.751

0.093 0.130 0.184 0.141

0.061 0.104 0.086 0.070

0.046 0.066 0.055 0.046

0.765 0.294 1.518 1.012

0.104 0.141 0.238 0.172

0.061 0.116 0.101 0.080

0.046 0.072 0.063 0.051

1.144 0.334 2.672 1.646

0.126 0.171 0.318 0.219

0.061 0.141 0.127 0.093

0.046 0.080 0.075 0.057

0.620 0.346 1.230 0.961

0.099 0.151 0.197 0.138

0.061 0.122 0.093 0.072

0.046 0.074 0.057 0.047

1.057 0.408 2.287 1.282

0.115 0.204 0.278 0.195

0.061 0.162 0.115 0.092

0.046 0.085 0.065 0.052

1.714 0.498 4.391 2.561

0.154 0.252 0.412 0.292

0.068 0.212 0.148 0.113

0.047 0.099 0.084 0.066

0.684 0.356 1.462 0.796

0.098 0.170 0.219 0.148

0.061 0.146 0.093 0.072

0.046 0.068 0.059 0.048

1.322 2.810 0.479 0.568 2.751 5.057 1.576 3.570

0.122 0.183 0.237 0.315 0.314 0.443 0.218 0.320

0.061 0.069 0.219 0.260 0.124 0.164 0.092 0.124

0.046 0.050 0.091 0.123 0.071 0.088 0.057 0.071

IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 1, 1996 MARCH

TABLE 4 MSE{<,l?} with Outliers

[ I 4 = 1 .O]

Shape Parameters of the Outliers,q

2.0 3.0 4.0

Number of Outliers Estimation

n Method 1 2 3 1 2 3 1 2 3

rjBoot 0.261 0.401 0.618 0.344 0.740 1.459 0.427 1.178 2.665 5 GnOnB 0.271 0.396 0.605 0.388 0.676 1.356 0.574 1.417 2.334

rjMIe 0.362 0.606 0.934 0.562 1.238 2.333 0.741 2.128 4.323 rjBlue 0.490 0.796 1.250 0.755 1.605 3.082 1.065 2.816 5.680

fBoot 0.120 0.144 0.189 0.135 0.209 0.326 0.156 0.288 0.507 10 rjnonB 0.129 0.178 0.243 0.153 0.258 0.399 0.157 0.336 0.603

fMle 0.144 0.195 0.269 0.189 0.330 0.525 0.219 0.477 0.868 ijBlue 0.164 0.225 0.314 0.211 0.384 0.633 0.264 0.552 1.020

fjBoot 0.076 0.084 0.101 0.082 0.111 0.153 0.088 0.136 0.231 15 ijnonB 0.113 0.124 0.154 0.113 0.151 0.231 0.125 0.157 0.265

f M l e 0.087 0.112 0.137 0.108 0.160 0.240 0.126 0.219 0.371 rjBlue 0.097 0.120 0.159 0.115 0.184 0.274 0.135 0.252 0.414

rjBoot 0.058 0.062 0.074 0.059 0.078 0.105 0.064 0.094 0.145 20 fnOnB 0.086 0.100 0.125 0.106 0.117 0.163 0.096 0.147 0.177

{Mle 0.065 0.075 0.094 0.076 0.103 0.148 0.083 0.135 0.210 fBlue 0.068 0.080 0.100 0.077 0.113 0.164 0.088 0.146 0.237

TABLE 5 MSE{<,::'} with Outliers

[for 14 = 2.01

Shape Parameters of the Outliers,q

2.0 3.0 4.0

Number of Outliers Estimation

1 2 3 1 2 3 1 2 3 n Method

0.018 0.032 0.059 0.028 0.079 0.169 0.041 0.139 0.310 5 fnOnB 0.062 0.076 0.097 0.069 0.109 0.188 0.071 0.152 0.328

fMle 0.069 0.100 0.141 .0.092 0.172 0.297 0.118 0.258 0.475 0.084 0.123 0.177 0.117 0.221 0.368 0.154 0.337 0.597

0,008 0.011 0.018 0.010 0.022 0.042 0.014 0.036 0.071 10 rjnonB 0.030 0.033 0.039 0.031 0.042 0.062 0.031 0.052 0.089

{Mle 0.032 0.039 0.049 0.036 0.057 0.088 0.044 0.078 0.133 0.033 0.042 0.056 0.040 0.066 0.102 0.048 0.091 0.154

0.004 0.006 0.009 0.005 0.010 0.019 0.006 0.015 0.032 l5 GnmB 0.022 0.022 0.023 0.022 0.023 0.028 0.022 0.024 0.032

0.020 0.023 0.028 0.023 0.032 0.045 0.026 0.041 0.066 0.021 0.025 0.031 0.024 0.036 0.050 0.027 0.047 0.072

0.003 0.004 0.006 0.004 0.007 0.013 0.004 0.011 0.022

{Mle 0.015 0.017 0.019 0.016 0.021 0.030 0.018 0.027 0.039 0.015 0.018 0.021 0.017 0.023 0.031 0.019 0.030 0.042

