On a test for generalized upper truncated Weibull distributions

Statistics & Probability North-Holland

October

Letters 12 (1991) 273-279

1991

On a test for generalized upper truncated Weibull distributions Servet Martinez Departamento de Ingenieria Santiago, Chile

Matemcitica,

Facultad de Ciencias Fisicas y Matemciticas,

Unioersidad

de Chile, Casilla I 70-4 Correo _z,

Fernando Quintana Departamenio de Probabilidad y Estadisiica, Santiago 22, Chile

Facultad de Matemciticas,

Pontificia

Universidad

Cat6lica de Chile, Casilla 6177,

Received June 1990 Revised August 1990

Absfracr: We study upper truncated Weibull random variables with density given by ga.s,Jt) = /3&-’ exp( - Pt*)(l - exp( - p@))-’ for 0 d t d 7 (T is the truncation parameter), 6 > 0 and /3 E W. Denoting by B, d and 7^ the maximum likelihood estimators we show that sign(p) = sign(f -G,,), where G, = (l/n)X~=1(~/~)8. It is also shown that 4&(f - GiB=‘)) converges to a normalized Gaussian. This result is then used to provide a test for the hypothesis /3 = 0.

Keywords: Weibull

distribution,

upper

truncation

parameter,

maximum

likelihood

estimator,

spacings.

1. Introduction In this paper we deal with the upper truncated Weibull distribution, whose density function is given by gp,B,Jt)

= p’ts-’ eep’s 1 - e-PT6

for 0 < t < 7.

0.1)

The parameter j3 takes arbitrary values in 88. For j3 = 0 density (1.1) reduces to g,.a,T(i)=F

forOGtg7.

This family has been studied in relation to seismological models. In several works it has been argued that the case j3 > 0 fits seismological data for earthquakes of magnitude 2 7.0 (see Brillinger, 1982; Nishenko, 1985). When P < 0 the truncated Weibull distribution gives a higher weight to the values around the truncation parameter. This family has been used in the study of the historical seismological sequence in the central region of Chile, taking into account only the earthquakes of magnitude 2 8.0 (see Martinez et al., 1986). A more detailed description as well as some results concerning this family have been previously developed in Martinez and Salinas (1989). Partially

financed

0167-7152/91/$03.50

by grant

FONDECYT

0553, 88-90.

0 1991 - Elsevier Science Publishers

B.V. All rights reserved

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STATISTICS & PROBABILITY

In this work we propose a test for the null hypothesis density (1.1). A preliminary result towards this purpose sign of p^ is the same as that of : - G,,, where G,=;

i

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p = 0, based on a random sample T,, . . . , T, from is given in Proposition 2.1, which asserts that the

(;)8,

0.2)

i=l p^, 8, 7^ being the corresponding which concerns the asymptotic fi = 0). Thus in Theorem 3.1 we standard Gaussian distribution.

maximum likelihood estimators. In Section 3 we establish our main result, behaviour of i - G,(O) (being G,,(‘) the variable G, under hypothesis Ho: show that the asymptotic null distribution of statistic 4J?;;;($ - GZ’O’)is a This provides a basis to set up a test for the null hypothesis fi = 0.

2. Truncated Weibull distribution The random variable we shall deal with will be positive and upper truncated. The density h, will often depend on some real parameter /? and will be properly defined for p # 0. The values for p = 0 will be defined by a continuity argument through ho = lim, _ ,A,. We denote by d, the indicator function of the set D, i.e. d,(x) = 1 if x E D and 0 otherwise. For A > 0, p E R we denote by XP,A an exponential upper truncated random variable with parameters p and A, whose density is given by

(2.1) When

/3 + 0 we get a uniform

distribution

on [0, A]:

&4(x) = ~~o&3,Ab)= The expected

value of XP,A is

and

E( XO,A) = jimoiE( XP,A) = iA. Let sign(u)

= 1 if u > 0, - 1 if u -C 0, and 0 if u = 0. It is easy to verify that

sign(/?)=sign

(1l-E

(2.3)

(9)).

For 6 > 0 the random variable ( Xp,0)1’6 gives all the weight to the interval distribution of the random variable

[0, 71, where

7 = A”‘.

