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A Note on a simple testfunction for the Weibulldistribution locationparameterDick B. Schafer aa Body Engineering Office , Ford MotorCompany , Room 2079, Dearborn , Michigan ,48121 , USAPublished online: 22 Dec 2011.

To cite this article: Dick B. Schafer (1975) A Note on a simple test function forthe Weibull distribution location parameter, Scandinavian Actuarial Journal,1975:1, 1-5, DOI: 10.1080/03461238.1975.10405072

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Scand. Actuarial J. 1975: 1-5

A Note on a Simple test function for the WeibullDistribution Location Parameter

By Dick B. Schafer

Abstract

In this short note some of the results of Dubey (1963) on a simple test functionfor the location parameter of a negative exponential distribution are extendedto cover the case of the three-parameter Weibull distribution.

1. Introduction

S. D. Dubey (1963) investigated the properties of the simple test function

s= (ta - GO)/(fb - fa) (1.1)

(1.2)

(1.3)

where f l <... <fa<... <fb <... < t n is an ordered random sample from thenegative exponential distribution with probability density function (p.d.f)

f(f) = {(I/()) exp ~ -(f-G)/(J); f>G, (»O0, otherwise.

G, the location parameter, is referred to as the guarantee time in Dubey (1963).(J is the scale parameter. The hypothesis being tested is H o: G = Go against theone-sided composite alternatives HI: G < Go or HI: G > Go, or, the two-sidedcomposite alternative HI: G =1= Go. Dubey (1963) derived the p.d.f. of Sunderthe null hypothesis, the power functions for G < Go and G =1= Go as well as themoments of S.

This paper investigates a similar test function under the more generalassumption that the samples are from the three-parameter Weibull distributionwith p.d.f.

t ={(b/(J)(f-G)8-1 exp (-(t-G)8/(J); c-o. b, (»Of( ) 0, otherwise.

If the shape parameter, b, equals one, (1.3) reduces to (1.2). Under the moregeneral assumption that O<b< 00, the power function for GIS Go, will bederived as well as the p.d.I, of Sin (1.1). In each case some of the analogousresults of Dubey (1963) will be derived as special cases.

2. The decision procedure and the power function

In the decision procedure, (J is assumed to be unknown and b is assumed tobe known. If Ho: G=Go, HI: G=GI =1= Go, the decision function is

1 - 753821 Scand. Actuarial J. 1975

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2 D. B. Schafer

{O, if k1<s<k2

.p(s)= 1, otherwise.

.p(s) is the probability of rejecting Ho. s = «;- GO)/(tb- ta) with t« and tbbeing the ath and bth order statistics in a simple random sample from thethree-parameter Weibull distribution of (1.3). The critical values, k1, k 2;

0~kl<k2<oo, are chosen so that l-a=H:f(s)ds, where f(s) is the p.d.I.of sunder Ho•

The power function, P(G1) , will now be derived for the case of G1=!=Go.If G1=!=Go,

P(G1)= 1- P(k1«ta -GO)/(tb- ta)<k2).

The joint p.d.f, of ta and tb is

f(ta, tb) = C(ta)f(tb)F(ta)a-1 [1 - F(tbW- b[F(tb) - F(ta)]b-a-l

(2.1)

(2.2)

(2.3)

where C=n!/«a-l)!(b-a-l)!(n-b)!) and for l=l, ...,a, ...,b, ...,n; F(t/)is the c.d.I, of f(t/) for the unordered sample and is given as

F(t) = {I -exp ( -:- (t/- GI)"/O), t/ > GI ,

/ 0, otherwise,

The statement in (2.1) is modified to the equivalent statement

P(GI) = 1-P{{ta - GO)/k2«tb- ta)<(ta - GO)/k1). (2.4)

Expanding the terms in (2.2) as well as utilizing the transformations Ya= ta- Goand Yb = tb- ta changes (2.2) to

b-a-Ia-lf(Ya,Yb)=C 2: 2: (b-~-I)(ail)(-1)1+1(MO)2(Ya+Go-Glt-1

1~O j~O

X (Yb +Ya+ Go -G1)'H exp [ - (b - a + i - j) (ya+ Go - GI)"jO]

+[- (n -b +j+ l)(Yb+Ya+ Go -Glt/O]. (2.5)

Consequently, (2.5) is given by

fOO [fValkl ]P(GI) = I-P(Ya/k2<Yb<Ya/kl)= 1- f(Ya,Yb)dYb dYa,L Valk, (2.6)

where L=max. (0, G1-GO) ' since the limits of Ya and Yb are independent.Integrating over Yb and using (3.3), which is given later in this paper, thepower function is

P(GI) = C b-fl ail e-f-l) eil) ( -1)l+1fOOg(Ya){h2(Ya) - hl(Ya)}dYa,1~O I~O L

g(Ya)= (15/0)(n - b +j + 1)-1 (Ya+ GO - G1)6-1 exp [- (b - a + i - j)

x (y" + GO - GI )6/0],

hp(Ya) =exp [- (n - b +j + l)(Ya Wp+ Go - G1t/0],p = 1,2, (2.7)

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Simple test function 3

where Wp = 1+k;l. Approximate solutions to (2.7) are provided by numerous,standard quadrature methods. For the specific case of <'5 = 1 and k l =0,(2.7) reduces to

b-a-l a-I e-f-l) (ajl) ( _1)1+1 k 2 exp [- (n - a + i + 1)(Go- GJ)/OJP(Gl ) = 1- C 2: 2: . - .. - - .