$Boot

{Blue

+Boot

{Blue

{Boot

fMle

{Blue

{Boot 20 i j n o n ~ 0.020 0.020 0.022 0.020 0.021 0.026 0.020 0.023 0.029

+Blue

SEKVYOKOYAMA: ROBUST PARAMETER-ESTIMATION FOR THE 2-PARAMETER WEIBULL DISTRIBUTION

TABLE 6 MSE(fj,;:’) with Outliers

[for e = 3.01

Shape Parameters of the Outliers,q ~

2.0 3.0 4.0

Number of Outliers Estimation

n Method 1 2 3 1 2 3 1 2 3

{Boot

5 {nonB

GMIe

{Blue

$Boot 10 GnonB

{Mle

{Blue

$Boot

15 GnonB

{Mle

$Blue

{Boot 2o fnonB

$Mle

{Blue

0.004 0.030 0.029 0.034

0.001 0.013 0.013 0.015

0.001 0.010 0.009 0.009

0.001 0.009 0.007 0.007

0.01 1 0.032 0.039 0.048

0.003 0.015 0.016 0.018

0.001 0.010 0.010 0.010

0.001 0.009 0.007 0.007

0.023 0.040 0.054 0.066

0.006 0.016 0.020 0.023

0.003 0.010 0.012 0.012

0.002 0.009 0.008 0.009

0.009 0.030 0.038 0.045

0.003 0.014 0.016 0.017

0.001 0.010 0.010 0.010

0.001 0.009 0.007 0.007

0.032 0.043 0.065 0.082

0,008 0.018 0.023 0.027

0.003 0.010 0.013 0.015

0.002 0.010 0.009 0.009

0.070 0.069 0.106 0.131

0.017 0.025 0.034 0.040

0.008 0.012 0.018 0.020

0.005 0.011 0.012 0.013

0.015 0.031 0.046 0.058

0.005 0.014 0.018 0.020

0.001 0.010 0.01 1 0.01 1

0.001 0.009 0.008 0.008

0.058 0.059 0.097 0.121

0.015 0.020 0.031 0.036

0.006 0.010 0.017 0.019

0.004 0.010 0.011 0.012

0.125 0.114 0.172 0.206

0.029 0.034 0.052 0.059

0.014 0.014 0.026 0.029

0.010 0.011 0.016 0.017

ACKNOWLEDGMENT

We are grateful to the Managing Editor, Associate Editor, and the referees for their comments & suggestions. We especial- ly thank Prof. S.P. Mukherjee of Calcutta University for reading the manuscript and making many helpful suggestions. The research for this paper was partially supported by GRANT-IN- AID from Teikyo University of Technology.

REFERENCES

J.L. Bain, C.E. Antle, “Estimation of parameters in the Weibull distribu- tion”, Technometrics, vol 9, 1967, pp 621-627. A.C. Cohen, “Maximum likelihood estimation in the Weibull distribu- tion based on complete and on censored samples”, Technometrics, vol

B. Efron, “Bootstrap methods: Another look at the jackknife”, Annals ofStatistics, vol 7, 1979, pp 1-26. B. Efron, “The jackknife, the bootstrap and other resampling plans”, SIAM monograph 38, 1982, CBMS-NSF. D.I. Gibbons, L.C. Vance, “A simulation study of estimation for the 2-parameter Weibull distribution”, IEEE Trans. Reliability, vol R-30, 1981 Apr, pp 61-66. A.J. Gross, D. Lurie, “Monte Carlo comparisons of parameter estimators of the 2-parameter Weibull distribution”, IEEE Trans. Reliability, vol R-26, 1977 Dec, pp 356-358.

7, 1965, pp 579-588.

[7] H.L. Harter, “The use of sample quasi-ranges in estimating population standard deviation”, Annals of Mathematical Statistics, vol 30, 1959,

[8] J. Lieblein, M. Zelen, “Statistical investigation of the fatigue life of deep- groove ball bearings”, J . Research Nat’l Bureau of Standards, vol 57,

[9] T. Seki, S. Yokoyama, “A note on the application of the bootstrap method”, Technical Report No. 8, 1991, Musashi Institute of Technology.

[lo] T. Seki, T. Takashina, S. Yokoyama, “Bootstrap for the normal parameters”, Communications in Statistics - Series B, vol 22, 1993, pp 191-203.

pp 980-999.

1956, pp 273-316.

AUTHORS

Tetsurou Seki; Dept. of Information Systems; Teikyo University of Technology; 2289-23 Uruido, Ichihara-shi, Chiba 290-01 JAPAN.

Tetsurou Seki (S’89, M’91) is a research associate in the Department of Information Systems, Teikyo University of Technology. He received his BE (1986) in Industrial Engineering from Chiba Institute of Technology, and his MS (1988) in Statistics from Musashi Institute of Technology. He is pursuing his PhD in Statistics at Keio University.

Dr. Shin-ichiro Yokoyama; Dept. Industrial Engineering; Musashi Institute of Technology; 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158 JAPAN.

Shin-ichiro Yokoyama (M’88) received his PhD (1982) from the Dep’t of Industrial Engineering at the Tokyo Institute of Technology in Tokyo. He was a visiting research associate at William E. Simon Graduate School of Business Administration, University of Rochester in 1985-1986. His current research interests include quality control, safety management, reliability theory and its applications. He is Associate Professor in the Dep’t of Industrial Engineer- ing, Musashi Institute of Technology. He is a Member of IEEE.

Manuscript received 1995 September 17

Publisher Item Identifier S 001 8-9529(96)02248-8 4 T R b