The

Tp,s.* = (X,QY will be called an upper gp,s,&)

274

truncated

Weibull

= ‘;r:_e;l’“dp.rl(‘).

with parameters

p, S > 0 and 7 > 0. Its density

is given by

(2.4)

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When

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October 1991

p = 0 we get iwl = -y&l.T1W

g0,8,,(t) = /3imog ~ p,dt>

(2.5)

From its definition, it follows that (Tp,s,7)s has an upper truncated exponential distribution with parameters p and TV. In the case p = 0 this distribution is uniform on the interval [0, ~~1. From (2.3) we also derive

sign(b)=sign(+-lE(+)‘).

(2.6)

Let now T=(T,,..., T,) be a random j3 # 0, the likelihood function is L(T.

p

3 Denote

s 3 27

)

=

and that the following S k

of size n from an upper

truncated

Weibull

(ps)“(n:1,7;s*d,o,,,(I;)) exP(-PZ=,C) maximum

likelihood

estimators.

TB,s,7. For

(2.7)

(1 - exp( -pT”))”

by p^, s^, 7^ the corresponding +=max{7;:

sample

It is easily checked

that

i=l,...,n}

(2.8)

equalities

hold:

T’ + np^$(exp(

p^+“) - l)-’

= n,

(2.9)

i=l

(2.10) In the case j3 = 0 the likelihood

function

is

(2.11) Denote and

by i(O) and f(O) the maximum

likelihood

estimators

for this case. We get +(O)= 8, as given by (2.8),

gog(;))j’.

i(O)=-n(

It is now easy to see that s^(‘) 1s also the limiting To study the sign of p^ we shall use the statistic

(2.12) value of s^ in (2.10) as p^ tends to zero.

(2.13) and the modified

version -

lim G,,. a-0

(2.14)

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Note that G, is the natural counterpart of the expected value in (2.6) replacing parameters r and S by their corresponding maximum likelihood estimators. Intuitively, one would expect that G,, is a consistent estimator of the above mentioned expectation. The first result establishes an equality analogous to (2.6), but written for estimators: Proposition 2.1. Let @ f 0. Then sign(p)

= sign(i

- G,,).

(2.15)

Proof. Put 5 = fi?“, so sign( p^) = sign( [). It is easy to prove that equation G,, = ~(0 for 5 # 0, with

(2.9) is equivalent

G(t)=;-&.

(2.16)

We are then led to prove that sign(t) From (eS - l)[ > 0,

= sign(i

- G(5)) for 5 Z 0.

’ 4

if and only if

2(e’ - 1) - 2.$ > (e5 - l)t,

&C> ’ !z

if and only if

2(eE - 1) > (e’ + 1)E.

+(t>

to the equality

or equivalently,

Put p( 6) = 2(es - l)/(e[ We have shown

+ 1) - 5.

(2.17)

that (p(t) > i if and only if p(E) > 0. Besides,

p’(E)=

p(O) = 0 and

-[(et-l)/(e6+1)12.

Hence p’(O) = 0 and p’(5) -C 0 for E,# 0. Then sign( - p(t)) if .$ < 0 and (p(t) < i if 6 > 0, and (2.15) follows. •I

= sign([).

We have thus proved

that G(E) > :

Remarks. (1) Note that lims4a+(5) = 4, so that we can properly define +(O) = i. Besides, through the same continuity argument we get 6 = 0 if and only if (p(E) = i. However, the equality Gi’“’ = +(O) is not verified. In fact, the statistic $ - GA” takes positive and negative values. Its distribution will be useful to propose a test for the hypothesis j3 = 0. (2) Despite the clear analogy between equations (2.6) and (2.15), the last equality is not obvious. In fact, the proof of Proposition 2.1 is not related to any probabilistic aspect, the main fact being the likelihood equation (2.9). On the other hand, there is no apparent way to analitically solve equations (2.9) and (2.10), so that the exact distributions of G,, is unknown.

3. Asymptotic result In this section

we prove the following

Theorem 3.1. Under the hypothesis fi(i-G,“‘)%Z where the distribution 216

asymptotic

result:

j3 = 0, GL’O’in (2.14) satisfies

asn+cc of Z is N(0, A).

(3.1)

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Proof. Introduce the random variable ? which denotes i.e. q= 7^. Define the family of n - 1 random variables if i < L,

T

Tl, =

T.

i

October 1991

i such that q attains

the index

the maximum

value,

for i=l,...,n-1.

if i > T,

If1

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We have gcw

$ + +‘i1

Gn’o)=

.

i=l

The maximum

likelihood

estimator

when hypothesis

p = 0 is assumed

is

-1

go’=

(?ji’=-~(~+(~)

$log I; ii_,

_n

i

.

so

gogj q-‘.