I~O 1-0

(2.8)for Gl-GO<O and, to

b-a-l a-IP(Gl)=l-C 2: 2:

I~O I~O

e-f-l)(ajl)( -lri k 2 exp [ - (n - b +j + l)(Go- Gl)/OkJ (29)X (n-b+j+l)[(n-b+j+l)+kln-a+i+l)J .

for Gl - Go > O. 2.8 and 2.9 are algebraically equivalent to the cases given byDubey (1963), pages 13 and 14 respectively.

3. The p.d.f. of sunder Ho

For G = Gl = Gothe joint p.d.f. of ta and tb remains the same as that defined in(2.2) through (2.4). Again by expanding the terms in (2.2) and utilizingYa= ta- Go and Yb= tb- ta results now in

b-a-la-lf(Ya,Yb)=C 2: 2: e-j-l)(ajl)(-1)i+I«)/O)2y~-1(Yb+Ya)6-l

1-0 1-0

X exp [ - (b - a - j + i)y~/O - (n - b +j + l)(yb +Yat/O]. (3.1)

Letting S = Ya/Yb and t = Ya and integrating over t results in

b-a-la-lf(s) = C 2: 2: (b-j-l)(ajl)( - 1)i+1

I~O I~O

(3.2)<'5(S-l + 1)6-1

X s2[(b_ a-j + i)+ (S-l + 1)6(n - b +j+ 1)]2'

Then, for example, if Ho:G = Go, HI: G> Go and setting k l = 0,

0:= P((ta - GO)/(tb- ta» k 21G= Go) = f:/(S)dS= 1- C b~~l :~:

b-a-l (a-l ( )i+1 (n - b +j + 1)-1X( i ) I) -1 [en _ b +j + 1)(1 + k

2-1)6+ (b - a - j +or (3.3)

To demonstrate that (3.3) equals 0 for k 2= 00, the following result fromDubey (1963) is required;

b-a-l a-I 1C " "(b-a-l)(a-l)(_l)i+1 = 1

~L. L., j i 1'._ L I s I 1 \. r.; _ I ,! I 1 \. ,1-0 I~O

and (3.3) takes on this form when k 2 = 00.

(3.4)

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4 D. B. Schafer

By setting ~=1, the expression in (3.2) reduces to

b-a-l a-l 1 (3 5)"" "" (b-a-l) (a-l) ( 1)I+J 2' .f(s)=C £., £., 1 1 - [(n-b+j+l)+s(n-a+i+l)]

1=0 1-0

which is the form given by Dubey (1963).

4. Determining critical values

For any specified level of ex, the critical value, k~, appropriate for anyWeibull variable Z with shape parameter ~z, can be derived from the criticalvalue, kg, appropriate for any Weibull variable, X and for that specified levelof ex. The notation z,...., W(Gz, ()z,~z) means that Z is a Weibull distributedrandom variable with location parameter - 00 < Gz < 00, scale parameter,o<()z< 00, and shape parameter, O<~z<oo. The following relations establishthe desired method.

[Z a- Gz k]

P Zb-Za ~or:( a

[rz - G )bz k bZ]

_ a Z ;:;, s; g

- P (Zb - Gztz - (Za - Gz)bz ""or"" (kg+ 1)"Z - k:z

=p[~~or:( k~z ]Yb-Ya (kg + 1)"z-k~z

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

[(X - G )"x kbz ]_ a X ;:;, ~ g

-p (Xb-Gxtz-(Xa-Gx)"X ""or"" (kg + 1)"Z-kgz

_p[Xa-Gx;:;, «; kt ]- Xb-Xa :;--or"" (kg+ 1)t/>-kt .

In these relations z-: W(Gz, ()z, ~z),Y""" W(O, 1, 1), X,...., W(Gx,()x,~x), - 00 «a;Gx<oo,O<()z,()x<oo,O<~z,~x<oo, and eP=~z/~x, The relations of (4.1) to(4.2) and (4.4) to (4.5) are based upon a direct algebraic manipulation. Therelation of (4.2) to (4.3) is based upon the fact that the transformationY = (Z - Gz)bz/()z reduces the three-parameter Weibull distribution to the unitWeibull distribution,

{exp (-Y), Y>O

fey) = 0, otherwise.

The basis of (4.3) to (4.4), in turn, is from the relation Y = (X - Gx)"x/()x, whereZ and X are any three-parameter Weibull random variables.

The purpose of the relations (4.1) to (4.5) is that if for some combination ofn, a and b, a critical value of k l or k 2, say kg is obtained from the results of (3.2)and (3.3) for a specific value of the shape parameter, ~z, the appropriatecritical values, k~, for all other values of ~, say ~x, can be calculated from the

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Simple test function 5

computationally expedient relation k~=kt/[(kl1+1)<P-kt]. Dubey (1963) givesextensive tables of critical values for b= 1. These can be used to generate newvalues of k~ of sufficient accuracy for most practical purposes by settingbz=1 in (4.1) to (4.5).

Acknowledgement

The author is grateful to a referee for detecting several mistakes and generally improving thequality of the paper.

Reference

Dubey, S. D. (1963). A generalization of a simple test function for guarantee time associated withthe exponential failure law. Skandinaoisk Aktuarietidskrift 46, 1-24.

Recieved July 1973

Dick B. SchaferRoom 2079Body Engineering OfficeFord Motor CompanyDearborn, Michigan 48121, USA

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