*Supported by the U.S. Army Research Office, Durham, undeT Contract No.
DAHC04-71-C-0042.
nCmJSTNfSS OF CERTAIN TESTS OF CENSORING OF EXTRENE SAr~?l E VALUES
I I': SOME EXACT RESULTS FOn EXPONENTIAL POPULATIOf!:;*
by
N. L. Johnson
Department of Statistics
University of North CaroUna at Chapet 11Ht
Institute of Statistics Mimeo Series No. 940
July 1974
ROBUSTr-JESS OF CERT/\Hl TESTS OF CE~!SfJ~ING OF E)(TRFJ~E SAP1PLE V;'-.LUES - II
SOME EXP,CT RESULTS FOR EXPONENTIAL POPULATIONS
by
N. L. Johnson
University of North
1.
Ca1~oZina
at Chapel Hill
Introduction
In [1] we have given general formulae for evaluating the effect of an
incorrect choice of population distribution on certain tests of sample censoring.
(For definitions, etc. see [2][3].)
These forrm.llae apply "'hen the form of population distribution in known,
but certain parameters
~
possibly incorrect values
!'.1hen
~*
~
have to be estimated.
are used for
~e
formulae apply when
Q.
has to be estiMated, there will usually be random variation in
Tne overall effect of this can be evaluated by taking expected values of
the formulae in [1] with respect to suitable distributions for
l~e
~*
shall suppose throughout the paper that there is sD.:r.1pling from a
population in which the measured character has an absolutely continuous
distl'ib~ltion.
2.
Gen£ral Discussion
As in [1] and earlier papers we suppose
r
values
Xl
$
X2
$ ••• $
Xr
are available and wish to test whether they are a complete random sample, or
cerna from a complete random sample of size
So
smallest and
h~'P~thesis
by
sr
(sO + r + Sr) , from which the
largest values have been censored.
We denote this It'st
cOTresronds to the case of no
censo~ing.
-2-
The critical regions of the tests described in
[1]~2]
are all defined in
terms of the probability integral transforms
X
el)
f(xle)dx
Yr
'"
of the minimum and maximum values
density function of
X.
=
Xl' X
r
J_: fexl~)dx
respectively.
fexl~)
Here
If, in fact, an incorrect value of
~
is the
is used -
e* , say - then instead of calculating Yl , Yr we actually calculate
X.
(2)
Y;
How provided
= f_:
f (x I~) > 0
unique function
h * (Y.)
J
fexIQ*)dx
for all
of Y. .
J
(j
x and
= 1,
Q,
r) .
(1) and (2) define
So when we use a critical region
Y*
j
as a
w(k)
defined by
(C~k)
is a constant which \'!ou1d give lOO<x% significance level if !lO,O
were valid and the correct
region
*
w(k)
2 were used) we are actually using a critical
defined by
~ C (k)
(3)
in terms of the correct
(under
<X
Yl
and
Yr
The latter have joint density function
-3(4)
For a given
region
*
w(l,:)
~ * • t~e power
8(1<:)(so, sr lSZ* )
is the integral of (4)
of the test with critical
over the region
(5)
The unconcitional, overall power
respect to variation in Q*
of
S(k) (sO' sr)
is the expected value, with
S(k) (sO' sr IQ * ).
~1!. ?~r·~t~yl1ir ~1)e ~ftual
significance level is
(6)
8(k) (0, 0) = r(r - I)E[
_
J* J. (Yr
a, if C(k)
a
2
c.YI dYrJ
w(k)
(e * ,of course appears inplicitly in
equal to
- yl / -
*
w(k)
.)
TI1is will not in general be
is calculated in accordance with the formulae in f.2].
It will, in particular cases, be possible to choose
significance level is
density function of
so that the overall
a, but the necessary formulae will depend on the
Y., and on that of Q* , while the values in (
As a concrete example we consider the case of normal variation.
In [1] it was shown that if
~
=
(~,
0)
and
r;:;-; -1
2 -1)
f(x I~, 0) = (ov2'IT)
exp ( - (x - 0 2 (20-)
then
*
SCsO' srl~ * , 0)
depends only on
) do not.