!y!= _[

(3.2)

Define Z(,, = ( qi,/+)’

and

Rci) = -lOg( Z(i)).

(3.3)

We have ( ~i,/?)b’o’

= ( Z~ij)B’O”s = exp( -R,,$“)/S).

From (3.2) and (3.3),

GA’)= i + i ‘i’exp i=l

Consider

now the random

j=l

variables i= l,...,

Wi= -log(q/r)“, By definition

(3.4)

-PzR~~,/~~‘~R,~, (

n.

(3.5)

of [ w=min{w:

i=l,...,

rz}.

From (3.3), (3.5) and (3.6) the following l4$&i, =

w$

i I%+, - I4$

if i < L, if i > L,

(3.6) equality

is easily shown:

fori=l,...,n-1.

(3 -7)

The assumption j3 = 0 implies that random variable (T/r)* is uniformly distributed on [0, 11. Hence (B$: i= l,..., n) are i.i.d. Exp(l), where Exp(1) is the exponential distribution with mean 1. Then, it is 277

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easy to show that the set of random variables ( Rcij: i = 1,. . . , n - 1) given by (3.7) are also i.i.d. Exp(1). Using this last fact and a classical result on Spacings Theory (see Pyke, 1965) it follows that

!

R,,,/

n-l c Rcj,:

i=l,...,n-1

_.

j=l

has the same distribution as that of a sequence of n - 1 spacings (Di: i = 1,. , n - 1) determined by n - 2 independent uniform random variables on [0, l] (if we denote by U,, . . . , U,,_, the ordered sequence of uniform random variables and we define U, = 0, U,_, = 1, we put Di = U, - Ui_,). The distribution of G,“’ coincides with that of

(3.8) We have

(3.9) Le Cam’s theorem &

(see Le Cam, 1958; Pyke, 1965) implies

;$’ (e-“Q I 1

distribution

of

_ +)

is N(0, Var(e- ‘) - Cov(eeasily obtained. q

“, V)*), where V is exponentially

Based upon statistic

2.1 and Theorem

Proposition

that the asymptotic

3.1 we propose

distributed,

with mean

to test the hypothesis

1. From

this, (3.1) is

Ho: /? = 0 by means

of

J48n(+ - G,(O)). If it behaves like a distribution N(0, 1) we accept Ho, if not we reject. The critical test, for a given significance level a, is

region of this asymptotic

(3.10) being Z1_a,Z the percentil 1 - fa of N(0, 1). The one-sided hypotheses Hh: /? > 0 and H;/: fl < 0 can be analogously tested. It seems interesting to know the properties of the power function of these tests. For finite samples one could do a simulation study including a few values of /3, 6, and 7. For large samples one could study the general asymptotic distribution of G,,. Once this distribution is obtained, the power of a size cx test is readily obtained.

Acknowledgements

We are indebted to Professor Guido E. de1 Pino for indicating us the papers (1965), and for valuable suggestions for the final version of this work. 278

of Le Cam (1958) and Pyke

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1991

References Brillinger, D.R. (1982), Seismic risk analysis: Some statistical aspects, Earthquake Predict. Res. 1, 183-195. Le Cam, L. (1958), Un theortme sur la division dun intervalle par des points pris au hasard, Publ. Inst. Statist. Uniu. Paris. .Vol. 7, pp. 7-16. Martinez, S. and V.H. Salinas (1989), Distribuciones Exponencial y Weibull Truncadas: Propiedades y Estimation de Parametros, Preprint, Univ. de Chile (Santiago, Chile). Martinez, S., V.H. Salinas and E. Scheming (1986). Historical seismicity: A probability law for the total elastic energy

and times between occurrences, Seismic Risk Workshop, International Physics Center (Bogota, Colombia). [In Spanish.] Nishenko, S.P. (1985), Seismic potential for large and great interplate earthquakes along the Chilean and Peruvian margins of South America: A quantitative reapprisal, J. Geoph. Res. 9O(B5) 3589-3615. Pyke, R. (1965), Spacings, J. Roy. Statist. Sot. Ser. B 27, 395-449.

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