-4-
fy,
If
~
* and
a
*2
* = (~ * -
~)/a
e: *
and
= a * /a
are calculated from an independent random sample of size
N,
as the sample mean and variance respectively then .'
.{i)
(ii)
(iii)
fy,
* and e: * are independent
fy,*
l
lJ(D, N- )
is distributed
e: *
is distributed
(N - 1)
which gives useful practical indicatL.
-J.?
XNl
This information is sufficient to enable us to calculate
(7)
the expectation being with respect to variation of
~*
and
a*
Although this may be evaluated explicitly using numerical cubature
applied to
8(5 , srl~ *
0
,
*
a)
Hhich is itself quite a complicated function,
the calculations are quite heavy even with the assista!lce of an electronic
computer.
(It is hoped to provide some numerical tables in the third report
in this series.)
It is rather easier to use a two-entry table of values of
*
S(sO' srl~ * , a)
and (very roughly) average with respect to the distributions
af
~
* ar.d a *
A further possible complication is that it may be necessary to estiJ':)ate
Xr actually available, so that ~* and
are no longer mutually independent. !"e do not consider this
Q, from the r
values
Xl"'"
(Y , Yr )
I
situation in the present paper, except for a brief
discussio~
in the final
-5-
section,
3,
Exponential Populations
T,~e
main aim of the present paper is to evaluate performance of the three
tests described in ; [2] when applied to a population in which the (unorthodox)
X's
have an exponential distribution with expected value
pose
6
e, l'!e will sup-
to be estinated as the arithmetic mean of a random sample (indepen-
dent of the
r
values to be analysis) of size
a* /a
(3)
n, so that:
is distributed as
Under these conditions it is possible to get explicit formulae for properties
of the tests which are relatively simple (as compared, for example, with the
normal case).
This discussion will, it is hoped, help to make clearer the general
description in Section 2.
TIle population density function is
t0 a
'
pondIng
1)l
e have
e-1 exp (-x/a)
(e,
x > 0)
corres-
10 2 d'Istrluutlon.
'h
.
~X2
Y
=
x
0
I
e-
1
exp(-xla)dx
=1
- e
-x/e
(X > 0) ,
Eence
(9)
y* = 1
*
_ e- x/ a
= 1 _ (1 _ Y) e/a
The critical regions to be used in testing for
*
extre~e censorin~
(1) form
below (2) symmetrically or (3) in general, are (see -[2]) respectively'
-6-
Yl ~ C(1)
ex
(10.1) (Below)
Yl (l - Yr ) ~ C(2)
ex
(10.2) (Symmetric)
(10.3) (General)
Yl
+
(1 - Y ) ~ C (3)
ex
r
8 * , is used for
If an incorrect value,
e ,
t~e r~gions
actually
use~
are, respectively
(11.1)
1 - (1 -
(11. 2)
{l -
(11. 3)
1 - (1 -
4.
(1
Y )8/e * ~ C(l)
1
ex
_ Y )8/e * } (1 _ y )e/e * ~ C(2)
1
r
ex
y )8/e *
1
*
(1 - yr )8/e ~
+
c (3)
ex
Censoring from Below
The inequality (11.1) can also he written
(11.1)1
\l1herc we put
T =
(12)
Since; given H
SO,5 r
parameters
50
+
1, r
+
e* /e .
the distribution of
sr we have
Yl
is stancard beta with
-7-
=I
where
Ip(a, y)
= {R(a,
(l-C
(1) T (r
) ..
+
sr' So
+
1)
a
y)}-l JP t a - l (1 - t)y-l dt
o
is the incomplete beta
function ratio.
In particular the conditional significance level is
(14)
The unconditional significance level is
~[QIJ ( 1 (0
ole*)]
= F[(l _ C~l))rT]
)'
,"<
""
::.
= E[
(15)
= {I
(I -
C~I))h~,r'''
- rN- l log(l _ C(l))}-N .
In order to make the significance level equal to
e(l)
CL
Le.
(16)
(= c (1) (N) , say) to satisfy
a
J
a
Ct,
l'le
J'1Ust choose
-8-
If the value
eel)
ex
= 1 - a llr , which is appropriate when e is known, is
used then the overall significance level is actually
-1
(1 - N
log ex) -N
...
(17)
As would be expected this tends to
ex
as
The power with respect to
B(1) (sO' 0)
= E[
N tends to infinity.
(i.e. actual censoring from below) is
I
(1) T (r, sf) + 1) ]
(I-C)
ex
.I
(r+s ) I
= (r-l) oI sOl
(18)
Inserting the value for
cance level equal to
(19)
(0)
~(l) sO'
D
C (1)
ex
ex) l·,re obtain
from (16) (vhich gives overall signifi-
_ (r+sO) I
So
J. [so]
. -1 {I
\' ( 1)
()
- (r-l)ls " j~1) j
r+J
O
Table 1 gives values of
8(1) (sO' 0)
for
+
(l+J·r- 1 ) (N - 11 N_ 11 }-~1 •
"'-
r = 5(5)25, rr = 5,1.0,25,50
and
So.1 = 2 (2)8 with ex = o. as. These are pm-lers when the actual significance
level is 0.C5.
Table 2 gives the actual significance levels and powers obtained if the value eel) = 1 - ex l / r is used. In this case
ex
-9-
(
)
S(l) sO,O
(20)
(Some values of
5.
=
~o (-l)j !~O]
(r+J·) -lU_ (l+J·r -1)N- 1
loga}-r1
J
(r+s o) !
(r-1)lsO!
l
j=O
S(l (sO' sr)
with
Symrr~trical_ and ~eneral Censoring
sr ~ 0 are also given.)
Similar calculations for the tests with critical regions (11.2) and (11.3)
are complicated by the fact that both
Y
l
and
Y
r
appear in the inequalities.
In these cases we use the fact that the joint censity functions of 21
and Z = 1 - Y under
is
r
=1
- Yl
r
s
(21)
Z
r
*
(3(k) (sO' Sr1e)
and so the conditional power
$
r
1)
is obtained by integrating
(21) over tp-e regions
(22.2)
(1 - zle/e *Jzre/e *
~ C(2)
a
for
k
=2
(syr~etrica1
*
+ Ze/e
c (3)
for
k
=3
(general censoring)
censoring)
or
e/e *'.
(22.3) 1 - ;Zl
r
a
The resultant value has in each case, to be averaged over the distribution
e*
(or equivalently of T
fiakine the transformations
(23)
= e* /e
).
v. = z.a/e
J
J
*
(j = 1,2)
we have
-10-
The regions of integration are:
(24.2)
For
k = 2: (l - v )v ~ e(2)
1 r
a
(24.3)
For
k = 3: 1
- vI
+
.
:>
0
v ~ C(3)
r
a
vr
~
0
~
vr
~
vI
~
1
~
vI
1
~
.
The overall power is obtained by taking the expected value with respect
to the distribution of
T, which is
e2I'!)
-1 2
X2N
By performing the integration first with respect to
respect to
T, and t1:en with
v r ' we can reduce the triple integral to a finite sum of single
integrals.
TIle integrand in (23) is
= (vlvr )
So
~
-1 2
T
L
r-2
~
L (-1)
..
1+)
· 0 ·J= 0
1=
[so] (r- 2) vI(
.
.
. . 1)T
T+1-J-
v
1)
(sr
r
The expected value of this with respect to variation in
(v.v)-l
1
r
o r-2 f
.. s
S
]
L L (_1)]+)
.0 [r:2JCW)-'
+j+1)T
T is
?
tl)
. -0 ). --0..
1-
x
(25)
x [1 - r(l {(r + i
- j - 1)
log v. + (s
1
r
+ j
+ 1)
log v }]-n!+2) .
r
-11-
(Using the relation
E[ X~ exp(Ax?')J
= v(v
2A)-~-2 for A < ~
+ 2)(1 -
TIle indefinite integral of (25) with respect to
So r-2
L L
(_l)i+j
(5
i=O j=O
[1·. p-l {(r + i - j
-1)
logv
r
.)
is
v
r
+ 1) -1 x
+ j
+ (sr + j + 1) lOgVr}]-(N+ 1)
l
(26)
cence
'f'!"
(27.2)
[1 -
n- l (r + sr +
l
U_ll- [(sr +j +l)log
\.Jhen k
=3
[~~::J.(.,.~-j
, the range of integration for
and the range of integration for
Eence
V~]-N }
i) log
vI
is
-1) log vI
v
1 - C~3)
is
r
$
vI
n" tN+ i) dVJ.
vI - (1_C(3))$V $V
a
r 1
$
1 .
-12(r+so+s r )!
So
\ r-2
\
= s !(r-2)!s! ,f. .f. (-1)
o
r 1=0 J=()
[SO] (r- 2)
.+
, J,
l
J'
j
.
' ) -1
(sr + J + 1
x
x [
-r
J.
1 _C(3)
a
6. Comments on Tables
Note that in all cases, for
So
decreases.
So
+ sr
This is to be expected, since this test is designed to
alternative hypotheses of the hype
e(l)
a
=1
- a l/r
the power,
If
Table I shows that the use of
As might be expected, this effect decreases as
e(l)(N)
a
Only for
N+
00
N increases.
is used so that the actual significance level is
So
the power increase for
As
d~tect
as critical limit increases the significance level, and also
2 shows that the power increases with
Table 1.
constant, the power decreases as
+
sr
N.
a , Table
This is not always the case in
sufficiently big, and
So
not too
s~all,
does
N in Table 1.
the values in both Tables tend to the values for the case when
the population distribution is known, (which do not depend on the actual distribution, so long as it is known and continuous).
I would like to express my gratitude to Mr. Kerry L, Lee for programming
the computation of extensive tables of which these are a part.
-13-
7. No Prior Estimate of Parameter Available
to be valid)
Ip. this case we may (assuming' the null hypothesis
estiI!'.ate
HO,O
e
* , the arithreetic mean of the
bye
is valid then
e*
are not independent of
is distributed as
e*
(2r)
r
-1
available
X's.
If
2
ex;..r
, but X1 ,X 2 "."Xr
In fact
(the series terminates at the m-th term when
1 - mk < 0 ~ 1 - (n - l)K ).
In this case, the actual significance level for
k
=1
(i.e. when
testing for censoring from below) is
pr[Y~
>
c~l)IHQ,O]
= Pr[X l (re*)
(28)
>
_r- 1 log(l _ ~~1))]
= rU +r -llOg(l_C~l)) }r-l_ [~J {l +2r -llog (l-C (1~ n r - 1+ •••
It is possible to choose
0.
to make the actual significance level equal to
but '.. . e cannot give an explicit formula, as in (16).
REFEREi'~CES
[1]
[2]
[3]
Johnson, N.L. (1973) Robustness of certain tests of censoring of extreme
sample values, University of North Carolina Mimeo Series No. 866.
Johnson, N.L. (1970), A general purpose test of censoring of sample
extreme values, (In Essays in ProbabiZity and Statistics (S.N. Roy
Memorial Volwne) ), Chapel Hill, University of North Carolina Press,
pp. 379 - 384.
Johnson, N.L. (1971), Comparison of some tests of sample censoring of
extre~e values, Austral. J. Statist.~ 13, pp. 1 - 6.
-14-
Power w.r.t. H
sO,sr
r=
N=
TABLE 1
-1r
of .(11.1) with eel)
= 1 - a (a
a
~
0.05)
5
5
10
25
50
So Sr
r=
N=
0
0
.0956
.0728
.0590
.0545
2
1
0
1
.3396
.1810
.3572
.1438
.3334
.1192
.3245
.1106
4
3
0
1
.6287
.4350
.6237
.4035
.6211
.3793
.6204
.3699
6
5
4
0
1
2
.7789
.6348
.4681
.7955
.6316
.4385
.8109
.6309
.4150
.8174
.6309
.4056
8
7
6
0
1
2
.8673
.7679
.6398
.8919
.7858
.6383
.9124
.8032
.6394
.9205
.8109
.6403
10
5
10
25
so
15
5
10
25
50
So Sr
0
0
2
1
0
1
.4285
.2218
.4003
.1853
.3796
.1605
.3719
.1517
.4433
.2379
.4167
.2019
.3973
.1774
.3901
.1686
4
3
0
1
.7005
.5321
.7050
.5140
.7101
.5007
.7124
.4957
.7271
.5695
.7345
.5564
.7418
.5472
.7449
.5439
6
5
4
0
1
2
.8546
.7516
.6121
.8754
.7638
.6050
.8925
.7755
.6008
.8993
.7805
.5994
.8800
.7924
.6672
.9006
.8082
.6677
.9169
.8221
.6697
.9231
.8278
.6708
8
7
6
0
1
2
.9311
.8748
.7900
.9514
.8979
.8082
.9657
.9165
.8248
.9707
.9238
.8318
.9497
.9073
.8399
.9668
.9287
.8606
.9779
.9448
.8782
.9816
.9507
.8852
See r=5 values
See r=5 values
(cont. )
-15TABLE 1 (cont. )
r=
N=
20
5
10
25
25
50
5
10
25
50
So
5
0
0
2
1
0
1
.4511
.2465
.4253
.2109
.4067
.1865
.3997
.1778
.4559
.2519
.4307
.2165
.4124
.1922
.4056
.1836
4
3
0
1
.7409
.5891
.7496
.5786
.7579
.5716
.7614
.5690
.7493
.6013
.7588
.5923
.7676
.5865
.7713
.5845
6
5
4
0
1
2
.8925
.8128
.6957
.9127
.8298
.6998
.9282
.8444
.7045
.9340
.8503
.7067
.8999
.8251
.7132
.9197
.3426
.7192
.93<17
.8574
.7254
.9402
.8633
.7282
8
0
1
2
.9582
.9224
.8639
.9734
.9423
.8849
.9829
.9567
.9019
.9860
.9618
.9085
.9630
.9310
.8778
.9770
7
6
.9498
.8936
.9856
.9630
.9150
.9883
.9677
.9213
r
See r=S values
See r=5 values
TABLE 2
(1)
-1
-lIN -I)} ( =0.05)
Power \'l.r.t. H
of (11.1) with C
=1-exp{-r il(a
a.
sO,sr
r=
N=
5
5
10
25
SO
So
5
2
0
1
.2408
.0964
.2736
.0993
.2973
.1008
.3059
.1013
3
0
1
.4396
.2678
.5188
.3075
.5764
.3370
.5975
.3480
6
5
4
0
1
2
.5965
.4368
.2874
.7027
.5200
.3329
.7746
.5826
.3677
.7996
.6062
.3810
8
0
1
2
.7099
.5751
.4346
,8216
.6355
.5210
.8885
.7634
.5881
.9096
7
1
4
6
r
(cent. )
.7~13
.6139
-16TABLE 2 (cont. )
r=
N=
10
15
5
10
25
50
5
10
25
50
So
5
2
1
0
1
.2731
.1241
.3132
.1327
.3420
.1385
.3526
.1405
.2858
.1355
.3286
.1465
.3594
.1541
.3706
.1568
4
3
0
1
.5174
.3537
.6077
.4133
.6701
.4371
.6923
.4733
.5484
.3895
.6414
.4562
.7042
.5043
.7261
.5219
6
5
4
0
1
2
.7008
.5682
.4212
.8054
.6689
.4981
.3678
.7376
.5548
.8877
.7617
.5759
.7402
.6204
.4794
.8402
.7229
.5656
.8965
.7893
.6270
.9137
.8118
.6492
0
1
2
.8196
.7253
.6090
.9107
.8325
.7173
.9542
.8946
.7898
.9659
.9137
.8147
.8575
.7793
.6782
.9362
.8776
.7851
.9700
.9784
.9437
.8723
5
10
25
SO
5
10
25
50
8
7
6
r
r=
N=
So
20
5
.9289
.8509
25
r
2
0
1
1
.2926
.1417
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